Nonlinear Partial Differential Equations and Their Applications (Studies in Mathematics and its Applications, Vol 31)

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS Coll~ge de France S e m i n a r Volume XIV

STUDIES IN MATHEMATICS AND ITS APPLICATIONS

VOLUME

31

Editors: Minnesota P . G . C I A R L E T , Paris P.L. L I O N S , Paris H.A. VAN DER V O R S T , Utrecht D.N. A R N O L D ,

Editors Emeriti: Paris G. P A P A N I C O L A O U , New York H. F U J I T A , Tokyo H.B. K E L L E R , Pasadena J.L. L I O N S *

ELSEVIER AMSTERDAM - BOSTON - LONDON - NEW YORK - OXFORD - PARIS SAN D I E G O - SAN F R A N C I S C O - S I N G A P O R E - S Y D N E Y - TOKYO

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS Coll~ge de France Seminar Volume XIV Editors DOINA CIORANESCU C e n t r e N a t i o n a l de la R e c h e r c h e S c i e n t i f i q u e L a b o r a t o i r e J.L. L i o n s U n i v e r s i t 6 P i e r r e et M a r i e C u r i e Paris, F r a n c e and JACQUES-LOUIS LIONS * C o l l ~ g e de F r a n c e Paris, F r a n c e

2002 ELSEVIER AMSTERDAM - BOSTON - LONDON - NEW YORK - OXFORD - PARIS SAN D I E G O - SAN F R A N C I S C O - S I N G A P O R E - S Y D N E Y - TOKYO

E L S E V I E R SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 9 2002 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science via their homepage (http://www.elsevier.com) by selecting 'Customer support' and then 'Permissions'. Alternatively you can send an e-mail to: [email protected], or fax to: (+44) 1865 853333. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2002 Library o f Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.

ISBN: 0-444-51103-2 ISSN: 0168-2024 The paper used in this publication meets the requirements o f ANSI/NISO Z39.48-1992 (Permanence o f Paper). Printed in The Netherlands.

To the memory of Jacques-Louis Lions

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Contents Preface

ix

An introduction to critical points for integral functionals D. Arcoya and L. Boccardo A semigroup formulation for electromagnetic waves in dispersive dielectric media H.T. Banks and M. W. Buksas

13

Limite non visqueuse pour les fluides incompressibles axisym~triques J. Ben Ameur and R. Danchin

29

Global properties of some nonlinear parabolic equations M. Ben-Artzi A model for two coupled turbulent flows. Part I: analysis of the system C. Bernardi, T. Chac6n Rebollo, R. Lewandowski and F. Murat D~termination de conditions aux limites en mer ouverte par une m~thode de contrSle optimal F. Bosseur and P. Orenga

10

11

12

13

57

69

103

Effective diffusion in vanishing viscosity F. Campillo and A. Piatnitski

133

Vibration of a thin plate with a "rough" surface G. Chechkin and D. Cioranescu

147

Anisotropy and dispersion in rotating fluids J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

171

Integral equations and saddle point formulation for scattering problems F. Collino and B. Despres

193

Existence and uniqueness of a strong solution for nonhomogeneous micropolar fluids C. Conca, R. Gormaz, E. Ortega and M. Rojas

213

Homogenization of Dirichlet minimum problems with conductor type periodically distributed constraints R. De Arcangelis

243

Transport of trapped particles in a surface potential P. Degond

273

14

Diffusive energy balance models in climatology J.I. Diaz

15

Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems M. EUer, V. Isakov, G. Nakamura and D. Tataru

16

On the unstable spectrum of the Euler equation S. Friedlander

17

D~composition en profils des solutions de l'~quation des ondes semi lin~aire critique ~ l'ext~rieur d'un obstacle strictement convexe I. Gallagher and P. Gdrard

18

19

297

329 351

367

Upwind discretizations of a steady grade-two fluid model in two dimensions V. Girault and L.R. Scott

393

Stability of thin layer approximation of electromagnetic waves scattering by linear and non linear coatings H. Haddar and P. Joly

415

20

Remarques sur la limite a --~ 0 pour les fluides de grade 2 D. Iftimie

21

Remarks on the Kompaneets equation, a simplified model of the Fokker-Planck equation O. Kavian

469

Singular perturbations without limit in the energy space. Convergence and computation of the associated layers D. LeguiUon, E. Sanchez-Palencia and C. de Souza

489

22

23

Optimal design of gradient fields with applications to electrostatics R. Lipton and A.P. Velo

24

A blackbox reduced-basis output bound method for noncoercive linear problems Y. Maday, A. T Patera and D.V. Rovas

457

509

533

25

Simulation of flow in a glass tank V. Nefedov and R.M.M. Mattheij

26

Control localized on thin structures for semilinear parabolic equations P.A. Nguyen and J.-P. Raymond

591

Stabilit~ des ondes de choc de viscosit~ qui peuvent ~tre caract~ristiques D. Serre

647

27

571

Preface This volume is the 14 th and last one of the series "Nonlinear Partial Differential Equations and their Applications. Coll~ge de France Seminar", which published the texts of the lectures given at the seminars organized by Jacques-Louis Lions, from his election at the Coll~ge de France in 1973 until his retirement in 1998. It was one of the foremost seminars in nonlinear PDE's and their applications during that period. It is unfortunate that because of his untimely death, on May 17, 2001, Jacques-Louis Lions will not see its publication. This volume is dedicated to his memory.

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Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chap ter 1 A N I N T R O D U C T I O N TO CRITICAL P O I N T S FOR INTEGRAL FUNCTIONALS

D. ARCOYA AND L. BOCCARDO

1. Introduction The study of minima of functionals defined in spaces of functions may be considered one of the keystones of the mathematical analysis. Remind the efforts by the great mathematicians of the last and present century to develop sufficient conditions on the functional for the existence of minimum. This theory is deeply related with the existence of solutions of boundary value problems. Indeed, this connection is estabilished by the well-known Euler-Lagrange equations associated to the functional. However, there exist boundary value problems for which the associated functional is indefinite, i.e. it is unbounded from below and from above. This means that it has not global extrema and so we have to search the solutions of the problem among the critical points, i.e. the points for which the derivative of the functional vanishes.

From the abstract point of view there is a difference between the study of minima and of critical points. Indeed, for the existence of minima we need only assumptions on the functional. On the contrary, we point out that the results of existence of critical points involve additional hypotheses on the regularity of the functional to assure the existence of a derivative in some sense. This may explain why the theory of mimima handles classes of functionals with more general hypotheses of smoothness than the critical point theory. In some papers [4], [5], [6], we overcame this difference by developing a critical point theory for nondifferentiable functionals. We observe explicitely that our model functionals does not involve similar functions to the modulus. In fact, the nondifferentiability of the considered functionals is due to the introduction of some smooth Carathdodory function A(x, u) (as smooth as you_ want). Specifically, we consider here functionals or defined in W~

2

An introduction to critical points for integral functionals

(~"~ C I ~ N

open, N > 1) by

J(v) = f~ A(x, v)lVvl dx - / ~ F(x, v +) dx, v ~_ Wlo'2(a),

(1)

with 0 < a <_ A ( x , z ) < ~ < c~, IA'(x,z)l <_ ~/ and f ( x , z ) - F ' ( x , z ) (derivative respect to z) a subcritical Carath~odory function. Observe that J is only differentiable along directions of W 1'2 (fl)NL ~ (~), even for smooth functions A (see [11]). This note is devoted to present the critical point theory developed in [5] (see also [?, ?, ?, ?, ?, ?, ?]) for functionals which are not differentiable in all directions.

2. A mountain pass theorem for nondifferentiable functionals Our abstract setting for the functionals J that we study is given by the following asumption: (H) (X, I[" [[x) is a Banach space and Y c X is a subspace which is a normed space endowed with a norm I1" [[Y- Moreover, J : X > IR is a functional on X such that it is continuous in (Y, I1" [Ix + ]I" [[Y) and satisfies the following hypotheses: a) J has a directional derivative (J'(u), v) at each u E X through any direction v C Y. b) For fixed u C X, the function (J'(u),v) is linear in v c Y, and for fixed v E Y, the function (J'(u), v) is continuous in u c X. Due to the lack of regularity of the functional, some words are needed in order to establish our definition of critical points. D e f i n i t i o n 2.1 - A function u E X is called a critical point of J if

(J'(u), v) = 0, Vv c Y. In this framework a suitable version of the Ambrosetti-Rabinowitz Theorem has been proved in [5]. Specifically, Theorem

2.2

-

Assume (H) and that for e C Y,

c = inf max J(~/(t)) > -rEF rE[0,1]

C 1 ---

max {J(0), J(e)}

with F the set of the continuous paths 7 : [0, 1} * (Y, I]" IIx + II" Ilv), such that 7(0) = 0 and 7(1) = e. Suppose, in addition, that J satisfies the condition

D. Arcoya and L. Boccardo

3

(C) A n y sequence {un } in the Banach space Y satisfying for some {Kn } C ]R + and {en} ---* 0 the conditions

{J(un)} is bounded,

(2)

Ilun[fy ~ 2K~ Vn ~ IN,

(3)

< en

llvllY ] Kn + [[vllx

Vv e Y,

(4)

possesses a convergent subsequence in X . Then there exists a nonzero critical point u c Y of J such that J ( u ) = c.

Remarks 2.3. The proof of this theorem is done by dividing it into two steps. In the first one, only the geometric hypotheses are used to deduce the existence of a sequence {un} in Y satisfying for some {Kn} c IR + and {en} ~ 0~the conditions (2)-(4). The proof is then concluded by using condition (C). .

In this way, condition (C) can be considered as a compactness condition on the functional J, which substitutes in the nondifferentiable case the role done by the well-known Palais-Smale condition in the regular case Y = X. This compactness condition is connected with the coercivity of J extending the previous results for C 1 functionals in [11]. To be specifical in [7] we prove

T h e o r e m 2.4 - In addition to (H), assume that Y is dense in X and that g is continuous in X and bounded from below. I f J satisfies condition (C) then J is coercive, i.e., lin~l J ( u ) = c~ . II~llx--*~

3. A s i m p l e m o d e l The application of the abstract result quoted in the previous section to the study of the functional J defined in (1) is very technical. In particular, the verification of the condition (C). For this reason, we present here a simple but not natural functional which is not differentiable in wl'2(gt), but for which the verification of condition (C) has not technical difficulties like for

4

An introduction to critical points for integral functionals

functionals studied in [5], [6]. Specifically, we consider the functional J defined in W 1'2 (~t) by setting 1 J(v) = ~1 f a [Vv [2 dx + ~1 fa a(x, v)[Vv[ q dx - -~ f a ( v + )m dx ,

(5) wherel a > 0 such t h a t

<_ a(x, z) < for almost every x C f~ and z CIR. (a2) There exists 7 > 0 such t h a t

[a'~(x,z)l < ~/,

for almost every x E ~t, Vz > 0,

and a'(x,z) - 0, z < 0. (a3) Either

a(x, z) is increasing and concave with respect to z _> 0,

(6)

a(x, z) is decreasing and convex with respect to z :> 0.

(7)

or

Let X - W 1'2(~), endowed with the usual norm I[" I]; Y - W1'2(~t) N

L2/(2-q)(~t), endowed with the norm [[-I[Y - ] ] " ]]2/(2-q). By (al) and (a2) the functional J is continuous on X and satisfies (H). We point out t h a t X=Yonlyforq_
3.1 [ 5] - Assume (al - a 3 ) and 1 < q < 2 < m < 2*. Then the

y

(5)

(c).

Proof. Let {un} be a sequence in Y satisfying (2), (3) and (4) for some {Kn} C ]R + and let {~n} , 0. We prove t h a t {Un} is b o u n d e d in X. Indeed, taking v adding, we obtain

u,~ as test function in (4), multiplying (2) by m and

D. Arcoya and L. Boccardo

(m-ff

- 1 ) / ~ I W ~ l 2 dx - _q1 ~ , ( xaz,

+

q - 1

5

7~n)Unl~?~nlqdx a ( x , ~ ) l V ~ n l qdx

<_

C~ + ~n(2 + Ilunll)-

Hence, if a(x, z) satisfies (6), taking into account (al), it follows for some 6"2 > 0 that

( q --1 ) a ( x , Z ) - - q1a ,(x,z) z

ql [ a ( x , z)-alz(x,z)z]

:

+ (q-l-

~ ) a(x ,z) _> - C 2 ,

and thus we deduce

2 dx <_ C 2 / ~ IVUnl q dx nt- C 1 + s

-Jr-I1~11),

which implies, since q < 2, that the sequence {un} is bounded in WI'2(f~). On the other hand, if instead of (6), it holds (7), then that the sequence {un} is bounded follows easily from the fact that a 'z (x , z)z <_ O. Therefore the sequence {un} is bounded in X = wl'2(t2). Then there exist u E X and a subsequence (still denoted u~) such that u , converges weakly to u. Now let if Izi < k Tk(z) =

z

if [z I > k. and Ck(z) = z-

T k ( z ) , Vz ~ Ia,

Vk > O.

To conclude the proof, it suffices to prove

Step 1. {Tk(Un)}----~Tk(u) in Wol'2(f~), as n --~ cx~, for every k > R1. Step 2. For every ~ > 0, there exist ko > R1 and no E IN such that ]]Gk(Un)il < 5 for every k > ko and n > no. Indeed, Steps 1 and 2 imply that, given 5 > 0, there exist n l E IN and kl _> R1 such that

An introduction to critical points for integral functionals

It~n -- ~11

<--- I1~ --

Tk, (u)ll

+ IITk~ (u) - ~!i

IITk~ ( ~ ) - Tk~ (~)11 -4- IIGk~ (~)11 + ilTk~ (u) - ~11 <

35, V n > n l ,

i.e. {Un } is strongly convergent in W 1'2 (f t) to u E W 1'2 (f t).

Step 1. P u t t i n g wn,k = Tk(un) -- Tk(u) as test function in (4), we deduce

/a VUn "VWn,k dm + s a(z, un)lVu,,lq-2VUn . VW,~,kdz + with { s ~ }

-1 q

/o

a~' ( x , ~ ) w ~ , k l V ~ l

~ d~ _< ~

, e~. R e m a r k t h a t

a(x,~tn)[V?.tn]q-2VZtn'VWn,kdX >___~ a(x, Un)lVTk(?.t)lq-2VTk(u).VWn,kdx +f

,~l>k

a(x,

Un)lVuniq-2VUn

9VTk(u)dx

and the right hand side converges to zero. Moreover,

a'z(X,~)Wn,klVu~lqdx <_C1

IW~12Wndx

IITk(~)--Tk(~)llq/
Thus, it follows t h a t the sequence Tk(un) is convergent in W~'2(~) to Tk(u) for every k > 0.

Step 2. The assertion is easily proved by taking Gk(un) as test function in (4) and using (a3).

m

Thanks to the previous lemma, we can prove existence of a nontrivial critical point for the functional J. T h a t is, the existence of a weak solution of the quasilinear Dirichlet problem -At-

1 z'(X , ~t)lVUl q = I~l m-2 div (a(z, u)lVuiq-2Vu) + qa

u}

u C W01'2(a), u > 0 in ft 3.2 - A s s u m e ( a l - a 3 ) and 1 < q < 2 < m < 2*. Then the functional Y given by (5) has at least a positive critical point.

Theorem

D. Arcoya and L. Boccardo

7

Proof. We point out t h a t every nonzero critical point of J is positive. In fact, it is deduced taking Tk(u-) as test function (note t h a t u may not belong to Y, but Tk(u-) E Y). In order to show the existence of a nonzero critical point, and following the ideas of L e m m a 3.3 in [2], it is easy to check that u = 0 is a strict local minimum of J, t h a t is, there exist p, R such t h a t J(u)

u

> p > 0

for [[u[[ = R > 0.

(8)

Moreover, limltt__, ~ J(tpl) = - o c , being ~1 > 0 an eigenfunction associated to the first eigenvalue A1 of the homogeneous Dirichlet problem for the laplacian operator with L2-norm equal to one. Thus, there exists to > / ~ such t h a t J(topl) < 0. Thus, letting e = t0pl and considering the set F of the (continuous) paths ~/"[0, 1] ,

(wl'2(f~)N

L2/(2-q)(a), [[-[[-Jr-[[. [[2/(2-q))

which join 0 and e, i.e. such that ~(0) = 0 and 7(1) = t0pl, we observe that every ~ E F is continuous from [0, 1] to w l ' 2 ( f t ) , so that, by (8), for every -~ C F there exists { c [0, 1] such t h a t m

It7(t-)ll = R .

Hence c-

inf m a x J(~/(t)) _> p > m a x { J ( 0 ) , J(t0~al)} = 0. "r~r t~[0,1]

Then, taking into account L e m m a 3.1 and applying Theorem 2.2, we deduce the existence of a critical point u C w l ' 2 ( f t ) C l L2/(2-q)(ft), of J with J(u) = c > 0 and thus u r 0. m

4. Main examples The abstract theorem (with X = w l ' 2 ( f t ) and Y = W01'2(ft)N L ~ ( f t ) ) of the Section 2 is applied now to obtain nonnegative critical points of the functional J " W~'2(ft) ,, ~ IR t2 {+cxD} defined by

J(v) = /aA(x,v)[Vv[edx - faF(x,v+)dx, v c Wl'2(~),

(9)

i.e. nonnegative solutions of the b o u n d a r y value problem:

tt E 1wlO'2' (ft) n L ~2(ft), , -div(

A(x, u) Vu)

+

-~Az(x , u)]Vu !

=

Fu(x , u) - f(x, u)

}

(P)

8

An introduction to critical points for integral functionals

where f 9f~ x IR , IR is a C a r a t h 6 o d o r y function w i t h subcritical growth. It is clear t h a t for a solution u of ( P ) we are m e a n i n g u e W01'2(f~) cl L ~ ( f ~ )

/

fa A(x, u ) V u V v d x + ~1 fa A" (x,u)lVul2vdx - fa f(x, u) vdx

/

for every v C W 1'2 (f~) C'l L ~ ( a ) . T h e h y p o t h e s e s t h a t we a s s u m e on the C a r a t h 6 o d o r y coefficient A " f~ x IR --, IR are t h e following: (A1) T h e r e exists c~ > 0 such t h a t

c~ < A(x, z), for a l m o s t every x c f~ and z >_ 0. (A2) T h e r e exists R1 > 0 such t h a t for every z _> R1. (A3) T h e r e exist m > 2 a n d (m-2)2

Ct 1 ~>

Atz(X, z) >_ 0 for almost every x c f~,

0 such t h a t

A(x' z) - 2zA1(x'

_> c~

for almost every x E f~, z _> 0. Notice t h a t all a s s u m p t i o n s on A(x, z) are for z _> 0. In fact, since we are looking for n o n n e g a t i v e solutions of ( P ) we can a s s u m e w i t h o u t loss of generality t h a t A(x, z) is even on z. On the o t h e r h a n d , we will a s s u m e the following conditions on f(x, z)" ( f l ) T h e r e exist C1, C2 > 0 such t h a t

If(x, z)[ ~ witha+l<2*,

Cllz[ ~ + C2,

(2*-2N/(N-2)

a.e. x C a , if2
Vz C ]I~+,

and 2* - oo if 2 _> N).

(f2) T h e r e exists R2 > 0 such t h a t

o < .~F(~. z) _< zf(x. ~). for almost x C f~ and every z _> R2 (m is t h e s a m e as in (A3)). (f3)

f(x, Iz[) -- o([z]) at z - - 0 , uniformly in x C ft.

D. Arcoya and L. Boccardo

Theorem

4.1

9

-- Assume (Al-3), (f1-3) and that A(x,z)

lim z --+ + c ~

= O, unif. in x E ~.

(10)

Z a

Then the problem (P) has, at least, one nonnegative and nontrivial solution. R e m a r k s 4.2. 1. The above theorem is essentially contained in [5]. However, in t h a t paper it is assumed in addition t h a t A(x, z) is bounded from above and its derivative A'z(X , z) with respect to z is also bounded. In [7], we have seen t h a t these additional hypotheses are not necessary for the existence. 2. The general case of fianctionals

/~

fl(x, v, Vv) dx - J~ F(x, v +) dx, v E W0:'P(~), (p > 1)

could be also handled as in [5]. For simplicity reasons, we just present here the case p = 2, fl(x, v, Vv) - A(x, v)IVvl 2. 3. Some remarks about the meaning of (A3) and (f2) m a y be found in [5, L e m m a 3.2 and Remarks 3.1]. I

Proof of Theorem 4.1. For every n E IN, let h~ be a nondecreasing C 1 function in [0, oo) satisfying hn(s) = s, Vs E [0, n -

1],

h,~(s) <_ s, Vs E ( n - 1, n), h,~(s) - n ,

Vs > n.

Consider the coefficients A n ( x , z ) =- hn(A(x,z)), x E t2, z E JR. Clearly, An satisfies (A1-3) and, in addition, it is bounded from above with bounded ! derivative A n ( x , z) (with respect to z). In this way, if we define the func12 tionals g~ : W 0' (t2) > IR by setting

&(~) = j~ A~(., ~)lWl ~d.

: s

s+l

~+) d~, ~ c wl'~(a),

then using ( f : - 2 ) and (A3), it can be seen in a similar way to the one in Section 2 t h a t Jn satisfies (C). Indeed, we have

10

An introduction to critical points for integral functionals

L e m m a 4.3 - (Compactness condition) Assume (A1-3) and (fl-2). Then the functional J~ satisfies (C). Using in addition (f3) and following the ideas of [2], it is easily seen that Jn satisfies the geometrical hypotheses of Theorem 2.2. Consequently, by it, there exists a nontrivial and nonnegative solution un of the problem u. ~

Wo~'2(~t) n

/

L ~ (gt),

1 (X , u~ )lW.I = f(x, u+). - d i v ( A n ( x , Un) r u n ) + ~A~

/

(11)

with critical level n =__ inf

max Jn(7(t)), -~EF rE[0,1]

Jn(un)--c

where F - { 7 " [ 0 , 1] ~ Wol'2(f~)n L~ W 1,2(~t) n L ~ (Ft) is such that J~ (e~) < O. An(x, z) < A(x,z) and (10), we observe that

Jn(t~Pl) ~ J(t~91) =

l ts +1 [c~

= 0,7(1) = en}, en e Taking into account that

A(x,t~l) [~7r [2 d x ts-1

--

1 s+l

/

~+ldx]

< O,

for all t C [to, co) if to > 0 is large enough. This allows us to choose e~ - top1 (independent of n c IN). On the other hand, by the Mountain Pass geometry of J1 there exist 5, r > 0 such that Jn(V)

~ J l ( v ) ~ (~, VlI~II _< ~,

(i.e., roughly speaking, v - 0 is a strict local minimum of Jn, uniformly in n E IN). This implies that Cn ~ 5.

(12)

We claim that {un} is bounded in w l ' 2 ( ~ ) . An (x, z) <_A(x, z), we deduce

Indeed, using again that

Jn(Un) :

&(~)

=

inf max Jn(7(t)) ~EF tE[0,1] _< inf max J(7(t)) ~ r t~[0,1]

_<

max 3(ttopl ) - C1. t~[o,1]

Subtracting --1 (j~n(Un), un) = 0 we derive m

D. Arcoya and L. Boccardo

1

1

An(x, un)lVunl 2 dx - ~

11

A~(x , ?-tn)~t n [V?_t n 12 d x ~ 6 2

which, by (A3), implies that Ilu~ll is bounded, thus proving the claim. Then, passing to a subsequence, if necessary, we can assume that {Un } is weakly convergent to some u C wl'2(ft). Now, we prove that the sequence {u~} is bounded in L~(ft). Indeed, we can use v = Gk(un), k > R1, as test function in (11) to deduce that {u~ } satisfies

IIGk(un)]]~. <_C311ull~ [meas Ft(k)] [(2*-1)r-2*a]/2*r, for any r > 2* such that u C Lr(~). Hence there is a constant C 4 > 0 such that [[Un[[c ~ ~ 6 4 (see [14]). Since the sequence {un} is bounded in W01'2(~) and L ~ ( ~ ) and (fl), we yield that {un} is compact in WI'2(~t) by using the results of [9]. Therefore, {un} ....; u and u E W1'2(~)N L~(Ft) is a nonnegative critical point of Y. In addition, {J(un)} ~ Y(u) and we get from (12) that J(u) >_~ so that u ~: 0. Thus u is the weak nontrivial (and nonnegative) solution we were looking for. II Remark 4.4. We conclude by noting that in [7] the reader can find more existence results for nonlinearities f(x, z) which are different combinations of concave and convex functions in the quasilinear spirit of [1], [8]. A c k n o w l e d g m e n t . This paper was partially presented by the second author at the Coll@e de France Seminar (24.3.1995). Both authors would like to thank to the organizers of the Seminar for having given the opportunity of presenting their work.

References [1] Ambrosetti, A., Brezis, H. and Cerami, G., Combined effects of concave arid convex nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519-543. [2] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. [3] Arcoya, D. and Boccardo, L., Nontrivial solutions to some nonlinear equations via minimization. Variational Methods in Nonlinear Analysis, edited by A. Ambrosetti and K.C. Chang, 49-53, Gordon and Breach Publishers, 1995. [4] Arcoya, D. and Boccardo, L., A min-max theorem for multiple integrals of the Calculus of Variations and applications. Rend. Mat. Acc. Lincei, s. 9, v. 6, 29-35 (1995).

12

An introduction to critical points for integral functionals

[5] Arcoya, D. and Boccardo, L., Critical points for multiple integrals of Calculus of Variations. Arch. Rat. Mech. Anal. 134, 3(1996), 249-274. [6] Arcoya, D. and Boccardo, L., Some remarks on critical point theory for nondifferentiable functionals, to appear in NoDEA. [7] Arcoya, D. and Orsina, L., Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. TMA. 28 (1997), 1623-1632. [8] Boccardo, L., Escobedo, M. and Peral, I., A Dirichlet problem involving critical exponent. Nonlinear Anal. TMA., 24 (1995), 1639-1648. [9] Boccardo, L., Murat, F. and Puel, J.P., Existence de solutions faibles pour pour des ~quations quasilin~aires s croissance quadratique. Res. Notes in Mathematics 84, Pitman, 1983, 19-73. [10] Canino, A. and Degiovanni, M., Nonsmooth critical point theory and quasilinear elliptic equations, in Top. methods in differential equations and inclusions, Kluwer Academic Publisher, 1995. [11] Dacorogna, B., Direct Methods in the Calculus of Variations. SpringerVerlag, 1989. [12] Degiovanni, M. and Marzocchi, M., A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. 167 (1994), 73-100. [13] Pellacci, B. Critical points for non diferentiable functionals, Boll. U.M.I 11-B (1997), 733-749. [14] Stampacchia, G., Equations elliptiques du second ordre ~ coefficients discontinus. Les Presses de L'Universit~ du Montreal, 1966. David Arcoya Departamento de An~lisis Matem~otico Universidad de Granada 18071-Granada Spain E-mail: [email protected] Lucio Boccardo Dipartimento di Matematica Universit~ di Roma 1 Piazza A. Moro 2 00185 Roma Italy E-mail: [email protected]

Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 2 A SEMIGROUP FORMULATION FOR ELECTRMAGNETIC WAVES IN DISPERSIVE DIELECTRIC MEDIA

H.T. BANKS AND M.W. BUKSAS

1. Introduction In a forthcoming monograph [2] we have developed a theoretical and computational framework for electromagnetic interrogation of dispersive dielectric media. In that work we show that one can take a time domain variational or weak formulation of Maxwell's equations in dispersive materials and, in the context of inverse problems, use partially reflected polarized microwave pulses to determine both dielectric material properties and geometry of bodies (specifically for plane waves inpinging on slab geometries in paradyms which approximate far field interrogation). This is done in configurations involving either supraconductive reflecting back boundaries or acoustically generated virtual reflectors. The propagation and reflection of electromagnetic waves in dispersive .dielectric media is, of itself, an interesting topic of investigation. As we point out in the next section (and demonstrate computationally in [2]), the underlying dynamical systems are not typical of either standard parabolic or standard hyperbolic (even with the usual dissipation) systems and are hence of mathematical interest. In this short note, we consider the Maxwell system for rather general dispersive dielectric media and show that such systems, under appropriate conditions on the polarization law, generate Co semigroup solutions. These results are presented in the context of the 1dimensional interrogating systems developed in detail in [2] and we invite interested readers to consult that reference for more detailed discussions and development of the underlying model employed here.

2. Modeling of dispersiveness in dielectric media We begin with time domain Maxwell's equations in second order form (e.g.,

14

A semigroup formulation for electromagnetic waves

see [2]) for the electric field E = E(t, z) of 1-dimensional polarized waves

+ 115 + -1- & E0

c 2E"= - - -1L

-

E0

(1)

~0

where c = 1 is the speed of light in vacuum, J~ is the conduction current density, J8 is a source current density and P is the electric polarization of the dielectric medium. We assume very general constitutive material laws for the polarization and conductivity given by

P(t, z) - (gp 9 E)(t, z) =

/o

gp(t - s, z)E(s, z)dz

(2)

3~(t, z) -- (gc 9 E)(t, z) - ~o t g~(t - s, z)E(s, z)dz

(3)

where we have tacitly assumed that E(t, z) = 0 for t < 0 and that both gp((,z) and gc((,z) vanish for ( < 0. With these assumptions, the integrals in equations (2), (3) are equivalent to integration over all of (-c~, c~) ~nd thus are indeed convolutions. The displacement susceptibility kernel gp (also referred to as the dielectric response function(DRF)) and the conductivity susceptibility kernel gc introduce nonlocality in time in the polarization and conductivity relationships [1], [15] which is equivalent to frequency dependence of the dielctric permittivity E and conductivity a when using frequency domain approach. We assume that either P or de or both may contain instantaneous (local in time) components by introduction of 5 distributions in the kernels gp and/or gr respectively. A medium is dispersive if the phase velocity of plane waves propagating through it depends on the frequency of the waves [16, Chap.7], [10, Chap.8]. To determine the dispersive nature of a medium described by equations (1)-(3) we seek plane-wave solutions of the homogeneous analogue of (1) of the form E(t, z) = Eoe -i(~t+~z) which travel in the z direction and have wavelength A = 2w/~. The phase velocity Vp of these waves is the speed at which planes of constant phase move through the medium. In this case the argument w t - ~z is constant and dz

Vp : d--t = w/~.

(4)

Seeking plane wave solutions of the form Eoe -i(~t+~z) in (1) is equivalent to seeking solutions of the form Eoe +i~z in the frequency domain version of (1). Thus we use the Fourier transform in (1) and obtain ~d2

iw

eo

EO

H. T. Banks and M. W. Buksas

15

where we have ignored the source term d8 and where the overhat will represent the Fourier transform throughout. Since we see from (2) and (3) t h a t /5 _ t)p/~ and Jr = t)cE:, this can be written

c2/~,, + 002(1 + i~c + ~)__pp)/~_--O. 02s

(6)

s

We note t h a t (6) is the generalized Helmholtz equation [16, p. 271] +

-

0

(7)

with 00 2

= -~(1 ~

~00E0

)

(8)

E0

which has solutions/~(w, z) = Eoe +i~(~)z. It follows that the corresponding time domain solutions are our desired solutions of the form E(t, z) = Eoe -i(wt+i~z) where the wavenumber n = n(w) will in general depend on the frequency w. The equation (8) relating the frequency w and the wavenumber of propagating waves is known as the dispersion equation for the medium. In the case of vacuum or free space where t~p = g~ - 0 so t h a t n - w/c, we obtain the corresponding phase velocity vp = c = the speed of light as expected. More generally the phase velocity in a dielectric medium with conductivity and polarization is given by

=

/V/1 +

+

(9)

In light of (9) and the definition of a dispersive medium, we see t h a t if either [tc/w or ~p depend on w, we will have dispersiveness. Several special cases are worthy of note. For instantaneous conductivity, t h a t is, go(t, z) = aS(t) so t h a t (3) reduces to O h m ' s Law Jc = erE, we see t h a t the term i[t~/eow becomes icr/eow. Thus a medium with simple Ohm's Law conductivity will be dispersive (it is also dissipative in the usual sense since the conductivity term in (1) becomes --r For instantaneous polarization (often assumed in standard EO treatments of the Maxwell theory) we find gp(t,z) = eoXh(t), where X is the dielectric susceptibility constant and hence [lp/eo = X and the medium is not dispersive. One must turn to more complicated (and more realistic) models, such as those of Debye or Lorentz, to have a polarization based contribution to dispersiveness in a medium. For the usual Debye polarization model [11, p.386] one has

gp(t) -- e - t / r

eoo)/~',

t > O,

(10)

16

A s e m i g r o u p formulation for electromagnetic waves

where ~- is a relaxation parameter and es, e~ are familiar dielectric constants. In this case one finds 1 - iwz O p ( W ) - e 0 ( e s - eo~)[ 1 _[_T2~d2]

For the Lorentz model [16, p.496] we have gp(t) - e~

-t/2~ sin~ot,

t > 0,

(11)

Vo where ~0 - v/w02 - 1/4T2. In the frequency domain this yields

4T 2 -

-

and again we have a polarization based dispersive medium. Higher order (the Debye and Lorentz models correspond to first and second order, respectively, differential equation models for the polarization P - see [2] and the references therein) models, as well as combinations of such models also lead to dispersion in a medium. Thus, in summary we see that instantaneous conductivity but not instantaneous polarization yields dispersiveness in a medium. For a polarization contribution to dispersiveness one must include first or higher order polarization models (instantaneous polarization can be correctly viewed as zero order polarization dynamics). For our semigroup presentation in the next section we shall therefore consider the model (1) with instantaneous conductivity and a general (higher order) polarization model given by (2) with gp = g where the D R F g is assumed smooth in time (i.e., without loss of generality we can assume no instantaneous component for g). Such distributed parameter systems are of interest since they are neither simple hyperbolic nor parabolic in nature. For simple Ohm's Law conductivity and instantaneous polarization (or no polarization), the system (1) becomes a well understood dissipative or damped hyperbolic system for which a semigroup formulation can readily be found in the research literature on distributed parameter systems. However, for (1) with polarization based dispersiveness, we obtain a system with behavior of solutions that are neither standard hyperbolic (finite speed of wave propagation along characteristics) nor standard parabolic (infinite speed of propagation of disturbances). Indeed for (1) with either Debye or Lorentz polarization, rather fascinating solutions can be observed. These

H.T. Banks and M. W. Buksas

17

involve the formulation of so-called Brillouin and Sommerfeld precursors where a pulsed excitation (containing waves with a range of frequencies) evolves into waves propagated with different velocities which coalesce into wave "packets" (see Chapter 4 of [2] and [1] and the references therein for discussions of these phenomena). It is of both mathematical and practical interest to know whether these interesting systems can be described in a semigroup context. The potential advantages afforded by a semigroup formulation are widespread since there is a tremendous literature for control, estimation and identification, and stabilization of systems in a semigroup setting. Results for both stochastic and deterministic control methodologies (in both time domain and frequency domain) including open loop and feedback formulations are abundant [12], [3], [4], [11], [19]. In the next section we present a semigroup formulation of the system (1) with simple Ohm's Law conductivity along with general polarization based dispersiveness generated by polarization laws of the form (2). To be more precise, we take (1) for t > 0 and z C (0, 1) with Jc(t, z) = a ( z ) E ( t , z) where a(z) vanishes outside ~ c (0, 1]. The closed region ~t is some dielectric material region (e.g., a slab or several slab-like regions) containing instantaneous conductivity as well as non trivial polarization of the form (2) with gp(t,z) = g ( t , z ) vanishing outside z c ~t. Using this form of conductivity and polarization in (1), we obtain the system 1 1 E ( t , z) + - - ( ~ ( z ) + g(O, z ) ) E ( t , z) + --[~(0, z ) E ( t , z) 6-0

+

6-0

i)(t - s , z ) E ( s , z ) d s

- c2E"(t,z) - -1Js

(t, z).

(12)

~0

With this system we take boundary conditions (see [2] for details) that represent a total absorbing boundary at z = 0 and a supraconductive boundary at z = 1. This can be expressed by E(t, 0) - cE'(t, 0) = 0

(13)

E(t, 1 ) = 0 .

(14)

With the definitions a(t,z)

=

_l~(t,z), •0

=

~(z)=--lt~(0, z) ~0

+

s(t,z) - -1L(t,z),

6-0

EO

we can write equation (12) as + ",/E + ~ E + o~ 9 E - c2E '' = ,7

(15)

18

A semigroup formulation for electromagnetic waves

where a , E is the usual convolution a 9 E ( t , z) -

~0t a ( t -

s, z ) E ( s , z)ds.

(16)

One can use the boundary conditions ( 1 3 ) - (14) to write ( 1 5 ) i n weak or variational form so as to seek solutions t --4 E ( t ) in V - H~(0, 1) = {r C H~(0, 1 ) : r = 0} in a Gelfand triple setting V r H ~-~ V* with pivot space H = L2(0, 1). Under modest regularity assumptions on a, ~, T and fl, one can establish existence, uniqueness and continuous dependence (on initial conditions and input) of solutions. Details are given in Chapter 3 of [2].

3. A semigroup formulation We turn in this section to a semigroup formulation for the dispersive system (12)- (14) or equivalently, (13)- (15), with instantaneous conductivity and general (non instantaneous) polarization. For our development we assume t h a t 7, ~ c L ~ (0, 1) while a E L ~ ( ( 0 , T) • (0, 1)) and a, ~, 7 vanish outside Ft. We moreover assume that a can be written as ~(t, z) = O~l(t)ol2(z) where 0 < ~L _< c~2(z) < C~U on ~t C (0, 1] for positive constants a L , a V , with a2 vanishing outside ft. We assume t h a t t --~ a~ (t) is positive, monotone nonincreasing, and in H 1(0, T) so that &l (t) < 0. This monotonicity assumption is typical of the usual assumptions in displacement susceptibility kernels (e.g., see [9, p.102] or [15]). We shall return to discuss this monotonicity requirement further after our semigroup arguments of this section. We consider the term (16) given by

/0ta ( t -

s)E(s)ds =

i

a(t-

s)E(s)ds

oo

from (15) and note that it can be equivalently written

l

a(t - s)E(s)ds (x)

f

a ( - ~ ) E ( t + ~)d~ oo

a ( - ~ ) E ( t + ~)d~ -

G ( ~ ) E ( t + ~)d~

where G(~) _-- a ( - ~ ) . We denote GI(~) = a l ( - ~ ) so that G(~) = Gl(~)a2. The approximation is valid for r sufficiently large (r = oc is permitted) so t h a t a(t) .~ 0 for t > r. We observe at this point that (~1 (~) _ 0 with G1 (~) > 0 on ( - r , 0].

H.T. Banks and M.W. Buksas

19

As introduced in the previous section, we take V - H ~ ( 0 , 1 ) , H L2(0,1) with Y ~-+ H ~-~ Y*. We shall have use of H - L22(~), the space L2(~t) with weighting function a2, which is readily seen to be equivalent to L2(gt) due to the upper and lower bounds on a2 C L ~ ( ~ ) . We shall denote the restriction of functions r in L2(0, 1) to fl again by r and write' r e L2(gt) or r e L2~ (~t) whenever no confusion will result. Using the above definitions and approximating, we may write (15) as

E,(t) + 7F,(t) +/3E(t) +

f

G(~)E(t + ~)d~ - c2E ''(t) =

if(u).

(17)

r

Using an approach given in [5], [6], [14] and [7] for viscoelastic systems, we define an auxiliary variable w(t) in W - L ~ l ( - r , 0;/~) by w(t)(O) = E(t)-E(t+O),-r_~ 0 < 0. Since G(0, z) > 0 for 0 E ( - r , 0],z C ~ we may take as an inner product for W the weighted L 2 inner product

(rl, W ) w --

f

Gl(O)(U(O),w(O))FidO

=

r

f

~1(0) r

L

a2(z)u(O,z)w(O,z)dz

under which W is a Hilbert Space. We note that by our notational convention explained above, we have w(t) e W for any E ( t , z ) with E(.,.) e L~I ( - r , 0;H). Using a standard shift notation, we may write w(t) = E ( t ) - E ( t + O) = E ( t ) - Et(O) where Et(O) - E ( t + O) for - r _ 0 < 0. Adding and subtracting appropriate terms in (17), we find

E(t) + ~/E(t) + ;3E(t) +

f

G(()E(t)d( r

-c2E"(t) =

if(U)

f

G(() [E(t) - E t ( ( ) ] d ( r

or, equivalently

E(t) + ~/E + (~ + G l l ) E ( t ) -

f

G(~)w(t)(~)d~ - c2E"(t) - i f ( u )

(18)

r

where Gll (z) - f o r G(~)d~ - a2(z) f o r G1 (~)d~ and w(t)(~) - E(t) Et(~). We observe that Gll, like/3, is in L2(f~) and L2(0, 1). For our semigroup formulation, we consider (18) in the state space Z V x H • W - H~(0, 1) • L2(0, 1) x L ~ l ( - r , 0, H) with states (r r 7) ( E ( t ) , E ( t ) , w ( t ) ) - ( E ( t ) , i E ( t ) , E ( t ) - Et(.)). To define an infinitesimal generator, we begin by defining a fundamental set of component operators. Let A c s ]2") be defined by

(19)

20

A semigroup formulation for electromagnetic waves

where 50 is the Dirac operator 50r = r

Then we find

< -- ~iqh, ~>V*,V -- H q- <(1~ -[- Gll)r ~ 5 -

(20)

so that it is readily seen that 51 : V x V H(13 defined by

~l(d/), ~)

--

< --

(21)

.ff~, ~>V*,V

is symmetric, V continuous and V coercive (i.e., ~1 (r r for constants Ao and cl > 0). We also define operators B E s 12") and I22 c s

~_ Cllr

~01r

by

Br = - ~ r - ~r

(22)

so that ( - Be, r and, for ~ e W -

- (7r r

+ cr162

(23)

L~, ( - r , O; H), (Kr/)(z)- { 0

z e [0, 1]\ft

f o 0(r162

(24)

z e a.

Since G({, z) = 0 for z e [0, 1]\f~, we abuse notation and write this as R.

-

f c(~),7(~)< r

even though, strictly speaking, ~(~, z) is only defined for z c ft. With these definitions and notations, equation (18) can then be written

as

(E, e>~.~ + < - AE, r +

+ < - ~E, r

( - k ( E - E+), ~ > ~ . ~

- v-,v

or E,(t) -- fiE(t) + BIE(t) + k ( E ( t )

- E t) +

J(u)

in l;*.

(25)

We rewrite equation (25) as a first order system in the state r (E(t),lF(t), w(t)) where w(t) - E ( t ) - E t. To aid in this we introduce another operator D " dom D c W ~ W defined on dom D - {~ c H l ( - r , 0;/t)[~(0) - 0} by D r / ( O ) - N0~ ( 0 ) .

H.T. Banks and M.W. Buksas

21

We then observe that w(t) = E ( t ) - E t satisfies

Thus we may the equation

=

E(t) - E(t + O) - E(t) - DEt(O)

=

# ( t ) + D(E(t) - Et(O)) =/E(t) + Dw(t)(O).

formally rewrite (25) as a first order system and adjoin to it

@(t) -- D w ( t ) + #(t).

(26)

We then obtain the first order system for ((t) given by - .AC(u) + 7 ( u )

where A given by .4-

(

(27)

o i o) A 0

B I

K D

(28)

is defined on dom A -

{ (r r r/) E ZIr E ~J, r/E dom 7), ~ r + Br C 7-/}.

(29)

That is, A| - (r ~ r + Be + ~ , r + ~ ) for ~ - (r r ~) in dom ,4. The forcing function ~- in (27) is given by ~ = col(z,J,t). To argue that Jt is the infinitesimal generator of a C0-semigroup, we actually consider the system (27) on an equivlaent space Z1 = V1 x H x W where V1 is the space Y with equivalent inner product (r162 1 - ~1(r162 where ~1 is the sesquilinear form given in (21). Recall that ~1 is symmetric, V continuous and V coercive so that it is topologically equivalent to the V inner product. We are now ready to prove the following generation theorem. T h e o r e m - Suppose that '7,/3 C L ~ (0, 1), a e L ~ ((0, 1) • (0, 1)) with c~,~, "Y vanishing outside ~. We further assume that c~ can be written a ( t , z ) = c~l(t)a2(z) where a l e HI(0, T) with al(t) > 0, all(t) < 0, and 0 < O~L ~ O~2(Z) ~ OLU f o r positive constants C~L,C~U. Then the operator .A defined by (28), (29) is the infinitesimal generator of a Co-semigroup on Z1 and hence on the equivalent space Z. Proof. To prove this theorem, we use the Lumer-Phillips theorem ( [15, p. 14]). Since Z1 is a Hilbert space, it suffices to argue that for some A0, A - A~Z is dissipative in Z1 and 7 ~ ( ) ~ - A) = Z ~ for some A > 0, where 7 ~ ( ) ~ - A) is the range of A I - .4. We first argue dissipativeness.

A semigroup formulation for electromagnetic waves

22

Let 9 = (r r r/) c dom .4. Then -

(A~I,, ~}z~

(r r

+ (fi'r + B e +

= (r r

+ (fie + Be, r

--

a l (r all))- 0"1 ( d/), r

-

-<~r r

K'q,

r

+ (r + Dr/, ~/)w

+ (Krl, r

+ (r + D~7,rl}w

( B e , r ) H -t-" ( K ?'], r ) H -[- (r Jr- D

- c1r

= +
_< 1~1~1r + I(R~, r

+
~7, rl } yv

Dr/, ~}w

+ I(r + Dr/, w}wl.

(3ff

We consider estimates for the last two terms in (30) separately. From (24) we have

=

[

f

< <

-

Gl(O)(~(O),r r

Gl(O)l~7(O)lgilr -2

~

GI(O){[v(O)IH + 1r

f-I}dO

_< kllr/l~v+k21r Moreover,

f Gl(O)l(~,~(O))~ldO

I('r ~7)wI ~

r

_<

; al(0){~lr 1

2

1 2 }dO + ~l~(O)la

r

_< kalr

+k4lwl~v.

Finally, since GI(0) _~ 0, GI _~0, and 77 C dom D requires r/(0) - 0 , we may argue

(D~?, rl>w

-

-

-=

/oGl(O)(Drl(O), ~?(O))~IdO

/~

,. GI(0)

/~gd

,.

dl

~lw(0)l ~ft dO 1

( a l (O)~lv(O)l ~f - I ) d O -

/_~ r

7(0~(O)lv(O)la2 )dO

l f: r d1(0)177(0)[2ffId0

=

1 1 G~(0)~Iw(0)I z~ - GI(--r)2Iw(--r)I=H--2

<

0.

H. T. Banks and M. W. Buksas

23

Combining these estimates with (30), we obtain for ~ c dom ,4 171oo1r _<

nc (kl ~- k4)lf/[~v ~- (]~2 nt- k3)1r

a01r

which yields the desired dissipativeness in Z1. To establish the range statement, we must argue there exists some A > 0 such that for any given ~ - (#, v, ~) in Z, there exists ~ in dom A satisfying

( A I - A)G - O.

(31)

In view of the definition of ,4, the equation (31) is equivalent to the system

-Ar

+

(32)

- B)r - Rv =. -r

+

- D)V -

for (r r ~/) c dom .4, (#, u, ~) c Z = 1; x ~ x M2. The first equation is the same as r -- A r while the third can be written as ~ = ( A - D ) - I ( ~ + r = ( A - D ) -1(~ + A r These two equations can be substituted in the second to obtain an equation for r If this equation can be solved for r E V, then the first and third can then be solved for r and r/, respectively. The equation for r that must be solved is given by - A r + (A - B ) ( A r

#) - R(A - D ) - x ( { + A r

#) - v

or

[A2 _ AB - .4 - / ~ ( A - D ) - I A ] r = ( A - B ) # + v + / ~ ( A - D ) - I ( { - # ) . If we can invert (33) for r C V, then r = A r Ar is in dom D c W and Ar + B e

=

A2r

(33)

is in V, 7/= ( A - D ) - I [ ~ +

A # - v - / 2 / ( A -- D ) - I ( ~ + Ar

#)

=

is in H so t h a t (r r r/) is in dom ,4 and solves (32). Thus the range statement reduces to solving (33) for r C V. This in turn reduces to invertability of the operator A2 - AB - A - / ( ( A D)-IA. We first observe that ( A - 0 ) -1 -- ( 1 - e~')/A since ( A - D ) ( 1 - e ~~ - A while r/(O) = 1 - ~ ~ [~ + A r #] satisfies r/(O) = 0 and hence is in dom D.

A semigroup formulation for electromagnetic waves

24

Thus, for r c H,

/s

- D ) - 1/~ satisfies

(R(1 - e~~162r

-'--

al

(0)(1

-

e~~162 r

7"

_< k51r and -

r

:

+

=

((A 2 +

>

k6[r

r

+ H + A

lr

2

for A sufficiently large.

Hence for A sufficiently large we have ((A 2 - AB - f t . - / ~ ' ( A - D ) - I A ) r

AB)r162

-

((A 2 -

>__

k6]r

2 nt- C 1 [ r

--

Cllr

2 + (k6 -

r + ~ 1 ( r 1 6 2 - (/~'(A - D ) - I A r 1 6 2

-- ) ~ 0 [ r )~0 -

-- k 5 1 r

k5)[r

H"

Thus if we define the sesquilinear form a~(r 0) - ((A 2 - A B - A - / ~ ' ( A -

D)-IA)r

r

we see that for A sufficiently large, a~ is V coercive and hence, by the LaxMilgram lemma [20], it is invertible. It follows immediately that (33) is invertible for r C V. This completes the arguments to prove the Theorem. m Let S(t) denote the semigroup generated by ,4 so that solutions to (27) are given by

((t) -- S(t)(o +

S ( t - s).T'(f)Vf.

(34)

Solutions are clearly continuously dependent on initial data (0 and the nonhomogeneous perturbation 9r. The first component of ((t) is a generalized solution E(t) of (17). One can now argue that the solution agrees with the unique weak solution obtained in Chapter 3 of [2], by using the arguments in Chapter 4.4 of [8]. Briefly, one argues equivalence for sufficiently regular initial data and nonhomogeneous perturbation. Then density along with continuous dependence is used to extend the equivalence to more general data (see [8] for details).

H. T. Banks and M. W. Buksas

25

4. Concluding remarks In the previous section we presented a semigroup generation theorem under general conditions on the coefficients a,/3, V of (15). The only possibly restrictive condition involved a(t, z) - ~og(t,z) 1 .. = Oll(t)oL2(Z) where it is required that a l (t) > 0, dl (t) _< 0. We consider more closely the condition for some common polarization laws. For Debye polarization in a region ~t, we have a l (t) - ~ogp(t) 1 .. where gp is given in (10). T h a t is,

gp(t) = eo (% - E~) e _ t / ~ 7"

so that

1 .. (es - E ~ ) e_t/~ > 0 OLl(t) = ~ogp(t ) T3 and (t)

-

-

-

<

0.

T4

Thus Debye polarization satisfies the conditions of the generation theorem and the associated system generates a Co semigroup. For Lorentz polarization, we have (recall (11))

gp(t) = c~ e -t/2~ sin Lot ~0 and hence

1 gp(t) a l ( t ) - - Eo

Wp

uo

~

)

Wo 2 sinuot

]

7- cosuot .

We therefore see that it is not possible to conclude t h a t a l ( t ) > 0 or (~l(t) ~ 0 SO that our generation theorem does not guarantee a semigroup representation for systems with a Lorentz polarization law. In spite of this, we do believe that the Lorentz law does yield a system with a semigroup representation. We conjecture that the proof of the theorem we present can be modified to weaken the hypothesis on c~ so as to include Lorentz and other oscillatory (even order) polarization models. We are currently pursuing these ideas. In closing we point out that the general class of dielectric response functions consisting of a linear combination of decreasing exponentials (essentially multiple Debye mechanisms) suggested for glasseous materials by Hopkinson [15](see the discussion in [9, p.101-103] are included under the theory presented in this note.

26

A semigroup formulation for electromagnetic waves

A c k n o w l e d g m e n t . This research was supported in part by the Air Force Office of Scientific Research under grants AFOSR F49620-98-1-0180 and AFOSR F49620-95-1-0447 and the Department of Energy, under contract W-7405-ENG-36. The authors are grateful to Dr. Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB, San Antonio, TX, for his continued encouragement and numerous specific technical disscussions throughout the course of the research reported on here and in [2].

References [1] R. Albanese, J. Penn and R. Medina, Short-rise-time microwave pulse propagation through dispersive biological media, J. of Optical Society of America A, 6:1441-1446, 1989. [2] H.T. Banks, M.W. Buksas, and T. Lin, Electromagnetic Interrogation of Dielectric Materials, SIAM Frontiers in Applied Mathematics, SIAM, Philadelphia, 2000, to appear. [3] A. Bensoussan, G. DaPrato, M.C. Delfour, and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Birkhs Boston, 1992. [4] A. Bensoussan, G. DaPrato, M.C. Delfour, and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. II, Birkhs Boston, 1993. [5] H.T. Banks, R.H. Fabiano and Y. Wang, Estimation of Boltzmann damping coefficients in beam models, In COMCON Conf.on Stabilization of Flexible Structures, 13-35, New York, 1988, Optimization Software, Inc. [6] H.T. Banks, R.H. Fabiano and Y. Wang, Inverse problem techniques for beams with tip body and time hysteresis damping, Mat. Aplic. Comp., 8:101-118, 1989. [7] H.T. Banks, N.G. Medhin and Y. Zhang, A mathematical framework for curved active constrained layer structures: Wellposedness and approximation, Num. Func. Analysis Optim., 17:1-22, 1996. [8] H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control, Masson/J. Wiley, Paris/Chichester, 1996. [9] F. Bloom, Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory, Vol. 3 of SIAM Studies in Applied Math, SIAM, Philadelphia, 1981. [10] D.K. Cheng, Field and Wave Electromagnetics, Addison Wesley, Reading, MA, 1989. [11] R.F. Curtain and A.J. Pritchard, Infinite-Dimensional Linear Systems Theory, LN in Control and Info. Sci., 8, Springer Verlag, Berlin, 1978. [12] R.F. Curtain and H.J. Zwart, An Introduction to Infinite- Dimensional Linear Systems Theory, Springer Verlag, New York, 1995.

H. T. Banks and M. W. Buksas

27

[13] R.S. Elliott, Electromagnetics: History, Theory and Applications, IEEE Press, New York, 1993. [14] R.H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Analysis, 21: 374-393, 1990. [15] J. Hopkinson, The residual charge of the leyden jar, Phil. Trans. Roy. Soc. London, 167:599-626, 1877. [16] J.D. Jackson, Classical Electrodynamics, J. Wiley & Sons, New York, 2nd edition, 1975. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [18] J.R. Reitz, F.J. Melford and R.W. Christy, Foundations of Electromagnetic Theory, Addison Wesley, Reading, MA, 1993. [19] B. van Keulen, H ~ Control for Distributed Parameter Systems : A State Space Approach, Birkhs Boston, 1993. [20] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. H.T. Banks Center for Research in Scientific Computation NC State University Raleigh, NC. 27695-8205 USA E-mail: [email protected] M.W. Buksas Los Alamos National Laboratory T-CNLS MS B258 Los Alamos, NM. 87545 USA E-mail: [email protected]

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 3 LIMITE NON VISQUEUSE POUR LES FLUIDES INCOMPRESSIBLES AXISYMETRIQUES

J. BEN AMEUR and R. DANCHIN

R ~ s u m ~ . On s'int~resse s la limite non visqueuse des ~quations de NavierStokes incompressibles tridimensionnelles axisym~triques. On suppose que les donn~es initiales ont des propri~t~s de r~gularit6 stratifi~e de type poche de tourbillon. En utilisant la conservation du tourbillon divis~ par la distance l'axe de sym~trie (dans le cas non visqueux), on trouve des r~sultats de convergence pour tout temps similaires ~ ceux de la dimension deux. En particulier, on a convergence forte au sens de la r~gularit~ stratifi~e, et le gradient de la vitesse est born~ ind@endamment de la viscositY. Lorsque les donn~es initiales n'ont pas de r6gularit~ stratifi~e, on donne une majoration de la vitesse de convergence L 2 en fonction de la viscositY, tr~s proche de celle de [4] pour la dimension deux. A b s t r a c t . We are concerned with the inviscid limit for three-dimensional axisymmetric incompressible flows. The initial data are vortex patches or, more generally, have striated regularity. Using the conservation of the vorticity divided by the distance to the axis of symmetry (in the inviscid case), we gather global convergence results similar to those of dimension two, namely, strong convergence for striated regularity and uniform estimates for the gradient of the velocity. When initial data do not have striated vorticity, we give an upper bound depending on the viscosity for the speed of convergence in L 2 norms. This result is similar to the one stated in [4] for two-dimensional fluids.

Introduction Consid~rons le syst~me de Navier-Stokes incompressible en dimension d -- 3:

{ Otv~, + v~, . Vv~, - rave, = -Vp~,, div v, = 0, v.(0) = ~0,

(NS~)

30

Limite non visqueuse pour les fluides incompressibles axisymdtriques

oh ~, la viscosit6, est une constante strictement positive, la vitesse v~(t, x) est un champ de vecteurs sur ]R 3 d6pendant du temps t >_ 0 et la pression p ~ ( t , x ) est un scalaire. La variable d'espace x d6crit ]R 3 entier et on s'int6resse ~ l'6volution du fluide pour tout temps t positif. Formellement, pour v t e n d a n t vers z6ro, on obtient les 6quations d'Euler Otvo + vo . Vv0 = - V p 0 , div v0 = 0, vo(0) = v ~

(NSo)

I1 est bien connu que, pour un champ de vitesse v ~ "un peu mieux que lipschitzien" et u >_ 0, le syst6me ( N S ~ ) est localement bien pos6 et que v~ tend vers v0 fortement lorsque u tend vers z4ro. Notre r6sultat de r6f6rence, dfi ~ T. Kato, sera le suivant (voir [8])" T h 6 o r 6 m e 0.1 - Soit s > 5/2 et v ~ un champ de vecteurs ~ divergence nulle et ~ coefficients dans l'espace de Sobolev H~(]R3). Alors il existe un temps T > 0 tel que pour tout v > O, le syst~me ( N S ~ ) a d m e t t e une unique solution v~ dans C([0, T]; H ' ) n C I ( [ O , T ] ; g ~-2) et tel que de plus, v~ tende v e r s Vo d a n s C([0, T]; g ~) n el([0, T]; H s-2) l o r s q u e u t e n d v e r s O. Sans hypoth6se suppl6mentaire sur la donn6e initiale, la question de l'existence globale d'une solution r6guli6re reste ouverte mais on dispose du crit6re d'explosion suivant (voir [1] et [9])" T h 6 o r ~ m e 0.2 - Soit ~ > 0, s > 5/2 et v~ E C([0, T*[; H s) une solution de ( N S . ) . ' a p p a r t e n a n t pas ~ C([O,T*]; Hs). Soit w~ - rot v . le yecteur tourbillon associd au champ v . . Alors on a

~0T* II.~(t)llL~

dt-

+~.

Ce crit6re d'explosion permet bien 6videmment de retrouver l'existence globale de solutions r6guli6res en dimension deux. Nous nous int6ressons ici s des champs initiaux v ~ axisym6triques, c'ests de la forme

v o = ~o (~, z ) ~ + ~o (~, Z)~z oh nous avons adopt6 un syst6me de coordonn6es cylindriques (r, 9, z) et not6 er-(cosg,

sing, 0),

ee = ( - s i n g ,

cosg, 0),

ez = (0,0,1),

les trois vecteurs de base au point x = (r cos g, r sin 9, z).

J. Ben A m e u r and R. Danchin

31

Pour v ~ suffisamment r~guli~re, (NS~) conserve cette propri~t~ de sym~trie. Le tourbillon w~ se r~duit alors ~ w~ - w~,o(r, z)eo. En identifiant le vecteur tourbillon w~ au scalaire w~,0, on constate que la quantit~ d~f

a~, = w~,/r v~rifie

(Or +v~.V)a~

-

v( ~

+

~

3

= o.

r

Lorsque u - 0, la quantit~ av est visiblement transport~e par le riot r de v~, et si u > 0, l'op~rateur du second ordre apparaissant dans (T~) a le "bon signe". Ceci a permis b. M. Ukhovskfi et V. Yudovitch d ' ~ t a b l i r dans [14] le r~sultat suivant: T h 6 o r ~ m e 0.3 - Soit v ~ E (H 1(IR3))3 un champ de vitesse axisymgtfique divergence nulle. Notons w ~ le tourbillon initial. On suppose que w ~ w ~ E L2(]R3)ALC~(]R3). Alors, pour tout u >_ O, ( N S v ) admet une unique solution vv clans L ~ (JR +; L 2) telle que de plus w~ E L~oc(]R+; L 2 M L ~ ) et w~/r E L~176 L 2 N Lc~). Cette solution reste azisymdtrique pour tout temps. En combinant le th~or~me 0.3 avec les th~or~mes 0.1 et 0.2, il est ais~ de voir que si v ~ E H s (s > 5/2) et w ~ E L2(]R 3) N L~ alors les r~sultats du th~or~me 0.1 sont valables pour t o u t temps. De plus, la solution obtenue reste axisym~trique.

Dans les trois premieres parties de cet article, on s'int~resse plus particuli~ment k des donn~es initiales de type poche de tourbillon. L'~tude de ce genre de structures provient du cas bidimensionnel non visqueux. On parle de poche de tourbillon lorsque west la fonction caract~ristique d'un domaine born~ de ]R 2. Lorsque u - 0, le tourbillon est conserv~ par le riot de la solution. Un r~sultat de Yudovitch (voir [15]) nous assure alors que la structure de poche de tourbillon est stable pour tout temps: le domaine de d~part est simplement transport~ par le riot. Lorsque la fronti~re du domaine initial a une r~gularit~ h61d~rienne C r (r > I), J.-Y. Chemin montre que la solution v0 de (NSo) appartient c~ + Lloc(]R ;Lip) (off Lip d~signe l'espace des fonctions born~es et lipschitziennes), et que la r~gularit~ C r de la poche est pr~serv~e pour tout temps. Ceci r~sulte en fait de r~sultats bien plus g~n~raux de persistance de la r~gularit~ stratifi~e pour les fluides incompressibles (voir [3] et les r~f~rences jointes). Dans [5], le premier auteur s'est int~ress~ k la g~n~ralisation du r~sultat de J.-Y. Chemin pour les fluides faiblement visqueux. On obtient notamment le r~sultat suivant: T h ~ o r ~ m e 0.4 - Soit ~o un ouvert bornd de ]R 2 dont la fronti&re est une courbe simple de classe C T M (E E]0, l[). Soit v ~ le champ de vitesses

32

Limite non visqueuse pour les fluides incompressibles axisymdtriques

divergence nulle et s'annulant ~ l'infini, de tourbillon w ~ = leo. Alors, pour tout ~ >_ O, (NS~) admet une unique solution v, dans Lzo~ avec donnde initiale v ~ et ii existe une constante C ne d4pendant que de ~o telle que Vv C ]R +, Vt E ]R +, IlVv.(t)ll~,~ < Ce Ct. Notons ~2t,, le riot de v, ~ 1'instant t et f~t,~ = ~bt,,(f~~ On a les rdsultats suivants de convergence: (i) Pour tout e' < e, O~tt,~ est une courbe simple de classe C 1+~'. Plus prdcisdment, si ~yO E CI+e(S1; ]R 2) est une paramdtrisation rdguli~re de Of~~ et si l'on pose "y~(t) = ~t,~(~~ a/ors ~y~(t) est une paramdtrisation rdguliSre de O~tt,~. De plus ~y~ e Lzo~(~+; C 1+~' ($1,]R2)) uniformgment en v e t ~y~ t .d C 1+'' (SX, lo[ q. . t .d O. (ii) Notons (~tt,v)~-- {x e ]R 2, d(x, [~t,v) > h} et (~t ~ )h = {x C f~t,v, d(x, Of~t,~) > h}. Alors il existe une constante C ne ddpendant que de f~o et telle que pour v, t, h > O, on nit h2 exp(--4(eCt'--l))

]lCd0]]L 2 '

[Iw~(t)- lrh,, IlL2 ((r~F,~)h) _< 2 II

~

min

1, c(vt)X/2e2(~C~-X)h e - a-~', exp(-4(~c'-1))

.

Remarque 0.1. Pour des raisons techniques, nous avons ~t~ amends dans [5] ~ utiliser les espaces de Besov B~,o~ (2 < a < +oo) pour mesurer la r~gularit~ stratifi~e lorsque ~ > 0. L'apparente perte de r~gularit~ dans O~tt,~ n'est en fait due qu'b. l'utilisation d'espaces de HSlder dans l'~nonc~! du th~or~me 0.4. Elle n'a pas lieu pour des poches de tourbillon ~ fronti6re dans ~a~c~ R l+e * I En dimension trois, m6me pour un fluide non visqueux, il n'y a aucune chance pour que la structure de poche de tourbillon stricto sensu soit stable, m6me s temps petit" le tourbillon n'est pas constant le long des lignes de flot. Pour un fluide axisym6trique non visqueux, on sait cependant que la quantit~ a0 est conserv~e. Ceci a permis b. P. Gamblin, X. Saint-Raymond et P. Serfati de prouver des re!sultats globaux de persistance de structures stratifi6es (voir [7], [10] et [11]). Dans le cas visqueux en dimension quelconque et sans hypoth6se particuli~re de sym6trie, on dispose de r6sultats de convergence de (NS~) vers

J. Ben A m e u r and R. Danchin

33

(NSo) en un sens qui p%serve la %gularit~ stratifi~e de type Besov (voir [6]). Ces %sultats ne sont bien stir que locaux en temps. Nous nous proposons de montrer que dans le cas axisym6trique, les %sultats de [6] sont globaux. Lorsque la donn6e initiale est une poche de tourbillon axisymdtrique (i. e. w ~ = r l a o avec t2 ~ domaine axisym~trique fronti~re C1+~), nous prouvons en sus l'analogue du th6or~me 0.4, partie (ii).(~ ceci pros que w v e s t chang~ en w~/r). Enonqons le rdsultat de convergence que nous obtenons pour une telle donn6e initiale. T h ~ o r ~ m e 0.5 - Soit f~o c IR 3 un ouvert bornd ~ symdtrie axiale et frontibre de classe C 1+~ (e E]0, 1[). Soit v ~ l'unique champ de vitesses coefficients H 1, ~ divergence nulle et de tourbillon w ~ = rluo. Alors, pour tout v >_ O, ( N S ~ ) a d m e t une unique solution v , dans Llo~ 1) avec donnde initiMe v ~ et il existe une constante C ne ddpendant que de f~o, telle que Yv C ]R +, Vt c IR +, lIVv~(t)tlLor _< Ce Ct 89log(l+t). On a de plus les rdsultats de convergence suivants.

Soit

(i) Pour tout t >__ 0 et ~' < e, Oat,v est dans C l+e,. { f 0 = 0} une dquation non ddgdndrde de On ~ et ft,v = fo o ~t,-1 . AIors { ft,v = 0} est une dquation non ddgdndrde de Of~t,v, fv C Llo ~ (JR +; CX+e') uniformdment en v et f , tend vers fo dans Llo~176

C 1+~') 1orsque v tend vers O.

(ii) I1 existe une constante C ne ddpendant que de f~o et telle que si 1'on note z(t) d~f exp(--4((1 + t) ct 89 -- 1)), on ait pour tout v, t, h > O,

h2

iI w~(t) r

020

la,,~ [I a) 0

< 211TIIL~ min

/

1, C (v

~l/2e2((l+t)ct 89

h2 z(t))

-1)e-9-~

Dans la derni~re partie de ce travail, on abandonne les hypotheses de r6gularit6 stratifi6e pour les donn~es initiales et on s'int~resse k la vitesse de convergence des solutions v~ de ( N S , ) vers celle v0 de (NSo) ~ v ~ fix6e v~rifiant les hypotheses du th~or~me 0.3. Le gradient de v0 n'est alors pas n6cessairement born6 et peut exploser comme un logarithme au voisinage de

34

Limite non visqueuse pour les fluides incompressibles axisymdtriques

certains points. On peut cependant montrer que v~ converge vers v0 fortement dans LtoC~(]R+; L 2) avec une vitesse de convergence qui se comporte comme une puissance de u se d6gradant au cours du temps" T h ~ o r ~ m e 0.6 - Soit v ~ E H 1 un champ de vecteur axisymdtrique ~ divergence nuUe tel que w ~ w ~ c L 2 fq L ~ . Notons v~ la solution de ( N S u ) donnde par le thdor~me 0.3. Alors il existe une constante C universelle telle que pour tout temps T > 0 et viscositg v >_ 0 vdrifiant uTg2(T) <

2e-'~exp(CTIIv~176247176

avec f ( t ) = t~ logt,

g(t)=

I1 ~ I1,~

et C0=2+ on ait IlVu -- VOII L~(O,T;L 2) exp ( - C T l l v ~ II L2 f ( Co 4rTll~ ~ ll L~r ))

(0.1)

Si de plus V v E L 1(0, T; L ~ ) , alors

IIV--VoflL(O,r;L) _< ~

min 2

(T [[wO[]2

IIv~

2 g2

(r)d

(~

avec V ( t ) - fo IlVv( T, ")]]L~ dT. Rappelons que dans [4] J.-Y. Chemin prouve un r~sultat tr~s similaire pour les fluides bi-dimensionnels. Dans ce dernier cas cependant, la vitesse de convergence L'apparition de peut cependant de scaling (voir

est du type /] 89 et non pas ~ 89215189 log(l+T)). la puissance 7 ressemble certes ~t un art~fact technique. On la justifier dans une certaine mesure par des considerations la remarque A.1 de l'appendice).

Nous avons adopt~ le plan suivant" Dans la premiere partie, on rappelle la d~finition des espaces stratifies et conormaux construits sur des espaces de Besov selon [6]. Ceci permet

J. Ben Ameur and R. Danchin

35

d'~noncer un th~or~me g~n~ral de convergence de (NS~,) vers (NSo) dont ddcoule la partie (i) du th~or~me 0.5 (voir the~or~me 1.1). La seconde section est consacr~e ~ l'~tude de l'~quation (T~) v~rifi~e par a~. On y prouve un r~sultat de d~croissance exponentielle hors du support de la donn~e initiale transport~ par le flot. Ceci entra~ne l'assertion (ii) du th~orbme 0.5. Dans la troisi~me partie, on prouve le th~or~me 1.1. On indique les quelques modifications s faire s [6] pour avoir un r~sultat global en temps. La quatri~me partie est d6volue au th~or~me 0.6. On donne en appendice un lemme de r~gularisation utilis~ dans la partie 3 et des bornes sur la croissance temporelle des normes L p du tourbillon pour les ttuides axisym~triques. 1. R ~ s u l t a t s de c o n v e r g e n c e p o u r la r ~ g u l a r i t d s t r a t i f i ~ e 1.1. Espaces de B e s o v et espaces stratifies Dans toute cette partie, d d~signe un entier strictement positif. D~finis-sons un d~coupage en fr~quences dyadiques: la d~composition de LittlewoodPaley. P r o p o s i t i o n 1.1 - I1 existe deux applications radiales ?~valeurs darts [0, 1], X C C ~ ( B ) et ~ ~ Cg~(C) (avec C -

{ z ~ lRa [ 5/6 < [xt < 12/5} et s

B --- {X C ~ d

[ IX[ ~___6 / 5 } ) telles que" qEIN

On d6finit alors des op~rateurs ZXp et Sp de s C ~ ( N . f ) ) qui correspondent respectivement k des localisations en fr~quences voisines de 2p pour Ap et plus petites que 2p pour Sp. Plus pr~cis~ment, soit h = ~-- 1~ et h = 9c - 1X. On pose

Apu = 0 si p < - 2 ,

A - l u = )((D)u = h , u,

Avu = ~ ( 2 - P D ) u = 2pd f h(2Py)u(x - y ) d y si p > 0,

=

=

Z

= 2;" [

- y)dy.

,.]

q~p--1

D ~ f i n i t i o n 1.1 - Pour a c [1, +c~] et r C JR, on note Bar(IR d) l'ensemble des distributions u C S ~(~d) telles que

ll IIB: a2

q

2

Ii

llLo <

36

Limite non visqueuse pour les fluides incompressibles axisymdtriques

Remarque 1.1. On dispose d'une caractdrisation des espaces de Besov par diff4rence finie qui permet de faire le lien avec la d6finition habituelle des espaces de H61der (voir [131). En effet, pour r E]0, 1[ par exemple, B~(IR d) est l'ensemble des 416ments u de La(IR d) tels que

IlUlIL-

+

sup O
[17"hu - uilL"

< +oo

off

ThU(X) = u(x + h).

[hi"

R e m a r q u e 1.2. L'espace (B~ (]Rd), I1" IIBr est un Banach. C'est une alg~bre lorsque r > d/a. Dans le cas particulier off a = +oo, on note C r dd.~_fBr( 2 ~ * .o L'espace C ~ coincide avec l'espace de H61der usuel lorsque r e s t positif non r d entier et on a l'injection B~ ~-+ C - z . R e m a r q u e 1.3. On a l e classique r6sultat d'interpolation suivant: Vu e B~(IR d) n B~(IRd),

_< II

V0 e [0, 11,

,llBa~

1--O

.

Nous allons maintenant construire des espaces stratifi4s de type Besov. Les notations adopt4es ici sont celles de [6]. On les rappelle pour la commoditd du lecteur. Notations. Si X est un champ de vecteurs g coefficients et divergence dans Bar, on pose

IIXIIB::-d6f

sup l
IIx i IIBx

et

NX[IB;~ " d6j IlXll Bx + II div X IIBS.

On d6finit la d6riv6e d'une fonction born6e u selon un tel champ X en posant X ( x , D ) u = Oi(Xiu) - u d i v X. Si (fa)~eA est une famille d'616ments d'un espace de Banach (B, I1"IIB), on 6crit d6f

[[Ifl[[ B = sup [[A [[BAEA

Enfin, on notera A A B le produit vectoriel entre deux 616ments de ]R 3. D 6 f i n i t i o n 1.2 - Soit a E]3, oo[, e c]3/a, 1[ et m E ]hi*. On dit qu'une famille X - (Xi)l
I(X) d~__f inf

sup IX), A X~(x)[ 89 > 0.

xEIR 3 ~ # / ,

37

g. Ben A m e u r and R. Danchin

On note a/ors B ~ ( X ) l'espace des fonctions u borndes s u r ]R 3 et telles qua X ~ ( x , D ) u C B~-~(IR z) p o u r tout i ~ { 1 , - . - , m } , m u n i de la norme

(1)(

IlUl]B:, x d,~f I ( X )

-

IlullL~ IlIxillBi + [llX(x'D)ulllB:-'

)"

Comma cas particuliers d'espaces stratifi6s, nous allons d6finir des espaces conormaux par rapport ~ des surfaces $ rdgularitd dans une classe de Besov. 1 . 3 - Soit a c [3, +oc] et r > 3/a. On dira qu'un f e r m d de ]R 3 est une surface de classe B~ +1 s'il existe un voisinage V de F, une application f C B~+I(]R 3) telle qua V f ne s'annule pas sur V et E f - l ( 0 ) A V. On dira qu'un champ X ~ coe]~cients B~ est tangent h E et seulement si X . V fl ~ =_ O. On notera J r ( E ) l'ensemble des champs classe B~ tangents h E.

Dfifinition

et = si de

1.4 - Soit a E [3, +oo], e E]3/a, 1[ et F, une surface compacte de classe B 1+~. On ddfinit alors l'espace conormal B~,E associd ~ F, par

D~finition

B a,E e = {ucL

~ I VX e Ta~(E) , div(Xu) e B~~- 1} .

Pour terminer, signalons que l'espace B~,E contient la fonction r l~ off ~t est l'ouvert born~ d~limit6 par E. Plus g~n~ralement, on ale r6sultat suivant (voir [6]): P r o p o s i t i o n 1.2 - Soit ~ un ouvert bornd ~ frontibre B 1+~ avec e El3, +oc] et e C]3/a, 1[. Alors p o u r toute fonction r E B~, on a 1~r E Ba~,~. 1.2. U n r 6 s u l t a t de c o n v e r g e n c e p o u r la r 6 g u l a r i t 6 stratifi6e ou conormale Commenqons par d~finir la notion de champ de vecteurs transport~ par le riot. D ~ f i n i t i o n 1.5 - Soit v un champ de vecteurs lipschitzien et ~ son riot, c'est-~-dire l'unique solution de l'dquation t

r

x) = x +

Soit X ~ un champ de vecteurs de BEa. On dira qua X t est le champ transportd de X ~ par le riot de v ~ l'instant t si X t ( x ) -- X ~ 1 6 2 1 6 2

l~non~ons notre r~sultat de convergence de ( N S , ) vers ( N S o ) dans toute sa gdn~ralit~.

38

Limite non visqueuse pour les fluides incompressibles axisymdtriques

T h ~ o r ~ m e 1.1 - Soit a E]3, +oo[ e t e E]3/a, 1[. On suppose que le champ de vecteurs v ~ est ~ coemcients dans H 1(1R3), ~ divergence nulle et que son tourbillon co~ vdrifie co~ E L 2 ML~176 On suppose de plus que l'une des deux hypotheses suivantes est vdrifide: (~1) w ~ est dans B ~ ( X ~ avec champs de vecteurs.

(7/2)

(X~)l<)~<mfamille

(e,a)-substantielle de

coo est dans B~,ro off ~0 est une surface compacte de classe B l+e.

Alors pour tout u > O, le syst~me ( N S u ) admet une unique solution v~ dans Lzo~176 Lip) M C(]R+;H1). En outre, vv tend vers vo et Cv - Id tend vers ~bo - Icl dans Llo~c(lR+; C 1-~) pour tout rI > 0 et il existe une constante C ne d @ e n d a n t que des donndes initiales et telle que

V~ E ]R +, Vt C ]R +, IIVvv(t)llL~ _~ Ce ct 89log(l+t). Sous l'hypoth~se (7-/1), on a de plus le rdsultat suivant: Notons Xt,v la famille transportde p a r le riot de v~. Alors Xt,~ reste (e, a)-substantielle pour tout temps t ~_ 0 et wv(t) reste dans BEa(Xt,u) uniformdment en v. Enfin, Si e' < e, a/ors X ~ ( x , D)r X~,~ et div X~,~ tendent respectivement vers X ~ ( x , D)r Xo,x et div Xo,~ dans Llo~176 B~'), et X . , ~ ( x , D ) w . tend vers Xo,~(x,D)wo dans L~oc(IR+; B~'-I). Sous l'hypoth~se (7-/2), on a de plus le rdsultat suivant" Pour tout t ~_ 0 et u > O, Et,u d6_~_fCt,u(EO) est une surface compacte de classe B TM, et o~u(t) est dans B~,~.~, . Enfin, Zt,~ tend yers ~t,o au sens suwant. Soit { f o = 0} une dquation de classe B~ +E de p o. Posons ft,, ear f ~ r

Alors {ft,~ = 0} est une dquation de Et,~ et pour tout

e' < e , f~ tend vers fo dans L~o~(IR+; B[ +~') lorsque u tend vers O.

Remarque 1.4. L'hypoth~se v ~ E H 1 est automatiquement v~rifi~e d~s que ~o c L p N L ~ pour un p _< 6/5 (c'est le cas par exemple si coo c B ~ ( X ~ est support compact ou dans L 1). Remarque 1.5. Lorsque u = 0, la partie existence et r~gularit~ d'une solution pour le syst~me d'Euler s'~tend aux espaces de HSlder (voir [7], [10] et [11]). Dans le cas u > 0, la preuve du th~or~me 1.1 utilise des estimations uniformes en u pour les solutions d'une ~quation de la chaleur avec terme de convection. De telles estimations ne sont pas connues dans le cadre des espaces de HSlder (voir [5] section 4) mais sont vraies dans les Bar pour l < a < +oo. Remarque 1.6. Si (7-/2) est v~rifi~e, on peut construire une famille (e,a)substantielle X ~ constitute de 6 champs de vecteurs tangents ~ N o et telle que coo e B~ (X ~ (voir [6] partie 5).

J. Ben A m e u r and R. Danchin

39

Indiquons bri~vement pourquoi le (i) du th6or~me 0.5 d6coule du th~orbme 1.1. Supposons que w ~ = r l a o avec t2 ~ ouvert born6 de classe C I+E. Comme les fonctions de C 1+~ ~ support compact sont aussi dans t o u s l e s B I+E, on a w ~ c B~,oao pour tout a > 3/e d'apr~s la proposition 1.2. L'hypoth~se (7-/2) est donc v4rifi4e. On conclut en appliquant le th4or~me 1.1 puis en utilisant r4p4tition que B~ ~-. C e - ~ .

2. Un r~sultat de d4croissance exponentielle Dans cette partie, on 4tudie le syst~me

{(0~ + v. V)a - .(O~a + O~a + }O~a) = O, air=0 = a0~

(T~)

que v~rifie w~/r avec le champ de vitesses v -- v~. Nous nous int~ressons au cas oh a0 est la fonction caract~ristique d'un domaine born6 et off v est lipschitzien. Dans le th4or~me suivant, nous montrons qu'aux 4chelles spatiales grandes devant x/-~, la fonction a(t) est proche de la fonction caract6ristique du domaine transport6 par le riot. T h 4 o r ~ m e 2.1 -- Soit ~ > 0 et v E -"'Llo~(ltt+;Lip(ltt3))__ -^"~" - "~" un champ de vecteurs azisymdtrique ~ divergence nulIe. On suppose que a = a(t, r, z) vdrifie ( I v ) avec ao -- ao(r,z) C L2(]R3). N o t o n s Fo le support de ao, ~2 le flot de v et Ft = Ct(Fo). Posons (Ft)h = {x e ]R 3, d(x, Ft) > h}, (F~) h =

{~ e f~, d(~, OF~) > h} et V(t) - fo

a

[[a(tlllL2((Ft)~) <

IiVv(s)llL~

ds. Pour tout h, t > O, on

h 2

e -2-6~vt exp(-4V(t))

]]aO]lL2.

(2.1)

Dans le cas off ao est la fonction caractdristique d'un domaine bornd Fo, on a de plus

i[a(t) - 1F~ IlL2((FT)h)

(2.2) < 2 i]a01[L2 min

1

1+ C

e-9--6~vt exp(-4v(t))

off C est une constante universelle. Ddmonstration. Elle ressemble beaucoup ~ celle du th~or~me 1.1 de [5]. La principale diffgrence provient du terme du second ordre dans (T~) qui n'est plus un Laplacien "classique".

Limite non visqueuse pour les fluides incompressibles axisymdtriques

40

Par une mdthode d'dnergie, on trouve v t e ]R +

,

2 Ilao ILL.-.

Ila(t)ll 2L~ + 2~ "] ~ IlVa(s)lt 2L"- ds < Jo

(2.3)

Soit (I)o C C ~ ( I R 3) axisymdtrique (c'est-5~-dire ne ddpendant que de r et de z). On pose ~O(t,x) = (I)o(r si bien que ~ est conservde par le flot et reste axisymdtrique. Supposons dans un premier temps que v e Lto~176 (S(]R3)) 3) et a e Lzo~176 S(]R3)). D'apr~s (T.), ~ a vdrifie

(Ot+v.V-u(02+02+

3

O~))(~a)=-ua(O2+O2+-O~)~-2uV~.Va.

(2.4)

r

Supposons de plus que 9 est constante hors d'un compact (qui ddpend du temps). Dans ces conditions, ~a(t) est dans H 1. Rappelons par ailleurs que pour une fonction u = u(r, z), le Laplacien se rdduit s Au = O~u + O~u + O~u/r. En prenant le produit scalaire de (2.4) avec ~a(t) au sens L 2 et en se souvenant que div v = 0, on trouve ld

2

/m

2 dt ]l~a[[ L2 --u

3

2 ~ a ( A ( ~ a ) + -O~(~a))r dx a 2 ~ ( A ~ + -O~(~a)) dx - 2u

= -u 3

r

~ a V ~ . Vadx. 3

En intdgrant par parties, on obtient 2l ddt "" " 2 " + u[lV(~a)l] 2L 2 (__ /ma2 (lla2allL,] < u I[aV~]I 2 L 2 -

3

r --Or~2 dx )

(2.5)

Pour d~montrer (2.1), prenons (I) de la forme O(t,x) = e x p r avec r = fo(~2-1(t,x)) et f0 axisymdtrique constante hors d'un compact. On a visiblement IlaVOllL2 <_ IIVr I]a(~llL2. Pour majorer le terme deux), on dcrit

IR -a2- o ~ 3

r

2 dx

f.a'3 r

~0~

]<

2 ff0~r

2 dx (qui n'appara~t pas en dimension

/IR ~a2 (I)2 d x , 3

r

(2.6)

J. Ben A m e u r and R. Danchin

41

Une int6gration par parties donne

f0

+cr a2(I)2 dr = - 2

/0

ad)O~(a~)r dr

Injectons cette expression dans (2.6) puis appliquons l'in6galit~ de CauchySchwarz. On trouve

Is 3 -a2r- 0 , . (~2 d x

4 IlO~r

_< 4 Iio~r

IlaOIIi~ IlOr(aO)lli~, 2

2

2

lla~l15~. + IlO~(a~)llz~. 9

En revenant ~ (2.5), on obtient ld 2 2 dt/" "~'ll~al12L ~) < 5P ]tVr _

L:r

2 II(~allL~ ,

et le lemme de Gronwall donne donc

foilVr

II(~a)(t)llL~ ~ e5~'lW~

d~ll(~a)(O)llL~

A ce stade, on peut conclure exactement comme dans [5] en supposant d ' a b o r d a0 ~ support compact et en choisissant f0 = a min { R , d ( x , Fo)} avec R > 0 et a > 0 (convenablement %gularis6e), puis en faisant tendre R vers +oo et en prenant le "meilleur" a. Pour d~montrer (2.2), posons w ( t , x ) = a ( t , x ) - 1g,(x) et ~ ( t , x ) = 9o ( r avec ~o E Co~(Fo) axisym6trique. Remarquons que 1F~ vdrifie (Or + v . V)IF, = 0 et que Supp ~t est compact dans Ft, ce qui implique d2tOrw = ~tOra. On en d6duit donc que

(

Ot-[-v" V - - U Ant---Or r

=-uw

A+-0r r

((~W)

(2.7)

~-2uV~-Vw.

Une m6thode d'~nergie donne 1 d -2- dt

(ll~wll2L~)

2 2 + ~ IlV(+w)llL~ < ~ IlwV+ll L 2 -

-

/~

w2c3~(I)2 3

r

d~.

(2.8)

Soit ho > 0 et Xo E C ~ ( F o ) axisym6trique vatant 1 sur (F~)ho et ~ valeurs dans [0,1]. On impose de plus que IIV)iOlIL~ <_ Coho I oh Co est une constante universelle. On choisit alors (I)o = x2e/~ avec fo = ad(x, F~). Notons ft = fo o C t 1 et Xt = Xo o g2t 1.

Limite non visqueuse pour les fluides incompressibles axisymdtriques

42

En remarquant que I l v w ~ l l [ ~ trouve ii~Vr

~ ~'(~) et que I1~11~ -< 2 IlaoliL~, on

,~ < s IIx, vx,w s,, il .~ + 2 Iir --

L 2

,

-< 2e2V(t) (16C2oho 2e2~h~ Ilaoll 2L~ + c~2 II~wll 2L~)On a par ailleurs

dx = 4 3

r

w Xt O~Xte2A dx + 2 3

r

dx. 3

r Y I2

11

J

Remarquons que jf

2 2 dx. Xtw

En proc6dant comme pour majorer/1as a2(I)2r dx, on obtient donc

151 _~ 8 IIx

a,x, "

IIx,~llL~. IlO,(x,~)llL~,

(2.9)

Comme XtO~w = XtOra, on en d~duit que

ILl ~ 16Coho leV(t)e2ah~ IlaollL~ (lla,-allL~ + 2Coho leV(t) IlaollL~). (2.10) Pour le t e r m e / 2 , on 6crit simplement

II~1 ~ 4 IlO,f, llL~ IIr IIO~(~W)IIL~, <_ IIO,.(~w)il L2 ~ + 4a2e2V(t)I1~11 L~ 2.

(2.11)

En injectant les in6galit6s (2.10) et (2.11) dans (2.8), on conclut que 1 dt d 2v

2 < 2v(t) [[~WIIL 2 2 + 64C2oho2e2ahoe2V(t) [[aO[[L2 2 -- 6a2e

[[(I)WllL2

+16Coho leV(t)e2ah~ IlaollL~ IlO~allL~, d'ofi ld 2 dt

~

I1r

L~ ~ 6~'~2 e2y(,) IIr + 128vC2oho 2e2~h~

2

(2.12)

IlaollL~ + ,~2~ho ilOrall 2L 2.

d. Ben Ameur and R. Danchin Intdgrons (2.12) en temps. Grs

II'I'wl]2L2 < ~2~o Ila011L22 ( 1 + -

-

43

?~ (2.3), il vient

256Co2ho 2

U~ote2V(r)dT) +12~,

dv(-) ile~(~-)ll ~L~. dr.

Le lemme de Gronwall donne donc

L2

1))

__~e2c~ho lla0112L2 (e12cc2u for e2V(~.)dT

On peut alors conclure comme dans [5]. En effet, si l'on prend h0 = he-V(t)/2, on trouve

<_ Ilaoli2L2 e ~hr

(e12~2vtr

I1 ne reste plus qu'~ choisir c~ =

+ 256C2oe2V(t) (e12~2,te2v(.)_l)) 3h2a2

he-3V(t) 24ut

pour obtenir (2.2).

m

3. P r e u v e d u r 6 s u l t a t d e c o n v e r g e n c e p o u r l a r d g u l a r i t 6 strati-

fide o u e o n o r m a l e On traite d'abord le cas off l'hypothbse (~'~1) est vdrifide. On procbde en quatre 6tapes: 1) Preuve d'estimations dans les espaces stratifi6s pour les solutions de ( N S , ) lorsque la donnde initiale est rdguli~re. 2) Rdgularisation de la donnde initiale et preuve d'estimations uniformes pour la solution correspondante. 3) Convergence des solutions rdgularisdes vers la solution de (NSv). 4) Convergence de (NS~,) vers (NSo). P r e m i e r e 6 t a p e : I1 s'agit de prouver la proposition suivante. P r o p o s i t i o n 3.1 - Soit a c]3, +oc[ et e E]3/a, 1[. Soit v ~ C HC~(]R 3) un champ de vecteurs axisymdtrique ~ divergence nulle. On suppose en outre que a ~ d6j wO/r C L 2 N L ~ . Alors (NS~) a une unique solution globale v c C ( I R + ; H ~ ) . De plus si (X~)I
44

Limite non visqueuse pour les fluides incompressibles axisymdtriques

de champs de vecteurs a/ors la famille Xt transportde par le riot qd de v reste (e, a)-substantielle pour tout temps, et on a l e s estimations suivantes (pour une constante C ne ddpendant que de E et de a): Ii~(t)/rllL~ <_ II~~

II~'(t)llL= < II~'~

II~(t)llL~ < II~~ -

+ C IIv~

pou~ tout

+tllv~

p C [2, +cx~],

II~0IIL~,

s -(2 + II~~176 _

IIv~

(3.2)

+t II~~

II div x~,~lls~ <_ II div X~[IB ~e Cv(t),

(3.4)

IIx~,~ IIB~ + IIX~ (~, D)~(t)II B~ < c (11 div(X~ | ~~ [B~-' + ilX~llB~)eCW(~) -- \ II~011L~ eCW(t)

[[~(t)lls:,x, ~ CIl~~

(3.1)

(3.5) (3.6)

,:%V e C

v(t) =

~0 t IIW(~)IIL~ dT,

w(t)-- I1~O11~t + v(t),

f ( t ) = t~ logt.

En/~n~ on a

[[Vv(t)[[Lor <_ CLOeCtll'~

(

2+

IIo-,~JIL2nL~ II.~~

(3.7)

~t v e c

L ~ ~J (Jl~~

+ IIv~

+

[[~0[]B:,xO

II~~ IIL~nL~ + IIv~IIL~)"

Ddmonstration. L'existence et l'unicit6 d'une solution globale v axisym6trique v6rifiant entre autres w c Llo~c(]R+;L 2 0 L ~ ) et w/r C L~(]R+; L 2 0 L ~176r6sulte du th6or6me 0.3. Comme v ~ est de plus dans tous les H s, les th6or6mes 0.1 et 0.2 nous assurent que v 6 C(]R+; H~176 Les in6galit6s (3.1), (3.2) sont d6montr6es dans [14]. On renvoie au lemme A.2 pour la preuve de (3.3). Le fait que la famille Xt reste substantielle d6coule de la proposition 4.1 de [6]. I1 e n e s t de m6me pour les in6galit6s (3.4), (3.5)et (3.6).

45

J. B e n A m e u r and R. D a n c h i n

Pour prouver (3.7), on utilise l'estimation stationnaire suivante (voir par exemple [6] proposition 1.3 et les r6f~rences jointes)"

IlVv(t)IIL= <_

c ( I]~(t)IIL=~L= (e + log

II (t)ii 2

(3.8) 9

I1 suffit alors d'injecter les in~galit~s (3.1), (3.2),(3.3) et (3.6) dans (3.8) puis d'appliquer le lemme de Gronwall pour obtenir l'estimation souhait~e. D e u x i ~ m e ~ t a p e : On se donne une vitesse initiale v~rifiant l'hypoth~se (;H~) du th~or~me 1.1. On la %gularise comme dans [3] & l'aide de la d@omposition de Littlewood-Paley radiale donn~e dans la proposition 1.1: on pose v n0 - S~ v 0. C o m m e v o est dans L 2 , on a clairement v n0 E Hoo . De plus, gr&ce au lemme A.1 de l'appendice, on a pour p E [2, +co],

_< c ,,,,ll ~ r

Lp

ce qui permet d'appliquer la proposition 3.1. Pour tout u >_ 0, on a done construit une ( N S ~ ) qui correspond & la donn6e initiale v n. 0 la proposition 3.1 s'appliquent, mais les termes donn@ initiale done de n. I1 est en fait classique

[lS

~

a,X 0

__ It ~

a,X 0

solution %guli~re vn,. de Toutes les estimations de de droite d6pendent de la (voir [3] on [5]) que

.

On conclut done au % s u l t a t suivant.

Proposition 3.2 - La solution Vn,v vdrifie routes les e s t i m a t i o n s de la p r o p o s i t i o n 3.1 avec une c o n s t a n t e C i n d d p e n d a n t e de n et de t/. T r o i s i ~ m e ~ t a p e : La convergence de Vn,v vers v~ solution de ( N S v ) v~rifiant les propri~t@ de r6gularit~ stratifi~e voulues se fMt comme dans [5]. On commence par ~tablir que la suite est de Cauchy en petite norme (dans des espaces de H61der ~ indice n~gatif par exernple) puis on interpole avec les estimations uniforrnes de la proposition 3.2. La limite v~ v~rifie de plus toutes les estimations de la proposition 3.1 avec une constante C inddpendante de t/. Q u a t r i & m e ~ t a p e : I1 s'agit d'~tudier la limite non visqueuse de v~. D'apr~s l'estimation (0.2) du th~or~me 0.6, le champ Vn,, tend vers vn,0 dans L1o~(R+; L2). On utilise alors les estimations uniformes des ~tapes 2 et 3,

46

Limite non visqueuse pour les fluides incompressibles axisymbtriques

et les rdsultats de convergence de l'dtape 3 pour obtenir les rdsultats de convergence pour la rdgularitd stratifide. I1 reste k traiter le cas off (T/2) est vdrifide. D'apr~s la remarque 1.6, on salt ddjk que (7-/1) est vdrifide pour une famille X ~ (e,a)-substantielle bien choisie. On dispose donc d'une unique solution globale vdrifiant des propridtds de rdgularitd stratifide. En appliquant k cette solution les rdsultats de la partie 5 de [6], on obtient les propridtds de convergence voulues pour la rdgularitd conormale, m

4. C o n v e r g e n c e tifide

pour des donndes initiales sans rdgularitd stra-

Cette partie est consacrde ~ la preuve du thdor~me 0.6. Sous les hypotheses w ~ ~ E L 2 N L ~ , l'existence de solutions ayant les propri~t~s voulues est assurde par le thdor~me 0.3. En appliquant une ddf

mdthode d'dnergie ~ wv - v ~ - v0, on obtient d IIw~llL= 2 + ~ IlVw~IIL= 2 -< ~ IlWolIL~ ,,~w IIV II ,,L2 + I(t) 21 dt avec

(4.1)

r

I(t) = Jm ~ Iw,(t, ~)121Vvo(t, x)l d~. On constate par un calcul direct exploitant l'incompressibilitd du champ v0 que 11~TVOl]L2= IlwollL2. En revenant ~ (4.1), on trouve done 1 d

2

/2

2 dt [IWu]IL2 --< -4

2

IIcMO[]L2 + I.

(4.2)

Traitons d'abord le cas tr6s simple oh la solution vo de (NSo) est dens 1 + ;Lip). On a Lzoc(]R I -< [lw~ll2i~ rlVv0 ILL=(4.3) En injectant (4.3) dens (4.2) puis en appliquant le lemme de Gronwall, on en ddduit finalement que

IIw.(t,

L= -< ~ I1~o(~,')11 =L~ef= IIv~o(~')rl~- d+ d~. .)jl~ ~ fo ~

Sans hypothbse particulibre de symdtrie, une mdthode d'dnergie appliqude t~ l'dquation du tourbillon Ot~0 + vo. V~o = (~0" V)vo

J. Ben Ameur and R. Danchin

47

permet d'obtenir

lifo(t, ")IIL~ < I1~~

~s

>,. ~ ~..

(4.4)

Dans le cas axisym~trique, on dispose aussi de l'in6galit6 (A.7) de l'annexe. On en d~duit (0.2). Dor~navant, on ne suppose plus que Vv0 est borne. L'in~galit~ (0.1) se montre en adaptant la preuve de [4]. Fixons un r~el a _> 2. Rappelons que d'apr~s [12] pages 42 et 250, il existe une constante C universelle telle que

IlVvollLo -< Ca II~,ollLo.

(4.5)

En appliquant l'in~galit6 de HSlder pour majorer I, on obtient donc I < Ca

It~ooliLo ilw,,ll~.~-~_%

(4.6)

Sn combinant l'in~galit~ de Sobolev iiztiLo _< 4 IlVzllL~. (voir [2] page 162), l'in6galit~

i-~

IlzllL-~:, _< IlzllL~ ~ Ilzil;,~,

(4.7)

le f a r que IlVv~llL~ = ll~IIL~ et l'estimation (A.7), on obtient

ii~vll 2 ~

2-~-< c(ll ~~ !!~ + t I1~~II ~ Ii ~~ ~ ) ~_ ltw. 11.~.

Par interpolation entre (A.7) et (A.4), on a

II~(t.-)ll~ ~ llv~

(Co + t ll~~

--~ 0og(C0 + t il~~

1-~

En injectant cette derni6re in6galit6 dans (4.6) et en utilisant (4.7), on obtient finalement I(t) <_ Ca Ilv~

+~ (Co + t I[a~

log(Co + t [la~

~) IIw, l]~ ~ , (4.8)

d'ofi, en revenant 5~ (4.2) et en utilisant (A.7), d

2

v

~-~ LI~(t,.)li. _< ~(ll~~ ~+~

+ t tl~~

Jl~~

~ 2-~

+Ca I!~~I!~. /(Co + ~ I!~~ ~ ) E l ~ . It~ avec f (t) = t ~ log t.

(4.9)

Limite non visqueuse pour les ~uides incompressibles axisymdtriques

48

Fixons un 5 E]0, e - ~ [ et notons

Xs,v(t ) d__~ f

Ilw.(t,

.)[[2

Lz

+5.

L2

Le fait que w~ c C(]R+; L 2) et que w~(0) = 0 assure l'existence d'un temps T~,~ > 0 (que l'on peut supposer maximal) tel que x~,~ < e--~ sur

[0,T~,~[. Autorisons la "constante" a de l'in~galit~ (4.9) ~ d @ e n d r e du temps. On a sur l'intervalle [0, T~,~[:

(

x~,~(t) ~-~

ilvOllL~+tll~~

+ Ca(t)IIv~

)2

f(Co + t

II~~

~ (t).

Choisissons a(t) d=6 f - ~3 log x~,~,(t) (ce qui entrMne bien a(t) _> 2 puisque

xa,~ < e - ~ ) . Apr~s integration, il vient

( II~~ IIL2 + t ll~~ )2 IIv~ +CIIv~ /otf(Co + ~ll~~

~t xa,~(t) < ~ + 7

avec #(z) = - z log z. Comme la fonction z ~ - z log z est croissante sur ]0, e-l], et que nous avons x5,~, C]0, e - ~ [ sur [0, Ta,~,[, on peut appliquer le lemme d'Osgood l'in~galit~ ci-dessus (voir par exemple [4]). En r e m a r q u a n t que z ~ l o g ( - log z) est une primitive de #-1 sur ]0, e-l], on en d~duit que pour tout t c [0, T~,~[ tel que < e -1 on

a

(_

~t ([I~~

~o

)2)) _ log(_ logx6, (t))

/o'

J. Ben A m e u r and R. Danchin

49

2]

d'ofi

exp(--CllvOllL2tf(Co+TI]c~OllL~r

.t

x~.~(t) < ~ + -ff

ilv01lL, +tlI~~

En faisant tendre 6 vers 0 et en utilisant un argument de bootstrap, on obtient (0.1). II

Appendice On g~n~ralise le lemme 3.1 de [10] au cas de r~gularisations construites partir de troncatures qui ne sont pas ~ support compact. On peut ainsi, dans l'6tape 3 de la partie pr~c6dente, r~gulariser les donn~es initiales l'aide d'une d6composition de Littlewood-Paley radiale et r~utiliser telles quelles les estimations de [5] sur la r6gularit6 stratifi6e. L e m m e A.1 - Soit X E S(]R 3) radiale. On suppose que w est le vecteur tourbillon d'un champ de vitesse v axisymdtrique. Notons ~u (X) = n 3 x ( n x ) , O-)n = ~n $ (M, Ol = o2/r et O~n = COn/r Ofl r ddsigne la distance g l'axe. Alors il existe une constante C ne d @ e n d a n t que de X et telle que

llC~nllL~ ~ C ]lC~ltL, pour

1 <__p <__+o0.

Ddmonstration. Notons x = (Xl,X2, X3) - - (Xt, X3) et ~ = (0,0, x3). Le caract~re radial de )/ et les propri6t~s de sym~trie du vecteur tourbillon nous assurent que

f x~(~ - y)~(y)dy = 0

pour tout

x C 1[:{3.

On en d6duit donc que OLn(X) :

f

Cn ( X ) - ~

ly'l

( ~ n (X -- Y) -- ~)~n

(~_y)) ~

,,

,

(y) d y

J

+ / (1 - Cut,~,,ly'l ~ J V ~ ( X -- Y)~(Y) dy. N

oh l'on a not6 Cn(z) d~f llz,l<_n_l (z).

J

50

Limite non visqueuse pour les fluides incompressibles axisymbtriques Supposons qu'il existe une constante C ind@endante de n et telle que

max ( f lK~(x, y)l dx, / rK~(x, y)l dy) < C

pour

i -- 1,2.

(A.1)

Alors le lemme de Schur donne IlanllLp _< C IIc~llLp pour 1 _< p _< +oe. Montrons que (A.1) est effectivement v~rifi~e. En appliquant la formule de Taylor avec reste integral, on obtient les in~galit~s suivantes:

/01s

f IK~(x'y)ldy <<-r <__Cn(x)

3 lY'l l V X n ( 2 " + t ( x - x ) - y ) l d y d t '

(foolfiR3 1(2"+t(x-x)-Y)'l

IVxn(x+t(x-x)-Y)i

dydt

/o1/o~tlx' I ]~7x~(~+t(x-~)-y)ldy dt ) ,

+

<--s IVXn(Z)I]z'Idz + n-1 f•3 'VXn(Z)Idz, < I1(1 +

I.I)VxllL~

9

Comme V X est dans S(]R3), il existe une constante C2 telle que IVx(z)l <

c2 (1 + Iz'l)2(1 + Iz3l)2"

Utilisons cette in~galit~ et la formule de Taylor avec reste integral. I1 vient

f fI(~n(~,V)f dx <_ naly'[

folj~lx',
<- c2naly'] x

(/R

- y'), z3)l dz3 dx' dt,

e ) (1 + Ix3])2 dx3

(jfol ~z

(A.2)

1 ) ,,
Lorsque lY'] -< 2n-1, on ~crit simplement que (1 + n l y ' - t x ' l ) -2 _< 1. De l'in~galit~ (A.2), on d~duit alors ais~ment que

/I

K ~ ( ~ , y ) I d~ <_ C.

51

J. B e n A m e u r and R. D a n c h i n

Lorsque ly'l > 2n-1, on a visiblement [ y ' - t x ' ] >_ [y'[/2 pour tout t e [0, 1] et x E IRa tel que Ix'] _< n -1. De (A.2), on d6duit donc

nlY'l < C t. (1 + 2[Y'[) 2 -

IK l (x, Y)I dx < C

I1 ne reste plus maintenant qu's borner f ]Kn2(X, Y)[ dx et f IK~(x, Y)I dy ind~pendamment de n. Ceci se fait de la mani~re suivante"

f lK~(x,y)id~<_/(1-r

- y)ldx + --/(1-r

<__f

iXn(z)]dz+nf

I~(x - y)l d~,

i~'l Iz'liXn(z)Idz,

_< I[(1 + I.I)VxlIL1. Un calcul strictement analogue donne 6galement

lI

K e ( x, y)[ dy <__ ll(1 + I . l ) V x I I z l

.

A.2. Estimations pour le tourbillon Dans cette partie, on am61iore un peu les estimations de [14] sur la croissance en temps des normes L p du tourbillon. Nous obtenons le r6sultat suivant" L e m m e A . 2 - Soit v C L ~ ( O , T; H 1) un c h a m p de vecteurs axisymdtrique solution de ( N S ~ ) avec une donnde initiale v ~ C H 1 telle que w ~ C L 2 N L ~ e t a ~ d6f = r _1020 E L ~ . On suppose que ~ C L ~ ( 0 , T ; L 2 f3 L ~ ) e t a d~f = r - l w E L ~ ( 0 , T x IRa). Soit Co dgf 2 --l-

Itv~II-~

I1 existe une constante C universelle teUe que pour p C [2, +c~[, on ait

iiw(t)iILp < llw~

+ C p t Ilv~

2 lia~162 (Co + t Iia~

~-~,

(A.3)

et, dans le cas p = +oc,

+

_< II ~ c t ]iv~ 2 ]Ia~162

(Co+t

]]a~

log(Co+t Iia~

(A.4)

52

L i m i t e non visqueuse p o u r les ttuides incompressibles axisymdtriques

D d m o n s t r a t i o n . On reprend la preuve de [14] en utilisant "au mieux" les injections de Sobolev. On peut d'embl~e remarquer que, comme w C L 2 N L ~ et v c H 1, le champ v et son tourbillon w sont dans tous les L p pour 2 < p < +c~. I1 s'agit maintenant d'obtenir des in~galitds pour ces normes L p.

Consid~rons l'~quation du tourbillon en coordonn~es cylindriques: o~

+ v .w

- ~

(1

o~(~o~1

-

~

+ O2zz~

)

--

VrO~"

Multiplions cette ~quation par rwlwl p-2 et int~grons en (r,O,z). Une integration par partie permet de constater que le terme provenant de la viscosit6 est positif. Le membre de droite se majore ~ l'aide de l'indgalit~ de H61der. On obtient ld

p--1

d'oh, apr~s integration en temps,

..~(t. )11. ~ II~~

+ ~o ~ IIo~(~-, ")IIL~ IIv(~-, ")IIL~ d~.

(A.5)

En faisant tendre p vers l'infini dans (A.5), on obtient aussi

II~(t. )JJL~ ~ II~~

+ fo ~ II~(~, )IIL~ IIv(~-, )llL~ d~-.

(A.6)

Lorsque p - 2, le fait que IIv(t, ")[[L2 <-- [[vO/[L2 et que [[a(t, -) ILL~ < Ila0/IL ~ (voir [14]) assure que

lib(t. )ll. ~ II~,~

+tllv~

IIc~011L~

Traitons d'abord le cas 2 _< p _< 6. Par interpolation, on a

IIvlIL~ -< IlvllL~ ~ ~

IIvllLo~

3

En combinant avec l'injection de Sobolev suivante (voir [2] page 162)"

Ilvllz~ ~ 4 IlVVllz~.

(A.7)

d. Ben A m e u r and R. Danchin

53

et en remarquant que IlW[[L~ = IlVVllL2 , on obtient finalement _a 2

_a__a

p

2

2

Ilvllt, p -< 4 IlVllL~ I[w[]z2

p

9

(A.8)

I1 ne reste plus qu's reporter cette derni6re indgalit6 combin6e avec (A.7) dans (A.5). En utilisant l'indgalit6 d'dnergie IIv(t, ")ILL2 --< IIv~ on trouvo finalement

(

-

/

It~ollL,

3

3

(A.9) Traitons maintenant le cas 6 < p < +oc. D'apr6s [2], page 162, on a pour p > 3/2,

IlVIILp <- 3 IlVvli L a.~_. 3+p

(A.10)

Dans notre cas, 6 < p < +oo, done 2 < 3p/(3 + p) < 3. Comme, d'apr6s (4.5),

]lwll L ~

3t) < c II~llL ~+,

,

on peut appliquer (A.7) pour majorer IIVVllz~+. On en ddduit (A.3). Traitons maintenant le cas p = +oo. Avec les notations de la partie 1.1, on a

[lVllL~ <_ lIx(D)VllLOO + II(1 - x(D))VllLoo <_ C [lVllz2 + [1(1 - x(D))VllLOO . Pour majorer le dernier terme, on utilise l'indgalit6 de Sobolev suivante (voir [2] page 167)" '~/~ > 0, [l'/tllL~ '~ C E - 1 ItVUlIL3+~ .

En appliquant cette indgalit6 puis l'indgalit6 (4.5) ~ ( 1 - x ( D ) ) v , on trouve finalement que

IlVtlL~ < C

IlVllL~ +

3+e E

)

II~fiL~+~

On majore IlWllL3+~ 5, l'aide de (A.3). Aprbs quelques calculs, on aboutit

fi (t,

_< ll ~

+

E

pour Pour finir la preuve de l'in6galit6 (A.2), il suffit alors de choisir 3

E-IoN(C0

+ ll~~

t)

II

54

Limite non visqueuse pour les fluides incompressibles axisymdtriques

Remarque A.1. L'in6galit6 (A.3) est optimale parmi les in6galit6s

avec une constante C universelle et une constante C1 ne d~pendant que des donn6es initiales mais pas de la viscosit6: des consid6rations de scaling montrent que aq = 5 / 2 - 3/q est le "bon exposant". I1 suffit d'utiliser l'invariance de (NSv) par la transformation v(t,x) -, Av(A2t, Ax) pour le voir. Dans le cas L ~ , le "bon exposant" serait 5/2. Au logarithme pros, c'est ce que nous avons obtenu. Rappelons que dans [14], l'exposant trouv6 6tait 11/4.

R~fdrences bibliographiques [1] J. Beale, T. Kato et A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. in Mathematical Physics, 94 (1984), 61-66. [2] H. Br~zis, Analyse fonctionnelle. Th6orie et applications, Collection Math6matiques Appliqu6es pour la MMtrise, Masson, Paris, 1983. [3] J.-Y. Chemin, Fluides parfaits incompressibles, Ast6risque, 230 (1995). [4] J.-Y. Chemin, A remark on the inviscid limit for two-dimensionnal incompressible fluid, Comm. Part. Diff. Eq., 21 (1996), 1771-1779. [5] R. Danchin, Poches de tourbillon visqueuses, Journal de Math6matiques Pures et Appliqu~es, 76 (1997), 609-647. [6] R. Danchin, Persistance de structures g~om6triques et limite non visqueuse pour les fluides incompressibles en dimension quelconque, Bull. Soci~t~ Math6matique de France, 27 (1999), 179-227. [7] P. Gamblin et X. Saint Raymond, On three-dimensional vortex patches, Bull. Soci6t~ Math6matique de France, 123 (1995), 375-424. [8] T. Kato, Quasi-linear equations of evolution, with applications to partim differential equations, Lecture Notes in Mathematics, 448 (1975) , Springer-Verlag, 25-70. [9] T. Kato et G. Ponce, Commutator estimates and the Euler and NavierStokes equations, Comm. Pure and Appl. Math., 41 (1988), 891-907. [10] X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Euler system, Comm. Part. Diff. Eq., 19 (1994), 321-334. [11] P. Serfati, R~gularit~ stratifi~e et ~quation d'Euler ~ temps grand, Note aux Comptes-rendus Acad. Sci. Paris, 318 (1994), s6rie 1,925-928. [12] E. Stein, Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, Princeton University Press (1993). [13] H. Triebel, Theory of function spaces, Birkhauser, 1983.

J. Ben Ameur and R. Danchin

55

[14] M. Ukhovskii and V. Yudovitch: Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math.and Mechanics, 32 (1968), 59-69. [15] V. Yudovitch, Non stationary flows of an ideal incompressible fluid, Jurnal vychislitel'noj mat. i matematiceskoj fiziki, 3 (1963), 1032-1066. Jamel Ben Ameur D~partement de Math~matiques Universit~ de Sciences de Tunis 1060 Tunis Tunisie E-mail: j [email protected] Raphael Danchin Laboratoire d'Analyse Num~rique Universit~ Pierre et Marie Curie 175 rue du Chevaleret 75013 Paris France E-mail:[email protected]

This Page Intentionally Left Blank

S t u d i e s in M a t h e m a t i c s

a n d its A p p l i c a t i o n s , Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 4

GLOBAL PROPERTIES OF SOME NONLINEAR PARABOLIC EQUATIONS

M. BEN-ARTZI

1. Introduction In this paper we review some recent results concerning the class of nonlinear equations of evolution given by, ut-Au=#[Vu[

p, # c l t ( , p _ l ,

~(x, 0) = ~0(~), x e R ~.

(1.1) (1.2)

We denote

V "-- V x

--

0Xl,...,

OX n

,

n -

i--1

While equations of the type u t - A u = u p have been extensively studied (see e.g. [14, 15, 29] and references there), the same is not true for (1.1). We note t h a t most of the results mentioned in this paper apply to the more general case where the right-hand side of (1.1) is replaced by F ( V u ) , with suitable growth conditions on F. Thus, (1.1) can be viewed as a model for a "viscous Hamilton-Jacobi" equation. Indeed, this equation appears naturally in a variety of studies. Some examples include (a) The one-dimensional case n = 1. In this case the equation appears in the study of growth of surfaces and is called as the "generalized KPZ equation" [16, 17, 20, 21]. (b) Still in the one-dimensional case, we take # = - 1 and p = 2, thus obtaining the equation u t + u x2 = u x ~ . Differentiating with respect to x and setting v = u~, we get for v ( x , t) the equation vt + (v2)~ = v~x, which is the well-known Burgers equation.

58

Global properties of some nonlinear parabolic equations

(c) Consider the Navier-Stokes equations in the plane (n = 2), which in vorticity form can be written as

~, + (~. v_)~ - .A~, (~ is the vorticity 02u I - 01u 2 of the velocity field u = (u 1, u2)). Suppose we know in advance that lul is bounded. Then ~ satisfies the inequality

~ - . a ~ _< civil. Thus, the methods used in the study of (1.1) are also applicable in the case of the inequality. In the following sections we shall discuss the global well-posedness of (1.1) in various spaces and the decay properties of solutions as t --~ +c~.

2. Existence of global solutions Let C 2 (][~n) .__ C 2 ( ~ n ) N W 2'~176 (][~n), n a m e l y , the space of t w i c e c o n t i n u o u s l y differentiable functions with bounded derivatives. It was proved in [3] that C2(]R n) is a "persistence space" to classical solutions of (1.1). Namely, we have the theorem. Theorem

2.1 [31- Let Uo e C2(I~n). Then for any # e I~, p > 1, there

~xi~t~ ~ u , i q ~ c ~ i c ~ 1 ~olutio, to (1.1)- (1.2), ~uch that ~(., t) e C~(R n) for all t > 0 and the mapping UO E C 2 ( ~ n ) ~ ~t e C(~4_ , C ~ ( ~ n ) )

is continuous. Furthermore, principles:

the solution satisfies the following m a x i m u m - m i n i m u m

sup u(x, t) -- sup to(x)

xEIR,~ tE(O,T]

xEIR,~

inf u(x, t) -- inf t o ( x ) VT > O,

~ xEIl~n te(O,T]

[IVu( -, t)llL~(~--)

x EIR"

'

--< IlWolIL~(~), Vt ~ O.

(2.1)

(2.2)

In the proof, one shows that the solution exists in a time interval (0, T], where T depends only on IIVu011L~c(~tn). The inequality (2.2) then allows the continuation of the solution to [T, 2T], .... We remark that to prove (2.2)

M. Ben Artzi

59

the equation (1.1) is differentiated with respect to xj. Denoting uj = ~ we get 0

n

,

,

(2.3)

i=1

where

t#i(x, t) = #pfVui p-2 au e L ~ ( R " • (0, T] However, the solution uj to the linear parabolic equation (2.3) is not twice continuously differentiable hence some care must be taken in deducing (2.2) from the standard (linear) maximum principle. See the Appendix in [3] for details. Naturally, our next goal is to investigate the well-posedness of (1.1) in wider spaces of less regular functions, for instance, Lq(]~ ~) for suitable exponents q (possibly depending on p). To allow such solutions, the equation (1.1) is first cast in the integrM form,

u(x,t) =

J

JJ t

G(x-y,t)uo(y)dy+#

~.,

G(x-y,t-s),Vu(y, s)[Pdyds, (2.4)

0 ~"~

where G(x,t) = (47~t)-n/2 exp(-lx[2/4t) is the heat kernel. Taking Vx of Eq. (2.4) and using norms of the type

sup t~[[Vu(., t)[tL~(~), tE(0,T]

for suitable r, a, as in [29] one obtains the local (in time) existence of solutions to (2.4) in Lq(lt{n), for certain exponents. Then, by using the regularizing effect of the parabolic equation (2.4) (see also [9] for a direct argument) one shows that the solution u(.,t) E C2(R n) for t > 0, hence global existence follows from Theorem 2.1 above. As for uniqueness, we note that the solution was constructed by using "growth norms" of the type

sup t~llVu(., t)tlL-(~n).

te(0,T]

Thus, a contraction argument yields uniqueness using such norms (the "Kato-Fujita condition" [19]). However, an alternative approach as in [13] gives uniqueness for solutions in classes like C([O,T];Lq~Rn))r C((0, T]; Cb(Rn)). The exact exponents are summarized in the following theorem.

Global properties of some nonlinear parabolic equations

60 Theorem

2.2 [11] - For 1 < p < 2, let p-1 qc = n 2 _ p,

a n d take any q >_ m a x ( l , qc), q < oo (but q > 1 if qc = 1). Then, given any uo E Lq(I~ ~) (and a n y # E ]R), the equation (2.4) has a unique, global On time) solution u E C([0, oo), Lq(N~)). In p a r t i c u l a r we note t h a t if p >

:=

n+2 n+l

(2.5)

t h e n q~ > 1 and the e x p o n e n t q = 1 is outside the scope of T h e o r e m 2.2. Indeed, as the following claim shows, one cannot expect, for p > p~, to have solutions u of (2.4) for any u0 E LX (N~), even under the mildest a s s u m p t i o n s on u. In presenting the next claim, t h e r e is no a t t e m p t at achieving m a x i m a l generality. Claim

2.3 [12] - Let p > pc = (n + 2 ) / ( n + 1) and # = 1. Introduce, for 1 n+l 2(n+2

0<3<

1) p

the following function: vo(x)-

{l

xl -n+~,

o

lxl < 1, _> 1.

Ixl

Then, given any T > 0, there is no solution u(x, t) of (2.4) in (x, t) E R n x (0, T], where u0 = v~ and such that u E LP((0, T);

WI'p(]I~n)).

Proof. A s s u m e the existence of a solution u(x, t) with the above properties. Since T

/ ~ . IVu' pdxdt < ~176 0

given s > 0 there exists a sequence tj --. 0 such t h a t

~

IVu(x, tj)lPdx < Et;1,j - 1 , 2 , . . . , n

(2.6)

M. Ben Artzi

61

which implies, by the Sobolev inequality (observe that u _ 0 in view of

(2.4)), JfRn u(x'tj)p*dx

1 = p1 <---C(et-jl)~'p---2

1 n"

(2.7)

Take 0 < fl < 1 (to be specified later) and use H61der's inequality and (2.7) to get,

f

u(x, tj)dx < (C8t~-l)~ 9(Wnt~n)l--P1-'~,

(2.8)

ixl n+l

1

n ( 1 - 2-2) > p-

2(n + 1) n+2 '

so from (2.8),

//"

u(x, tj)dx < c-cl/pt; -~+2~n(+,~2+1) , j

= 1,2,...

(2.9)

Ixi ((n + 2)/(n + 1) we can choose fl < 1 such that '=

1 - - p+ 2 f l

n

+1

>0,

hence

u(x, tj)dx < csl/pt~. -~ 0

as

j --~ oe.

(2.10)

I~l_•(x, t). We have, for t > 0,

/ G(x - y, t)uo(y)dydx

f ~t(x,t)dx = I~l>t"

Ixl>t~ ~"

= /

/

I~l>t~ lvl< 89

<_ / 1~:i> 89

+ /

I

G(x-y,t)uo(y)dydx

I~l>t~Ivl> 89

G(~,t)d~'lluoliLl(i~.)+ / lyl> 89

uo(y)dy.

(2.11)

62

Global properties of some nonlinear parabofic equations

Since/3 < 1, we have

G((, t)d( = O(t N) as t --. 0, N = 1, 2 , . . . 1~l> 89 and, for u0 = v~

/

uo(y)dy = ( 1 - 2-~t~a))lUO[)Ll(R.),t < 1,

(2.13)

[yl> 89 so, since

II~(',t)llLl( -) /

= II

0[)Li(R-),

we conclude t h a t

~t(x, t)dx = 2-~tZ~[[Uo[ILl(~) 4-O(t N) as t --) O.

Setting t = tj and comparing with (2.10) we get, for j = 1 , 2 , . . . ,

C~I/pt~. ~ 2 -~ tj~ II 0111L (R-) +

O(t )

(2.14)

which is a contradiction by the choice of 5, since ~ < 89can be chosen such t h a t ~5 < 77. i

Remark 2.~. In view of the last claim, one m a y ask, in the case p > pc, # = 1, what is the set of initial d a t a uo(x) E LI(1R ~) for which a solution to (2.4) does exist. Theorem 2.2 implies t h a t this set contains all u0 E L I(R n) N Lq(Nn), q >_ qc, and in particular, all uo(x) E LI(IR ~) N L~ However, Claim 2.3 says this set is not all of L 1(Rn). The situation is still not clear for # = - 1 . On the other hand, i f # = 1 a n d p _ > 2, Claim 2.3 can be strengthened as it is shown in Proposition 2.5 below. P r o p o s i t i o n 2.5 [11] - Let u(x, t) be a classical solution of (1.1), with # = 1, p > 2, in a strip IR~ x (0, T). Assume that lim u(., t) = uo

t---)O

in Lloc(Rn).

Then exp(u0) C L~oc(Nn). Finally, while (for p > Pc, # = 1) existence is not guaranteed for all u0 C Lq(Rn), 1 < q < qc, uniqueness can also fail, as the following theorem shows:

M. Ben Artzi

63

T h e o r e m 2.6 [11] - Assume 2 > p > pc and let 1 <_ q < qc and # = 1. Then, for u0 = 0, there exists a positive solution u to (2.4). In Tact, u is self-similar, 2-p ~(~, t) = t - k U ( l ~ l t - 8 9 ), k = 2 ( p - 1)' ~h~

u = u(~) e c2([0,

~)).

Remark 2. 7. The case of a coupled system of equations of the type (1.1) was treated in [4]. m

3. F u r t h e r

extensions.

The case # =-1

We consider here some further results for solutions of Eq. (1.1) (or (2.4)) under the assumptions t h a t # = - 1 and u0 _ 0. The maximum-minimum principle guarantees t h a t the solution u is nonnegative and is majorized by the corresponding solution of the heat equation. In this case, the subcritical part of Theorem 2.2 has been extended by Benachour and Laurencot [5] to include positive bounded measures, as follows. Theorem

3.1 [ 5] - Let l
n+2 , n+l

#=-1

and uo e M + (R n) (= the space of positive bounded Borel measures). Then there exists a unique weak solution (in the sense of (2.4)) u such that,

U e C((0, (:x)); L1 (]l~n)) n L~oc((0, oc);

WI'p(~n)).

Remark 3.2 (a). We refer to [5] for a precise definition of a "weak solution". Also, for the uniqueness a "growth condition" (as t -~ 0) of the "KatoFujita" type is required, as in the discussion preceding Theorem 2.2 above. (b). The case p = 1 (and # = - 1 ) was treated in [7], by probabilistic methods, producing a spherically symmetric solution for any initial data uo(x) which is a "profiled" spherically symmetric bounded positive measure. (c). The more general equation ~,-

was treated in [23].

~

= -a(~)~(W)',

~0 > 0,

n

Global properties of some nonlinear parabolic equations

64

The supercritical case (p _> pc) is more difficult. Clearly, the method of proof of Claim 2.3 does not work here and the question whether or not the equation is well-posed in L I(R ~) remains an open problem. However, Benachour and Laurencot [5] have managed to prove the non-existence of "source-type" solutions, namely, solutions that converge (in the sense of distributions) to a multiple of the Delta-function. The exact formulation of the theorem is as follows. Theorem

u e L~176

i.

3.3 [5]- Let M > O,T > O, p > pc. There is no function Wl'v(Rn)) such that u t - A u = -[VulP

(R • (o, T) ) lim / u ( x t-*O

t)@dx = Mq~(o) Vq2 e C~~ ~

R ~

4. D e c a y

a s t -~ + c ~

Let us go back to classical (say, as in Theorem 2.1) solutions to (1.1), where we assume now that u0 _> 0 and # = - 1 . Then the solution u(x, t) is nonnegative and an integration of (1.1) shows that if, in addition, u0 E LI(]~ n) then u(., t) E L 1 for all t > 0 and the nonnegative function I(t) = f u(x, t)dx R~

is nonincreasing. Thus, the limit I ~ = lim I(t) >_ 0 always exists. It is t---. o o

interesting that the question whether or not I ~ = 0 is determined uniquely by p~ - (n + 2)/(n + 1), the same critical value as in the previous sections. We have the following theorem. T h e o r e m 4.1 [101 - Let 0 ~_ uo e C~(R n) N L l ( R n ) , u 0 r 0. Let u(x,t) be the solution to (1.1), with # = - 1 . Then Ioo > 0 r

n+2 = ~ . n+l

Remark ~.2. As was seen in Theorem 2.2, the well-posedness of (1.1) in L I(]~ n) was also linked to the same critical index p~. However, there is yet no direct argument connecting this well-posedness (essentially a short-time feature) with the long-time decay as expressed in Theorem 4.1. Remark ~4.3. In the case p < pc the equation is well-posed in LI(R~). Then, as in the discussion preceding Theorem 2.2, if 0 _< u0 c L 1(R ~) (and # = - 1 ) , it follows that u(., t) C C~(Rn)NLI(R n) for t > 0. Hence, Theorem 4.1 is applicable also, in the subcritical case, to all 0 _< u0 E L 1(Rn).

M. Ben Artzi

65

Remark ~.~ In the case p _< p~, the rate of decay of I(t) to zero becomes slower as p approaches p~. More precisely, let 1 < p < p~ and 2-p

2 ( p - 1)

n ~o

2

Then [101 I(t) < Ct -~ (for all sufficiently large t) implies u0 = 0. In particular, if p = p~ then I(t) cannot decay like t -~ for any c~ > 0. On the other hand, if p = 1 and u0 is compactly supported then, for some A, 0 > 0 we have (see [3]), sup exp(At~ < oc. 0_
References [1] N. Alaa, Solutions faibles d'~quations paraboliques quasilin~aires avec donn6es initiales mesures, Ann. Math. Blaise Pascal 3 (1996), 1-15. [2] L. Alfonsi and F. B. Weissler, Blow-up in R n for a parabolic equation with a damping nonlinear gradient term, in Progress in Nonlinear Differential Equations, (N.G. Lloyd et al Eds.), Birkhs 1992. [3] L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi equations, Nonlinear Anal. TMA 31 (1998), 621-628. [4] L. Amour and T. Raoux, Existence et d~croissance en n o r m e L 1 des solutions d'un syst~me parabolique semi-lin~aire, Note CRAS Paris, S~rie I , 329 (1999), 367-370. [5] S. Benachour and Ph. Laurencot, Global solutions to viscous HamiltonJacobi equation with irregular data, Comm. PDE 24 (1999), 1999-2021. [6] S. Benachour and Ph. Laurencot, "Solutions tr~s singuli~res" d'une 6quation parabolique non lin~aire avec absorption, Note CRAS Paris, S6rie I Math. 328 (1999), 215-220. [7] S. Benachour, B. Roynette and P. Vallois, Asymptotic estimates of solutions of u t - 1Au = -IVul in R+ x l[{d,d > 2, J. Func. Anal. 144

(1997), a01-a24. [8] M. Ben-Artzi, Global existence and decay for a nonlinear parabolic equation, Nonlinear Anal. TMA 19 (1992), 763-768. [9] M. Ben-Artzi, J. Goodman and A. Levy, Remarks on a nonlinear parabolic equation, Trans. Amer. Math. Soc. 352 (2000), 731-751.

66

[10] [11] [121 [13]

[14] [15] [161

[17] [ls] [19] [20] [21] [221 [231 [241

[26]

Global properties of some nonlinear parabolic equations

M. Ben-Artzi and H. Koch, Decay of mass for a semilinear parabolic equation, Comm. PDE 24 (1999), 869-881. M. Ben-Artzi, Ph. Souplet and F.B. Weissler, Sur la non-existence et la non-unicit@ des solutions du probl~me de Cauchy pour une ~quation parabolique semi-lin@aire, Note CRAS Paris, S@rieI, 329 (1999), 371-376. M. Ben-Artzi, Ph. Souplet and F.B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures et Appliqu@es, 81 (2002) to appear H. Brezis, Remarks on the preceding paper by M. Ben-Artzi, "Global solutions of two-dimensional Navier-Stokes and Euler Equations", Arch. Rat. Mech. Anal. 128 (1994), 359-360. H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. IX, Ser. 62 (1983), 73-97. H. Brezis, L. Peletier and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal. 95 (1986), 185-219. B. Gilding, M. Guedda and R. Kersner, The Cauchy problem for ut = Au + IVI q, Preprint 1998. M. Guedda, R. Kersner and L. Veron, On self-similar-type solutions to the generalized KPZ equation, Preprint 1998. A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167-189. T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Math. Univ. Padova 32 (1962), 243-260. J. Krug and H. Spohn, Universality classes for deterministic surface growth, Phys. Rev. A 38 (1988), 4271-4283. J. Krug and H. Spohn, Kinetic roughening of growing surfaces, in "Solids far from equilibrium", C. Godreche (Ed.), Cambridge Univ. Press, 1991, 479-582. P.L. Lions, Generalized solutions of Hamilton-Jacobi Equations, Pitman Research Notes in Mathematics, 69, 1982. R.G. Pinsky, Decay of mass for the equation ut = A u - a ( x ) u P l V u l q, J. Diff. Eqs. 165 (2000), 1-23 S. Snoussi, S. Tayachi and F.B. Weissler, Asymptotically self-similar global solutions of semilinear parabolic equations with nonlinear gradient terms, Proc. Royal Soc. Edinburgh, 129A (1999), 1291-1307. Ph. Souplet, R~sultats d'explosion en temps fini pour une ~quation de la chaleur non lin~aire, Note C. R. Acad. Sci. Paris, S~rie I. 321 (1995), 721-726. Ph. Souplet, Geometry of unbounded domains, Poincar@ inequalities and stability in semilinear parabolic equations, Comm. PDE, 24 (1999), 951-973.

M. Ben Artzi

[27] [2s] [29]

67

Ph. Souplet and F.B. Weissler, Poincar~'s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincar@, Analyse non lin@aire, 16, 3 (1999), 337-373. S. Tayachi, Forward self-similar solutions of a semilinear parabolic equation with a nonlinear gradient term, Diff. and Integral Equations, 9 (1996), 1107-1117. F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in L p, Indiana Univ. Math. J. 29 (1980), 79-102. M. Ben-Artzi Institute of Mathematics Hebrew University Jerusalem 91904 Israel E-mail: [email protected]

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 5

A MODEL FOR TWO COUPLED T U R B U L E N T FLUIDS PART I: ANALYSIS OF THE S Y S T E M

C. BERNARDI, T. CHACON REBOLLO, R. LEWANDOWSKI and F. MURAT

1. Introduction. This paper is devoted to the analysis of the following system which models two stationary turbulent fluids coupled by boundary conditions on the interface: - d i v (c~i(k~)Vu~) + g r a d p~ = f~

in Ft~, 1 < i < 2,

div ui = 0

in f~i, 1 _ < i _ < 2 ,

- d i v (Ti(k,)Vki) = a~(k,)[Vu,[ 2

inf~i, 1_
ui=O

onFi, 1
ki=O

o n F i , 1_
(1.1)

~ , ( k , ) On, U, - p ~ n , + ( ~ - ~ j ) lu, - u j i = o

on ~ k, = f ~ l - u ~ ! ~

l<_i#j<_2, o n F , 1_
where each triple (ui, ki, pi) is defined in the domain f~i, 1 _< i _< 2. In what follows, ~"~1 and ~2 stand for disjoint bounded domains in R d, d = 2 or 3, which are either convex or of class C 1'1. The generic point in R 2, resp. in R 3, is denoted by x = ( x , z ) , resp. x = ( x , y , z ) . We assume for simplicity that the interface F = Of~l n Of~2 coincides with the

A model for two coupled turbulent fluids

70

intersection of both ~tl and ~2 with the hyperplane z = 0, while ~tl and ft2 are contained in the half-spaces z > 0 and z < 0 respectively, see the following figure where each Fi is equal to 0fti \ F. Note also that, in the physical context, the heights of the domains are much smaller than their horizontal diameters.

F ga

Figure 1 The vector field ui stands for the velocity of a turbulent fluid in fti, pi represents its pressure and ki its turbulent kinetic energy (TKE in what follows). The quantity ai(ki) is the eddy viscosity, and we shall assume throughout this paper that the functions ai and "yi satisfy

{

a~CC~ 7~EC~

NL~(R) nL~

and and

VkclK, VkER,

a~(k)>_v,

1
v~(k)_L,,

1
(1.2)

for some positive constant v. The functions fi are given, with

fi c L2(~ti) d,

1 < i < 2.

(1.3)

System (1.1) is motivated by the coupling of two turbulent fluids Fi, i - 1 and 2, such as in the framework ocean/atmosphere or in the case of two layers of a stratified fluid (see e.g. [13, Chap. 1 ~z 3] or [17]). These fluids Fi are coupled through the interface condition in the sixth line of (1.1), on their common boundary F (which is supposed to be fixed). Indeed, we assume t h a t the so-called "rigid lid hypothesis" holds, an hypothesis which is standard in geophysics and oceanography. Actually, F is a mean interface and the values of ui, Pi and ki on F are in fact mean values of the velocity, pressure and TKE. So, the turbulent mixed layer of the two turbulent fluids is modelized by the sixth and seventh lines in (1.1) which summarize the

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

71

information related to a realistic interface ocean/atmosphere (see e.g. [13, w for more details about this modelization). The main goal of this paper is to prove the existence of a solution

(u~,ki,pi)l<~<2 of system (1.1) (see Corollary 5.3). We start by giving a sense to the equations. The two first lines of system (1.1) are the Stokes equations in fti, equipped with the eddy viscosity ai(ki) which is the quantity of interest. The third line is a scalar equation which allows one to compute the ki. Following the ideas of [15], one can write a weak mixed formulation of the first two lines (see also [17] for the case of coupling ocean/atmosphere without turbulence). The third line and the corresponding boundary conditions are more complex: the main difficulty comes from the fact that the right-hand side only belongs to L l(fti). In the case of homogeneous boundary conditions and with only one turbulent fluid, such type of equations has already been studied (see e.g. [4], [10], [13, Chap. 4], [14]). In those references, the equation for the TKE is taken in the renormalized sense of Lions and Murat (see [18], [20], [21]), or in the equivalent entropy sense of Benilan et al. (see [3]), and a priori estimates of Boccardo-Gallou~t type [5] are used. However, because of the boundary conditions at the interface F, this renormalization does not seem an easy way for the study of the TKE equations in the present problem, and one cannot hope to use directly the results of [5]. For this reason, we make Kirchoff's change of unknown in order to replace the operator div ('Yi(ki) V) by a simpler Laplace operator. We consider the corresponding new equation in the sense of transposition, according to the ideas of Stampacchia [23] and of Lions and Magenes [16, Chap. 2, w We are then able to prove the existence of a solution of the global system (1.1). Under some rather restrictive assumptions on the variations of the functions c~i, we prove a uniqueness result for the solution of (1.1). Some further regularity properties of this solution are also derived with the same assumptions, when the domains fti are rectangles. Most results of this paper have been announced in [1], however with a slightly different proof relying on Leray-Schauder fixed point theorem. The discretization of system (1.1) by spectral and finite element methods is presently under consideration, from both points of view of numerical analysis and experiments. It must also be observed that the present analysis can be extended to slightly different models, 9 by adding convection terms in the momentum equations for the velocities,

A model for two coupled turbulent fluids

72

9 by replacing the transmission conditions for the velocities on F (sixth line in (1.1)) by Manning's law ~.,(k,) On.~,,. + ( ~ , . - ~ j . ) I ~ - ~jl = 0

onr, 1 < _ i r

Uiv=O

onF, 1_i<2,

where u i , and Uiv respectively stand for the horizontal and vertical components of the velocities, 9 by adding the term due to Coriolis acceleration. An outline of the paper is as follows. In Section 2, we write a system which is equivalent to (1.1) through a change of unknowns. Sections 3 and 4 are devoted to the proof of the existence of a solution for the equations on the velocities and the turbulent energies, respectively. In Section 5, we state and establish our main result, namely t h a t the global system (1.1) admits a solution. In Section 6, we prove the conditional uniqueness result and, in Section 7, we derive the optimal regularity of this solution when the domains f~i are rectangles.

2. Transformation of the system. Let us define the functions Gi, 1 _< i _< 2, by k

G~(k) = ~o "7~(~)dg.

(2.1)

In view of (1.2), the Gi are increasing functions of class C 1, SO that they admit an inverse G~-1 from N into R. Moreover, the functions c)i, i = 1 and 2, defined by 5i = ai o G~ 1, (2.2) satisfy the same properties as the c~i, namely c)icC~

and

VgER,

&i(g)_>u,

1_
(2.3)

The idea is to introduce the new unknowns gi by

g i - Gi(ki),

1 _~ i _~ 2.

(2.4)

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

73

From the formula Vg~ = -yi(k~)Vk~, it is readily checked that system (1.1) is equivalent to

infti, 1 < i < 2 ,

' - d i v (5,(g,) V u , ) + g r a d p i = fi div ui = 0

inf~i, 1 < i < 2 ,

-~xe, = ,~,(e,)IW,,I 2

inf~i, 1 < i < 2 ,

ui=O

onFi, 1
gi--0

onFi, 1_
c~,(e,) o n , ~ , - p, ,~, + ( ~

- ~ j ) lu, - u j l = o

on F, ei -- G i ( l U l

(2.5)

- u2[ 2)

l
The interest is that now each function gi is a solution of a nearly standard Laplace equation. The goal of this paper is to prove the existence of a solution of (2.5), see Theorem 5.2 below. Throughout the paper, we use the spaces Lv(f~i), 1 _< p _< cx~, and the Sobolev spaces HS(fti) and H~(f~i) for any real number s, provided with the standard norm II " IIHs(n~) and semi-norm [. IHs(n,), together with their analogues on F. We also need the Sobolev spaces Wl,p(fti) 1 and the special space H0~0(F), defined e.g. in [16, Chap. 1, Th. 11.7].

3. The equations on the v e l o c i t i e s . Throughout this section, we assume t h a t the functions gl and g2 are given such t h a t gi C LI(Di), 1 _< i _ 2, (3.1) even if they turn out to be smoother in what follows. For 1 _< i < 2, we introduce the spaces X i --- { V i C

Hl(~i)d; vi -- 0 on Fi}.

(3.2)

We now write correctly (and not only formally) the equation on each velocity

74

A model for two coupled turbulent fluids

ui through its variational formulation:

Find (ui,pi), 1 _< i _< 2, in Xi • L2(f~i) such that, forl<_ir

w~ ~ x,,

[

~,(e~) v ~

J~

9w~ d~ + b~(~,p~)

(3.3)

i

~- ~F [Ui -- Uj] (Ui -- Uj) . vi dT = /~ Yi

. vi dx,

i

Vqi e L2(f2i),

b , ( u , , q i ) = O,

where the form bi(.,-) is defined by

b~(v~, qi) = - [ J~

q~(div vi) dx.

(3.a)

i

Note t h a t the first bilinear form in (3.3) depends on t?i. As standard for the Stokes problem, we consider the kernel

Vi = { vi c Xi; div vi - 0 in gti }, and we observe that, for each solution (u~, p~) of problem (3.3), the velocity ui is a solution of:

Find ui in Vi, 1 <_ i < 2, such that, for 1 < i ~ j < 2: Yvi C Vi,

/

5i(gi) V u i . V v i dx

(3.5)

i

s

i

The converse property relies on the following inf-sup condition of Babu~ka and Brezzi type. L e m m a 3.1. - For i = 1 and 2, there exists a constant ~i > 0 such that

Yqi e L2(f~i),

sup

bi(vi,q~)

> fl~ Ilq~llL~(~)-

(3.6)

Proof. The argument here is due to Boland and Nicolaides [6]. Let us write any function qi in L2(~i) as qi = qi + qi,

with

qi dx.

qi -- meas(f2i) i

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

75

Indeed, from the standard inf-sup condition [11, Chap. I, Cor. 2.4], there exists a function vi in Hlo(~i) d such that

and

div vi = - q i

[[Vi[[gl(f~i)d ~_~ ci [[qi[[L2(~i),

where the constant ci only depends on the geometry of f~i. On the other hand, there exist an open interval or disk 7) such that its closure is contained in the interior of F and positive real numbers si such that the cylinders C1 = ~DX]0,EI[ and C2 = T ) x ] - E2,0[ are contained in ~1 and ~2, respectively. If p denotes a smooth function that is positive on T) and vanishes on O:D, the function with horizontal components equal to zero and vertical component viz defined on the cylinder Ci by (in dimension d = 3 for instance)

~ ( ~ , y, z) = ~ p(x, y) ( + ~ - z), can be extended by zero into a function vi of Xi which satisfies

bi(vi,-qi) -- c~ I[qi[[22(f~,) and

[[Vi[[Hl(f~)d ~ C~t [[qi[[L2(f~,).

Finally, taking vi = vi + #i vi yields

bi(vi, qi) -IIq~ll~2(a,) + #ic~ Ilqili2L2(fl~) + #i b~(~i, qi)

> Ii~tl~(a,) + .~c~ II~ll ~

"

1

>-II~llL(~,) +,~(4 - 2

~-2

" ~2

)11~11~L~(a~)'

and also 1

!

So choosing #i = ~C i leads to the desired result.

m

The next corollary is now a direct consequence of Lemma 3.1, see [11, Chap. I, Lemma 4.1]. C o r o l l a r y 3.2. - For i = 1 and 2 and for any data f i in L2(~i) d, (i) for any solution (ui, pi) of problem (3.3), the velocity ui is a solution of problem (3.5), (ii) for any solution ui of probIem (3.5), there exists a unique pressure pi in L2(~i) such that the pair (ui,p~) is a solution of problem (3.3). We now prove the existence of a solution of problem (3.5). We begin with an a priori estimate.

A model for two coupled turbulent fluids

76

L e m m a 3.3. - For every g~ in Ll(f~i) and f~ in L2(f~) d, 1 <_ i <_ 2, any solution (Ul, U2) O1c problem (3.5) satisfies

IlullIH~(a~)~ + Ilu2llH~(a~)~ _< __c (l[flllL2(al)a /J Proof. give u

+ IlY211L~(a~).).

(3.7)

Taking v~ equal to u~ in (3.5), using (2.3) and summing up on i

(lul[2,(a,), + lu~l~,(~.).) + Llu, - ujl 3 d~ f2 9u2dx. 1

2

Since the term integrated on F is nonnegative, the desired estimate follows from the Cauchy-Schwarz and Poincar~-Friedrichs inequalities, m P r o p o s i t i o n 3.4. - For every g~ in L I ( ~ ) and f~ in L 2 ( ~ ) d, 1 _< i _< 2, problem (3.5) has a solution (ux, u2). Moreover, this solution satisfies (3.7).

Proof. Since both spaces V1 and V2 are separable (indeed, they are closed subspaces of the spaces Hl(f~i) which are separable), there exist increasing sequences of finite-dimensional Hilbert subspaces V/m of V/, i - 1 and 2, such that Vi= U vim, 1 < i < 2_. _ m>O I

We define a mapping ~,~ from V1m • V2m into itself by

V(Ul, u 2 ) e v ? • vF,

V(~l, ~ ) e Vlm • y~m,

1

(3.8)

2

+ f r I~1 - u~l (~1 - ~2)(~1 - ~ ) d ~

- - ~ fl " v l d X - ~ 1

f2 9v2dx, 2

where (-,-) stands for the scalar product on V1 x 1/2. Since the function &~ is bounded and the traces of functions in V/on F belong to H 89(F) d, hence

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

77

at least to L4(F) d from the Sobolev embedding (recall that d _< 3), each mapping ~,~ is well-defined and continuous on V1TMx V2m. Moreover, as for estimate (3.7), it is readily checked that

(~(~1,

u~), (Ul, ~ ) ) _> - (iuxl~,,(~). + I ~ 1 ~ , ( ~ , ) . ) - c (11~1 ii ~L'(~.)'+

~ 89(I~11~.(~.).+ lu21~.(~..).)89 II/211L'-(~.).)

So the right-hand side is nonnegative on the sphere of radius # defined by c (i]f1[12

1

Applying Brouwer's fixed point theorem (see e.g. [11, Chap. IV, Cor. 1.1]) yields the existence of a pair (u~, u ~ ) in V1m x V2~, with norm less than #, such that 9~(uF, ~) =0. Since the sequence (u~n, u ~ ) m is bounded by # in V1 • 112, there exists a subsequence which converges weakly to (Ul, u2) in V1 • 112. Using the compactness of the imbedding of H 89 into L3(F), we obtain that this pair (Ul, U2) is a solution of problem (3.5). 1 We conclude the study of problem (3.5) by a uniqueness result. P r o p o s i t i o n 3.5. - For every s in L](~ti) and fi in L2(gt~) d, 1 _< i _< 2, the solution (u], u2) of problem (3.5) is unique. Proof. Let (Ul, U2) and (~l,U2) be two solutions of this problem. By setting wi = ui - u i , 1 <_ i <_ 2, we observe that each wi belongs to Vi and satisfies for 1 < i < 2 Vv~ C V~,

f _ (~(g~) Vw~ 9Vv~ dx i

+ ] F ( l u i -- uji (ui - u j ) - I ~ i

-- u j l (ui -- u j ) ) vi dr = O.

Taking vi equal to wi in this equation and summing on i yield, since Oi is :>~,

+ jr (1~1 - u~l (u~ - u~) - i~1 - ~ 1 ( ~ -

~))

((Ul -- U2) -- (Ul -- U2)) d r < O.

A model for two coupled turbulent fluids

78

From the inequality, valid for all real numbers A and A, (1:~! ~ -

I~1:~)(~ -

~) > o,

we deduce that both ]Wi[Hl(f~,)4 are zero, whence the result,

(3.9)

m

Note as a conclusion of this section that, for any gi in L l(f~i) and fi in L2(f~i) d, system (3.3) has a unique solution ( ( u l , p l ) , (u2,p2)) in the space (X1 x L2(f~l)) x (X2 x L2(f12)), which is bounded as a function of the norms of the data fi. 4. T h e e q u a t i o n s o n t h e t u r b u l e n t

kinetic energies (TKE).

Throughout this section, we assume that the functions U l and u2 are given such that u~ E Hl(gt~) d, 1 < i _< 2. (4.1) The correct formulation of the equations on the TKE is by transposition, following the ideas of Stampacchia [23] and Lions and Magenes [16]. Let us first perform a formal computation. If ~i, 1 <_ i < 2, are functions in C~162 which vanish on 0f]i, we obtain by multiplying the third line of (2.5) by these functions and integrating twice by parts

- ~

gi A~i dx - - ~r Gi([Ul - u2'2) COn.~idT + /f~ ~i(gi) [Vui[2 ~i d~c. i

So, from now on, we look for a solution of the following problem:

Find gi in L2(f~i), 1 < i < 2, such that, for 1 <_ i < 2:

vv~ e g~(~,) n g](a,), - [ J .~ t 4

g~ A~i dx : - f_ Gi([Ul - u2[ 2) O~,~ai dT

(4.2)

JF

+[ J~t

~(e~)Iv~,l ~~ d~. i

Our existence proof relies on the fact that, since the domain ~i is convex or of class C 1,1, the Laplace operator L:i which, with data gi in H-I(f~), associates the solution ~i -- f-.igi in H1(~2i) of the problem { -Ag~i = gi ~ai - 0

in ~i, on O~i,

(4.3)

is continuous from L2(~i) into H2(~ti) (see e.g. [12, Thm. 3.2.1.2]). Combining this property with an interpolation argument [16, Chap. 1, Th. 5.1] leads to the following result.

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

79

L e m m a 4.1. - L e t t be a real number, 1 < t_< 2. For 1 _< i <_ 2, the operator/:~ is continuous from H t - 2 ( f ~ ) into H t ( f ~ ) and satisfies Vgi e Ht-2(f~i),

IIL~g, IIH,(~,) _< ~ IIg~llH'-~(~,),

(4.4)

for a constant c independent of t. We are now in a position to prove the a priori estimate. L e m m a 4.2. - L e t s b e a r e a l n u m b e r , 0 < s < 1. For 1 <_ i <_ 2 and for every pair (Ul,U2) in H1(~1) d • H1(~2) d, there exists a constant c depending on s and on the m a x i m u m of ai and ~/i such that every solution gi of equation (4.2) satisfies

[[ei[[Hs(f~i ) ~___C ([l~/,X[[~l(~l)d -~-[['U2[[~./l(~2)d). Proof.

(4.5)

Let us introduce the linear form

(~i(~i) lVui[2 9~i dx,

Fi ~i = - ~ r G i ( l u l - u212) On~9~idT + ~ i

and check that it is continuous on H2-~(f~). Firstly, since yi is bounded, it follows from the definition (2.1) of Gi that G ~ ( l U l - u212) _< ~ tul - u212.

Moreover the trace u l - u2 belongs to H 89(F) d, hence to L4(F) d thanks to the Sobolev imbedding in dimension d < 3. On the other hand, since F as a part of an hyperplane is smooth and 9~i belongs to H2-~(f~i), O~pi belongs to H 89 So we have

Similarly, since 5~ is bounded and ~ belongs to H 2 - S ( ~ ) , hence to L ~ ( ~ ) , we obtain 5i(gi)IVu, I2 9~idx I < c(I]ul[12H,(n~)~

]~

+ [[U2[[2H,(f~2)d)IIg~i]IL~(n,).

i

Combining all this leads to Fi ~i ~ c

(![u111~/1(~1) d + I]~.t211~/l(~2)d)11~gi[IH2-s(~'~,).

(4.6)

A model for two coupled turbulent fluids

80

When combined with (4.4) for t - 2 - s, this yields that the linear form Fi o Z:i is continuous on H - ~ ( ~ i ) . Note [16, Chap. 1, T h m 11.1] that, for 0 _< s < 1, the dual space of H~(~i) coincides with H - ~ ( ~ i ) and that these spaces are reflexive. So any function gi satisfying (4.2), hence

Vgi E H - ~ ( ~ i ) ,

< 9i, gi > = Fi o s gi, m

belongs to H ~ ( a i ) and satisfies (4.5).

From Lemma 4.2, we derive that equation (4.2) with the integral in the left-hand side replaced by the duality pairing is satisfied for any ~i in H 2-s (~ti) N H 1(~i). Proving the existence result is now easy, by using the same arguments as for Proposition 3.4, with a further regularization of the data. P r o p o s i t i o n 4.3. - For 1 <_ i N 2, and for any pair (ux, u2) in X 1 x X 2 , problem (4.2) has a solution gi. Moreover, this solution belongs to H~(~i), 1 for every s < g.

Proof.

We firstly consider the problem with more regular data (/~i, pi) in 1

Le(ai)

x

H0~0(F) 9

- A g i = &i(gi) hi

in f~i,

gi = 0

on Fi,

gi = Pi

on F.

(4.7)

Let us still denote by fli a function in H l(f~i), the trace of which vanishes on Fi and coincides with Pi on r'. The idea is to set g~ = gi - p i . So, we now consider the problem:

Find g~ in H l ( a i ) , 1 <_ i <_ 2, such that:

i

(4.8)

i

-- ~ , , V p i

9 V g i dx.

Since H l ( ~ i ) is separable, there exists an increasing sequence of finitedimensional Hilbert subspaces Z m of H l ( ~ i ) such that

Ho'(a,) = U

m>O

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

81

We then define a mapping ~ m from Z ~ into itself by

ve ~ e z y , v~, e z ? ,

(~''(~~

= ~

ve~ "v~ ~d~i

~ c~,(e~+ Pi) ~

g~ dx

i

(4.9)

4" f ~ V pi " V gi dx. It is readily checked that

where c only depends on the maximum of the function &i and on the Poincar6-Friedrichs constant. So, this quantity is nonnegative when the function go belongs to a sphere with appropriate radius

Using once more Brouwer's fixed point theorem (see e.g. [11, Chap. IV, Cor. 1.1]) yields the existence of a solution gOm of the equation ~im(g ~ -- 0 with norm less than #i. Since the sequence (Hi)m>_0 Om is bounded in H l ( ~ i ) , it admits a subsequence which tends to g0 weakly in Hl(Fti). Due to the compactness of H 1 (~'~i) into L2 (~2~), this subsequence converges strongly in L2 (~t~). Hence by the inverse Lebesgue theorem (see e.g. [7, Th. IV.9]), there exists a further subsequence, still denoted by (g~0 m )m>__0,which tends to g0 a.e. on ~-ti. Since the function &i is continuous and bounded, the subsequence (&i(g~ + pi)Hi)m>0 is dominated by c gi which belongs to L2(~ti), and tends to &i(g ~ + Pi)gi a.e. in ~ti, hence in L2(~ti). So, the function go is a solution of (4.8) and finally the function g~ = go + pi is a solution of (4.7). Since we have proved the existence of a solution of problem (4.7), we now pass to the case of real data. We approximate each ui in Xi, 1 <_ i <_ 2, by a sequence (un)n_>0 of C ~ ( ~ ) d n Xi which converges towards u~ in Hl(~-~i) d. Then, it is readily checked that the functions A~ = IVu~l 2, resp. m

1

p'~ = G i ( l u ~ - u~]2), belong to L2(Fti), resp. H~o(F ) (see [16, Chap. 1, Th. 11.7] for the characterization of this space). Let g~ be a solution of problem (4.7) with data .k~ = / ~ and p~ = p~. Since it belongs to Hl(~t~), by integration by parts, it is readily checked that it satisfies

V~i C H2(~ti) n H01(fti),

v~(lu? - u~[ ~) 0 ~

d~

i

+ fa a , ( e ? ) I w ? l 2 ~ d~. i

82

A model for two coupled turbulent fluids

Let now so be such that 0 < so < 89 From Lemma 4.2, the sequence (gni )n>0 is bounded in H s~ (f~i), so that it admits a subsequence which converges to gi weakly in HS~ (f~i) (and in fact in all H~(f~i), s < 89 From the compactness of H~~ into L2(f]i) and the boundedness of the function 5~, by the same arguments as previously, we deduce the existence of subsequences, still denoted by (fP)n>o and (U?)n~O,such that the sequence (6i(gp)IVU?[2)n~0 is dominated by a function of LI(f~) and tends to &~(g~)[Vui[ 2 a.e. in f~, hence in Ll(f~i). Similarly, there exists a subsequence (Gi([u~ - u~[2))n>o which tends to the term G~(]Ul - u 2 [ 2) in L2(r). So, the limit g~ is a solution of (4.2), which completes the proof. II .

Remark. Consider a solution gi of problem (4.2) which further belongs to Hl(f~i). Then, if the pair (ul, u2) belongs to W I ' ~ (f~l)d X W I ' ~ (f~2)d, it solves the following variational problem: Find s in Hl(fti), 1 <_ i <_ 2, with e i -- 0

01rl F i

&/la

e i -- ci(Itt

1 -

on r ,

(4.1o)

such that, for 1 <_ i <_ 2" Vg~,EHoI(f],),

V~.Vg~idx-/~

~ i

6i(g~),Vui,29~dx. i

Note that (4.10) is now a classical formulation, which could be discretized in an easy way.

5. T h e g l o b a l s y s t e m . Proving the existence of a solution for the full system (2.5) follows from the same arguments as previously, however for technical reasons we begin by working with a truncated problem. For each positive integer n, we introduce the function Tn defined from ]K onto R by -n if x _< - n , Tn(x)x if - n < x _ < n , n if x ~ n , and we consider the problem

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

- d i v (Si(~i) Vui) + g r a d pi = fi

int2i, 1 < i _ 2 ,

div ui = 0

i n , i , 1_
-~e~ = T. (a~(e,)Iv,,,I ~)

intii, 1
ui=0

o n F i , 1_
t~i=0

o n F i , 1_
a,(e,) O~,u, - p, ~

l<_i~j<_2,

onF, 1_
(ai([Ul -- U212))

--" I n

(5.1)

+ (u, - ~j) I~ - ujl = o

on F, ei

83

We first write its reduced variational formulation (where the word "reduced" means that the pressures pi do not appear in it)" Find ui in Vi, 1 <_i <__2, such that, for l <_i r j < 2: Vvi C Vi,

/fl (~i(gi) V u i . V v i d x i

Jc ~ ]ui -- uj , (ui -- uj ) . vi dT -- / ~

f i . vi dx. i

(5.2)

Find fi in H1 (~ti), 1 <_ i <_ 2, with ei -- 0

on Fi

and

ei -- Tn (Gi(lUl - u212))

on F,

such that, for 1 <_ i <_ 2"

~ ~ .1 (~), s ~ 4

~ ~ : s ~ (~l~)I~! ~) ~ ~ i

The next lemma states that problem (5.1) is well-posed. L e m m a 5.1. - For any positive integer n, for any pair ( f l , f2) in L 2 ( f~ l ) d • L2(fi2) d, problem (5.1) admits the variational formulation (5.2). System (5.2) h,~ ~ ~olutio~ (U1, U2) with ~ h U~ = (u~, e~) i~ V~• Mo~o~ the functions f l and s are nonnegative. The proof of this lemma is rather complex, it is performed in five steps.

84

A model for two coupled turbulent fluids

Proof (I). Liftings of traces. 1

For i = 1 and 2, we introduce the continuous lifting operator Li from H~o (F) into harmonic functions in Xi (note that the extension by zero of a function in H~0(F ) to O~ti belongs to H 89 Next, with each pair (Ul,U2) in X1 • X2, we associate a function pi(ul, u2) defined on ~i by

p~(u~, u2) = Tn (G~(IL~(Ul - u~)le)). 1

Since the trace of each ui on F belongs to H~0(F) d, the function gi = Li(Ul - u2) belongs to Hl(gti) d and satisfies

Let us now prove that the function have [23, Lemme 1.1]

Tn(G~(Ig~I2)) belongs

g r a d Tn (Gi(Igil2)) = 2Tnt (c,(Ig,

l=)) ~,(Ig, I~) g r a d

to H'(Fti). We

gi " gi-

Note that 7i is bounded and that g r a d gi belongs at most t o L2(~i) d2. Moreover, T'~ (Gi(Igil2)) is either 0 or 1, and, when it is not zero,

~, Ig~l 2 <

fo Ig'12~ ( ~ )

d~ =

G,(19~I ~) _< ~,

so that T~n(Gi(Igi]2))gi is bounded and Tnt (c,(Ig~I~)),-y~(Ig, I2) g r a d gi "gi belongs to L 2 ( ~ ) d. As a consequence, the function pi(ul, u2) belongs to H 1(~ti) and satisfies

llp~(Ul, u2)llH,(~,) <_ cn89(llulll~,(~,),,

+ llu211~,(~)~,) 89

Next we work with the new unknowns ( u i , [ . ~ HI(Qi).

~-

(5.3)

p ~ ( u l , u 2 ) ) i n V~ x

Proof (II). Existence of a solution for a finite-dimensional system. For i - 1 and 2, as in the proof of Proposition 3.4, we introduce an increasing sequence of finite-dimensional Hilbert subspaces Vim of V~, i = 1 and 2, such that V i - U vim 1O

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

85

and, as in the proof of Proposition 4.3, we introduce an increasing sequence of finite-dimensional Hilbert subspaces Z ~ of H l ( ~ i ) such that m>0 We now define the mapping (I)m from V1m x V2m into itself by

V(ui, ~ ) e y ~ x y s

v ( ~ , ~ ) e vim x v2~,

(,~(ui,u2),(v,,v2))

=

[

J .~Z1 +

al(el~ + pi(uT, u'~)) V u , /~

-

0

Vv, 9 dx

m

a2(gi + p2(ul , u ~ ) ) Vu2 9Vv2 dx 2

+ ./r lul - u21 (Ul - u2)(Vl - v2)dT

1

2

and, similarly, for i = 1 and 2, a mapping O im from Z ~ into itself by

ve~ e z ? , vg~ e z ? , (~im (,),g,) e~ - [

w ~ 9vg, ~ i

Tn (&i(~,0 + pi(Ul, u2))IVu, I~) g, d~

-- [ i

-4"-f

Vpi(ul, u2)

9Vgi

dx.

J ~-t i

Finally, we consider the mapping ~,~ from V1m x Z ~ x V2m x Z ~ defined by V(Ui,~0, U2,~ 0) e Vlm x Z T x V2m x Z~n, V(Vl,gl, v2,g2) E V1m x Z~n X V2m x Z~n,

(~m (~tl, ~0, U2 ' tO), (Vl, gl, V2, g2)) --- ((~m(Ul, U2), (Vl, V2)) -Jr- (~im(~0), gl) + (~2m(~20),g2).

Taking (Ul, U2) on the sphere with radius #, for the same # as in the proof of Proposition 3.4, yields that (I)m((UI, U2), (UI, U2)) is nonnegative. This combined with the arguments of the proof of Proposition 4.3 implies

( ~ (~1, ~0, ~,~0), (u,, ~, ~:, e0)) ->

2 ~ ( l e ? ! 2H , ( r ~ , ) i--1

-- (c'n

+

[Pi(Ul,

u~,)lH,(r,,))[g~

A model for two coupled turbulent fluids

86

So using the bound (5.3) for Ilpi(ul, u2)[Ig,(a~) and taking each go on the sphere with radius #i = c'n + cn 89#, we deduce that the previous quantity is nonnegative. Thus, applying Brouwer's fixed point theorem yields the existence of a solution

(~?, ~ , ,,~, e0~) of ~m(~,T e~ == , ~ = ) = 0 ,

(5.4)

,

which satisfies

IlU~IIH'(~,>~ + Ilu~IIH*(~=)~ ~ #,

Ile~

_< #i,

i = 1 and 2.

Since the sequence (u~, t ~ u ~ , ~m)m is bounded, there exists a subsequence, still denoted by (u~, gore, u~, t ~ for simplicity, which converges to (Ul,g~, U 2 , ~ ) weakly in V1 x Hl(f~l) x V2 x Hl(f~2). Next, for a fixed (vl, gl, v2, g2), we pass to the limit in problem (5.4).

Proof (III). The limit on the equations for the velocities. We start from the equation ~ m ( u ~ , u ~ ) -- 0. For 1 < i < 2, there exists a subsequence still denoted by (g~ which converges to go strongly in L2(fli), hence a.e. in f~i. On the other hand, due to the continuity of Li, the sequence (gm = Li(u~ - U~))m converges to Li(ul - u2) weakly in H l(f~i). Due to the formula grad p? = grad

Tn(Gi(]gml2)) = 2Tnt (Gi(Ig?I2)) "yi(Igml2) grad g ? - gm,

there exists another subsequence (p,(u~, U~))m which converges to the function p/(ul,u2) weakly in H1(~2/) d and strongly in L2(fl/) d. So, the corresponding subsequence (t om +pi(u~, U~))m converges to t ~ +pi(ul, u2) a.e. in fli and, since 5i is continuous and bounded, for all fixed vi in Xi, the sequence ((~i(g~m + pi(ur~, urn)) VVi)m tends to &i(t ~ + pi(ul, u2)) Vvi a.e. in fl/ and is bounded in L2(fl/) d2' hence converges strongly in L2(fl/) d=. This yields the convergence of the first two integrals in the definition of ~m(', "): for i = 1 and 2,

lim s

&i(g~ + pi(u~, u'~)) Vu m . Vvi dx i

-- /a (~i(eOi Jr- pi('Ul, U2) ) VUi " VVi dx. i

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

87

The convergence of the third one follows from the compact imbedding of H 1 (F) into L 3 (F)"

liInor IF lU~n -- u~nl (U~n -- U ~ ) ( V 1 -- V2)dT -- ~F lul -- u21 ( u l -- u2) (vl -- V2)dT. Combining all this implies that the desired equation is satisfied by (Ul, u2).

Proof (IV). A stronger convergence result. For i = 1 and 2, we start from the formula J

Si(g~

u ~ ) ) V ( u ' ~ - ui) 9V ( u m - u i ) d x

4 --" / a

(~i(~Om + p i ( U ~ '

U~)) VU m

9V U m

dx

i

- Om + pi(u~, Um -- 2 / a ai(~i 2 )) V U ~ 9Vui dx i

+

v~

~ i

Om

+

m

)1

From equation (5.4), the first term in the right-hand side is equal to

-ff_r ]u~ - u~] (u~ - u ~ ) . u'~ d~" + /~ fi . u m, i

and using once more the compactness of H 89(F) into L3(F) implies its convergence. The convergence of the second and third term follows from the weak convergence of (VU~')m in L2(~i) a2 and the strong convergence of ((~i(~ Om + pi(u~, u~)) ~Tvi)m in L2(~ti). So, we obtain t h a t

lirn ff.q ~i(~ ~ + pi(uF, u~')) V ( u ~ - ui) - V(u~' - ui) dx = O, 4

which yields the strong convergence of (u~')m towards ui in Hl(~ti) d.

Proof (If). The limit on the equations for the TKE. Next, we consider each equation ~m(g0m) = 0. As previously, there exists a subsequence (c%i(t~~ + pi(u~', U~)))m which tends to &i(g~ + pi(ul, u2)) strongly in L2(~i). From part IV of the proof, there also exists a subsequence (i~Zu~'12)m which tends to IEZui]2 strongly in L l ( ~ i ) . Hence, the

A model for two coupled turbulent fluids

88

sequence (&i(g ~ + pi(u~, u ~ ) ) I w T l ~ ) m converges a.e. in ~i and, since Tn is continuous and bounded, the sequence

+ p,(~7', ~))IV~?l~))~,

(Tn(~,(e ~

converges towards Tn(&i(g ~ + pi(ul, u 2 ) ) I W ~ l yields

~) strongly

in L2(f~i). This

- om + p,(~7, u~m ))IW?l ~) g, dx mlim ~ / ~ Tn (~,(e, i

-- /fl Tn (&i(go q_ p~(ul, u 2 ) ) I V u ,

I~) g~ dx.

i

Also, from the weak convergence of a subsequence ( p i ( u ~ , u ~ ) ) m Pi(Ul, U2) in H l ( ~ i ) d, we deduce

m

Vp~(u~, u 2 )- Vg, dx =

lim

/:

to

V p i ( u l , u2)- Vg, dx.

m - - - - ~ (:x3 i

i

So the desired equation is satisfied by t~i. Finally, the nonnegativity of the g~ follows from the standard maximum principle [7, Prop. IX.29]. We are now in a position to state the main result of this section. There also, we write the reduced variational formulation of system (2.5), where the equation on the t~i has now the same "transposed" form as in Section 4:

Find ui in Vi, 1 <_ i <_ 2, such that, for 1 <_ i ~ j <_ 2" Vv~ ~ V~,

/ a ~(e~) Vu~ . Vv~ dx i

--~ ~ [ui -- uj [ (ui -- uj) . vi dT ---- ~

f i . vi dx. i

Find gi in L2(~i), 1 _< i __ 2, such that, for 1 <_ i <_ 2"

- ~

(5.5)

g ~ A ~ , d x = - f r G , ( l U l - u212)On.,p, dv i

+/~ ~(e~)IW~l ~~ d~. "L

5.2. - For any f~ in L 2 ( ~ ) d, i = 1 or 2, problem (2.5) admits the formulation (5.5). System (5.5) has a solution (U1, [/2) with each

Theorem

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

89

U~ = (ui, g~) in Xi x L2(f~,). Moreover, each function t~, i = 1 and 2, is nonnegative and belongs to H~(f~i) for all s < 89 and this solution satisfies (3.7) and (4.5).

Proof.

For each integer n, let us now denote by (U~, U~), with U/~ = (u~, g~), a solution of problem (5.1) (its existence is proven in Lemma 5.1). As previously, see (3.7) and (4.5), it satisfies, for a fixed number s < 89and for a constant c independent of n, /]

IlgnllH.(a,) < c (llu~ll~l(al)~ + Ilu~li~l(a2)d),

i -- 1 and 2.

So, there exists a subsequence, still denoted by (u~, g~, u~, g~)n, which converges towards (ul,gl,U2,g2) weakly in V1 • H8(~1) x V2 • H8(~2). We must now prove that (u1,~1, u2, g2) satisfies (5.5), which is performed in three steps. 1) We start from the variational formulation, for i = 1 and 2"

Vv~ E V~,

~

( ~ ( ~ ) Vu~ 9Vv~ dx i

(5.6)

/+ ](u?

-

- uyl

= <

>.

Next, from the convergence properties of the sequence (g~)n, there exists a subsequence, still denoted by (gn)n, which converges to gi strongly in L2(~i) and a.e. in ~i. So, thanks to the continuity and boundedness of the function &~, for any fixed vr in V~, the sequence (~(g~') Vv~)~ tends to &i(g/) Vv/ a.e. in ~i and is bounded in L2(~i) d2 by c IlVv~IIL2(~.~)j2 , hence it converges strongly in L2(~i) d2. Since (Vu~)n converges to Vui weakly in L2(~i) d2, this yields

liIn f

(~(~)Vu~.Vv~

dx = [

5 ~ ( ~ ) V u ~ . V v ~ dx.

Moreover, due to the compactness of the embedding of H 89(F) into L~(r), there exists two subsequences, still denoted by (u'~)n and (U~)n, SO that ((u n - u ~ ) ] u n - u~I)~ converges to (ui - u j ) l u i - uj] strongly in n~ (F). Consequently, (Ul, u2) satisfies the first equation in (5.5) for i - 1 and 2. 2) Taking vi = u~ in (5.2) yields

1

2 n

---< f l , U ~ > Jr- < f2, U2 > ,

A model for two coupled turbulent fluids

90

so that passing to the limit yields

li~

1

J[]2 : < I'1, Ul > + < I'2, u2 > - J r ]ui - u2[ 3 dT.

We also derive from the first equation in (5.5) that

L ~i(~'l)]VUl'2dx+L 5i(~2)[Vu2[2dx 1

2

-- < f l , Ul > + < f2, U2 > -- IF In1 -- U2I3 dT, whence

lim f

n--+(X) y~-~ 1

&, (eT)IVuTI 2 d~ + f

J~"~ 2

&:,(G')IV"-'71~d~

#,

-/_

#-

1

<~1(<1)iv,,,I ~ <~ + / _ <~(<~)ivu~l ~ <~-

(5.7)

J~2

On the other hand, let us set: h~ = V/&i(t~)Vu~. Since the sequence (hn)n is bounded in L 2 ( ~ ) d2, there exists a subsequence, still denoted by (hn)n, which converges to h~ weakly in L2(gt~)d2. In order to identify h~, we introduce a function ~ in L2(~i) d2. Since the previous subsequence ( i ))n converges to V/&i(gi) a.e. in ~i, the subsequence (V/&i(f n) ~)n also converges t o 4~i(~i)(t9 a.e. in ~i and, since it is obviously bounded by a function in L2(fl~), it tends to V/&i(gi)~ in L 2 ( ~ ) d2. Since (VU'~)n converges to Vui weakly in L2(~i) d2, this yields

li~m~176 S, ~l&i(f'~)Vu:.sodx = j; yi&,('i)Vui.sodx, i

i

so that hi is equal to v/5;(t~i)Vui. Moreover, from the weak convergence of (h~)n, we deduce that

d~t

&i(f~)[Vu~] 2 dx,

&i(gi) 1~TUil2 dx ~ linm_inf I J~

z

z

and combining this inequality with (5.7) implies

lim f

~(~)IV~I: d~

- f

~,(t~)IV~I:d~,

~ - 1 ~n~ ~.

(~.S)

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

91

Equivalently, the sequence (V/&;(g n) Vu~)n tends to v/&i(gi)Vui strongly in L2(~i) d2, so that the sequence (&i(t~n) ]Vu~]2)n tends to &i(gi)]Vui[ 2 strongly in L 1(~i). 3) We observe that the solution g~, i -- 1 and 2, of the second equation in (5.2) also satisfies the "transposed" formulation, where ~i is a smooth enough function on ~ti"

-L

gn A~i dx = - f r T n ( G i ( l u ~ -

u~12))On~cflidT

i

+/~

Tn (&/(e n)

IW?l

dx.

i

The convergence of the last term follows from part 2) of the proof together with the definition of Tn. The convergence (G~(lu ~ -u~I2)) n can easily be deduced from the sublinearity of Gi. So, for i - 1 and 2, each gi satisfies the second part of (5.5), which ends the proof, m To conclude, we go back to the initial system (1.1) and we write its full variational formulation:

Find (u~,p~) in X~ x L2(f~), 1 < i < 2, such that, forl < _ i r Vvi C X~,

j f ai(ki) Vui . Vvi dx - / ~ i

pi(divvi)dx i

+ ~ [ui -- u j ] (ui -- u j ) . vi d7 -- / ~ f i . vi dx. i

Yq~ e L2(fl~),

q~(divu~) dx = O,

- L

(5.9)

i

Find ki in L2(f~i), 1 < i < 2, such that, for 1 < i < 2" V~i e H2(ai) n Hi(hi),

-s

4

c,(k,)

dx = - J; c,(lu, - u212) On, i d"r

+ fa

i

Here, the argument is due to [24]. C o r o l l a r y 5.3. - For any fi in L2(~i) d, i = 1 or 2, system (1.1) admits the formulation (5.9). System (5.9) has a solution ('Wl, W2) with each Wi --

92

A m o d a l for two c o u p l e d t u r b u l e n t fluids

(u~,p~,k~) in X~ x L2(f~) x L2(f~i). Moreover, each function k~, i = 1 and 1 2, is n o n n e g a t i v e and belongs to HS(f~i) for all s < ~. Proof. Since the existence of pi in L2(~i) is a consequence of Lemma 3.1, it suffices to check t h a t the mapping" g H k - G~-l(g) is continuous from HS(gti) into itself. This follows by an interpolation argument: indeed, it is continuous from L 2 ( ~ ) into itself and from Hl(Fti) into itself thanks to the inequalities k < /2- 1 e,

6. A u n i q u e n e s s

lVkl ~

result for smooth

/2--1

IVel.

solutions.

The aim of this section is to prove t h a t any solution of system (1.1) which is slightly more regular than in the existence theorem, is unique when a further condition holds: the data must be small enough in comparison of the relative variation of the functions ~i. So, we assume that, for some data fi in L2(~i) d, system (1.1) has two solutions ( u i , p i , ki)i=l,2 a n d (ui,Pi,~i)i=l,2, we define the corresponding functions gi - G i ( k i ) and g~ - G i ( k i ) as in Section 2 and we set: w i = ui - u i ,

m i -- gi - gi,

i=1,2.

(6.1)

We assume t h a t the functions c~i are continuously differentiable with bounded derivatives and, for simplicity, we also introduce the notation u* = max sup sup { a i ( k ) , ~ / i ( k ) } , i=1,2 kER

u ' = max sup la'i(k)l. i--1,2 kER

-'i is smaller than -~. '' Note t h a t the m a x i m u m of c~ In the next two lemmas, we treat separately the equations on the velocities and on the turbulent energies. L e m m a 6.1. - A s s u m e that the pair (Ul,U2) belongs to WI,P(~I)d x 1_~_1~ = 1. T h e W I ' P ( ~ 2 ) d for a real number p > 2, and let q be such that: -~ following e s t i m a t e holds for the f u n c t i o n s w i , i - 1 and 2"

/,it

_

1

,

,

w,,,,(~2)d) 7

(6.2) 1

(pIm, IIL q ( a l )

. "~-lira=liLt(a=)

9

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

93

Proof. Subtracting formulation (3.5) for ui and ui, we obtain MVi C Vi,

/ a 5i(gi) V w i

9Vvi dx

i

q- ~ ( u i -- uj) 'ui -- uj] vi dT -- ~F(~i -- ~j) ]~i -- ~jl vi dT

i

Since wi belongs to Vi, we take each vi equal to wi, we sum up on i and we observe from (a.9) that, as in the proof of Proposition 3.5, the quantity that is integrated on F is nonnegative. This yields 2 "--

2 "r

i=1

i

whence 2

2

i--1

i--1

We also have

/]r _< --Ilm /2

Ila (e ) -

llLq(n,),

so that the desired estimate follows by applying HSlder's inequality.

I

L e m m a 6.2. - Assume that the pair (~1,u2) belongs to WI,p(~I) d • 1 WI'p(f~2)d for a real number p > 2, and let q be such that: 1p + 1 -~ = g.

Let s be a real number, 0 < s < 89 The following estimate holds for the functions mi, i 1 and 2: =

/j*

[lmillHs(~,~) < c - -// ( [ l f l l l 2 = ( a l ) a

q-Ilf2112=(a=)a) } 1

(llWll]~_/l(~l)d +

/J -

q-I]W2[l~_/l(~2)d ) ~

(6.3)

II 2

Proof. Subtracting formulation (4.2) for gi and t~i gives

--s

miA~oidx---~(Gi(lux-

u212)- G~(l~a-~212)) 0 ~ , ~ d r

i

+ ~ ((~i(gi)]Vuil u - &i(-[i)lVuii 2) pi dx. i

A model for two coupled turbulent fluids

94

As in the proof of L e m m a 4.2, relying on the analogue of (4.6), with any function g~ in H - S ( ~ i ) , we associate the function ~i = s defined by (4.3) and we recall that

II0~llL=(r) + sup I~(~)1 ~ ~ IIg~llH-~(~,), for a constant ci only depending on f~i. We now estimate the right-hand side of the previous equation for such a ~i. We observe that

IG~(lul - u2] 2) - G~(]~I - ~2J2)1 ~ v* ]ul -4- u2 +

-f- U2I ]Wl

~1

--

W21,

so t h a t denoting by c~ the norm of the embedding of H l ( ~ i ) into L4(f~i), we obtain

l j (r (G~(lu~ - u2l 2) - G~(I~I - ~212)) On~, dT I 2

<_c~ t 2 / ] . (~-~ lu~l~,(~,)~ + I~1~(~)~) 89 i=1 2

i--1

The norms ]uil~/l(n,),~ and we have

I~,1~(~,)~

are bounded from (3.7). Similarly,

_

b~ t p

12

whence

I ~ (~(e~)1ruff ~ ,t,

~(~)IW,

I~) ~, d~l

2

Ci (L]* ( E

2

I~J'il~l(~"~'t)d +

]Ui]2HI(~,,,)d)l ( E IWil~HI(~,)d) 1

i--1

i--1

~' .

) ,~(~,.) IIm~llL~(a.,) IIg~llH-s(a,).

Combining all this yields the desired result. We are now in a position to prove the uniqueness result.

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

95

T h e o r e m 6.3. - Assume that the data fi, i = 1 and 2, belong to L2(~i) d. If system (1.1) admits a solution (W1, W2), with W~ - ( ~ , ~ , ki), such that the ui, i = 1 and 2, belong to Wl'P(f~i)d for a real number p > 2d and that the following condition holds for appropriate constants c and d c -~, X X + c '

~

(11fl II2

L2

2 )a ) 89 < 1 (~1 )a -4- Ill2 IIL~(~ 1

with

(6.4)

X = (ll~ll]~l,,,(a,)a + i1~2 ]lpWl,P(~2)d ) "P

then it is the unique solution of system (1.1) in the sense that the pair (Wl, W2) is the unique solution of problem (5.9). Proof. Since p is > 2d, there exists an s < 1 such that HS(fli) is imbedded in Lq(f~i). So replacing the Ilmillnq(a,) and IlmillL~(a,) in (6.2) and (6.3), respectively, by an appropriate constant times IlmillHs(n,), summing up the estimate (6.3) on i and inserting (6.2) in the sum, we derive from (6.4) that both mi cancel. The complete uniqueness result follows, m V ! So, the uniqueness mainly depends the parameter -~ which represents the variation of the function 6i. When the ai are constant, we recover in the previous theorem the nearly obvious and unconditional uniqueness result which also follows from Proposition 3.5.

7. Some further regularity properties in d i m e n s i o n 2. W~e intend to study the regularity of the solution of system (1.1) in the simple case of dimension d - 2, when the fli are rectangles. Indeed this geometry is the key one for most discretizations. We need several lemmas. The first one deals with the Stokes problem with mixed boundary conditions, and results from [22, Cor. 4.2], extended to less smooth data thanks to the arguments in [9, Chap. 8]. From now on, we set" so = 1.5946.

Lemma

(7.1)

7.1. - For i = 1 and 2, if the domain f~i is a rectangle, the

96

A m o d e l for two coupled t u r b u l e n t fluids

m a p p i n g : f~ ~-~ (v~, q~), where (vi, q~) is the solution o f t h e S t o k e s p r o b l e m -Avi

+ g r a d qi = f i

in ~ i ,

div vi = 0

in ~ i ,

vi --- 0

on Fi,

Om vi - qi n i = 0

on F,

(7.2)

is c o n t i n u o u s from H S - 2 ( ~ ) 2 into H ~ ( ~ ) 2 x H ~ - 1(~i) for all s, 3 < s <_ So.

The second lemma extends this result to the case where the Neumann boundary conditions are not zero. L e m m a 7.2. - For i = 1 and 2, if the d o m a i n ~ i is a rectangle, the S t o k e s m a p p i n g $i: (fi,g~) H (v~,q~), where (v~,q~) is the s o l u t i o n o f the S t o k e s problem - A v i + g r a d qi = f i in ~ i , div vi = 0

in s ,

vi = 0

on Fi,

On~, Vi -- qi n i -- gi

on F,

(7.3)

is c o n t i n u o u s from H s - 2 ( Q i ) 2 • H S - ~ (F) 2 into H S ( Q i ) 2 • S s - l ( ~ i ) s, 3 < s < s o .

for all

Proof. We want to construct a divergence-free function wi and a function ri with the required regularity such that:

wi=0

onFi

and

On~wi-rini=gi

onF,

and to apply the previous Lemma 7.1 to the pair (vi - w i , qi - ri). So, denoting by gix and g~z the components of gi, we look for a function w i of the form c u r l ~, so that the desired boundary conditions are written ~2 =On,~,~ = O

and

0~

onFi = +

(g~z - r~)(~)d~,

0~V2 = + ( - g ~ )

on r,

C. Bernardi, T. Chacon ReboHo, R. Lewandowski and F. Murat

97

where (a, 0) stands for the left endpoint of F and the sign + depends on i. The idea is to take: ri =

1

meas F

giz(x) dx

(such a constant is well-defined since g~ belongs to L I(F)). Indeed, thanks to this choice, the right compatibility conditions are satisfied at the corners of f~i. So, the existence of a function r in H~+l(Fti) follows from [2, Thm. 2.d.2], together with the desired bound on its norm. This, combined with Lemma 7.1, yields the result. I Let us now check how the previous results extend to smaller values of s, namely 1 < s < 3 C o r o l l a r y 7.3. - For i = 1 and 2, /f the domain f~i is a rectangle, the Stokes mapping s ( f i , g i ) ~ (vi,qi), where (vi, qi) is the solution of (7.3), is continuous from (Xg-~) ' x H~-~ (F) 2 into HS(ai) 2 x H~-l(f~i) for a1I s, 1 < s < ~, where (X2-~) ' stands for the dual space of

(7.4)

X2i -~ = {vi c H2-~(ai)2; vi = 0 on r i } .

Proof.

From Lemma 7.2, the mapping is continuous from H~~ 2x x Hs~ When writing the variational formulation of problem (7.3), we observe that it is also continuous from

H~o-~(r) 2 into H~~

X~ x (H0[o(F)) '2 ( t h e ' denotes the dual space) into H i ( a ) 2 x L2(a). So, if 1

f

H~~ 2 is dense in X~ and if H~~ ( F ) i s dense in (H~0(F)), applying the main theorem of interpolation [16, Chap. 1, Th. 5.1] yields that the mapping is continuous from Fs x G~ into g s ( f ~ ) z x H ~ - l ( f ~ ) , with:

F~

1 :

<1o,

=

with

0= so-s so - 1

So, it remains to check the density results and to characterize the spaces Fs and Gs. 1 ~)(a) is dense in H2-S~ Since :D(f~)2 is 1) First, since 2 - so is < 7, contained in Xi, Xi is dense in H2-~~ 2, so that H~~ 2 is dense in X~. Moreover the following characterization holds [16, Chap. 1, Th. 6.2]"

Fs =

(IXi, H 2-s~

(~i)2]1--0

)'

9

98

A model for two coupled turbulent fluids

And the interpolate space in the previous line coincides with X 2-~. This follows from one-dimensional interpolation results, combined with the tensorization property (where the rectangle fti is the product of the two intervals A~: and A~) X 2-~ = Ho2-~(A~:; L2(AZ)) n L2(A~:; H2-~(A~)), where H.2-~(A~)stands for the space of functions in H2-S(Af)vanishing in the upper endpoint. 2) Next, we observe that H~0(F ) is dense in L2(r)

which

is dense in

H~-~o(r), so that H~o-~(r) is dense in (Ho0(F))' and the space Gs is well-defined. Moreover, applying [16, Chap. 1, Th. 6.2] gives G~ =

( [H~0(F) 2,H~ -~~

)' .

Next, relying on [16, Chap. 1, Rem. 12.6], we observe that the interpolate 1

sp~ce H~-~(F) since s is > 1. This ends the proof,

38

coincides with H~

(F), which coincides with m

We also need some regularity properties of problem (4.2) when the functions U l and u2 are smoother than in the existence result of Theorem 5.2. 3 Fori-lor2, andfor L e m m a 7.4. - Let s be a real number, 1 < s <_ 3" any pair (Ul, u2) in HS(f~l) 2 x H~(ft2) 2, the solution g~ of problem (4.2) belongs to HS(f~i) and satisfies, for a constant ~i only depending on fti,

(7.5) The same properties holds for 3 < s < 2 if the functions q/i are continuously differentiable with bounded derivatives Proof. Thanks to Lemma 4.1, it suffices to check that 5i(gi)IVui[ 2 belongs to HS-2(f~i) and that G~(lUl - u212), extended by zero on Fi, belongs to H s - l ( of~ ).

1) On one hand, the functions ui belong to H~(a~) 2, hence to W 1' ~---~2.~(fh) 2 by the Sobolev imbedding theorem. So, ]Vu~[ 2 belongs to L r~-l.~(ft~) and, since c)i is bounded, the same property holds for c)~(g~)IVu~l 2. 3 For s _> 3, we obtain that I belongs to L2(f~), hence 3 since the imbedding of H3-2~(f~i) into to HS-2(fti). For s <_ 3,

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

99

L s71~(fti) yields the imbedding of L ~--~-;~~(Fti) into H2~-3(f~), we derive that &~(gi) IVui[ 2 belongs to H2~-3(ft~), hence to H~-2(ft~). Moreover, in both cases, we have the following estimate II,~(e~) IVu~l~llH~-~(a,) < c Ilu~ll ~ --

H ~ ( ~ ) 2

"

(7.6)

2) On the other hand, since G i ( [ U 1 - - U212) vanishes at the two endpoints of F, it suffices to check that it belongs to H~- 89(F). First, we observe that Ul - u 2 belongs to HS- 89(F) 2 and, since HS- 89(F) is an algebra [12, Thm 1.4.4.2], It1 - u 2 1 2 also belongs to H~- 89(F). When s is _< g,3 relying on [24] and using similar arguments as for Corollary 5.3, we derive that Gi([Ul - u2[ 2) belongs to g s - 8 9(F) and satisfies I l a ~ ( l ~ - u212)llH~- 89(r) <- ~ (!1~111 ~

(7.7)

The same result is derived when s is > ~, by applying the same arguments to the derivative of Gi(lUl - u212). This ends the proof. I We are in a position to prove the main result of this section. T h e o r e m 7.5. - A s s u m e that the domains ~1 and f~2 are rectangles and that the functions c~i and 7i, i = 1 and 2, are continuously differentiable with bounded derivatives. Let fi, i = 1 and 2, belong to L2(fti) 2, and let ( W l , W 2 ) b~ ~ ~ol~tio, o ~ y ~ t e m (1.1), ~ / t h W, = (~,~,p,,k,). S~ th~ u,, i = 1 and 2, belong to Hs*(Fti) 2 for some s* > 1, the Wi, i = 1 and 2, belong to Hs~ d x H~~ x H~~ Proof. Let (W1, W2) be any solution of problem (1.1). The proof relies on a boot-strap argument and is performed in two steps. 1) From the assumption on the u~, each pair (ui,pi) belongs to H ~*(Fti) 2 • H ~ * - I ( ~ ) . So, applying Lemma 7.4 yields that each g~ belongs to H ~*( ~ ) . Finally, from [24], the function ~(~.~) p~ also belongs to Hs*-l(f~i). 2) Assume now that both triples (ui, ~ p', gi) belong to Ht(ft~)2 x H t - l ( f t i ) z Ht(f~i), for a real number t, 1 < t _< so. Then the pair (ui, qi), P~ with qi - a,(e~), is a solution of the Stokes problem

A model for two coupled turbulent fluids

100

-Aui + grad

a;(e,)

f~ + qi -- &~(g~)

&(&) V g i

9V u i

-

a~(e,)

&(g,)2 V g i qi

in ~ti,

div ui = 0

in ~i,

ui -- 0

on F i,

(7.8)

On,

-

=

1

(u,

-

uj)l

-

on F,

In this equation, Vii and qi belongs to H t - l ( ~ i ) , together with each component of Vui. Since H t - l ( ~ i ) is included in L~-~-2~(~i), the righthand side in the first line of (7.8) belongs to L 2~--~-~(gti) 2, hence to L2(~i) 2 when t is _> 3 and to (X3-2t) ' when t is < 3. It is also clear that the right-hand side in the fourth line of (7.8) belongs to Ht- 89(F) 2, hence to H2t-~ (F) 2. So using Corollary 7.3 (or Lemma 7.2) yields that (ui, qi) belongs to H t*(~i) 2 • Ht'-x(~ti), with t* = m i n { 2 t - 1, s0}, and applying Lemma 7.4 yields that gi belongs to H t* (~ti). Iterating n times this argument, where n is the smallest integer such that s* + n ( s * - 1 ) i s :> so, we obtain that the triple (ui,qi,gi), hence (ui,pi,gi), belongs to HS~ • H~~ • H~~ Then, ki also belongs to HS~ (~i).

Remark.

A more technical proof, relying on Meyers' argument [19] (see also [8]), allows for replacing, in the statement of Theorem 7.5, the assumption "the ui belong to H s~ (~i) 2'' by the modified one "the ki (or gi) belong to H s* (gti)", also for any s* > 1, however this new assumption seems stronger. Of course, these regularity properties can be extended to any convex polygons ~i, since the corresponding values of So can easily be computed from [22]. By a boot-strap argument, they also hold in the case where convection terms are added in the system. But, when the interface conditions are replaced by Manning's law, the regularity of the solution of the basic Stokes problem, a fortiori of the present system, seems unknown.

References [1]

[2]

C. Bernardi, T. Chacon, R. Lewandowski and F. Murat, Existence d'une solution pour un module de deux fluides turbulents couples, C. R. Acad. Sc. Paris 328 s~rie I (1999), 993-998. C. Bernardi, M. Dauge and Y. Maday, Polynomials in weighted Sobolev spaces: basics and trace liftings, Internal Report 92039, Laboratoire d'Analyse Num~rique, Universit~ Pierre et Marie Curie, Paris (1992).

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

[3]

[4] [5] [6] [7] IS] [9] [10] [11] [12] [13]

[14] [15] [16] [17] [is] [19] [20]

101

P. Benilan, L. Boccardo, T. Gallou~t, R. Gariepy, M. Pierre and J. L. Vasquez, An L 1 theory of existence and uniqueness of nonlineai elliptic equations, Ann. Scuola Norm. Sup. Pisa C1. Sci. 22 (1995) 241-273. D. Blanchard and H. Redwane, Solutions renormalis~es d'~quatiom paraboliques ~ deux nonlin~arit~s, C. R. Acad. Sc. Paris 319 s~rie 1 (1994), 831-835. L. Boccardo and T. Gallou~t, Nonlinear elliptic and parabolic equatiom involving measure data, J. Funct. Anal. 87 (1989), 149-169. J. Boland and R. Nicolaides, Stability of finite elements under divergence constraints, SIAM J. Numer. Anal. 20 (1983), 722-731. H. Brezis, Analyse fonctionnelle, Collection "Math~matiques appliqu~es pour la ma~trise", Masson (1983). S. Clain and R. Touzani, Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients, Mod~l. Math. et Anal. Num~r. 31 (1997), 845-870. M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341, Springer-Verlag (1988). T. Gallou~t and R. Herbin, Existence of a solution to a coupled elliptic system, Applied Maths Letters 2 (1994), 49-55. V. Girault and P.-A. Raviart, Finite Element Methods for the NavierStokes Equations, Theory and Algorithms, Springer-Verlag (1986). P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (1985). R. Lewandowski, Analyse math~matique et oc~anographie, Collection "Recherches en Math~matiques Appliqu~es", Masson (1997). R. Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Analysis TMA 28 (1997), 393-417. J.-L. Lions, Quelques m~thodes de r~solution des probl~mes aux limites non lin~aires, Dunod & Gauthier-Villars (1969). J.-L. Lions and E. Magenes, Probl~mes aux limites non homog~nes et applications, Vol. 1, Dunod (1968). J.-L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean, Computational Mechanics Advances 1 (1993), 1-120. P.-L. Lions and F. Murat, Solutions renormalis~es d'~quations elliptique~ non lin~aires, to appear. N.G. Meyers, An LP-estimate for the gradient of solutions of second ordeI elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa 17 (1963), 189206. F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Internal Report 93023, Laboratoire d'Analyse Num~rique, Universit~ Pierre et Marie Curie, Paris (1993).

102

A model for two coupled turbulent fluids

[21] F. Murat, l~quations elliptiques non lin~aires avec second membre L 1 ou mesure, Actes du 26~me Congr~s National d'Analyse Num~rique, Les Karellis, France (1994), A12-A24. [22] M. Orlt and A.-M. Ss Regularity of viscous Navier-Stokes flows in nonsmooth domains, Proc. Conf. Boundary Value Problems and Integral Equations in Nonsmooth Domains, M. Costabel, M. Dauge et S. Nicaise eds., Lecture Notes in Pure and Applied Mathematics 167, Dekker (1995), 185-201. [23] G. Stampacchia, l~quations elliptiques du second ordre ~ coefficients discontinus, Presses de l'Universit~ de Montreal (1965). [24] L. Tartar, Interpolation non lin~aire et r~gularit~, J. Functional Analysis 9 (1972), 469-489. Christine Bernardi, Francois Murat Laboratoire Jacques-Louis Lions C.N.R.S. & Universit~ Pierre et Marie Curie Boke postale 187 4 place Jussieu 75252 Paris Cedex 05 France E-mail: [email protected], [email protected] Tomas Chac6n Rebollo Departamento de Ecuaciones Diferenciales y Ans Universidad de Sevilla Tarfia s/n 41012 Sevilla Spain E-mail: [email protected]

Numerico

Roger Lewandowski Equipe de M~canique, IRMAR Campus de Beaulieu Universit~ de Rennes 1 35042 Rennes Cedex 03 France E-mail: lewandow@maths, univ-rennes 1. fr Research partially supported by Spanish Government, MAR97-1055-C02-02 Grant.

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chap ter 6 D E T E R M I N A T I O N DE C O N D I T I O N S A U X LIMITES EN MER OUVERTE PAR UNE M E T H O D E DE CONTROLE OPTIMAL

F. BOSSEUR ET P. ORENGA

R@sum~. Nous pr@sentons dans ce travail le principe d'une m@thode num@rique adapt@e k la d@termination de conditions aux limites en mer ouverte d'un probl~me de shallow water. Cette m~thode repose essentiellement sur l'utilisation de la th@orie du contrSle optimal et de l'assimilation de donn@es. On se propose de reconstituer des conditions aux limites k partir d'un ensemble de mesures disponibles k l'int@rieur du domaine d'@tude. Le contrSle est effectu@ sur les conditions aux limites de la vitesse aux fronti~res ouvertes, en consid@rant des observations ponctuelles. Si au niveau th@orique, nous avons montr@ des r@sultats d'existence dans le cas lin@aire, nous n'avons p u l e faire dans le cas non lin@aire. Nous donnons des r@sultats num@riques dans le cas lin@aire et non lin@aire, tout d'abord sur une g@om@trie simplifi@e pour justifier la m@thode num@rique, puis sur le cas r~el d'une baie.

A b s t r a c t . We present here a numerical method adapted to the boundary conditions in open seas for a shallow water problem. Essentially, this method depends on the use of optimal control and data acquisition theory. We propose to reconstruct the boundary conditions from a set of available measurements taken in the interior of the region under study. The control is made on the velocity boundary condition on open boundaries, in considering observations at isolated points. If on the theoretical level we showed existence in the linear case, we could not achieve it in the nonlinear case. We present numerical results in the linear and nonlinear cases, first in a simplified geometry in order to justify the numerical method, then in the real case of a bay.

104

D d t e r m i n a t i o n de c o n d i t i o n s a u x f i m i t e s en m e r o u v e r t e . . .

1. Introduction Nous ~tudions dans ce travail un module de type sh a l l o w water qui, dans le cas d'une mer peu stratifi~e ou peu profonde, s'av~re suffisant pour representer la dynamique des fluides et peut constituer une ~tape pr~alable une ~tude plus approfondie des ph~nom~nes par un module tridimensionnel. Plusieurs ~tudes th~oriques et num~riques ont ~t~ faites sur ce module (Orenga, 1991; Chatelon-Orenga, 1997; Bisgambiglia, 1989). Pour notre part, nous nous proposons d'~tudier le cas de domaines avec des fronti~res ouvertes oh se pose souvent le probl~me de la connaissance des conditions aux limites sur la vitesse u. Celles-ci sont g~n~ralement ~valu~es grs ~ des donn~es exp~rimentales issues de campagnes de mesures, ou des donn~es calcul~es ~ l'aide d'un module de plus grande emprise. Dans le cas de la baie de Calvi, des mesures in situ ont pu ~tre effectu~es (Norro, 1995) grace ~ l'implantation voisine de la station de recherche oc~anographique STARESO. Toutefois, celles-ci sont en nombre insuffisant et ne nous donnent que peu de renseignements sur les conditions aux limites en mer ouverte ~ adjoindre au syst~me. De m~me, les r~sultats issus des programmes de recherche tels que Medalpex qui donnent des renseignements sur la circulation g~n~rale en M~diterran~e, ne peuvent prendre en compte les ph~nom~nes ~ l'~chelle de la baie. C'est dans le but de lever ces ind~terminations et d'utiliser au mieux les donn~es disponibles, que nous nous proposons d'utiliser des m~thodes d'assimi-lation de donn~es, d~j~ utilis~es avec succ~s dans de nombreux domaines, comme par exemple les pr~visions m~t~orologiques. Le probl~me est r~solu par la m~thode de Galerkin. Afin d'obtenir des conditions aux limites homogbnes, on effectue le changement de variable v - u - w, off u est la solution et w un rel~vement de la condition aux limites. Le contr61e est alors effectu~ sur w, en utilisant des donn~es ponctuelles issues de mesures effectu~es in situ ou de r~sultats de modules ~ plus grande ~chelle. Dans la section 2., nous rappelons bri~vement les ~quations du module utilis~. Dans la section 3., nous donnons les ~quations v~rifi~es par le contr61e optimal dont nous d~montrons, dans le cas lin~aire, l'existence et l'unicit~ dans la section 4. Les sections 5. et 6. sont, quant ~ elles, consacr~es ~ l'analyse num~rique des ~quations. Nous pr~sentons dans la section 7. un certain nombre de r~sultats obtenus sur une g~om~trie simplifi~e, constitute par un carr~ de cSt~ unitaire et qui nous a surtout permis de tester l'efficacit~ du code impl~ment~, puis sur le cas plus concret de la baie de Calvi (Corse), off l'on constate des ph~nom~nes d'~rosion des cStes dfis principalement ~ la modification des courants apr~s l'implantation de structures en mer (extension du p o r t , . . . ) .

F. Bosseur et P. Orenga

105

2. Equations du module Dans un module oc@anique complet, sont prises en consid@ration des variables biochimiques (concentrations chimiques, biomasses, ...) et hydrodynamiques (salinit@, temp@rature, vitesse et masse volumique). Dans notre cas, nous nous sommes limit,s g l'@tude des variables hydrodynamiques du syst~me. Pour d@crire l'@volution de ces variables, nous utilisons des modules de m@canique des fluides g~ophysiques qui se distinguent des probl~mes classiques d'@coulement, tels que ceux de Navier-Stokes, par les dimensions des domaines, les @chelles temporelles, la loi de conservation de la masse, la faible profondeur et surtout les conditions aux limites. A partir des @quations g@n@rales de conservation et des diff@rentes hypotheses simplificatrices li@es aux propri@t@s des fluides g@ophysiques (Nihoul, 1977), on @tablit les @quations du module tridimensionnel. Les ~quations du module que nous utilisons sont quant & elles obtenues en int@grant les @quations du module tridimensionnel non-stratifi~ sur la verticale.

x

u(t,x)

a

Fig. 2.1" Domaine d'dtude

9 ~'f repr@sente les c6tes ( u . n - 0), " 7e les fronti~res ouvertes ( u . n - G(x, t)). De plus, pour les fronti~res ouvertes, on distingue les parties de la fronti~re off le fluide est entrant (G(x,t) < 0) que l'on notera ~/-, des parties oh celui-ci est sortant (G(x,t) > 0), alors not~es ~/+. Cette diff@renciation est justifi@e par la n@cessit@ de fixer la hauteur d'eau sur la partie de la fronti~re o5 le fluide est entrant (Chatelon-Orenga, 1997). On d@signe par u(x, t), la

106

D d t e r m i n a t i o n de c o n d i t i o n s a u x limites en mer o u v e r t e . . .

fonction de f~x]0, T[ $ valeur dans 1t(2 reprdsentant la vitesse ~ l'intdrieur du fiuide et h ( x , t ) , la fonction de F~x]0, T[ ~ valeur dans ll{ ddsignant la hauteur de la colonne d'eau. On note 9 l'opdrateur c~, qui $ u = (ui,u2) E ]1{2 fait correspondre ( - u 2 , u l ) E I~ 2 ,

9 l'opdrateur R o t , qui ~ une fonction scalaire q(x, y) fait correspondre la fonction vectorielle Rot q - (Oq _ oO_~z) Oy ~ 9 l'opdrateur rot, qui ~ une fonction vectorielle u = (Ul, u2) fait correspondre la fonction scalaire rot u - ( ~Ox _ Oul ) Oy et on introduit Q

=

ax]0,T[,

r~

=

~•

r~-

=

~-•

2+

=

3, + x ] 0 , T [ .

Les 6quations du problbme de s h a l l o w water s'dcrivent ou _ A A u + ~1 V u 2 + rot u a ( u ) -5( +Du + wa(u) + gVh = f, u.n = a ( x , t),

rot u = O, ~(t = 0) = ~ 0 ( ~ ) , Oh ~-7 + div (uh) - 0, h = #(x, t),

dans Q, sur ~, sur ~, dans f~,

(2.1)

dans Q, sur E - , clans 9t.

h ( t = O) = h o ( x ) ,

Pour la rdsolution numdrique des dquations du module, et notamment pour pouvoir utiliser la base spdciale dans la mdthode de Galerkin, nous effectuons un changement de variable de mani~re ~ nous ramener ~ des conditions aux limites homogbnes. Ainsi, on pose ~ -- V'k-W~

off w = Vp,

(2.2)

avec p solution du probl~me -Ap--O Op

dans f~j2 (2.3)

-G

sur "7,

Gdg=O

107

F. B o s s e u r et P. O r e n g a

Les @quations du module se r@~crivent, en rempla~ant u par (v + w) + ~1 V v 2 + grad ( v . w ) + rot v a ( v ) + rot v a ( w ) + D v + w a ( v ) + g V h __ f Ow 1Vw 2 _ Dw - wa(w)

~t - A A v

-

ot

v . n -- O,

rot v - O, v(t

-

o) =

-

w(t

-

o),

Oh ~-? + div ( v h ) - - div (wh), h = #(x, t), h ( t = O) - ho(x),

dans Q, sur ~, sur E, dans f~,

(2.4)

dans Q, sur ~ - , dans ft.

Des r@sultats th@oriques d'existence et de r@gularit@ de solutions de ce probl~me sont donn@s dans Chatelon-Orenga (1997). Dans la suite, oh les r@sultats th@oriques sont d@montr@s dans le cas lin@aire, nous consid@rons le probl~me (7)) lin@aris@ not@ (P)l , -5~ ov _ A A v

+ Dv + wa(v) + gVh _ f

m

ow Ot

v . n - 0,

rot v -

0,

v ( t - O) - u o ( x ) - w ( t - 0), Oh -~ + h div v - O, h ( t - 0) - h0(x),

Dw-

wa(w)

dans Q, sur E, sur E, dans f~,

(2.5)

dans Q, dans f~.

off h repr@sente la hauteur moyenne sur le domaine, ne d@pendant pas du temps.

3. L e m o d 6 1 e

adjoint

Le principe de base du contr61e optimal est la minimisation d'une fonction cofit J mesurant les @carts entre la solution calcul@e et un ensemble d'observations disponibles. Ainsi, si l'on note X, l'espace des contr61es (g@n@ralement un espace de Hilbert), Xod, un convexe ferm@ de X et w E X, la variable de contr61e, le probl~me s'@crit t r o u v e r Wo E X~d r d a l i s a n t

inf

wE Xoa

J(w).

108

Ddtermination de conditions aux limites en mer ouverte...

En pratique, la minimisation de la fonctionnelle J(w) n~cessite la connaissance du gradient de J par rapport aux variables de contr61e. Parmi les diff~rentes m~thodes de d~termination de ce gradient, l'utilisation des ~quations adjointes du module lin~aire tangent semble ~tre la plus int~ressante, notamment au niveau num~rique (Lions, 1968; Talagrand-Courtier, 1987). Puisque l'on consid~re ici des observations ponctuelles, il est essentiel d'exiger comme condition pr~alable que les fonctions recherch~es soient au moins continues sur le domaine d'~tude. Dans notre cas, off la dimension du domaine est ~gale ~ deux, le th~or~me de plongement de Sobolev montre que ces fonctions doivent ~tre recherch~es naturellement dans H2(~). Compte tenu du type de rel~vement considerS, l'espace des contr61es est donc donn~ par L2(0, T; W ) , off W-

{ wEH2(~)2;w-Vp,

Ap-0,

0p

~n-0Sur3'I

}

9

Consid~rons X l , . . . X m , des points de ~, off m repr~sente le nombre d'observa-tions disponibles. On d~finit la fonctionnelle J(w) par

g(w) - I I c ( v + w ) - u~ll H = + ~IIwlIL~-(o,T;W) = ,

(3.1)

off C est l'op~rateur d'observabilit~ (d~fini ci-apr~s) et H - (L2(I~)) m l'espace des observations. Si les donn~es du probl~me sont suffisamment r~guli~res, il a ~t~ d~montr~ dans Chatelon-Orenga (1997) que la vitesse v appartient au moins

L~ (0, T; v), o~ V - {~p E H2(fl)2; ~.n - 0, rot ~ - 0 sur 3'}. L'op~rateur d'observabilit~ C est donc d~fini par

c

(L2(~)) m

L: (0, T; v u w) v

,

)

v(~m) Notons enfin la vitesse d~sir~e,

Ud

"-"

Ud E

H, sous la forme

lu,Xl,i idll

109

F. Bosseur et P. Orenga

Alors, (3.1) se r66crit sous la forme

m

T

J(w) - ~-~ fo Iv(xj ' t;w) +w(xJ) -Udj(t)12dt j=l

+ llwll L2(O,T;W)"

(3.2)

Expression du probl~me adjoint Les 6quations adjointes sont obtenues formellement en multipliant les 6qua-tions du module lin6aire tangent I par les variables adjointes, not6es v* et h*, puis en int~grant sur Q = ~ • T[. On obtient alors le probl~me (P*) suivant: or* A A v * - v. div v* - w div v * + rot (va(v* ) ) ot + rot ( w a ( v * ) ) + v* a ( r o t v) - h grad h* - w a ( v * ) + Dv* --- Ejm=l (V(X j, t; W) + W(Xj) -- ~tdj (t)) @ (~(X -- X j) v*.rt = O, rot v* = O, = T) = 0 ,

dans Q, sur ~, sur E, dans ft.

Oh* ot v. grad h* - w. grad h* - g div v* - 0, h* = O, h* (t = T) = 0,

dans Q, sur E +, dans f~. (3.3) oh, (v(xJ, t; w) + w ( x j ) - Ud~ (t)) | 5(x -- xJ) est la distribution d6finie par

r --+

jr0 T

(v(x j, t; w) + w(xj) - Ud~(t))r

j, t)dt,

(

qui est une forme lin6aire continue sur L 2 0, T; (H2(f~) 2)

r e

).

Dans le cas off l'on consid~re le probl~me (P) lin6aris6, les 6quations adjointes s'6crivent

-Or* ~

A A v * - h grad h* - w a ( v * )

Ej=I (V( xj, (P*)t

+ Dv*

-

t; W) + W(Xj) -- Udj (t)) @ e l t a ( x -- x j)

V*.n = O, rot v* = O, v*(t= T)=0,

Oh* , ot g div v - 0 , h* (t = T) = 0,

dans Q, sur ~, sur ~, dans f~. dans Q, dans f~. (3.4)

1Obtenues en diff6renciant les 6quations (2.4) par rapport ~ la variable de contr61e.

110

Ddtermination de conditions aux limites en mer ouverte...

Les propri6t6s de l'op6rateur adjoint nous permettent alors d'6tablir l'6quation v~rifi6e par le minimum local de la fonctionnelle J, qui s'e~crit 2, 'v'0 E L2(0, T; W)

I Ov*

+ v div v* + w div v* + rot va(v*) - Dv* + wa(v*) - h* grad h + C*(C(v + w) - Ud) -~- s

O~

/ L2(O,T;W ' ),L2(O,T;W)

"- O

(3.5) dans le cas non lin~aire, et

I Ov* --Oi

Dv* + wa(v*) \

+ C*(C(v + w) - Ud) + eAww, O) =0 / L2(O,T;W ' ),L2(O,T;W) (3.6) dans le cas lin~aire.

4. Conditions d'existence et d'unicit~ du contrSle optimal darts le cas lin~aire Avec les notations introduites au chapitre precedent, on a, T h ~ o r ~ m e 4.1 - On suppose que J(w) est donn~e par (3.1) et que G E H 1 (0, T; H~ ('y)). Le contrdle optimal w E L2(0, T; W) est caractdrisd de mani~re unique par les probl~mes (7~)~, (7)*)~ et l'dquation (3.6), avec

v E (L 2 (0, T; V) N H 1 (0, T; L2(~-~)2)), h E L (x)(0, T; H 1(~'~)), v* E L 2 (0, T; L2(fl)2), h* E L 2 (0, T; L2(gt)).

Ddmonstration. Les r~sultats d'existence et d'unicit~ du probl~me (P)t ont ~t~ d~montr~s dans Orenga (1991). Pour d~montrer que le probl~me adjoint admet une solution unique, on op~re par transposition (Lions-Magenes, 1972).

2Off AW repr~sente l'isomorphisme canonique de L2(0, T; W) sur L2(0, T; W').

F. Bosseur et P. Orenga

111

Pour simplifier les notations, on considbre, dans les dquations (3.4), les constantes du probl~me dgales s un. Le syst~me se rddcrit alors

Ov* Ot

m

(v(x j, t; w) + w(x j) - Udj (t)) | 5(x -- xJ), (4.1)

Av* - Vh* - ~ j=l

Oh* div v* = 0, Ot v*.n = 0, rot v* = 0 sur ~/,

(4.3)

v* (t = T) = 0, h* (t = T) = 0.

(4.4)

(4.2)

Introduisons l'espace .= = {r C L2 (O, T; H2 (a) 2) AHI(O,T;L2(gt)2); r

= 0, rot r = 0 sur 7, r

= 0},

et le probl~me

or

A r + V ~ = F,

(4.5)

Ot O~ + div r = 0, Ot r

(4.6)

= 0, rot r = 0 sur 7, r

= o,

(4.7)

= 0.

(4.s)

On a le rdsultat suivant: Si F e L 2 (0, T; L2(a) 2) alors le probl~me (4.5)-(4.8) admet une solution unique dans =. En effet, l'application T ddfinie par

or off ~ vdrifie: 0~ 0--t- + d iv r - 0, est un isomorphisme de ~ dans n 2 (0, T; L2(ft)2). Par transposition, on en ddduit qu'il existe v* e L 2 (0, T; L2(Ft) 2) et h* e L2(O,T; L2(f~)) uniques, vdrifiant

/; ( V*

-~Or _ A r + V ~

)oh. dQ - M ( r

- ~ - + div v* - O,

112

D6termination

de c o n d i t i o n s a u x l i m i t e s en m e r o u v e r t e . . .

off: m

T

~..!

r... !

d@finit une forme lin@aire continue sur :. et off - est le dual de = Si e > 0, il y a existence et unicit~ du contr61e optimal w (Lions, 1969), caract@ris@ par

-~-Dv*+wa(v

*) +

--Ud

)+Aw ,8} W

L2(O,T;W'),L~(O,T; W)

=0,

m

for all 0 E L2(0, T; W).

5. Principe de la m@thode de r@solution 5.1. R~solution des @quations du module et des ~quations adjointes La m@thode utilis@e pour la r@solution des @quations du module et des @quations adjointes est bas@e sur l'utilisation de la m@thode de Galerkin. On transforme, par troncature d'une base de l'espace consid@r@, le syst~me d'@quations aux d@riv@es partielles par un syst~me d'@quations diff@rentielles ordinaires dont les inconnues sont les projections de la solution du probl~me approch@ sur la base. Dans le cadre de ce travail, nous avons utilis@ la base sp@ciale dont les propri@t@s ont @t@mises en @vidence dans Orenga (1992). En particulier, on a le r@sultat suivant: Thdor~me - S o i e n t V - {u C L2(f~) 2,div u E L 2 ( ~ ) , r o t u E L2(f~);u.n - 0 sur "7}, Ho(div 0, rot 0) - {u E L2(f~) 2, div u - 0, rot u - 0; u . n - 0 sur 7}, I'1, (') la n o r m e et le p r o d u i t scalaire d a n s O n c o n s i d b r e les p r o b l ~ m e s

-Au(Pl)

u . n -- 0

rot u - - 0

(7'2)

-ApAp grad p . n - 0

dans f~ sur -7

Au

L2(~)

ou

L2(~)2.

dans f~ sur .7 sur -),

(7'3)

q- A - q 0 - #q

dans f~ sur 7

113

F. Bosseur et P. Orenga

et les p problbmes - - A r -- 0 (P4,)

/'--1

r--0

dans f~ sur 7i sur 7j

avec j ~ i avec i = 1 , . . . , p et j = 1 , . . . , p . On a alors 9 Si (A, p) est solution de (P2), a/ors (A, grad p) est solution de (Pl). 9 Si (#, q) est solution de (P3), alors (#,Rot q) est solution de (Pl). Cette propridtd montre en particulier 1'existence de solutions de (Pl ) divergence nulle. 9 Si ri est solution de (7~4~), alors (0, Rot ri) est solution de (Pl). 9 Si ~ est simplement connexe, alors 0 n'est pas valeur propre de (Pl), sinon l'espace propre associd ~ la valeur propre 0 est 1'espace

H0(div 0, rot 0), de dimension p, engendrd par les p solutions des problbmes (7)4~). 9 Soit {pi, i E N), un ensemble de solutions de (~2) formant une base orthogonale de L2(~), soit { q j , j E N}, un ensemble de solutions de

fo m..t

b. e o thogo..1

n (a)

soit

les

p solutions inddpendantes des prob1~mes (P4~). A10rs 1'ensemble des grad pi, Rot qj et Rot rk forme une base orthogonMe de L2(f~) 2 et

de V.

5.2. Approximation du gradient de la fonctionnelle Nous consid~rons ici le cas d'observations ponctuelles, off les observations sont connues en tout temps mais en un nombre restreint de points du domaine. Rappelons que l'espace des contr61es est donn~ par L2(0, T; W), la vitesse d~sir~e est telle que" Ud E (L 2 (I~)) m, l'op~rateur d'observabilit~ est not~ par: C" L 2 (0, T; Y U W) ~

(L 2 (I~)) m.

On introduit l'espace: (

Op )

Ddtermination de conditions aux limites en mer ouverte...

114

Soit 7-/', le dual de 7/. On identifie 7-I et ~ ' , et on a

W ~__~7-l = 7-l' ,__+W '. On va d~terminer le minimum de J(w) par une m~thode it~rative. On ~crit l'~quation v~rifi~e par le minimum de la faqon suivante:

s (W' (9) L2(O,T;W ) -

;or. --Oi

)

(5.1)

Dr* + coa(v*), 0 L~(O,T;W'),L~-(O,T;W)

La m~thode consiste ~ consid~rer le second membre de l'~quation (5.1) fonction de w ~ l'it~ration pr~c~dente. Ainsi on va calculer w n ~ la n i~me iteration par

~(~. 0).~o T;w , + (~w~. ~0)(~.~.)~ : _(~o --

--~

1_ ~. ~0)(~(~./~ L2(O,T;W,),L2(O,T;W )

Dans le cas non lin~aire, on r~sout de la m~me mani~re

e(W,9)L2(O,T;W) + ( w d i v V*,91L2(O,T;W,),L2(O,T;W ) + (Cw, Cg) (L2(R))m -

(C v - u d ,

)

C9 (L2(R))m--

-~+vdivv*+rotva(v*)

- Dv* + wa(v*) - h* grad h, 0~

/ L2 (0,T; W' ),L2(O,T;W)

. (5.2)

Or, la r~solution de (5.1) ou de (5.2) n~cessite l'introduction de l'op~rateur C*, op~rateur adjoint de l'op~rateur C. En effet, on doit ~crire

et

(~v _ ~, ~ 0 ) ( ~ . ~ ~ =/~./~v - u~/. 0/~o ~;~ ~ ~o ~;~ , La difficult~ d'obtenir une caract~risation num~rique de l'op~rateur C*, nous a conduit ~ nous orienter vers une m~thode de r~solution plus appropri~e.

115

F. Bosseur et P. Orenga

Nous proposons donc de r~soudre les ~quations (5.1) ou (5.2) par une m6thode d'approximation variationnelle, en utilisant une base de fonctions de l'espace W. C h o i x de la b a s e de W Nous devons tout d ' a b o r d d~finir un produit scalaire sur W; pour cela, on utilise l'6quivalence des normes suivantes (Dautray-Lions, 1988):

IlWllHk+l(a)2 et

IIIwl[]

off

jflwrfr-(llwll 2

L~(a) 2

+

[Idiv

~112Sk (,)~. + [[rot w II2

H

k (~)~. +

II~.n II 2. ~ §

1

(~)

Ce r~sultat nous permet de munir W du produit scalaire

puisque, d'apr~s les propri6t6s de l'espace W, div w = rot w = 0. En outre, si w = Vp est une fonction de W, alors p v~rifie le probl~me Ap-

0

p = #

dans sur %

Op

~-~n - 0

(5.4)

sur "Ys

oh # C H~ (%). Consid6rons h pr6sent les solutions (i des problbmes de N e u m a n n suivants: A~i - 0

dans

~i = ~bi

sur %

O~i = 0

(5.5)

sur 7f

oo

dans lesquels { r } i=1 est une base de H~ (%) et v~rifie:

~r

i-0,

'v'i E N.

116

Ddtermination de conditions aux limites en mer ouverte...

On d~finit alors l'application

7~" H~ ('7) ~

Ha(f~)

qui est lin~aire et continue (Girault-Raviart, 1979). Avec ces notations, on a le L e m m e 5.1 - {V~i}i=l forme une base de W. D~monstration. Soit

0p r E ~(P E H3(Ft);Ap-- O, ~nn - 0 s u r '7.f~. ./ I1 existe # E H~ (%), tel que r {r

oo

~tant une base de H~ (%), il existe #n,v~rifiant #n -- E

air

Hi(~)

> ~,

i--1

et d'apr~s la continuit~ de l'application 7~: n

7~

H3

- r - y~ a~ i=1

(a) > O,

o~ fi = 7~r

et O vdrifie O-#

sur%

O0 ~ n --0

sur011

De plus 0 = ACn

> A @ = 0.

D'apr~s le r~sultat d'unicit~ de la solution du probl~me (5.5), on en oo d~duit que O - ~ et donc que {fi }i=1 est une base de

(pEH 3(~);Ap-0,

op

~ n - 0 S u r f f f ~ ")

(x)

En consequence, {V~ci}i=l est une base de W. D~montrons alors le L e m m e 5.2 - {V(i }i=1 forme une base de Ill.

II

F. Bosseur et P. Orenga

117

Ddmonstration. Soit f - V~ E 7-/, telle que (f, V ~ i ) n - 0

Vie N

On a (f, V ~ i ) n -- s

VrI.V~ci

Or

soit (f, X;7~r / --

0~ ~~ r

ViEN

-- 0

(X)

{r }i:1 6tant une base de H~ (%) donc de L2(7), on en d~duit que o~ - 0 sur %. Finalement, ~1 v~rifie Arl - 0 et ~ = 0 sur 3', d'ofi on d~duit que r] - 0 et f - 0, ce qui d~montre que {V~i }oo /=1 est une base de 7-/. 1 On va maintenant approcher w par w~ E L2(0, T; W~), off Wr est le r sous-espace engend% par les r fonctions {0i V~i}i=l ; on raisonne dans le cas lin~aire, le cas non lin~aire ne posant pas plus de difficult6s. On note -

f (v*, h* ) -

Or*

(5.6)

Ot ~ Dv * + wa(v *),

Wr : E

#k Ok,

k--1

et avec i - 1 , . . . , r, l'~quation (5.1) s'~crit

+Z k=l

O (xj) O (xj)

-

j=l m

-- E (V -- Ud) (Xj).Oi(Xj) -- (f (v*, h*), Oi) L2(Q)2

.

(5.7)

j--1

6. Mise e n oeuvre num~rique 6.1. Notations On note 3d le maillage du domaine, Mi, i - 1 . . . N1, les points de l'int~rieur de f~ de coordonn6es (Xi, Yi) et Fj, j - 1 . . . N2, les points fronti~re, de coordonn6es (xj, yj).

--

i--1 ['j

} j--l"

118

D d t e r m i n a t i o n de conditions a u x limites en mer o u v e r t e . . .

6.2. Calcul des vecteurs de la base propre et de la base de W Nous avons vu ~ la section 5.1 que les ~l~ments de la base propre sont obtenus par r~solution des probl~mes (P2), (/)3), (P4,). Dans le cas d'un carr~ de c6t~ unitaire, les solutions de ces probl~mes sont connues de mani~re analytique. Dans le cas d'un domaine quelconque, les solutions sont calcul~es l'aide du logiciel Modulef, par la m~thode des ~l~ments finis. Les ~l~ments utilis~s sont de type hermite ~ trois degr~s de libertY, ce qui nous permet d'avoir acc~s, en t o u s l e s points du maillage, ~ la valeur nume~rique de la fonction solution ainsi que de ses d~riv~es premieres en x et en y (VidrascuGeorges, 1990). Pour construire num(~riquement la base de Wr, on r~sout les r probl~mes scalaires suivants: /X~ - 0 1 ~(xj,yj)

-

0

si j - a, off ( x j , y j ) E % si j ~ a,

La r(~solution num~rique de ces probl~mes est, comme pr~c~demment, effectu~e par la me~thode des ~l~ments finis ~ l'aide du logiciel Modulef, en utilisant les m~mes ~l~ments.

6.3. R~solution des ~quations du module et des ~quations adjointes Pour r~soudre les probl~mes (2.4) et (3.4), nous utilisons la m~thode de Galerkin, associ~e~ la base propre d~finie s la section 5.1. On transforme ainsi les syst~mes d'~quations aux d~riv~es partielles en des syst~mes d'~quations diff~rentielles ordinaires. Ces syst~mes sont alors r~solus par la m~thode d'Adams implicite, initialis~e par la m~thode d'Euler implicite.

6.4. M~thode de calcul du minimum La principale difficult~ pour le calcul du minimum provient de la presence du terme (w, ~)L2(o,T;W) dans les (~quations (5.1) et (5.2). Or, nous avons vu en (5.3) que l'on pouvait munir W du produit scalaire

(e,,,

+

Pour calculer la valeur du produit scalaire de H~ (~,), on utilise alors le r(~sultat suivant (Dautray-Lions, 1988)"

F. Bosseur et P. Orenga

119

Soit f, une fonction pdriodique ddfinie sur ]0, a[; on ddfinit la norme de l'espace de Sobolev H~ (0, a) 1

Ilsll~ -

~ (1 + j~)~lcJ~l ~

(6.1)

jeZ

o~ Cj -

f (x).e -i aI" J~ dx.

I

fO a

On associe alors ~ (6.1) le produit scalaire suivant: (f,g)-(E(I+J2)~CJ(f).-CJJ(g)), s

(6.2)

jeZ

off f et g sont deux fonctions p4riodiques d4finies sur ]0, a[. Dans notre cas, off la fronti~re % est constitu6e par la r4union de q segments de longueur ai, les fonctions (Ok.n) sont d6finies sur ]0, ai[ et sont nulles au bord. On a

cj(ok.n)

-~,q f a i

(O~.nl(x).~-~2,r jx ax

i--1 Jo

= - E i

(Ok.n)(x). sin U j x dx,

i---1

et -

a, -

.

o,

dx

i--1

=

i

(ai)

(Ol.n)(x). sin 27r j x dx.

i=1

Au total, le produit scalaire s'exprime donc sous la forme

(Ok.n, Ol.n) H~ ('),) = E

~(1+

j2)~[(~oa~(Ok.n)(x).sin(2----~jx)dx ) a i 9

(81.n)(x).sin - - j x dx

.

ai L'4quation (5.7) se pr6sente alors comme un systbme alg4brique de r 6quations, off les r inconnues sont les #k et m repr6sente le nombre de points d'observation, {xj, j = 1 , . . . , m}.

Ddtermination de conditions aux limites en mer ouverte...

120

Ce syst~me matriciel, du type

A X = B, d'inconnues #1

#2 Z

__

9

#r est alors r~solu par la m~thode du pivot de Gauss. Une lois connues les valeurs des coefficients #k (k = 1 , . . . , r), on reconstruit la solution approch~e

Wr -- ~

#k Ok,

k--1

qui correspond ~ la valeur approch~e de w pour l'it~ration suivante de l'algorithme de contr61e. En effet, on doit donc, ~ ce niveau, calculer les nouvelles valeurs de v, h, v*, h* correspondant ~ cette nouvelle valeur de wr et ainsi de suite jusqu's convergence de la m~thode (Fig. 6.1).

6.5. Convergence de l'algorithme de calcul du m i n i m u m Au chapitre 3., nous avons donn~ l'expression de la diff~rentielle de la fonctionnelle, qui s'~crit dans le cas lin~aire 1 j , (wl , O) -

*,h* 1 + C* ( C v -

+ C* C w

+ eAw w, O)L2(O,T;W,),L2(O,T;W )

(6.3)

off A W est l'isomorphisme canonique de L:(0, T; W) sur L2(0, T; W ' ) et f ( v * , h * ) est d~finie comme en (5.6). Pour d~montrer, dans le cas lin~aire la convergence de l'algorithme de calcul du minimum, on raisonne par r~currence 3. 9 On se fixe w ~ quelconque et on calcule (v ~ h ~ et (v *~ h *~ en r~solvant les probl~mes (T')~ et (P*)~. 3Nous donnons un r~sultat de convergence uniquement dans le cas lin~aire, car dans le cas non lin~aire, nous n'avons pas de r~sultats concernant la convexit~ de la fonctionnelle.

F. Bosseur et P. Orenga

121

Calcul par Modulef de w k ----X7pk, avec p~ solution de --A pk = 0 V pk.n

= G k

d a r t s ~2 sur y

~1~~1 ~

Calcul

wk+l

de

Calcu1 de

I

v , . J ( ~ 9)

N~gatif

Test ......

vwJ(.,9(~

y

Fig. 6.1: Schdma gdn6ral de la rdsolution

[

D6termination de conditions aux limites en mer ouverte...

122

1) Si w ~ v6rifie 1 j, , < (w~ O> - 0

, , V0 e L 2(0 T; W)

(6.4)

alors, d'apr~s le th~or~me 4.1, w ~ est le contr61e optimal. 2) Si w ~ ne v~rifie pas (6.4), on d~termine w I par

(C*C + ~ A ~ ) w I - - ( f ( v * ~

h "~ + C * ( C v ~ - u ~ ) ) .

L'op~rateur

T-

(C*C + eAw) " W---+ W',

~tant continu et elliptique, on peut ~crire

~, -_~-l(~(v,O,h,o)+c,(cvO_u~)). Utilisons (6.3) pour donner une expression de <J' (w~ w 1 - w~

il vient

L2<J' ( w~ ) , wl - w~ > --
*0) -Jr-C* ( C v 0 - Ud) +

Tw~

I -

WO>L2(O,T,W,),L2(O,T;W )

- (s (v,O, h,o) + c* (CvO _ u~) + T~O,

_~-, (s(v.O ~.o)+ ~. (~o _~)) _ ~o>~o,~;~,,,,~o,~ ' = <S(v,O, h,o) + c* (CvO_ u~) + :r~o,

_,-l(S(~.o ~.o)+ ~.(~vO_ ~)+,~o)> '

<_ - K

IIS(v*~

h,o) + C* (CvO -

L2(O,T;W , ),L2(O,T;W)

~ ,,~) + T~O IJ--(o,~;~')

(off K est une constante positive) <0. Dans le cas lin~aire, la fonctionnelle ~tant convexe, on en d#duit que J ( w 1) < J(w~ et on calcule (v 1 , h I) et (v* 1, h* 1) en r~solvant les probl~mes (P)e et (P*)~. 9 1) Si w n-1 v~rifie (6.4), alors d'apr~s le th~or~me 4.1 w n-1 est le

contr61e optimal. 2) Si w ~-1 ne v~rifie pas (6.4), on d~termine w ~ par:

w~ - - T - l ( f (v*~

M~

) + C* (CvO-l _ u~) ) "

F. Bosseur et P. Orenga

123

Utilisons comme pr~c~demment l'~quation (6.3) pour donner une expression de ( J ' (w ~-1), w n - w ~-1 ); il vient

! ( j ' (wn-1) w n _ W n - l ) 2 - bigl < f (v *n-l, h *~-1) + C* (Cv n-1 - Ud) + T w n - l , --"
(V*n-1 , h *n-l)

W n -- wn--1 ) L2(O,T;W'),L2(O,T;W) --[--C* ( C v n-1 - Ud) --~ T w n - l ,

-T-l (f (v*n-l, h*n-l)

-~- C* ( C v n - l -

Ud) + T w n - 1 )

)L2(O,T;W,),L2(O,T;W)

-KIIf(v*n-1 h •n-l) --~-C*(Cv n-1 --Ud) "~- ~'wn--ll[ 2 __ ~ L2(O,T;W ~) (off K est une constante positive) <0. Pour la m~me raison que pr~c~demment, on en d~duit que

J(w n) < g ( w n - i ) , et on calcule (v n, h ~) et (v *n, h *n) en r~solvant les probl~mes ( P ) l et (P*)t. On obtient ainsi une suite r~elle { J(w n) }n' positive et d~croissante; elle converge donc vers x E ~, off x v~rifie x = inf J (w n). Wn Mais J ~tant convexe, il existe w unique qui r~alise le minimum, soit

x = J(w). On en d~duit que

w n ---+ w,

faiblement dans L2(0, T; W).

7. R ~ s t f l t a t s n u m d r i q u e s

7'.1. Cas analytique Pour valider la m~thode num~rique impl~ment~e, nous avons effectu6 des tests sur une g~om~trie simplifi~e, constitute par un carr~ de c6t~ unitaire. Dans ce cas, le champ de vitesse d~sir~e n'est pas constitu~ par un ensemble de mesures, mais est issu de la r~solution des ~quations du probl~me de shallow water en consid~rant un rel~vement donn~, Wd. Ceci nous a permis de comparer non seulement le champ de vitesse calcul~ ~ la solution choisie comme r~f~rence (Ud), mais ~galement le rel~vement calcul~ par la m~thode de contr61e au champ Wd. Le sous domaine ~ l'int~rieur duquel la vitesse Ud est suppos~e connue a ~t~ d6fini de deux mani~res diff~rentes"

124

D d t e r m i n a t i o n de conditions a u x limites e n m e r o u v e r t e . . .

9 En consid@rant des points dispers@s de manihre r@guli~re ~ l'int@rieur du domaine mais avec un pas d'espace beaucoup plus grand, ce qui correspond au cas oh les donn~es sont issues de r@sultats de modhles de plus grande emprise. 9 En consid@rant des points ~ l'int@rieur d'un rectangle inclus dans le domaine et qui correspond physiquement ~ un ensemble de mesures effectu@es dans une zone particuli~re. Dans le premier cas, nous avons pu @tudier l'influence du nombre de donn@es disponibles sur la "qualit@" de la solution obtenue. Ceci montre qu'un nombre m i n i m u m de points de mesure est n~cessaire pour reconstituer un champ de vitesse en accord avec la r~alit@ physique. Le deuxihme cas nous a permis d'~tudier l'influence de la localisation des points de mesure ~ l'int@rieur du domaine. En particulier, il nous a ainsi ~t@ possible de v@rifier que les meilleurs r@sultats sont obtenus quand les observations sont situ@es au centre du domaine et r~parties sur un axe orthogonal au courant principal. Les tests num@riques effectu@s sur le carr@ ne sont pas pr@sent@s ici oh nous donnons uniquement quelques r@sultats significatifs obtenus sur un cas r@el. 7.2. C a s r@el Une deuxi~me s@rie de tests a ensuite @t@ effectu@e sur une g@om@trie plus complexe, repr@sentant la baie de Calvi. Le choix de ce site a @t@motiv@ de par la proximit@ de la station de recherches oc~anographiques STARESO, ce qui a permis la r@alisation de nombreuses campagnes de mesures; en parallele, des simulations num@riques ont @galement @t@ effectu@es par Norro (1995). Nous avons donc utilis@ les r@sultats de ces simulations pour reconstruire les conditions aux limites sur les fronti~res ouvertes. Dans ce cas, oh le champ de vitesse est connu dans tout le domaine, le sous domaine dans lequel la vitesse d@sir@e est suppos@e connue a ~t@ d~fini de la m@me mani~re que dans le cas du carr@. En outre, la m@thode utilis@e @tant de type it~ratif, nous avons dfi nous fixer une valeur de d@marrage; pour les tests que nous pr@sentons ici, nous sommes partis de conditions aux limites nulles aux bords. La figure 7.1 repr~sente un champ de vitesse @tabli ~ partir d'une simulation effectu@e par Norro (1995); il s'agit du champ que nous avons utilis@ comme r~f@rence. Dans un premier temps, nous avons suppos@ ce champ connu sur la moiti@ des points du maillage. Les r@sultats obtenus apr~s 20 it@rations

F. Bosseur et P. Orenga

125

de la m@thode de contr51e sont pr@sent@s aux figures 7.2. La figure 7.25, repr@sentant la vitesse correspondant au rel~vement obtenu par le code d'optimisation, montre une bonne ad@quation de la solution avec le champ de r@f@rence, ce qui est confirm@ ~ la figure 7.2c, oh est repr@sent@e l'@volution de l'erreur relative aux points de mesure: -

=

Les figures 7.3 repr@sentent quant ~ elles les r@sultats obtenus en consid~rant la vitesse d@sir@e connue seulement en un point sur vingt du maillage 4. On s'aper~oit alors sur la figure 7.35 que la vitesse calcul~e dans ce cas ne correspond plus au champ de r@f@rence de la figure 7.1, ce qui est v~rifi@ sur la figure 7.3c en repr@sentant l'@volution de l'erreur 71 en fonction du nombre d'it@rations. Ces remarques, ainsi que la figure 7.4 oh est repr@sent@e l'~volution de l'erreur C o: r]2 - sup

lUdj I

,

en fonction du nombre d'it@rations, confirment les r@sultats pr@alablement obtenus sur le carr@ que nous rappelions dans la section 7.1, ~ savoir qu'en dessous d'un nombre minimum de points de mesure, la solution calcul@e n'est plus en accord avec le champ de r~f@rence. Enfin, nous pr@sentons aux figures 7.5, les r~sultats num@riques obtenus dans le cas non lin@aire. Dans les m@mes conditions que pour le cas de la figure 7.2 (solution d@sir@e connue en un point sur deux du domaine), nous remarquons que les r@sultats sont semblables s ceux obtenus dans le cas lin@aire, ce qui n'est pas @tonnant car dans le cas des @coulements g@ophysiques, les termes de Coriolis, de gradient de pression et de viscosit@ sont pr@pond@rants.

Conclusion Le but de notre travail est la mise en oeuvre d'une m@thode num~rique adapt@e & la d@termination de conditions aux limites e n m e r ouverte d'un probl~me de shallow water. Les premiers tests effectu~s sur le carr~ se sont r@v@l@sencourageants. En particulier, dans ce cas simplifi@, il nous a @t@ possible de reconstituer 4Sur la figure, les points du mail!age off la vitesse est connue sont repr@sent~s par des croix.

126

D ~ t e r m i n a t i o n de c o n d i t i o n s a u x l i m i t e s en m e r o u v e r t e . . .

des conditions aux limites assez complexes avec une precision tout ~ fait convenable. En outre, la m~thode de Galerkin s'est averse bien adapt~e ~ la r~solution des ~quations adjointes ainsi qu'au calcul du minimum de la fonctionnelle et nous a permis de conserver des temps de calcul raisonnables. Dans un deuxi~me temps, nous avons appliqu~ le code num~rique un domaine plus proche de la r~alit~ repr~sentant la baie de Calvi, qui correspond bien aux objectifs que nous nous sommes fixes, de par sa taille et sa situation g~ographique. Les r~sultats obtenus dans ce cas se sont ~galement montr~s encourageants. Actuellement, nous travaillons ~ une ~volution du code num~rique en vue d'effectuer un contr61e simultan~ sur la vitesse et la hauteur. Cette ~tude nous para[t ~tre justifi~e par la n~cessit~ (Chatelon-Orenga, 1997) de se fixer la valeur de la hauteur sur la partie de la fronti~re oh le fluide est entrant. En outre, l'~volution r~cente des techniques de mesure, en particulier l'utilisation du satellite, devrait nous permettre d'obtenir des mesures ponctuelles d'~l~vation dans le domaine que nous utiliserons pour reconstituer la hauteur sur la fronti~re en utilisant le code d'optimisation. Enfin, dans le cas non lin~aire, il serait int~ressant d'~tudier le cas de petits domaines oh les termes de Coriolis, de gradient de pression et de viscosit~ sont moins importants.

iiiii.......

0

Vmax=

I 2 O. 1 5 7 7 0 6 7 1 E + O 0

27

Fig. 7.1" Vitesse ddsirde.

F. Bosseur et P. Orenga

, : : : : :,

lo . . . . . . . . .

:, ,,,,,..,........

:.......

~ . . . . . . . . .

V m a x = o. 1 3 6 3 0 7 0 3 E + 0 0

Fig. 7.2a: Relbvement calcul~ en consid6rant la vitesse connue en un point sur deux.

!i!!iii:::.: ~:::::;:.:!:~-#{~ilt!

, o

:"- : :-:.? 1 2 Vmax= 0.14467617E+00

Fig. 7.2b: Vitesse correspondante.

127

128

D 6 t e r m i n a t i o n de c o n d i t i o n s a u x l i m i t e s en m e r o u v e r t e . . .

!

!

!0

20

2

Fig.7.2c"

E v o l u t i o n de l ' e r r e u r ~

II~j II (L=(R))

=

2

\

m

!

en f o n c t i o n du n o m b r e d ' i t 6 r a t i o n s .

"!i>:..:.:., .........

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.

.

'

.

.-

i:i:i:i:'.:.:i::.:::" ." " i i i i i':.:..:......::: ' " . .

0 Vmax=

1 0.24570774E--01

2

27

Fig.7.3a: E v o l u t i o n R e l b v e m e n t calcul6 en c o n s i d & a n t la v i t e s s e c o n n u e en un p o i n t s u r v i n g t .

F. Bosseur et P. Orenga

129

tttttt*. * , , .- - . ~ - ~ . , . tt~ ~." , § ""N.

_ . .

o

.

,

,

.-.

(."

1 2 Vmax= 0.42615213E---01

-

2~

Fig.7.3b: Vitesse correspondante.

oss:

032 o31

o ~sl 0.271 0.261 0.251

Fig.7.3c: Evolution de l'erreur r/~ =

ILir

II~dj II9.(L2(R))~

en fonction du nombre d'it&ations.

130

D ~ t e r m i n a t i o n de c o n d i t i o n s a u x l i m i t e s en m e r o u v e r t e . . .

,,,,\

.......................................................

Fig.7.4" E v o l u t i o n de l'erreur 7/2 - sup IIlu(xJ)-u~l

lu~jl

en fonction du n o m b r e d'itdrations.

9 !~:~.::~..

: ', : : : : : : : , ' . ' . ' . , .,...... . : , : , : , : .',',.,...,...,. ' 9

.',..:,',:,:,:.:.

O Vmax=

'....

1 2 O. 1 3 7 " 7 1 8 5 2 E + O 0

27

Fig.7.5a: R e l ~ v e m e n t calculd en c o n s i d & a n t la vitesse connue en un p o i n t sur deux.

F. Bosseur et P. Orenga

131

i::.::-.::.:.

. . . . . . . . . .

i

. . . . . . . . .

~

. . . . . .

27

V m a x = 0.146848531:;-+00

Fig.7.Sb"

Vitesse correspondante.

o12L _o 2

Fig.7.5c:

Evolution de l'erreur ~11 -

Ilu~j II(L2(~)) 2

en fonction du nombre d'it~rations.

m

132

D d t e r m i n a t i o n de c o n d i t i o n s a u x f i m i t e s en

mer o u v e r t e . . .

References Bisgambiglia P., Traitement num(Mque et informatique de la mod~lisation spectrale. Th~se de Doctorat, Universit~ de Corse 1989. Chatelon F.J., Orenga P., On a non-homogeneous shallow water problem, Math. Modelling and Num. Anal.,31, 1 (1997), 27-55. Dautray R., Lions J.L., Analyse math~matique et calcul num~rique pour les sciences et les techniques. Masson, Paris 1988. Girault V., Raviart P.A., Finite elements approximation of the NavierStockes equations. Springer-Verlag 1979. Lions J.L., Contr61e optimal de syst~mes gouvern~s par des ~quations aux d~rive~es partielles. Dunod 1968. Lions J.L., Magenes E., Probl~mes aux limites non homog~nes et applications. Dunod 1972. Nihoul J.C.J., ModUles mathe~matiques et Dynamique de l'environnement. Elsevier Publ., Liege 1977. Norro A., Etude pluridisciplinaire d'un milieu c6tier. Approches exp~rimentale et de mod~lisation de la baie de Calvi (Corse). Th~se de Doctorat, Universit~ de Liege 1995. Orenga P., Analyse de quelques probl~mes en oc~anographie physique. Th~se d'Habilitation, Universit~ de Corse 1991. Orenga P., Construction d'une base sp~ciale pour la r~solution de quelques probl~mes d'oc~anographie physique en dimension deux, CRAS, 314 (1) (1992), 587-590. Orenga P., Un th~or~me d'existence de solutions d'un probl~me de shallow water, Arch. Rational Mech. Anal., Springer-Verlag, 130 (1995), 183204. Talagrand O., Courtier P., Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory, Q.J.R. Meteorol. Soc., 113 (1987), 1311-1328. Vidrascu M., Georges P.L., Guide d'utilisation et normes de programmation Modulef, I.N.R.I.A, Le Chesnay 1990. Frederic Bosseur et Pierre Orenga Universit~ de Corse Pascal Paoli SDEM-URA CNRS 2053 Campus Grossetti 20250 Corte France E-mail : [email protected], [email protected]

Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 7 EFFECTIVE D I F F U S I O N IN V A N I S H I N G V I S C O S I T Y

F. CAMPILLO AND A. PIATNITSKI

1. Introduction In the present article we consider the asymptotics of the effective diffusion for elliptic operators with vanishing diffusion and with potential first order terms, the potential being a statistically homogeneous field. The homogenization problems for singular perturbed operators have many important applications, among them fluid dynamics in porous media Bear [5] or groundwater pollution Fried [11]. In the recent years, various such questions were considered in detail for the operator with divergencefree vector fields. Many interesting asymptotics were constructed for the periodic case ( Bensoussan-Lions-Papanicolaou [7], Fannjiang-Papanicolaou [9]) - - and then in the random case ( Avellaneda-Majda [1, 2, 4], CarmonaXu [8], Fannjiang-Papanicolaou [10]). In contrast to divergence-free case, where the effective diffusion is usually much greater than the initial one (see, for instance, Fannjiang-Papanicolaou [9]), we typically have in case of potential vector field the exponential decay of the effective coefficients. For operators with periodic coefficients this phenomenon was investigated in Kozlov [14] and Kozlov-Piatnitski [15] where the logarithmic asymptotics of effective coefficient was found in terms of "Morse properties" of the potential on the torus of periodicity. The operators whose first order terms are not potential but show in a way similar behavior, were considered in Kozlov-Piatnitski [16], where the logarithmic asymptotics of the effective diffusion was established. In the present work, we study a particular case of operators with random potential first order terms. Namely, we assume that the potential is a random perturbation of a given periodic function. Considering this random perturbation, we assume that it does not change essentially the topological structure of the initial potential. This allows us to use the results from the percolation theory and to find the required asymptotics in terms of the proper percolation levels.

Effective diffusion in vanishing viscosity

134

All the exact assumptions are provided in Section 1. Then, in Section 2, we prove the general result on asymptotic behavior of homogenized coefficients. One of the key condition of this statement is non-explicit. In Section 3 we present a couple of sufficient conditions expressed in explicit terms.

2. The setup Let us consider a potential on IR2 (with orthonormal basis {el,e2}) of the form U - U0 + U1 where U0 is a deterministic smooth potential which is supposed to be periodic with with period 1 in each coordinate directions. We denote the cell of periodicity [0, 1]2 by [3 and identify it with the 2D torus T ; U1 is an isotropic random field, it represents a small random perturbation of Uo. If S denotes the rotation matrix of angle 7r/2, we suppose that:

(i) Uo(S x) = Uo(x) for all x E T, (ii) the distribution of U1 is invariant with respect to any integer shift of IR2 and to S: law(Ul(S x)) - l a w ( U 1 (x)) for all x e 7",

(iii) there exists 7o > 0 such that ]Ul(x)l _< 70, for all x E ]R 2, a.s. (iv) there is p > 0 such that any a(Ul(X),X E G1)-measurable random variable 771 and a(U1 (x), x C G2)-measurable 772 are independet whenever dist(G1, G2) > p; here G 1 and G 2 are arbitrary subsets of ]R2. The two first conditions ensure the isotropy of the effective media. Under condition (i), the potential Uo has a specific structure: in the simplest case other cases rely on the same arguments the minimum number of degenerate points that Uo could admit on T, i.e. points x such that VUo(x) - 0, is four: one minimum point Xmin, one maximum point Xmax, and two saddle points x.. In ]R 2 minimum points, maximum points, and saddle points will be denoted Xmin, Xmax, and xs respectively. Without loss of generality we may assume that in ]R 2, the set of maximum points is Xmax = Z 2, then the set of minimum points should col ~), 1 incide with Xmi n -- Z 2 + (~, and the set of saddle points with Xs = Z 2 + (0, 1 ) U Z 2 + (1,0), (see Figure 1). The case of a more general potential Uo(x), having more singular points, including minimum points, could be treated with the same reasoning. We make the following hypothesis:

F. Campillo and A. Piatnitski

135

Figure 1: Example of structure for the potential Uo.

H y p o t h e s i s 1 - For ali m i n i m u m point is Xm~.), let us consider: OL(Xmin,ei)

----

inf

Xmi n

(the set o f m i n i m u m points

sup U ( X ( t ) ) , i = 1, 2,

X(')eX(Xmin.e~) O
(1)

X(Xmin,ei) is

the set of functions [0, 1] 9 t ~-~ X ( t ) E 1R2 such that Xmin, X(1) -- Xmin + ei, and which are h o m o t h e t i c to [0, 1] 3 t Xo(t) ~ / R 2 \ Xmax defined as follows: X o ( t ) - Xm,. + t e i . We suppose that {a(Xmi., ei); Xmin E Xmin, i -- 1, 2} is a family of indewhere

X(0)

--"

p e n d e n t r a n d o m variables.

This last assumption is rather non explicit, a couple of sufficient conditions that ensure the above independence are supplied in Section 4. Consider the following homogenization problem: i

ZXu (x) + 1 Vz[_U(z)]]z= = . V x u 6 ( x ) _ f ( x ) , x e Q , u ~ e Hi(Q)

(2)

for some bounded domain Q in IR 2. We assume first that the viscosity parameter # is small and fixed, and pass to the limit as e $ 0. Then we study the asymptotic behavior of the effective (scalar) diffusivity a(#) as # $ 0.

Effective diffusion in vanishing viscosity

136

3. Effective diffusion In this section we show that, under Hypothesis 1, the logarithmic limit of the effective diffusivity a(#) can be found in terms of a proper critical percolation level of the potential U. After standard transformations, Equation (2) reads:

#eU(~)/t' i=1,2 E ~X/ 0 ( e-U(-~)/# ~ OXi 0 US(X) )

-- f ( x )

(from now on we will always suppose that u s E H~(Q)). Multiply each term of this last equation by e-U(~ )/~ so that:

O ( u(~ )l~ O

.

i= 1,2

I

-

(3)

OXi

Without loss of generality, we can assume that: ess

Then, for any # > 0,

inf U(x)=0. ze]R 2

(4)

e-V(~ )/~ f --" ~(#) f weakly in L2(Q) as ~ $ 0, where: ~(#) zx ~ e_U(~)l~ dx.

Moreover # log ~(#) -+ 0 as # $ 0. For each #, the family of operators appearing in (3) is coercive, uniformly in e. Thus, it suffices to homogenize the following P D E :

p ~

~

i=1,2

eU(~)/~

re(x)

-- ~(p)f(x).

(5)

OXi

Then, v s ~ u s as e $ 0 (in the sense that these functions have the same limit in Hi(Q) as c $ 0). Clearly, we can omit both factors # and/~(g) for a while; we end up with the equation:

i=1,2

OXi

v (x)

- f(z).

(6)

According to Jikov et al [12], under above conditions, Equation (6) admits the effective diffusion matrix E(#) which is isotropic: E(#) = a(#) I and the

F. Campillo and A. Piatnitski

137

scalar effective diffusion coefficient is supplied by the following variational problem:

a(#) - liminf e$0

inf

f~ eU(~ )/~ IVv(x)l 2 dx.

(7)

~ e HI(~) v ( 0 , .) _-- o

~(1, .) = 1

3.1. Lower bound Let [:]1 and [:32 be two neighbor cells, let say that D2 - [ - ] l - k e l and x m1i n E [31, X2m~C [12 the corresponding minimum points of U0. We introduce the following random open set: {x

9 U(x) <

and the events: 9 A ~ 9 the set of w such that there is a path connecting Xminl and x 2m,.n 1 el) and which is included in G~ (w). which belongs to X (Xmin, 9 A~ 9the set of w such that there is a smooth curve from A'(Xlm,n,el) of length not greater than n such that its 1-neighborhood is included in G~ (w). In our case, A ~ could be also defined as the event: the set of co such that G~ (w) A ([:11 t2 [:12) contains Xlm,~ and X2mi. Clearly Un>oA~ - A ~ and, hence: lim P ( d ~ ) = P ( A ~

(8)

nJ'oo

It is also obvious that under our assumptions on the structure of U, the 1 2 xmi~) above events are independent for different pairs ( x rain, of neighbor minimum points. Consider standard bond percolation model using minimum points of U0 as sites, and let pc be the critical probability of the appearance of the infinite 1 We define the critical value r/c as follows: cluster" Pc - 5"

p(d~

1 - -~ ,

or, if such a r/c does not exist: r / c - inf{r/; P ( d ~ < 51 } -

sup{r/; P ( d ~ > 89

(9)

138

Effective diffusion in vanishing viscosity

This last equality is, in fact, an additional assumption which is supposed to be fulfilled later on. 1 Thus, using (8) , P(An~+.y) > For all ~/> 0 small enough, P ( A ~ > 7" n !2 for sufficiently large n. We fix such a n and denote it by n o ; we also denote p0 We say that a bond

-

no

P(A,c+~

1 Xmin) 2 (Xmin, is

).

open if the corresponding w belongs

1 X 2mi.)" A~:+~( X min,

to the set As proved in Kesten [13], for almost all realizations and for all sufficiently large N, the square [0, N] 2 contains at least c(p ~ N mutually non intersecting channels connecting left and right sides of the square. Finally, we arrive at the following conclusion: Conclusion. For sufficiently large N, [0, N] 2 contains at least c(p ~ N 1 -pipes connecting left and right sides mutually non intersecting smooth n-~ of the square such that along each of these pipes: U(y) < tic + '7.

(10)

Denote the above pipes by Q 1 , . . . , Q k ( N ) , k ( N ) > c(p ~ N. Without loss of generality we assume that for any function x ~ u(x) such that u(0, x2) - 0, u(N, x2) - 1 we have: Q m

Ou (y_____~d) y > 1 1 0~. - 2 no

(here I is a variable directed along the pipe after rescaling). Indeed, taking a smooth pipe included in Q m and choosing, if necessary, a larger value of no, one can achieve the above lower bound. After rescaling x = ~ y, ~ = l / N , we find: Q

On(x) 1 1 Og dx >_ -~ ~

withQ~-~Qm.

By the Shwartz inequality: e2

1 1 < (n~ 2 4 -

[/o

~

On(x) dx 0~

/o

< IQ~ml -

IVu(x)l 2 d x .

Thus, Q

IVu(x)l 2 dx > ~ 1 1 - 4 (n~ 2 cl(p~ "

F. Campillo and A. Piatnitski

139

Summing up over m leads to: k(N)

f IV(x)l

m--1

e

> c(p ~

1

1

c(p ~

4 (n~ 2 cl(p~

Q~m

1

Cl (pO) (2 Tt~ 2 "

From (10), we have:

s

IW(z)ldx

>_ k~) /s m--1

e-U(~)/t* lVu(x)l 2 dx

Q~ k(N)

->

e-(n~+'~)/~' E

/Q

m--1

> e_(n~+.y)ll, c(p~ -

cl(p~

IVu(x)12

dx

~m

1 (2 n~ 2"

Using Definition (7) of a(#), and taking into account the fact that -y is an arbitrary positive number, we obtain: lim inf # log a(#) > - ~ c . itS0

3.2. U p p e r b o u n d Let o1 and [:]2 be two neighbor cells, let say that [:]2 - [:]1 + el, and the corresponding maximum points of U0. We introduce the random set:

1 x E O1 ~ X 2max E [-12 X ma

and the events" 9 B ~ the set of w such that there is a path connecting x 1 and x 2 1 which belongs to A'( X =,~x, el) and which is included in G + (co). max

max

9 B~ the set of w such that there is a smooth curve of length not greater than n such that its ! - n e i g h b o r h o o d is included in G + (w) n

o

n

"

Comparing this setting with the one used for the proof of the lower bound, one can easily see that: ~7c - max{r/;

P(B ~

1 - minIr/; p ( B o ) > ~}. 1 < ~}

Effective diffusion in vanishing viscosity

140

Thus, for any small positive 7 we have: 1

P(B~

> -~.

This implies the existence of no = no(7) > 0 such that: 1

nO

P(Bn~-'~) > 2" We use the notation pO - P(Bnc_~ no ). In the same way as above one can assert that for sufficiently large N, the square [0, g ] 2 contains at least c(p ~ g mutually non intersecting smooth 1n o pipes connecting bottom and top sides of the square such that along each of these pipes:

U(x) > nc - 7. We consider a specific test function ~ such that: (i) ~(0, x2) - 0 and ~(1, x2) - 1, ~ is continuous,

(ii) ~ is constant between any pair of channels (pipes), and also between {X;Xl = 0} and the first pipe from one side, and between the last pipe and {x; Xl = 1} from the other side,

(iii) crossing each channel (pipe), ~ makes a jump of amplitude 1/ (c(p ~ N) ; inside a channel ~ is linear in the direction orthogonal to the curve that forms the channels. Hence [V~[ < no/c(p~ inf v E HI([:])

1/N, we get:

and letting c -

[ eU(: )/" IVv(x)[2 dx

JD

<

[ eV(~)/" IW(x)l 2 dx

-

L

JD

v(O, .) = o v ( 1 , .) -- 1

e -U(~)/t' [VO(x)] 2 dx annels

<

n._____~ e_(n_~)l~

-

c~(p~

=

C(7) e -(~~

here we also used the fact that [V~[ - 0 outside the channels. Back to Definition (7) of a(#), we get a(#) < C ( 7 ) e -(nc-'Y)/~. Taking the lim-sup as # $ 0, we find: lim sup # log a(#) _< -~c + 7. tt$0

F. Campillo and A. Piatnitstd

141

Since 7 is an arbitrary positive number, this relation implies" lim sup It log a(It) < - r / c . tt$0

3.3. M a i n r e s u l t 2 - Under above assumptions, in particular Hypothesis 1, the logbehavior of the effective diffusion a(#) in the small viscosity case is given

Theorem

by: lim # log a ( # ) u$o

-

-~lc

where Tic is the critical value given by (9).

4. Hypothesis 1" Sufficient conditions In this section we provide two different sufficient conditions for validity of Hypothesis 1. 3. - Let the random field U1 be equal to 0 everywhere in the vicinity of the level set s = {x : Uo(x) = Uo(x~)} except for some neighborhoods of the saddle points. Then, under the assumptions (i)-(iv) of Section 2, for sut~ciently small 70 and p the random variables a(xmi., ei), Xm~n E Xm~n, i = 1, 2, are independent.

Lemma

2 for each one Proof. Each periodic cell has two saddle points x 1 and x~, of these saddle points x i` (see Figure 2) we denote by x~i-n - Xmin and i+ Xmin -- Xm~n+ ei the two neighbor m i n i m u m points, symbol + corresponds to the greater value of one of the coordinates. Similarly, by x~+x - Xmax and x i ' - -- Xmax -- e2 (if i -- 1) -- Xm~ -- el (if i -- 2) we denote the neighbor m a x i m u m points. We begin by constructing a periodic family of sufficiently small neighborhoods Q(x~) of saddle points x~ E X~ t h a t possesses the following properties (see Figure 2 and Figure 3)" max

(i) for all x~ E X~, Q(x,) is a smooth domain t h a t contains no singular points except x, ;

(ii) the a - a l g e b r a s generated by {Ul(x), x E Q(x~)}, x~ E X~, are independent;

142

Effective diffusion in vanishing viscosity

Figure 2: Sufficient condition, Lemma 3. (example of Figure 1).

Uo(x)=Uo(xs)+ f3 Uo(x)_ Uo(xs) . ~ ,

F1

.

,///

9

!F4

G Figure 3: Zoom on point x s1 in Figure 2 with level lines {x ; Uo(x) - U0(x~)} and {x ; Uo(x) - Uo(x~) + ~}, and the decomposition F1 U F2 U F3 U F4 of

OQ(~.).

F. Campillo and A. Piatnitski

143

(iii) there exists ~ > 0 such that, for all x. E Xs, the following decomposition is valid: OQ(x.) - Uj=IFj, 4 where Fj are connected components of OQ(x~) such that Uo(x) > Uo(x~) + ~ if x E F1 U F3, and min Uo(x) < Uo(x.) - fl,

xEF2

max

xEF2UF4

min Uo(x) < Uo(x.) - fl,

xEF4

Uo(x) ~_ Uo(x~) + ~ ;

(iv) for all xs E Xs" if x E OQ(x.) and Uo(xs) - ~ ~_ Uo(x) ~_ Uo(x~) + j3 then U1 (x) - O; (v) all the trajectories of the equation 2 - - V U 0 ( x ) starting at F2, are attracted with x~-~ - Xmi~ while the trajectories starting at F4 are i+ attracted with Xmin. Under the above assumptions on U0 and U1 the said neighborhoods do evidently exist if/~ is small enough. We are going to show now that for V0 < /~/2 the random variables a(Xmi., ei) are independent. To this end we consider arbitrary two neighbor minimum points Xm~. and xmi~ + ei and a minimizing sequence of curves { ~ ( . ) } such that 995(0) = Xmin, ~ ( 1 ) = Xmi~ + ei, ~ E X(Xmin,ei) and max U(~(t)) ~_ C~(Xmi~,e i ) + (f.

0~t~l

Due to the structure of Uo and the choice of Q(x~), the intersection of ~(.) with Q(x~) is nontrivial for all sufficiently small 5. It is also clear that ~5 only intersects OQ(x~) at the points located at F2 U F4. Denote T1 -- max{t; ~5(t) E F2},

T2 -- min{t > 71; ~ ( t ) E F4}.

Now one can replace the segments {~(t) ; 0 < t < T1} and {~p(t) ; T2 _< t _ 1} by the new ones in such a way that the curve ~5(.) obtained is continuous, still belongs to A'(Xmi~, ei) and satisfies the estimates:

U(~(t)) < a(Xmi.,ei),

for all t < T1 and t > T2.

Thus O~(Xmin,el) only depends on {Ul(x); x E Q(x~)}, and the statement of the lemma follows. D The proof of the next assertion is similar to that of the preceding lemma and will be omitted.

144

Effective diffusion in vanishing viscosity

L e m m a 4 - Let Ul (x) be statistically homogeneous field (whose distributions are invariant w.r.t, any shifts) supported by Lipschitz functions, and suppose that

[Vl(x)[ ~ ~0, IV1(xl) -Vl(X2)[ ~ ")/l[X1 - x2[, x, x l , x 2 e 1~2, and that a{U1 (x); x e G 1} and a{U1 (x); x e G 2} are independent whenever dist(G 1, G 2) > p. Then for sufficiently small V0, "/1 and p the random variables a(Xmi., ei) are independent.

References [1] M. Avellaneda and A.J. Majda, Mathematical models with exact renormalization for turbulent transport. Communications in Mathematical Physics, 131 (1990), 381-429. [2] M. Avellaneda and A.J. Majda, Superdiffusion in nearly stratified flows. J. Stat. Phys., 69(3-4) (1992), 689-729. [3] M. Avellaneda and A.J. Majda, Application of an approximate R - N - G theory, to a model for turbulent transport, with exact renormalization. In Turbulence in fluid flows. A dynamical systems approach, The IMA volumes in Mathematics and its Applications, G.R. Sell et al., editors, 1-31, Springer Verlag New York, 1993. [4] M. Avellaneda and A.J. Majda, Simple examples with features of renormalization for turbulent transport. Phil. Trans. R. Soc. Lond. A, 346

(1994), 205-233. [5] J.J. Bear, Dynamicsof Fluids in Porous Media. Elsevier, New York, 1972. [6] A.Yu. Belyaev and Ya.R. Efendiev, Homogenization of the stokes equations with a random potential. Math. Notes, 59, 4 (1996), 361-372. [7] A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and its Applications. North-Holland, 1978. [8] R.A. Carmona and L. Xu, Homogenization for time dependent 2-D incompressible Gaussian flows. Preprint. [9] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows. SIAM Journal on Applied Mathematics, 54:333-408,. [I0] A. Fannjiang and G. Papanicolaou, Diffusion in turbulence. Probability Theory and Related Fields, 105 (1994), 279-334. [ii] J. Fried, Groundwater Pollution. Elsevier, Amsterdam, 1975.

F. Campillo and A. Piatnitski

[12]

145

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer Verlag, 1994. [13] H. Kesten, Percolation Theory for Mathematicians, volume 2 of Progress in Probability. Birkh/iuser, Boston, 1982. [14] S. Kozlov, Geometric aspects of homogenization, Russian Mathematical Surveys, 44, 2 (1989), 91-144. [15] S.M. Kozlov and A.L. Pyatnitskii, Averaging on a background of vanishing viscosity. Math. USSR Sbornik, 70, 1 (1991), 241-261. [16] A.L. Pyatnitski and S.M. Kozlov, Homogenization and vanishing viscosity. In B. Grigelionis, editor, Probability Theory and Mathematical Statistics. Proc. Fifth Vilnius Conference, 1989, 330-339. VSP/Mokslas, 1990.

Fabien Campillo SYSDYS, INRIA/LATP, CMI 39 rue F.Joliot-Curie 13453 Marseille Cedex France E-mail: [email protected] Andrey Piatnitski Narvik University College HiN Department of Mathematics P.O. Box 385 8505 Narvik Norway and Lebedev Physical Institute Russian Academy of Science Leninski Prospect 53 Moscow 117333 Russia E-mail: [email protected]

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Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 8

VIBRATION OF A THIN PLATE WITH A "ROUGH" SURFACE

G. CHECHKIN AND D. CIORANESCU

1. Introduction Rough surface problems have attracted much attention in the context of wave propagation and scattering (see, for example, [1], [2], [5] and [19]). The frictional behavior of deformable bodies depends explicitly on the structure of contact surfaces. Micro-characteristics such as the roughness of the contact surface or material properties near the surface, influence the large scale behavior. The asymptotic analysis of these problems was treated for instance, in [4] and [31], while different homogenization problems in domains with rapidly oscillating boundary were considered in for instance in [6], [17], [18], [19], [29] and [30]. Based on classical Kirchhoff and Reissner plates theories, new models for bending of plates with rapidly varying thickness was proposed in [20]-[23] (see also [7], [16], [24], [25] and [33]). In these papers, motivated by the development of structural optimization (see for instance, [26]) the authors have studied symmetric, linearly elastic plates with thickness of order s varying on a length scale of order sv. They restricted their attention to locally periodic plates, and to loads transverse to their midplanes. The transverse loads and the symmetry allow, in particular, to reduce the limit problem to one fourth-order equation. There are three different regimes, depending on whether U < 1 (the case of relatively slow variation of the thickness), U = 1 (when the variation is on the same scale as the mean thickness), or U :> 1 (the case of relatively fast variation of the thickness). For each case, an effective rigidity tensor M ~ relating the bending moment to midplane curvature of the limit plate obtained as s --~ 0, was determined. In the limit problem, the vertical displacement of the midplane solves a

148

Vibration of a thin plate with a rough surface

fourth-order equation of the form

-

a,/3,-y,5= 1

The methods developed in [20]-[23] relied essentially on the symmetry of plates. In the present paper we drop out the symmetry assumption, and consider a nonsymmetric locally periodic plate with oscillating boundary. In contrast to the case of plates with symmetric geometry, the limit equations in the nonsymmetric case cannot be decomposed into equations describing the vertical displcement and equations describing the horizontal ones. Due to the absence of a symmetry in the initial problem, a nontrivial interrelation between the leading terms of the asymptotic expansion do appear. In particular, the limit problem is characterized by the coupling of vertical and horizontal displacements. In Section 1 we introduce necessary notation, define a class of thin domains and formulate the problem. In Section 2 we prove a priori estimates for the solution and investigate the asymptotic behavior of the moments and means of stresses. Then we introduce a family of auxiliary problems and obtain by a formal asymptotic expansion method the homogenized system. In the last section we prove a convergence theorem which justifies the results from Section 2. 1. S t a t e m e n t of the problem Let ft c ]Ra be a domain of thickness e, ~--{:;C I 0
1
0

with e = ks, where e is a small parameter. Suppose we are given a smooth nonpositive function p = p(~, ~), with ~ ' - {~1, ~2} and ~ - {x l, x2}, ~ being I-periodic in ~'. We introduce the domain II~ by setting A

1 •

a, 0 < z2 < b, s~(2, z_) < z3 < 0}. E

The domain f~e (the plate with a "rough" surface) is defined as follows: A

G. Chechkin and D. Cioranescu

149

Introduce now the surfaces we =

{,x xC~tE,

(~ , x)} , A

xa=sp

w -{x

I xe~e,

we={x

IxeFt~, x3=e}=

x3=0}={xl0<xl {xI0<xl


x3=0},


x3=e},

= {x I x e

and denote by p ~ = ( SI, u~, v~) the unit outward normal to w e, by n = (n l, n2, n3) - (0, 0, 1) the unit outward normal vector to we and by n e = (n~, n~, 0) the unit outward normal to Se.

Figure 1. The plate ~e By construction, the surface w e is the lower face of the plate occupying the domain Re, while we is its upper face. The surface Se is the the lateral boundary of the plate. Observe that w e is the "rough" oscillating part of the boundary 0Fte of the plate. The surface w introduced above, is the lower face of the domain ft; as a m a t t e r of fact, it is the mid surface of the thin domain ftl. For simplicity, we will suppose that p is compactly supported in w, uniformly in ~ and such that !~1 < k for any ~ and ~. N o t a t i o n . In the sequel, the Einstein summation convention on repeated indices is adopted. As a rule, throughout the paper, Greek indices a,/3, 7, 5, p take values in {1,2} while Latin indices i, j, k, l, h, p, q take values in {1,2,3}. The current point in ~e is x = (Xl,X2, X3), while the current point on the surface w e is denoted ~ = (Xl, x2, x3).

Vibration of a thin plate with a rough surface

150

We consider the following boundary-value problem in linearized elasticity:

L[u e] - ~ ue = 0

0 (

m kt

0~ OxL ]

= - ff

in fie

on SE,

Aa I Oue -x/n3-0

(1)

on We, A

AktOue

e

b-77~~ =

ge(.-:-,

7

)

on we

,

where the unknown is the displacement u e = (u~, u~, u~). In system (1), d k l = dkZ(x) are (3 • 3)--matrices, whose elements aik~ = kl aij (x) are bounded measurable functions satisfying the usual symmetries in elasticity aijkl (x) : ajilk (x) = aki/"(x) a.e. in f~e. (2) We suppose also t h a t there exist A1, A2 6 IR, 0 < A1 < A2 such that ~1 ?~ik ?~ik ~_ a ikj l ( x )

a.e. in Fte 9

?']ik lljl _< • 2 ~ik ?]ik

(3)

for any symmetric matrix (ri)i,j with real elements. The first boundary condition in (1) means that the plate is clamped on the lateral boundary S~. Suppose we are given fe _ i f ( x ) , f f = (f~, f ~ , f ~ ) , the volume density of applied body forces and ge _ gE (~, ~) where gE _ (g~, g~, g~), the density of surface forces, satisfying the following assumption: A

{

f~(x) = O(e),

f~(x) = O(e 2)

A

for x e f t e ,

A

e X g.(~. 7) - o(~:) ,

3 7X ) - - O ( e 3) f o r x e g~(~.

WE .

(4)

Now, let us introduce the strain tensor

~(~)

- ~

+~

~

.

(5)

and the stress tensor

aijeik(uE). W i t h this notation, one can rewrite (1) in the equivalent form '

O0"ij

- -O--~x (u e) -- f~

ue=0

e

in f~e,

on Se,

aia(ue)n3 - - 0

(6) on

~0 e , A

on

we.

G. Chechkin and D. Cioranescu

151

D e f i n i t i o n 1 - The vector function u E e [Hl(f~, S~)] 3 is a weak solution of the problem (1) if the integral identity

/

A k Z ( x ) ~

o.,

Ox~

7 )' v(')) d. OJ e

(7)

+

holds

for

any v e [Hl(f~E, SE)] 3.

Here, as usual, [Hl(f~, SE)]3 is the the closure with respect to the norm of the Sobolev space [Hl(gt6)] 3, of the set of [C~176 3 - vector functions, vanishing in a neighbourhood of S~ (see for instance, [32])" [Hl(~e, Se)] 3 - {u E [Hl(~tE)]3 I u -- 0 on Se}, endowed with the usual gradient norm

[[UII[HI(~'~,S~)]3 = [lVull[L2(~6)]3x3. In problem (5) we also used the notation 3

(u, v)

=

~-~ ui vi,

for u, v C [L(f~)] 3.

i--1

Using definition (5) of the strain tensor, the left-hand side of (7) can be rewritten in the form Ou ~ Ov dx = A kz(x) Oxz Oxk

kl aij(x)eik(u E) ejz(v) dx.

Using Korn's inequality (see [10], [27] and [28]) and the Lax-Milgram theorem, one can prove the existence and the uniqueness of a weak solution of problem (1) (or (6) or (7)). Moreover the following estimate takes place"

_< K~ (lif~ltfL~(~)j. + ligllIL~(~-)I'), where K1 and K2 are independent of e (see, for example [27] or [30]).

(8)

Vibration of a thin plate with a rough surface

152

Let us transform the domain ~tE by means of the dilatation in the Xadirection A Z

"---

~. X,

Z3 :"-

X3 E

~ .

Denote by ~tE the transformed domain. Under this dilatation, any vector-function is changed as follows: we introduce a new vector-function tI/(Zl, Z2, Z3) instead of vector-function ~b(Xl, x2, x3) by 1 ~I/c~(Zl, Z2, Z3) -- 7~b~(Zl, z2, r tI/3(Zl, Z2, Z3) --"

~b3(Zl, z2, eZ3).

We will also use the notation

{

A

~ = {zC O~tE I za=k},

that are the transformed of respectively, w e, we and SE. Let us denote by U E, F E and G E the transformed of u E, fE and gE respectively. Then, in the new coordinates problem (6) becomes e e

or

~) O Z fl

o ~ ( u ~)

+ +

or

o ~ ( u ~)

OZfl

UE = 0

~) (~Z 3

=-eF~ = -F~

in f~E, in ~E,

OZ 3

on S,

aia(U E) n 3 - O

(9)

on we, on ~,

1 /I+~=IV~I = (~') 1 + [ V ~ [ 2 G~ ~ ' 7

EO'3fl(U e) l]"'~ -'[- o-33(U e) /2"~ -- 7

on ~E,

where

r

~) = a'J(z)r O tz

kl

kl

~ ~) + _2a3 ~ -~(z)r

1 3j + -fia~,(z)r

~)

(10)

G. Chechkin and D. Cioranescu

153

Assumption (4) reads

{

F~(z) = O(1),

(z)

c~ ~"7

F~(z) = O(s2),

= o(~),

(11)

c:, ~, ?- = o(~).

Denote

F~(z) := F~.(z),

F~(z) := 1 F~(z),

1

~-

a.(~, ~).= 7 a~(~-, ~'),

^ c~(~, ~) .= ~1 G~(~, ~').

Definition 2 - The vector-function U E C [HI(Fte, S)] 3 is a weak solution of problem (9) if the integral identity

o~,(z)

=

Ova(z)

l + ] V z ~ t 2 G~ ~ , -

_

v~(z)dz

(12) /

1 /'i+E21~7z~i 2

+

~" _z) vn(z) d S

V

+ f [~F~(z) v.(z) + F~(z) v~(z)] ez holds for any v E [H 1( ~ , S)] 3.

2. Preliminary results 2.1. A priori e s t i m a t e s From variational formulation (7), due to the coerciveness condition (3), to Friedrichs' inequality and keeping in mind assumptions (4), we derive the following a priori estimate"

/ (lvu~(x)l 2 + IVu~(x)l 2 + IVu~(x)!2) ~x < ~ gt~

~2,

Vibration of a thin plate with a rough surface

154

whence, in the new variables,

S[~2(OU~(z))2+~2(OU~(z)) Oz2 (OU~(Z)oz3 ).. 2]dz O, . ..

O,

) +(

) .z

+/[(OUJ(z)2+O(z,oUj(z)) + s'2-1OUJ(z))2]Oz.

dz < K3 ~2,

9t~ from which, the following a priori estimates are straightforward:

ile~fl(UE)[IL,('fi.) <_ K4,

(13)

Ilea3(US)llL2('~) <_E K4,

(14)

Ile33(U~)l]L,('~.) _< e 2 K4.

(15)

Then, by using (2) and (3), one immediately has

]lO'ij(UZ)llL2('~e) <_K4.

(16)

Now, let ~ • be the domain defined by ~+={z]0

-k
Let us extend oij(U E) by zero in ~+ and still denote oij(U E) this extension. Then, from estimate (16) and Rellich theorem it follows that there exists a subsequence {~'} such that, as ~' -+ 0, aij ( U ~' ) ~

0 a~j

weakly in L2(~+).

(17)

2.2. M e a n s a n d m o m e n t s To obtain the limit equations as ~ --~ 0, we introduce the moments and the means of stresses. Definition 3-

The means of stresses and of its moments are defined re-

spectively, by k

M~Z(~) -

f

~(~,~z)

z3 a~Z dz3,

(18)

155

G. Chechkin and D. Cioranescu k

k

N~z(z) =

a,~ dz3,

'~(~) = 7

~,(~,~ )

a<~3 dz3.

(19)

~(~, ~1

Remark. Note that due to (16), M~Z N ~ and Q~ belong to L2(w) Z3

Let v(~)

E

[C~(cd)] 3 and take V(z) = f v(~) dz3, as test function in (12) 0

to obtain

ay~(z) fl. '

=

(')

I + [ V ~ ! 2 G~ ~,~

V~(z) dz

02 r

+-

~

1 + lVz~l 2 a~ ~, -~ V3(z) as

+ S [r

F3(z)V3(z)] dz.

Keeping in mind (11) and (16), and using the fact that

S71

._

tO e

we

have

1 + IV z~l ~

h(-~.,,) ., = J (l+.,lv:~l'

h(~, ~) d~,

f.J

that, as r --~ O,

i ai3 (U ~) v~ dz ~ 0,

whence

~o = o

(20)

Taking (0,0, v) with v E H~(w), as a test function in (12), and recalling

Vibration of a thin plate with a rough surface

156

definition (19), we get

i

1iv/1 + ~'lV:,oi'C~ (~,~7) v($) d2

QS(z-) Ov(2) ozn d~=-~-5

02

k

1

(21)

Due to hypotheses (11) and using Lemma 5 from [9] (see also [3]), one obtains the estimate

S

i2

Az

(2")]v( d2I C1 Ilvll..(.).

where

0'3(2")= i ql + IVr

~')1~ c~(~, ~-) ,t~L

T

and T={~

90<~

<1}.

(22)

Now, observe that the second integral in the right-hand side of (21) can be rewritten as follows: k

k

1

(Z3) F3(z)

dz3) v(~) d~,

(z3) is a characteristic function of ~ , i.e.,

where X[r

A

xtr

(zz)=

i f ~ ~,

0

if - k < z 3 < p

0

if ~'~ w.

By Lemma 5 from [9], we have the estimate k

r

--k

<_z3<_k, zCw,

1

2",

,

zEw,

(23)

G. Chechkin and D. Cioranescu

157

X(z) -- f X[~o(z,~"),k](Z3) d~'.

(24)

where

T

Consequently, when passing to the limit in (21), and recalling (20), we obtain the limit system k

o -- ~3(~) + - - ~ zo Qf~(z')

f

X(z) F3(z) dz3 for ~ E w ,

-k 0 r Q~n~=O

(25)

on 0w.

Here QO(~) denotes the weak L2(w)-limit of Q~(~') as ~ --, 0. Let now take (Vl, v2, 0) with v,~ E Hi(w) as a test function in (12). We get (see definition (19)),

Ov~(~s163 Oz~ {M

-sl f V / i

+"~ l v ~ l

2 G~ ( ~~,) ~

vo(~) d~

o.2

k

+/( j' A

This, together with (25), leads to the system

k X(z) F~(z) dz3

for ~ E w,

-k

(26)

0 r = O on Ow, N~znz

where N~

") is the weak L2(w)- limit of N~Z($) as ~ ~ 0, and

T

Finally, let v~ C H~(w) and take (Z3Vl, Z3V2, 0) aS test function in (12).

Vibration of a thin plate with a rough surface

158

Using defitions (18) and (19), it follows that

Oz~ O2

k

: I( i z,,: (z),z,)v.<,,,, {,a) A A ~(.,:-) A

E

(,., A

O3

As above, bearing in mind hypotheses (11) and using Lemma 5 from [9], we have [~1V/1 _lt_E2lVzCp]2 G~ (~', ~z) ~(~, ~ ) - G~ (z')] v~ (~) d~ _< C 3 e IlvllHl(w) ,

O2 where

v~(~) =

J ~/i+ Iv~(~,~)l 2 co(~, ~) ~(~, ~) d~.

T

Recalling the definition of the set T (see (22)), the first integral in the right-hand side of (27) can be rewritten in the form k O2

k

A A ~(.,-~)

O2 --]g

where X[~(%.,~l,k] was defined in (23). Then, again by Lemma 5 from [9], we have the estimate k S

[J

O2

--k

Z3 (~(~[r

z~')'k]( z 3 ) -

X(z))Fci(z)dz3]v.(~)d~ C4~ llvllHl(O2),

where X was defined by (24). Passing to the limit as e -+ 0 in (27), we obtain k

o ~- + QO.(~) - ~ ( ~ ) ---~z0 M~f~(z)

+J -k

M~

= 0

on Ow,

z3 X(z) F(~(z) dz3 in w,

(28)

G. Chechkin and D. Cioranescu

159

where M~ is the weak L 2 ( w ) - limit of the moment M~Z, as ~ ~ 0. To finish this subsection, observe that the functions N~Z and M ~ be rewritten in the form k

can

k

i~(~.~).kl(za) ~.~(u ~) 7

(29)

dz~.

A A

~(z,"-)

-k

r

respectively, k

f

ML~(~) =

k

Z3 cro~/3(Ur dz3 -- / z3 X[~o(~.,_~),k] (z3) o'af~(U e) dz3. (30) -k

A A

~(z,~)

2.3. A u x i l i a r y p r o b l e m s To understand what M ~/~ ~ and N ~/~ O are, we have to know the limit of cr~z(Ue). To do so, we will introduce a formal asymptotic expansion of the function U e of the form A

Ue(z) = U~

A

A

Z nt- E2U2 (z, 7) Z @ ~3U3 (z,-)~ z nt-... .-Jr-~Yl(z, 7)

(31)

We substitute (31) for U E into all the equations in (9) and equate the powerlike terms of e. We do not detail here the computations since they are obvious (and rather technical). From the first equations in (9), taking into account that U 1, U 2 and U 3 are independent of ~3, we obtain an infinite chain of equations. When looking for the coefficients of e 2 in (9), we obtain t h a t U1 satisfies

02UJ

3c~ ~ ~"/c~(z) 0~02U1 0~----~+~3~ (z) Oz~ 0 ~

.~

02U1 3~ 02~ -1-a33 (z) -- 0 (32) 0~, Oz~ Oz~ Oz3

-+-a33 ( z ) - - - ~ -

'

and

73

02 U1 +a33~(z)------02U~ + .~3( z ) ~02U~

~ ( Z ) o~.~o~----~

Oz~ o~

a~

o~.~Oz~

33

nt-a33(z)

02U~

Oz~ Oz~

--

0. (33)

160

Vibration of a thin plate with a rough surface

The next expressions (coefficients of gl in (8)) are 02 U~ + a3~(z) 02 U~ + a,~Z (z ) ~,~ a3t3 (z) 0~, Ozo Oz3 0zz 2

o~u~ + a3~(z ~,c~ ) 02 U32 + a3~(z) Oz 3 0~13 0~.~0 ~

~o~(z) Ol.O...~ 3U~1 + 02U13 + a3~ Oz.~0 ~

~ (Z) ( 0 ~Uvo t a53

02U~ Oz~Oz3 + Oz~ Oz3

2 .~

q- a33 (Z)

02 Ur + U~ 0~ 0~ 0~.~ 0 ~

+ a,~3 (z) 2

U.~ + 0 ~ Oz3 0 ~ Oz3

(34)

0 U.~ o~u] ) ~ o~u~ 3~ o~u~ OZ30Z3 q- OZ.~OZ3 -Jr-a33 (Z) 0~.y OZ3 -t- a33 (z) Oz3 0z3

=0, and 02U1 a3/~(z) c9~.~Oze

o~u] d a~Z(z) + a33~(z)Oz30zz - ~

~3 + a3/~(z)

U~ + 02U] O{Z Oz.~0~

"t3

~3(z) ( 0 2U.yo

4 aa3

2

Oza Oz3

~3 + a33(z)

(

o2u~1

o~u~

o~a o ~ + o~ o~

)

+ a3z Y3(Z) o~u~ -4- a33~(z) o~u~ 0~.~0 ~ Oz3 0~

+ 02U~

q a53(z) 2

Oz.r Oz3

O~U] 0 U~ OzaOz3 + OZ7 0Z 3

Uy -kO~a Oz3 0~.~ Oz3

(35)

~a O~U~ aa o~u~ -~- a33 (Z) 0~.~ OZ3 + a33(z) Oz30Z 3 =0.

The treatement of boundary conditions in (9) is more delicate since they contain the terms e

and

1 + [V zqal2 G~ ~',~-

e2 ~/1 + r l + l V z p l 2 G3(~,7).

It is easy to see that, if IV~9)($, ~)1-J= O, then A

A

~lXyz~(~, 7) 1 z

1+

2

11 + IV~(~,~')12I

A

1+

IVz~(~,-~)1

IV~(~, ~)1

~

~2

~_~

(v~(~,~), v~(~,#))

#)13v/1

I +~

G. Chechkin and D. Cioranescu

161

A

At the points where [V~p(2, ~)1 = 0, one has A

z

~lv=~(~, 7)1 z 2 1 + Ivz~(~, 7)1

2

1+

A

= O(1).

This allows us to equate the power-like terms of e for the terms coming from the boundary &e in (9). Remembering the transformation of the normal to ~e by means of the change of variables (Zl, z2, z3) --~ ( ~ , ~2, z3), we end up with the following expressions:

[ ~oou~ ~ou~].. ~ou'~ a3z ~ + a3/~Oz3 J U/~ + a33 ~-s ~ = 0 [a3~ OU1

33

ou11--

~30u1--

+ a3r O~zaJ u} + a33 ~

4 = 0

on &e,

(36)

on &e,

(37)

and also

[ a s~ OU~

OU~ "~ a s~ OU1

cgU~

OU1

OU~ '~

a'~3 0U~

OU~'~

"2- ("~-~z~+ Oz~ ] + -"2"-(--~-~ -t- -~ ) -t- a'~ (-~3z3 + -~z~ ] OU2

0U23][

OU~

+ a3~ - ~ + a~ Oz3 j ~ + a~ 723 + --~ (-~z~ + -~z~J a~a OU1

cgU~

~o~(OUr

(38)

OU1

+--~-(-~5 + - ~ )+a33 \-~~z3+ ~z~ ) + a33 ~

+ a33 Oz3 .I vfi = O,

and

(39)

~31 ( oul

ou~

~ ( ovl

ou~

+ a~3~ \-~"5 + " ~ ) + a33,,.-~-~z3+ -~-z~) .y3 OU~ + a3333OU~ ] ~,~ = O, -t-- a33 ( z ) ~ . ~ " -~

0z3 J

162

Vibration of a thin plate with a rough surface

L e m m a 1 - T h e vector-function U ~ = U~

gO(z) :

( --Z 3 0ou(~) Z 1 q- Ul(Z'),

has the form

ou(~) + 32(~), u(~)),

--Z3 0Z 2

where U e H2(w) and bl e Hi(w)] 2 Proof. Estimate (16) implies

I~

_< K4~ 2.

(40)

Substituting expansion (31) of U~ into (40), and passing afterwards to the limit as E --. 0, one obtains, due to (17), OU~ = 0 . Oz3

This means that ~=U,

where u-u(~)

with U E H l(w).

Similarly, estimates (13)-(15) yield oe o Oz3

ov o t

Oz~

-0,

and hence,

O

_ z3

OU(s + Us(S) (0zc~

with Us e Hi(w).

Following [11] (see also [12], [13], [14] and [15]) one gets successively U c H2(w),

UIo~ -

OU

=0, Ow

and this ends the proof of the lemma,

m

163

G. Chechkin and D. Cioranescu

From equations (32)-(35) and boundary conditions (36)-(39), we are led to look for U 1 = (U 1, U2~, U31) in the form

1

v l(z, 7) = [v?~ (z, ;-) + 5 z~ [apq(Z, ~ O)]jl 2

-1

3/3 , 0)] O2U(z') ajc~(~

-

[ _~_ ~ / 3 U~(z,-~) =

V2 ~ (z,

z 7)

(Z

Oz~oz~

]

-- Z3 [a33(~. 0)] j-11 aJ3,(~-. O) eo~ (/4) , a

~-) + ~ z32 [a3q3(2,0)] ~2

ajo~ ( z, 0)

7)

(41)

Oz~ Oz z

33 A 0)] j-: aja 3/Y(Z", O)] ea~ (lA(), + [ "W; ~ (z, Z -- z3 [(apq(Z, Z

v~ (z, ;) =o. and for U 2 = (U 2, U 2, U2) in the form A

z

u~i (z, 7) =o, Z

u~(~, ;) :0,

2[ 21 U~(z, 7) = ~ , ( z , ;) + ~ d +

[W 2 t~(z ~)z

[apq(Z. ~ 0)]

-- Z3

(42)

--1 3/3 ]02U(~') a j . (~, 0)

OZo~OZ~ [a33(~",0)] _1 a3~(~",O)] e~z(b/) ^

j3

j3

"

In these formulas, for each a and p. the vector-function V ~p = V~P(z. ~/~) satisfies in the domain WT = w x T (see(22)). the following equation: as~

-2 (~' 0) a~ a~, + a~, a~z + az,(~, 0) a~v:" 0z30~

"yi

a53 (~

+ -5-'

' O)

( 02V~p o2g;p ) 0~6 0Z3

~ (~, o) ~

+ a3fl

+

0~,.,/ Oz 3

+

,~i

a33

(S, O)

02V~:~p OZ30z3

(43)

a ~ ~ o) ~

O~.y 0 ~ -[- 3~k ,

+ a33(~ .,/i , O) 02V2P

0 ~ Oz3 +

Oz 3 0 ~

P' , O) a 3i t2 O) 02VffP = ba3(~

33 t ,

Oz30z3

with the boundary conditions on OwT

a;~

av~,,

or2"

;,

av:~

.......

+ or;; ) .....

(44) 3i + az~(~,o)O v;~ ]

pi

Oz3 juk = z3 b.k(~, O) uk.

164

Vibration of a thin plate with a rough surface

Also, for each a and p, the vector-function W `~p = W'~P(z,~/s) from formulas (41) and (42), satisfies in WT the equation

a~z (~, O) 2

0~ 0~

+ --~

) 0G 0~

0~50z3

+ a~' ~/i

+

02W$P

+ a3z

0~/Oz3

Oz3 0 ~

a33

Oz30z3

(45)

0~, 0 ~ + a~(~, O) Oz~ 0 ~ t,.

02W~p

3i

O2W:p :

+ a3~(z, o) OG Oz3 + a33(~, 0) 0z3 0~3

0

with the boundary conditions

[a l

OW:" OW;"

OW;" OW:" (46)

+ a33~('~, O) OW3Poz3 ] ~'k - - OX~('~, O) ~'k

Remark. Observe that with systems (43)-(44) and (45)-(46), we have determined the first three terms of expansion (31). The complete characterization of its first term, the "homogenized solution" will be given in the next section.

3. Limit problems The aim of this section is to state the "homogenization" result showing that the limit function U ~ is a good approximation of the "physical" displacement U E, solution of system (9). Actually, one has the following result: T h e o r e m 1 - Let U E E [HI(f~E,S)] 3 be the solution of system (9) and assume that hypothesis (11) holds true. Then as ~ --~ O, one has the convergence U E --~ U ~ weakly in [H 1(fie, S)] 3. The limit function U ~ - (U~), U ~ U~ has the form

U~

- u(~),

U~

- -z~

ou(~) Oza

+ u.(~),

G. Chechkin and D. Cioranescu with U ~ H2o(Ca) and bl e

S

[Ho~(W)]2 and satisfying the limit problems

02 U

~.~(u) M.~(~')]

-

-

f

165

02V d'Z Oz, Oz~

02v

[Mb(~) o..,o'.u0., _ M~(~) ~(u)]

Oz. Oz~

r

k

,z.]v ,,+S.ov

: S [/,o,~.. 03

~ k

od

k

-

J'"

zzF ~ X(z) dz3 ~z~ d'5-03

m k

(47) ..

I

.v d'Z, Vv E H3(w )

G~ ~

Og

and S

02 U ]

Od

02U

Oz~,Oza M?,(;) r

+S [A'4~

-

r

d~

k

v ~ e [yl(~.)l ~

where b ~ ('2, z~ )

~ --- aa.),

('2, z~ ) -

~ aj(Z'~Z3)

~ ~ (apq('Z, Z3))ji- ' ai.7(z',z3 ),

k 3

--k k

-k

T k

-k T

--"

(48)

Vibration of a thin plate with a rough surface

166

Proof. Substituting (31) into definition (10) leads to o , j ( u ~) = b~{(~. ~z~) ~.~(u) - b'A(~. ~z~) z~

02U Oz~ Ozp

(49)

02U

+ ~j(y".) Oz~ Oz, + ~,j(w",)~.,(u) + o(~). where aij(V ~p) and aij(W ~p) are defined respectively, by

..

oW .

ou2.

aij(V") = --~ aSi (',ez3)( 0r

-~- a3i

.J

ow"

ov20

+ 0 ~ ) + a3i "j (~' ez3)( Oz3 + 0r

)

Oz 3

ow~,

ow2"

.~j(~, ~ )

~,j(w ~") = --~-(~. asi ez3) ( o~ + oG ) + a3, 3J(~.sz~)

ow~o + ow2~ ) (Oz3

oG

s

ow2 p

+ aai

Oz3

Using formula (49) we can write the function a c~fl ~ (see also the technique introduced in [7] and [8]) in the form

02U

0

~

~Oz~ + (Oz~ ~,,(w~))

(u~ + (~(v~))

~o,(u). (50)

Recall now formulas (29) and (30), satisfied respectively by the moments N~Z and M ~z. Using (50) in these formulas and passing to the limit, we get

02U NOn = A ~ e~5(bl) - B ~ Oz~ Ozs' 02U M~ - C ~ e~5(bl) - D ~ Oz~ Ozs' where b ~ is the following matrix: b~

~5

- a., - a

~

and k

-k

k

-k T

33 -1 35

(%~)ji a ~ ,

(51)

G. Chechkin and D. Cioranescu k

167

k

X[~(z,~-),k](z3)d~ dz3, -k

-k T

k

k

z3 a~z(W ~) X[~,(z,~),k](z3)d~dz3, --k

--k T

k

k Z3 ~

--k

( V'T5 ) )~[T(z,~'),k] ( z 3 ) d ~ d z 3 .

-k T

One easily can check, that the matrix b ~ is positive definite. To prove the assertions of the theorem, we follow along the lines of the proof from [7] and [8] (see also [14] and [15]). Let v e H2(w) and multiply (28) by Ov/Oz~ and (25) by v. Passing to the limit and using (51), we obtain (47). Similarly, multiplying (26) by w e [Hi(w)] 2, we deduce (48) and this ends the proof of the theorem.

References [1] Beckmann P., Spizzichino A., The scattering of electromagnetic waves from rough surfaces, New York, Pergamon Press 1963. [2] Belyaev A.G., Mikheev A.G., Shamaev A.S., . Plane wave diffraction by a rapidly oscillating surface, Zh. Vychisl. Mat. i Mat. Fiz., 32, 8 (1992), 1258-1272. (English translation Comput. Math. Math. Phys., 32, 8 (1992), 1121-1133. [3] Belyaev, A. G., Piatnitski, A.L., and Chechkin, G.A., Asymptotic behavior of solution for boundary-value problem in a perforated domain with oscillating boundary, Siberian Math. Journal, 39, 4 (1998), 621-644. [4] Bouchitte G., Lidouh A. and Suquet P., Homog~n~isation de fronti~re pour la mod~lisation du contact entre un corps d~formable non lin@aire et un corps rigide. C. R. Acad. Sci. Paris, Ser. I, 313 (1991), 967-972. [5] Brizzi R., Chalot J.P., Homog~n~isation de fronti~re, Ricerche di Mat. 46, 2 (1997), 341-387. [6] Butazzo G. and Kohn R., 1986. Reinforcement by a thin layer with oscillating thickness, Appl. Math. and Optimiz., 16, 3 (1987), 247-261. [7] Caillerie D., The effect of a thin inclusion of high rigidity in an elastic body, Math. Methods Applied Sci. , 2 (1980), 251-270 [8] Caillerie D., Thin elastic and periodic plates, Math. Methods Appl. Sci., 6 (1984), 159-191

168

[91 [10] [111

[121 [13] [141 [15] [161 [17]

[18] [19]

[20] [21] [22]

[231

Vibration of a thin plate with a rough surface

Chechkin G. A., Friedman A. and Piatnitski A. L., The boundary value problem in domains with very rapidly oscillating boundary, Journal of Math. Anal. and Applic., 231, 1 (1999), 213-234. Chechkin G.A.and Pichugina E.A., Weighted Korn's inequality for a thin plate with a rough surface. Russian Journal of Math. Physics, 7, 3 (2000), 279-287. Ciarlet P. G.and Destuynder P., A justification of the two-dimensional linear plate model, J. M~canique, 18 (1979), 315-344 Ciarlet P.G., Plates and junctions in elastic multi-structures: an asymptotic analysis. R.M.A., Vol. 14., Masson and Springer-Verlag 1990. Ciarlet, P.G., Mathematical elasticity. Volume II. Theory of plates., Studies in Mathematics and its Applications, Vol. 27, Elsevier 1997. Cioranescu D. and Saint Jean Paulin J., Asymptotic analysis of elastic wireworks, Preprint of Universit~ Pierre et Marie Curie, R 89008, Paris (1989). Cioranescu D. and Saint Jean Paulin J., Homogenization of reticulated structures, Applied Mathematical Sciences 136, Springer-Verlag New York 1999. Constanda C., Lobo M. and Per~z E., On the bending of plates with transverse shear deformation and mixed periodic boundary conditions, Math. Methods Appl. Sci., 18, 5 (1995), 337-344. Damlamian A. and Vogelius M., Homogenization limits of the equations of elasticity in thin domains, IMA Preprint Series 170. Minneapolis: Institute for Mathematics and its Applications, 1985. Gaudiello, A., Homogenization of an elliptic transmission problem, Advances in Math. Sci. and Appl., Gakkotosho, 5, 2 (1995), 639-657. Kohler W., Papanicolaou G., Varadhan S., Boundary and interface problems in regions with very rough boundaries, in Multiple scattering and waves in random media, Chow P.L., Kohler W.E., Papanicolaou G.C. eds., Amsterdam, North-Holland (1981), 165-197. Kohn R. and Vogelius M., A new model for thin plates with rapidly varying thickness, J. Solid Struct., 20, 4 (1984), 333-350 Kohn R. and Vogelius M., 1985. A new model for thin plates with rapidly varying thickness. II. A convergence proof. Quart. of Appl. Math., XLIII, 1 (1985), 1-22 Kohn R. and Vogelius M., 1986. A new model for thin plates with rapidly varying thickness. III. Comparison of different scalings, Quart. Appl. Math., XLIV, 1 (1986), 35-48 Kohn R. and Vogelius M., 1986. Thin plates with rapidly varying thickness, and their relation to structural optimization, in Homogenization and effective moduli of materials and media, J. L. Ericksen, D. Kinderlehrer, R. Kohn and J.-L. Lions eds., Springer-Verlag (1986), 126-149

G. Chechkin and D. Cioranescu

169

[24] Kolpakov A.G., Thin elastic plates with periodic structure and internal unilateral contact conditions, Prikl. Mekh. Tekh. Fizika, 5 (1991), 136142. [25] Lewifiski T. and Telega J.J., Plates, laminates and shells. Asymptotic analysis and homogenization, World Scientific 2000. [26] Lurie K.A., Cherkaev A.V. and Fedorov A.V., Regularization of optimal design problems for bars and plates, I and II, J. Opt. Th. Appl., 37 (1982), 499-543. [27] Nazarov S.A., On the accuracy of asymptotic approximations for longitudinal deformations of a thin plate, Mod41isation Math. et Analyse Num4rique, M2AN, 2 (1996), 185-213. [28] Nazarov S.A., Korn's inequalities for junctions of spatial bodies and thin rods, Math. Methods Appl. Sci. 20, 3 (1997), 219-243. [29] Nevard J. and Keller J.B., 1997. Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math. 57, 6 (1997), 1660-1686. [30] Oleinik O. A., Shamaev A. S. and Yosifian G. A., Mathematical problems in elasticity and homogenization, Amsterdam, North-Holland 1992. [31] Sanchez-Palencia E. and Suquet P., Friction and homogenization of a boundary, in Free Boundary Problems: Theory and applications, A. Fasano and M. Primicerio eds., London, Pitman (1983), 561-571. [32] Sobolev S.L., Some applications of functional analysis in mathematical physics. Third Edition. Translations of Mathematical Monographs Serie, Volume 90, AMS Press 1991. [33] Vlasov B.F., On equations of plates bending, Izv. Akad. Nauk. SSSR O. M. N., 12 (1957), 57-60. Gregory A. Chechkin Department of Differential Equations Faculty of Mechanics and Mathematics Moscow State University Moscow 119899 Russia E-mail: Doina Cioranescu Laboratoire Jacques-Louis Lions C.N.R.S and Universit4 Pierre et Marie Curie Boite postale 187 4 pl ace J ussie u 75252 Paris Cedex 05 France

E-maih cioran~ann.j ussieu, fr

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S t u d i e s in M a t h e m a t i c s a n d its A p p l i c a t i o n s , Vol. 31

D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 9 ANISOTROPY AND DISPERSION IN ROTATING FLUIDS

J.-Y. CHEMIN, B. DESJARDINS, I. GALLAGHER AND E. GRENIER

1. Introduction In this paper, our aim is to study dispersion phenomena occuring in singular perturbation models of fluid dynamics in IR3. We focus on the small Rossby number limit of solutions to the incompressible Navier-Stokes equations, and are going to study dispersion phenomena in the following system:

I

Otu ~ + u ~ . V u ~ _ u A u ~ +

U~X e3 div u ~

(RFe)

U~t=O

=

- V p ~ in

= 0 in "-- ?-to,

]R3

IR3

where e3 = (0, 0, 1), ~ > 0 denotes the Rossby number and ~ > 0 is the viscosity. We refer to [12] and [24] for a physical justification of the model. It is well known that dispersive effects are of great importance in the study of non linear partial differential equations. Historically, the use of dispersive effects in partial differential equations appeared in the study of the wave equation with the proof of the Strichartz estimates. The idea is that despite the fact that the wave equation is time reversible, it induces a time-decay in L p norms, of course for p greater than 2. But those decaying properties also provide smoothing effects; it has been the beginning of a long series of works (see for instance [4], [14], [17]-[22]) where those smoothing effects are used to improve, in non linear wave equations, the classical results of wellposedness (by classical results, we mean results proved with energy estimates). The same has been used in the framework of non linear Schrhdinger equations. Let us have a quick look at the way Strichartz estimates are proved in the case of the wave equation. The wave equation can be reduced to the following system:

Otu:e i iID[u+ =

0

in

IR x IRd,

with

[D[a = ~i~-l([~]~(~)).

(1.1)

Anisotropy and dispersion in rotating fluids

172

So the solution is of the form

u(t) -- JC--1 (eitl~l~/+ (~)

e-i,l~l~- (~)~/

ar -

(1.2)

where .7" denotes the Fourier transform on ]R d. Let us suppose that the support of the Fourier transform of the initial data 7 + and 7 - is included in a fixed ring C of IR d. The Strichartz estimate is based on the so-called "dispersive estimate"

II (t)llz (n )

C t-'~

(IIw+IIL,(n

)+ IIW-IIL,(n )).

(1.3)

Then functional analysis arguments (see for instance [11]) imply that

II --IIL=(n d))

IlUllLP( ,Lq( ")) <-

for suitable p and q, which is clearly a decay property. Using scaling arguments and Sobolev embeddings, it is easily proved that, for s strictly greater t h a n (d + 1)/2,

CT(II +IIH
LRFU~ d.ef p(u ~ • ez) is skew-symmetric implies t h a t if the initial data u0 is an element of the space L2(IR3), then we obtain a sequence u ~ of Leray's weak solutions, uniformly bounded in the space L ~ ( I R + ; L2(IR3)) N L2(IR+;/2/1(IR3)). We have noted/2/s the homogeneous Sobolev space of order s, defined as the def closure of ~)(]R 3) for I[" [[/~s, where IluII/~s - II [~c]s~(~)I]L2(IR3), and ~ is the Fourier transform of u. The aim of this study is to analyze that system in the limit when goes to zero. Formally, one expects the solutions to converge to elements of the kernel of the penalization operator. This leads us to the well-known bidimensional Navier-Stokes system

{ Ota+~'. V h ~ - vAhe (NS2D)

-(vhp, o)

div~

-

0

ult=o

=

u0,

in

IR 2

in

IR 2

d.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

173

where Ah -- 012 + 022, Vh = (01, oq2), U' : (Ul,U2) denotes the first two components of u, and g0 is a bidimensional, divergence free vector field. Finally, it will be useful in the following to define the solutions of the free linear equations associated with (RFe)" {

( F R F ~)

0 tWF~ - t A w S +

1

--SLRF WF~ -W~ it=0

f

---

in

IR3

W0~

where w0 is a divergence free, tridimensional vector field. In the spirit of the work of S. Ukai (see [26]) and of two of the authors (see [8]), we shall use dispersive effects in the system (FRF~) to prove strong convergence results. T h e o r e m 1 - The set u E converges to 0 in L2oc(IR+, Lq (IR3)), for any q c

]2,6[. Remark. A similar theorem can be stated for the compressible Navier-Stokes equations for isentropic gases in the low Mach number limit (see [8]). Since the limit of the system ( R F ~) is the bidimensional Navier-Stokes equation ( N S 2 D ) , it is natural to consider initial data of the type u0 = g0 + w0 where ~0 depends only on two variables. T h e o r e m 2 - Let uo be a tridimensional divergence free vector field, such that ~o = ~o + ~o,

with

~o ~ (L~(Ia~)) ~

~nd

~o ~

H ~ ( I a ~)

,

with go and wo divergence free. Then there exists eo .> 0 such that for aft e _< eo, t h e r e / s a unique globM solution u ~ to the system (RF~), which satisfies ~

- ~ ~ G~(~ § H-~ (la~)) c~ L~(~ § H~ Oa~)),

where g and w; are the respectively the solutions of ( N S 2 D ) and ( F R F ~) associated respectively with go and wo. The statement of Theorem 2 shows clearly that anisotropic phenomena are involved in the study of ( R F ~). The solutions of the linearized equation

Otv + LRFv -- usAv = 0 are given in a similar way to (1.2): we have

in

]R 3

(1.4)

Anisotropy and dispersion in rotating fluids

174

and it is proved in this paper that operators of the following type, where ~3 is seen as a parameter, e

:kit

~

((2--Ah)l/2

--r,tr

~(-,~a)

satisfy (anisotropic) Strichartz-type estimates. The structure of the paper is as follows. In Section 2, we prove some basic Strichartz estimates (see (2.6)) for systems of the type (1.4); they will be used to prove Theorem 1. Although these Strichartz estimates are not explicitly anisotropic, their proof requires an anisotropic decomposition of the frequency space, with respect to the vertical frequency variable ~3. That

~a

is due to the fact that the phase ] ~ is almost constant when ~3 is almost equal to 0, or when 1~31 is much larger than I~hl, so in particular is almost stationary. So, there is no hope of getting a dispersive effect in that case. In Section 3, we prove anisotropic Strichartz-type estimates, which take into account the viscosity effect. These estimates are of the type

'

--

t~

for any function whose Fourier transform is supported in the set 2JC A (IR 2 x 2k[a,b]), with a < 89and b > 2, and (j, k) C 7/2 . In Section 4, we prove Theorem 2. The method used is classical in the study of incompressible Navier-Stokes equations, once the dispersive estimates have been obtained.

2. C o n v e r g e n c e

o f L e r a y ' s solutions

The aim of this part is to prove Theorem 1. We first derive Strichartz type estimates for Coriolis force and then apply it to (RF~). 2.1. S t r i c h a r t z t y p e e s t i m a t e s for Coriolis force In order to prove Theorem 1, we need to derive dispersion estimates for wave equations related to the rotating Navier-Stokes model (RFe). The proof follows closely the lines of [11] and [19]. The linearized version of ( R F ~) reads

Otv+!vxea-~Av+Vp-O, s

divv Vlt=0

= =

0, in IRt • IR3, V0 in IR a ,

(2 1)

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

175

which yields in Fourier variables ( E IR 3

Ot~ + ul~[2~ - ~a~ ~1~!x2 ~

--0

in IRt x IR~,

and

.. Vlt=o - v 0.. .

(2.2)

Hence we are led to studying

g~(t) " f H / ~

]'(~)e+it~ -vt~l~12+i='~d~

=/:

f(y)e+it~ -~'tc[~l~+i(=-y)'~ d~dy, ~x~

(2.3)

A

first considering the case when f is supported in C~,R for some r < R, defined by

Cr,R = {~ C jR3/]~3] ~ r and ]~] _< R}.

(2.4)

Let us introduce

K(t, ~, z) dej/IR~ @(~)eita(~'z)+iz'(--T[~]2d[' where a(~, z) def__~-~'~3 and r is a function of :D(IR 3 \ {0}), which of course depends on r and R, which is radial with respect to the horizontal variable ~h = (~1,~2) and whose value is 1 near Cr,R. L e m m a 1 - For any (r, R) such that 0 < r < R, there exists a constant C~,R

such that IlK(t, "i-, .) ][L~(~a) _< C~,R min{1, t- 89}e -~2~.

Proof. The proof follows the lines of a stationary phase method, in a very simple way. First, using the rotation invariance in (~1,~2), we restrict to the case when z2 = 0. Next, denoting a(~) = -O~2a(~ ) = ~2~3/1~13, we introduce the following differential operator: s

def

1

1 + ta2(~)

(1 + i ~ ( ~ ) 0 ~ )

which acts on the ~2 variable, and satisfies ~_,(e ira) -- e ita. Integrating by parts, we obtain

~2

176

Anisotropy and dispersion in rotating fluids

Easy computations then yield rE(r

--

( 1 +ltc~ 2 --i(O~2a) (11)-- t~2) tc~2 + 2

ia

r

12

so using the fact that ~ is in a fixed ring of IR 3, and r C D(IR3), we get []K(t T,.)I]L~ < C r '

-

'

Re-rill2/IR

d~2 1 + t~'

which proves Lemma 1.

!

Remark. The above lemma can be extended to more general functions a. Indeed, in the case when the phase a is non stationary on supp ~b • {z /Ix[ < p}, i.e. V~a # 0, multiple integrations by parts yield arbitrary polynomial time decay, whereas if ~b is supported in a neighborhood of points ~0(z) such that Va(~o(z),z) 0 and D~a(~o(z),z) has j non-zero eigenvalues, then such a stationary phase argument yields a time decay of order t -j/2 For instance, in the case of acoustic waves occuring in the low Mach number limit of compressible flows, we obtain a decay in t ~-1 2 , which is closely related to Strichartz' estimate for the wave equation (see [11]). :

Lemma 1 yields the following corollary in a standard way (see [11]). C o r o l l a r y 1 - For any constants r and in (2.4). Then a constant Cr, R exists such fields such that Supp wo U Supp f c the linear equation ( F R F E) with forcing A

IIw%IIL4( +,L ) __ G,R

R, let C~,R be the domain deigned that if wo and f (t, .) are two vector C~R and if w ~ is the solution of term f and initial data wo, then

(llwollL

+ ]IflIL'(]R+,L2)).

(2.6)

2.2. P r o o f of T h e o r e m 1 We are now able to prove Theorem 1. Let us define the function u~ def

X(~)u

~, where IDI = (_A) 89 and X is a function in T)(IR), whose value def 3 1 is 1 near the origin. By Sobolev embeddings and stating a . . . . , we can write q 2 -

_<

CIl

C

(2.7)

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

177

Now let us estimate u~. We have

Otu~R - uAu~R + 1p(u~ R x e3) = f~, 5with

f~(t) def = - X (~-~) P div (u ~ | u~)(t), which implies, by Duhamel's formula, t h a t

UcR = et(uA+a(D))uO,R nt- ~0 t e (t- t')(uA+a(D)) f~(t') dt', where a(~)v def ~3 = z[~12~,, x v. Now, using the Berstein inequalities, we have, for any r > 0,

Jlet(~a+~(D))uO,RliL~(~:) < C(R2r) 89Ilet(uATa(D))UO,RIIL2(iR3), where D3 = -i03; so with Corollary 1, we have

Ilet(uA+a(D))UO,RIIL4(L~(lRa)) < Ilet(UA+a(D))x --~ UO,RI[L4(L~c(IR3)) + C r t(vA+a(D)) I d - x

--~

UO,R

L~.(L~ (IR3)) <_ CT88(R2~)+ [l~ollL~ + ~+ C~,RII~0[IL~, where to simplify, we have noted L~(nq(IRd)) et('A+~(D))uO,R goes to zero in L~(L~176 Moreover, we have

< -

-

(2.S) LP([O,T];Lq(IRd)). So,

R(~R 2)+11~o11~L2(IR3)"

Then, Corollary 1 yields, as for (2.8),

_< CR(rR 2) 89188Iluoll2L2(~3) + ~1 C~,RTlluoll 2L2. Using (2.8), we get

+ CR(rR2)89188

2

88

RTt[uoll 2

Anisotropy and dispersion in rotating fluids

178

;From this, we deduce t h a t for a fixed R, lim u~ = 0

~-.--.-~0

in

L~o~(IR+'L~(IR3)) '

'

which, with (2.7), finally implies that u~ ~ 0

in

L~o~(IR+;Lq(IR3))

Yq E12,6[,

and this proves Theorem 1.

3. Anisotropic Strichartz estimates In this section, we are going to prove anisotropic, and viscous, Strichartz estimates, which will enable us in the following section to prove Theorem 2. Let us first introduce some notation: for all z E IR 3 we set x = (Xh,X3) with Xh = (Xl ' x2) ' and denote L p,q for 1 ~< p, q < c~, the space defined Xh ~X3 ' by the norm

def Let us also recall some definitions of Littlewood-Paley theory (we refer for instance to [6] for details)" Aj (resp. Sj) denote the Littlewood-Paley spectral localization operators, defined in the following way" Vj E 77,

def

Aj = ~(2-JIDI)

and

Sj-

E

Aj,,

(3.1)

j'~j--1

where p E 7?(IR), is such p(s) = 0 near 0. p(s)-I

if

1

~_< Isl_
VtEIR\{0},

~p(2-Jt)=l.

jEZ In the rest of the paper, No E IN is a fixed constant, depending only on p, such that if IJ - J ' l -> No, then supp p(2 -j-) N supp p(2 - j ' . ) = 0. As we are interested in proving anisotropic estimates, we are also led to introduce an anisotropic Littlewood-Paley decomposition, in the spirit of [16]. To do so, let us define the operators A~ and S~ by

def VJ~E7"], A~--99(2-kD3)

and

S~=

A~,. k~Kk-1

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

179

T h e o r e m 3 - Let ~ be the operator defined in Equation (2.3); for any p E [1, co] and for any c~ > 0,there exists a constant C such that, for any vector field f, we have < C2J(}-})(e22J)4,(~+ a) IIAyfIIL2(1R3),

(3.2)

_< c2J(1-} ) min {1, ( e 2 2 J ) + 2 + (y-k) }

• II/Xj/X~fllL~(~).

(3.3)

Proof. Like all Strichartz-type inequalities, (3.2) and (3.3) are consequences of the following dispersive estimates:

L e m m a 2 - For any function f, we have

IIAjA~O(~,0)SIIL~(~) < Cmin {2k-J,7- 89} 23Je-C~

IISIILI(~), (3.4)

and

v ~,2 _< Cmin {,1 T- 89 (j-k) } 22Je-C~ II/',j/XkG(~, O)f llL~h.~ where

,

~3 2f({) ) . g(r, O)f dej.)F_l {~eiT N-oI~I

First, let us remark that inequality (3.4) is nothing but Lemma 1 with a precise control of the constant. Proof of Lemma 2. We start by the proof of (3.4), and (3.5) will be obtained by very similar methods. We can write AjA~G(~-, O)f(x) = Kj,k(T, O) 9f ( x ) , with

Kj,k(T, O,X)

def j~a

eix'~+iT~T-Ol~le99(2-Jl~])~(2-k~3) d~,

(3.6)

hence all we have to prove is that

IIKj,k(',-, O,.)IIL~(~)_< ~C e_CO22J23j

T2

(3.7)

Anisotropy and dispersion in rotating fluids

180

Indeed, (3.7) implies that C _6022J ]]AjA~(r,O)f[IL~(IRS ) <_ ---we 23Jllfl[L~(ias),

(3.8)

T~

and we also have (3.9) Moreover, Bernstein's inequality yields (3.10) so with (3.9), we have

<_ Ce -C~

23j2k-jll/llL,(~

).

(3.11)

Inequality (3.11) associated with (3.8) implies that (3.4) is proved, except for the fact that we still have to prove (3.7). Using the rotation invariance of Kj,k, we have

(0,C2J)

Then we write

I~j,k(7,rl, y)

\JIR 2

~(2-Jl~l)~(2-k~3) d~2d~3)d~l.

Kj,k(T, O,x) - 23J~[j,k('r, 22Jo, 2Jx),

de_f/13

(o,c)

with

eim~l (/IR eir~'-nl~12~(]~])cP(2J-k~3) d~2d~3) d~l. 2

As in the proof of Lemma 1 in Section 2, we shall integrate by parts in the following integral: (1)(~1 ~3) def f ire3 JIR

2

~(1r d~2.

To do so, let us introduce the operator s defined as

Va, s

def

1 1 + Ta2({)

(a + ia(~)O~2a)

An easy computation yields

with a(~) =

-O~2a(~ ) =

~2~3/1~13

J.-Y. Chemin, B. Desjardins, I. GMlagher and E. Grenier

181

which implies immediately that

But we can compute, for any function ~, as in (2.5),

*~ (~(~)) =

(

1

_i(O~2c~) l - T a 2 )

1+ ~

(1 + ~ ) 2

ia ~(~) - 1 ~: ~ 2 0 ~ ( ~ ) ,

so, using the fact that ( is in a fixed ring of IR 3, t h a t T ~ D(IR), we get, after elementary computations, -1+

2 2e

"

Finally, we have

IIKj k(T, ?7, .)IIL~(IR 3) '

<~ -

-<

ce-c,f
Ce_C,r__} f~

_

d~2d~3

k-j
_

1 dr + r

~

6

~

2 2 3

(3.12)

So the lemma is proved in the case (3.4). The proof of (3.5) is exactly along the same lines, only we keep the variable ~3 as a parameter in the computations above. In other words, let us define

Then we have llAjA~(T,

O)fllL~,23 <

C]]/j,~(7, 0,-, ")IIL~' ~

Ilft]z~,~ a ,

where we have used a Young inequality in the horizontal variables and H61der's inequality in the vertical one, associated with Plancherel's formula. Now, all we need is an estimate of Ij k(7, O,., .) in r . ~ ' ~ which is obtained exactly as in the isotropic case above: we have

with

Ij k (7", ?'], Yh, ~3) def

];.

Anisotropy and dispersion in rotating ftuids

182

An identical computation as for (3.12) yields

< Ce-C~ ~(2J-k43)

IIb,~ (~, ,, Yh, 4a) IIL~'r

-


< C~-C'~- 89162162

%r2

L~

2

~3

~

1

< Ce-Cn2(3-k)T-~. So finally we have obtained that

IIAjA~G(~,O)fll L x ~,~ < h ,x 3 --

ce-CO2~22J2(J--k)r- 89

(3.13)

Now to conclude to (3.5), we write similar estimates as (3.10),(3.11)" we have, by Sobolev embeddings,

IIAj ANG(~, O)YllL~,y~

C2 y e-CO2~J IIAj A~,f IIL~.(IR~) c2=J ~ - c o ~ iizxj A~,/i Inr which with (3.13) implies that

IIAjAVk{~(7", O)f[IL~.2 <_Ce -C~

2 2j min {--1~ 2(J-k), 1 } llfllL~

Xh ,m3

7" ~

,m3

The lemma is proved.

II End of the woof of Theorem 3. To prove the result, we are going to use duality arguments, based on the isotropic estimate (3.2). In the following, we shall call B d__ef{~I/ E ~)(IR3), ][~llL~'(LX(Ia~)) < 1}. (3.14) The following set of equalities is standard:

IIAjG~flIL~(L~(fi3)) = ~e~sup/fit+ < AjG~(t)f, ~(t) > dt = ,I, et~sup (27r) - 3 / ~ + xfi s qp(2-Jlc~l)e-i}~-'tl~12]'(c~)~(t,~) dtd~ = sup (27r) -3 E

~B

k<_j

ffi

+x

fis fj,k(~)~j,k(t, ~)e -i-}~-~'tl~12 dtd~,

where we have defined fj,k = Aj Avkf and ~j,k = Aj A~ tlI. So it follows that

II~jG~flIL~.(L~(~)) <-- sup (271")-3 E Ilfj,kllL~(Ia~) (3.15) +

L2(fi

3)

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

183

But we have e

-L

utl~!ff~j,k(t,~)dt ~ IiL2(IR ~ 3)

lel

II/~ + ~

~ e-i'gt ~3-vtl(]2~lj,k( , ~)eirs~3_ ,~, ~1 ~I2 ~j,k (s, ~) dtdsd~

+)2xlR s

+)2

2 k-j, !t -~89 si 89} 23j e-~(t+s)22Jdtds,

xmin

(3.16)

where we have used (3.4). So IIAy~Sf[I 2L I ( L ~ (IR3)) < c ~ -

23Yllfj,kll L2(IR 2 3)

-

}<_j

x ~(IR

min

2t~-j

+)~

s~

e --u(t +s)22J dtds.

'l t -- sl 89

(3.17)

O n t h e one h a n d , t h e u p p e r b o u n d (3.17) can be e s t i m a t e d d i r e c t l y in t h e

following way:

On the other hand, we can write

23J/IR +•

1 ~------e -u(t+s)22j dtds = ,~f~x~t It~-ls~j 89e-'(t' +s') dt~ds p,

I t - sl}

(3.19)

which finally implies

ll/~

e ~~~'~,

~

F~j,k(t,~) dt II~

+

_< C2k-J2 -j min {1, (e22j)12 (j-k) }.

L2(IRa)

Using the fact that Ilmjm~fll~(n~) = tlAJfll~-(n~), k we

have -~-

{

IIAyG.flI51(L~(~)) <_ C2-~IIAjIIIL~(~) ~ 2 ~ min 1, (a22J) 882 s 2 k<j

}

o

Anisotropy and dispersion in rotating fluids

184

Finally we find, for any r > 1, E

2

2

(E22J)4AT,

(1

k<_j which means in other words that for any a > 0, choosing r = 1 + c~,

II/XjG~flIL~(Lo~(~)) <_C~2-~llAjflli~(~)(Z22J)'('i+'~).

(3.20)

Now, to get the L~ estimate, we just have to interpolate (3.20) with

IIAjG~flIL~(L~.(~)) <_CIIAYflIL~(~),

(3.21)

which, for all p E [1, oc] and for all a > 0, and with 1/q = 1 / 2 - 1/2p yields 9

1

IIAjG2flIL~(L~(~)) ~ C~2-~ (E22j) 4p(l-t-a)ii Ajfllz~.(~). Then Bernstein's inequality implies the result in the isotropic case. We are finally left with the anisotropic case. To prove estimate (3.3), the computation is identical to the one leading to estimate (3.2): we just have to change the set B defined in (3.14) into B' def { ~ C T)(IR 3) II~IIL~ '

~2

< 1}

( L = ~ ,.~3) - -

9

Then estimate (3.16) becomes, with (3.5),

II/

,

e ~ +

~j,k(t,~)dt

II

L 2 ( ] R 3)

<- J(~+ )2 IJl.~j,k(._.q,

.)][

Lx h1.2'x3]]l~j,k(t , ")1] Lit h2,x3

~ 892(J-k) } • min { 1, It - s] 89 22j e -v(t+s)22j dtds. So finally, using (3.18) and (3.19) we have

IIAjA~flILl(L~.23) <_C m i n {1, (a22J) 88 ~ } 2-J]IAjA~fl]L2(~3). The end of the proof is then exactly that of the isotropic case, and we obtain estimate (3.3). So Theorem 3 is proved, m

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

4. G l o b a l e x i s t e n c e a n d c o n v e r g e n c e

of smooth

185

solutions

Let us recall that we have noted w re for the solution of the linear equation

Otw;

uAw;+lLRr

~l) e

=

s We Flt--O

0

W0.

Classical results on the heat equation imply that vt e IR +,

2 89 ) IIw;(t)ll~z + 2 . j~0 t I1~ ~ (~)11~

Moreover, it is obvious that lim (Id - S N~)SNwo v = Wo N--~+c~ N3 --~-cx)

in

/2I 1(IRa).

Consequently, for any r / > 0, there exist two integers N and N3 such t h a t if w~m e d e f (Id v = - S N3)SNW~, then, for all t C IR +

]]W; (t) -- W e (t) lt 2 1 j~0t II~ ~ (~) - w ~ (~)[I 2 F.~ /:/~ (IRa) + 2~,

ds < rl. (4.1)

Now let us define w e d e f u E _ 12 - - w e where ~ is the solution of the bidimensional Navier-Stokes equation. Then the function w e satisfies

I

1 Otw e _ u A w e + - L R r w e s

W~t=0

=

Q ( ~ , ~ + 2~ + 2~;m )

=

+Q(w;.m, ~ . ~ ) + 2 Q(~, w~.~ ) ( I d - S N ( I d - S N~))~0, v

(4.2)

where Q(a, b) - Ae,m(D)amb ~, Ae,m(D) is a Fourier multiplier of order 1. Classical results on the incompressible bidimensional Navier-Stokes equations (see [2al for i n s t a n c e ) i m p l y t h a t vt e n~ + ,

lt~(t)ll~ 2 ( ~ ) + 2 . ~0 t I I W ( ~ ) t I :L2(IR 2) d~ = II~0II~ (IR2).

(4.3)

It follows from (4.1) that Theorem 2 is proved, if we show that (4.2) is well posed, for ~ small enough, in the space C~(IR +,/:/ 89(IR3))NL2 (IR +,/2/~ (IR3)), and also that wZ~O

in

L~(~.~+,I4 89

(4.4)

Anisotropy and dispersion in rotating fluids

186

Let us now apply the operators Aj defined earlier (see (3.1)), to equation (4.2), and write an energy estimate in L2(IRa). In order to estimate the terms in (4.2) containing the bidimensional function ~ , which is less smooth than w e, we use Lemma 5.1 of [9]: it states that the product of any tridimensional function by ~ is half a derivative smoother than if ~ were tridimensional; the setting in [9] is t h a t of the torus, but the results extend unchanged to the whole space. Let us recall the rules of product in Sobolev spaces, which will be used extensively throughout the proofs:

d

Vs, t < z='s+t Ys, t <

1, s + t

> o,

~llabll/:/-+'.-~ (IR~) ....

> O,

Ilabllz~.,+,-,(~) _

<_ C.,~llallg~(~.)llbllm(~.),

C,,tllallH,(~)llbllH,(~).

We shall also use the classical fact (see [6]) that

Vd > 1, IlvllI4s(~ta) ~ 112J~ll/XjVllL~(~.)lle~(z),

IlvzxjvllL~(w) _> ~2JllZXjvllc~(~.). Now let us apply operator Ay to equation (4.2) satisfied by w e. We then take the scalar product in L2(IR a) with Ajw e, and obtain ld

2

2 dt "' llZajwe~t)ll2 a ~ L2(IR 3) -}- Ct~22J[[Ajwe(t)]]L~(lR3) <_C c~(t)2z~lfF~(t)Jiii_ 89 + C c}(t)2JI]FE(t)]]I:I_I(IR3)[]Ajwe(t)[IL2(IR3) +Cc~(t) 2z ]]we(t)l]Hl(~ a) (]]we(t)[[I~l(~t3)+l]ft(t)l[[Z 89 (~{2)) ]]AJwe(t)]]L2(IR3), (4.5) where we have written F e def 2 Q ( w e w e ?

) -t- Q ( w ;

F rn.

e 7n ~ W

),

and

/ff,e def 2 Q ( U w e

F'm

z

~

F rn

) "

Here like throughout the paper, c~(t) is a sequence in g2(7]), of norm 1, and C denotes any "universal" constant. To simplify the notation of the space-time norms, we shall write

L~r([tS(IRd)) =LP([O,T]; HS(IRd))

and

LP(H~(IRd))

=LP(]R+;

[-[S(]Rd)).

Observe that estimate (4.5) has exactly the same form as estimate (B.8) of [10] (proof of Lemma B. 1), with the following correspondance of notation: a -

~1,

q = j,

A -

w e,

IB(t)l~+{

- II~(t)llu{(~),

and

W-

F e.

J.-Y. Chemin, B. Desjardins, L Gallagher and E. Grenier

187

The only difference with estimate (B.8) is the presence of an extra forcing term, F ~, but it is an easy adaptation of the computations of [10] to see that as in L e m m a B.1 of [10], we have

X~(t)e -C04(t) <

CIIF~IIL~(H- 89 +c(x~)~(t),

(4.6)

and

X~(t) + XL(t)

<

CltF~IIL~(H_ 89

+ CIIF~llL4(H-I(U~))

+CX[(t)(X~(t) + U4(t)),

(4.7)

where-are have noted

X~(t) XL(t)-

=

X ~ ( t ) - IlwC[IL~(~r~(]Ra)),

IlwxllL4(Hl(~a)),

Jrw~llLT(/~ 89

and

Un(t)- II~IIL~(H89

So to conclude, all we need is an estimate on F E in L2(/:/- } (IR3)) and on fie in L4(/2/-1(IR3)). These estimates are given in the following proposition, the proof of which is postponed for a while. Proposition

1 - Under the assumptions of Theorem 2, we have

[IF~IIL~(H_+<~:)) _< f / l ( ~ ) x [ ( t ) -t- r/2(s),

and

IIP~IIL4(H_,(~)) _< na(~),

with lim ~1~ : O, for i E {1 2 3}. ~---+0

~

Let us now conclude the proof of the theorem. Estimate (4.6), along with Proposition 1, implies t h a t

X~ (t)(e-C04(t) with rl(s) def = 712(s)+ •3(s). in (4.3), implies that

vt > 0 __

,

--

71

(6)) <_ C(X~)2(t)n< ~](~),

But the energy equality in (NS2D) recalled

02(t) < c.-211~ot1~ __

(IR2),

so let us choose s0 > 0 such t h a t ~ s ~ ~0~

_

1

(

(~2)

) 9

Anisotropy and dispersion in rotating fluids

188

We get, for any ~ < ~0 x~(t)

<_

c~c~-=""~

~, ((x~)~(t)

+ ,~(~)) .

So we have, for any ~ < E0,

X 4 (t) < Ce C~-=ll~~ Finally, from inequalities (4.6) and (4.7), we obtain lim w e -- 0

in the space

L~(/:/ 89(IR3)) N L2(/:/~ (IR3)).

e--+0

The global well-posedness of (4.2) for ~ small enough follows by classical arguments; this concludes the proof of the theorem, m

Proof of Proposition 1. Let us recall that

F e - 2 Q ( w e,weF,,~) +Q(w~m,w~.e ) and

F e-2Q(~,w

Ew).

It is here that we are going to use the Strichartz estimates proved in Section 3, namely estimates (3.2) and (3.3); this is the reason why we got rid of the very low and very high frequencies of the linear solution above" the function w Frrt, e has frequencies such that 2Na <

I~l ~ 2N

and

2 N~

< l~3l,

so estimates (3.2) and (3.3) simply become respectively

I]/'XjW;.. IIL~(L~(IRa)) <_ C(g22N)l/(4Pa)]lAjWO,mlIL2(IRa),

Vc~ > 1,

(4.8)

and

II~;.,[Ig~ ~ 9 (L:=h,

) _<

C(c22N) 1/4pIJw0,mllL=(~),

(4.9)

where we have omitted the dependence of the constants on N and N3, which are fixed once and for all. Let us start by estimating Q(w~m , w~,r~ + w e) in (IRZ)). We are going to use the paraproduct algorithm introduced by J.-M. Bony in [5]" we have, with the notation introduced in (3.1), and with the usual summation convention, Q(c, b) - Ae,m (D)(T~u b m + Tb,,,.c e ), (4.10)

L~(H-89

with

T'b - E Sj+2c Ajb J

and

Tbc -- E ~j-lb Ajc. J

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

189

We can write, for any c~ > 1 and according to (4.8),

2-~IIAjAe ,m(D) T'o-i~~ b m l l L ~ ( L ~ ( ~ ) ) _
y~ IlSj'W;~IIL~(L~(~))IIAj'blIL~(L=(~))2 ~

j ' >_j- No

[IAj.wollL~(~)2 ~ IIAj'bilL~(L~(~))

E

c~ N3 <j"

j'>j--No <_min{j'--l,N} jtt_

<

Ce[IWO'mllL2(IR3)

2~2

E 9t

N

~ I]AJ'blIL4(L~(IR3))

.

3 >3-No

Na<_j"<_min{j'-l,N}

<_ C~IIWo,mll5~(~O) ~ 2-N/22 j' ilAj,bliL~rr162 j ' >_j-No

~',

where C~ denotes a constant multiplied by any positive power of ~2 2N. So we have obtained, using Young's inequality,

[IAem(D)T'w: z,, (w ~ + w~m)'~[[ 2L2T (/2/, __

<JJ om

As to the term

(

89(IR3))

+ ilwell

L 2

) .(4.11)

Tb,,~w~'~, let us define Fm

We have

Cj <_ C

E

2~]ISj,blIL$(L2(~3))IIAj,w~mlIL?r(L~(~t3))

]j-j'l~No Na <_j'<_N

< C~

~

2%llAj,,wollL~(~)llSj, b[lz4(L2(~3))

Ij-j'l<_No Na <_j'<_N ~_

Ce]IWO,ml]L2(IRa)]]b]IL4(I~I(IR3)).

So with (4.11), we obtain finally

IIQ(<~ ,' w~ +w~Fm )il L~r ~ ( psob--l (lRa)) < __

C~]]wo,m[l~2(~3)(IIwo,m[[~ 2 ( ~ 3 ) + IIw~l[2L,~ (tdi I (IR3)) )

9

(4.12)

190

Anisotropy and dispersion in rotating fluids

Now, to end the proof, we need to estimate the term Q(w grn,' ~ g)" it is with this estimate in mind that we proved an anisotropic Strichartz estimate in Section 3. We have simply IIAe,m(D)wFm

-'

)

_< Cll e,~ = ~m ]lL~,(L2 (iRa)) <_

C II oIIL .< =)IIWo,milL:< =),

according to (4.9). This estimate, along with (4.12), proves Proposition 1.

References [1] A. Babin, A. Mahalov, and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. of Mechanics, 15 (1996), 291-300. [2] A. Babin, A. Mahalov, and B. Nicolaenko, Resonances and regularity for Boussinesq equations, Russ. J. of Math. Physics, 4 (1996), 417-428. [3] A. Babin, A. Mahalov, B. Nicolaenko, and Y. Zhou, On the asymptotic regimes and the strongly stratified limit of rotating Boussinesq equations, Journal of Theoretical and Comp. Fluid Dynamics, 9 (1997), 223-251. [4] H. Bahouri and J.-Y. Chemin, Equations d'ondes quasilin~aires et estimation de Strichartz, Pr@ublication du Laboratoire d'Analyse Num~rique, to appear in American Journal of Mathematics, 121 (1999). [5] J.-M. Bony, Calcul symbolique et propagation des singularit@ pour les ~quations aux d~riv@s partielles non lin~aires, Annales de l'Ecole Normale Sup~rieure, 14 (1981), 209-246. [6] J.-Y. Chemin, Fluides parfaits incompressibles, Ast~risque, 230 (1995). [7] J.-Y. Chemin, A propos d'un probl~me de p~nalisation de type antisym~trique, Journal Math. Pures Appl., 76 (1997), 739-755. [8] B. Desjardins and E. Grenier, Low Mach number limit of compressible flows in the whole space, Proc. Royal Society of London, 455 (1999), 2271-2279. [9] I. Gallagher, The Tridimensional Navier-Stokes equations with almost bidimensional data: stability, uniqueness and life span, International Math. Research Notices, 18 (1997), 919-935. [10] I. Gallagher, Applications of Schochet's methods to parabolic equations, Journal Math. Pures Appl., 77 (1998), 989-1054. [11] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, Journal of Functional Anal., 133 (1995), 50-68. [121 H.P. Greenspan, The theory of rotating fluids, Cambridge monographs on mechanics and applied mathematics (1969).

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

[13]

[14] [15] [16]

[17] [ls] [19] [20] [21]

[22] [23]

[24] [25] [26]

191

E. Grenier, Oscillatory perturbations of the Navier Stokes equations, Journal Math. Pures Appl., 76 (1997), 477-498. L. HSrmander, Lectures on nonlinear hyperbolic differential equations, Math. and Applications, 26, Springer (1996). D. Iftimie, La r~solution des ~quations de Navier-Stokes dans des domaines minces et la limite quasig@ostrophique, Th~se de l'Universit@ Paris 6 (1997). D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces, Revista Matematica Ibero-Americana, 15 (1999), 1-36. F. John, Existence for large times of strict solutions of non linear wave equations in three space dimensions, Comm. Pure and Appl. Math., 40 (1987), 79--109. L. Kapitanski, Some generalization of the Strichartz-Brenner inequality, Leningrad Math. Journal, 1 (1990), 693-721. M. Keel and T. Tao, Endpoint Strichartz estimates, American Journal of Mathematics, 120 (1998), 955-980. S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure and Appl. Math., 33 (1980), 43-101. S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation Comm. Pure and Appl. Math., 38 (1985), 321-332. S. Klainerman and M. Machedon, Remark on Strichartz type inequalites (with an appendix of J. Bourgain and D. Tataru), International Math. Research Notices, 5 (1996), 201-220. P.-L. Lions, Mathematical topics in fluid dynamics, Vol. 1, Incompressible models, Oxford University Press (1998). J. Pedlosky, Geophysical fluid dynamics, Springer (1979). S. Schochet, Fast singular limits of hyperbolic PDEs, J. Diff. Equ. 114 (1994), 476-512. S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ. 26, no. 2 (1986), 323-331. Jean-Yves Chemin Centre de Math~matiques Ecole Polytechnique 91128 -Palaiseau France E-mail: chemin@math, polytechnique, fr Benoit Desjardins DMA- ENS, 45 rue d'Ulm 75230 Paris Cedex 05 and

192

Anisotropy and dispersion in rotating fluids CEA/DIF/DCSA B.P. 12 91680 Bruy~res le Ch~tel France E-mail: desj [email protected] Isabelle Gallagher D~partement de Math~matiques Universit~ de Paris-Sud 91405 Orsay Cedex and Centre de Math~matiques ]~cole Polytechnique 91128 Palaiseau Cedex, France Emmanuel Grenier UMPA (CNRS UMR 5669 E.N.S. Lyon 46 all~e d'Italie 69364 Lyon Cedex 07 France E-mail: [email protected], fr

Studies in Mathematics a n d its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 10

INTEGRAL EQUATIONS A N D SADDLE POINT FORMULATION FOR SCATTERING PROBLEMS

F. COLLINO AND B. DESPRES

1. Introduction In this work we deal with integral equations for solving obstacle scattering by time-harmonic electromagnetic waves. We show that it is possible to design a system of integral equations with a positive spectrum. This kind of system has been introduced in [8], [9], in the 2D acoustic case and in [7] for Maxwell's equations. The system of equations may be viewed as coming from the minimization of a positive quadratic functional. This approach seems to be new. It can be either derived from some more or less simple manipulations of classical integral operators. One interesting feature of the new system of equations, comparing to the classical theory, is that the functional framework is L 2 based. It comes directly from the construction of the quadratic functional. A natural penalisation procedure may be used to get more coercivity on the multiplier, even if the inf-sup condition is already true for the weak formulation. The strong coercivity properties of this system allow to get a global and coherent framework for the introduction of numerical iterative algorithms for the numerical computation of the discrete solutions, as it is the case for many coercive problems : Jacobi algorithms, conjugate gradient algorithms, and so on, may be introduced with comprehensive proof of convergence [10]. The outline of the paper is the following. After some definitions we derive the system of equations by minimization of a natural quadratic functional under constraint. It explains the presence of a Lagrange multiplier in the unknowns. After that, we discuss well posedness of the system, introduce penalisation, and propose various algorithms to solve the system. Finally some properties of the integral operators are proved, based on the analysis of Calderon projectors.

194

Integral equations and saddle point formulations for scattering...

2. S o m e d e f i n i t i o n s In this work the x and y notations may denote either the space variable needed to define the integral operators x E R 3 and y E R 3, or the pair unknown-multiplier (x,y) c X x Y where X and Y or the appropriate functional spaces. Natural interpretation of these notations is enough, and no confusion is possible. Let D - c R 3 a bounded domain with regular boundary F, n(x) the unit normal vector to F directed into D + c R 3, the exterior domain of D - . The problem we address is the determination of the out-coming electromagnetic field scattered by F. Let k > 0 be the wavenumber and Z0 > 0 the impedance of the vacuum. The unknown field is a solution to (1), (2) where

V A E + - i k Z o H + = 0,

in D + (1)

V A H + + i k Z o l E + = O, in D + is the Maxwell's system and lim Ixl Ixl-~

x _E+) Z o g + A 7--; IXl

-0,

(2)

is the Sommerfeld radiation condition at infinity. System (1-2) must be supplemented by a boundary condition on F. Let us assume for a while that the boundary condition is an absorbing boundary condition

n(x) A (E~r (x) A n(x)) + Zo(H?r(x ) A n(x)) = G in.

(3)

Other boundary conditions will been introduced in section . Following [5] We define the classical integral operators T a n d / ~ defined from F to D +

TJ(x)

=

1

(

~ V~A V~A

jfrG(x,y)J(y)dr(y) ) (4)

RJ(x)

= -V~ A frG(x, y)J(y) dr(y),

or, in another form,

TJ(x) [(J(x)

-

k J f r ( G(~, y)J(y)+ VV~G(~, y)V'. J(y) ) dr(y) 1

= frV~G(x, y) A J(y) dr(y).

(5)

F. Collino and B. Despres

195

Here V t- J(y) denotes the surface divergence of J. The kernel G(x, y) is the radiating Green function for the Helmholtz equation

G(x, y) = expiklx-YJ 4 ~ l x - Yi"

(6)

Passing to the limit x --+ F, x E D - we define the classical surface to surface integral operators

TJ(x) = lim n(x) A (iPJ(y)) A n(x)) y----->x

(7)

[

ln ]

Now we define the 2 x 2 integral operator S by

s =

K-

lnA

T

(s)

'

To go further we introduce the decomposition in real and imaginary part of the Green function G(x, y)

C(x, y) = c~

- Yl) + isin(kl x - Yl)

4~1x - yE

(9)

4~j~ - yf

We obtain a similar decomposition for S in real and imaginary parts S= T+iR,

(10)

with T

--

i

Kr - ~nA

Tr

'

R

=

Ifi

Ti

.

(II)

From a mathematical point of view Ti, Tr, K~, Ki are symmetric, n(x)A is antisymmetric and Gi(x, y) is a regular kernel. So R appears as symmetric and regularizing while T* the adjoint of T satisfies [ T-

T* = I I =

0--n(x)A] --n(x)A 0

(12) "

Next we derive the following decomposition of R. If d is a direction of the unit sphere S 2 and (J(x),M(x)) are two fields on F we define the far field operator

a~J(a) = ~

a A (J(x) A a ) e - i k •

(13)

Integral equations and saddle point formulations for scattering...

196

We combine this into a new far field operator

M(x)

M

It may be proved t h a t

(a) - a ~ J ( a ) - ia A a ~ M ( a ) . (14)

R = (Am) * A ~

(15)

3. D e r i v a t i o n of the s y s t e m The idea is to consider a set of solutions of the Maxwell's system

VAE-ikZoH-O,

in D + (16)

V A H + i k Z o l E = O, in D +, with the following expansion at infinity _~

e iklxl _ o ~ ( ~ ) + e -iklxl ~.

Ixl ~

E(x)

eiklxl ZoH(~)

Ixl a~(~), Ixl

x

~,

(~ - VI

e_elxl

_out

Ixl (~A "~ (~))

-~

~

~ (Ixl ~A

) (17)

~n a~(~)), I~l-~ ~ ,

where u~-~ ~ , H) and a~in(E,H) are some tangential fields on the unit sphere, and the convergence e --~ e ~ holds in the sense lim 1 /R le(x) -- e~(x)l 2 dx = 0. R-,c~ -R <_lxI<_2n

(18)

In other words, we consider all the in and out-coming electromagnetic fields on D +. In general, we are only interested in out-coming fields but nevertheless we begin by embedding the problem in this larger set of solutions. For technical reasons, we will assume that all these fields possess tangential traces on F of square integrable modulus. Indeed, we will prove further that such electromagnetic fields satisfying in addition to lim R1 / B R--,~

IEI2 + Z~

< oc,

(19)

R

where BR -- {IxJ < R , x E D+}, have the asymptotic behavior (17) with aOUt, a imn in L 2 (TS2). Let

{ G~n - G~n(E, H) - n(x) A (E/r A n(x)) + Z o Y / r A n(x) (20) "JrP~ _ u r~-~

H) - - n ( x ) A (E/r A n(x)) + Z o g / r A n(x)

F. Collino and B. Despres

and G in s o m e functional

I

197

given tangential field of L2(TF). We consider the following

I(E,H)-

1 in(E , H)II~ + ~1 IIG~Ut(E,H))I z2 ~[[Gr

out (E, g)[[~ + I[a c xin +[la~ ~ (E, H)[]~

m

(21)

~e (C~n(E, H) ~ G : E E n )

where (., .) and [[.][ denotes respectively the inner product and norm in the following spaces Definition

1

-

Let L2(TF) be the space of tangent fields

L2(TF) = {r e L2(F) a, ~.n = 0)} c L2(1-')3.

We define the functional spaces X = L2(TF) x L2(TF)

and (with same definition for the tangent space) Z = L 2 (TS2). This functional has the following feature. T h e o r e m 1 - The minimum of I(E, H) in the space of all out-coming and in-coming electro-magnetic fields E, H, with tangential traces on F of square integrable modulus and growth behavior (19), is reached by the solution of the following problem

V A E + - ikZoH + = 0 V A H + + ikZolE + =0 n(x) A (E?r(x) A n(x)) + Zo(H?r(x ) A n(x)) = G in ai~ (E + , g +) = 0

in D +, in D +, on F, at infinity.

(22)

This result means that it is possible to relax both the condition at infinity and the boundary condition, and to recover them through the minimization process. The condition at infinity is treated exactly like the boundary condition on F. The key point of the proof rests on an isometry lemma, which is exactly equivalent to the unitarity of the scattering matrix in scattering theory [11]. L e m m a 1 ( I s o m e t r y L e m m a ) - Let (E, H) some electromagnetic field satisfying (17) with tangential traces on F of square integrable modulus. Then, the following equality holds

-

1 in(E,H)[[2 +][ in (E,H)[[z2 ~[[Gr acx~ 1 ~ ~-, + l r " ~ IIz-

(23)

198

Integral equations and saddle point formulations for scattering...

To prove this lemma, we introduce Br = {Ixl _< ~,~ e D + } for some large r. The outward normal is denoted as u. So u = - n ( x ) on the interior boundary of Br that coincides with F. We begin with ~ (E/r(X)A u ( x ) ) + Zo(H/r(X) A u(x))l 2 -

B,.

fo =

Br

I-u(x) A (E/r(X)A u ( x ) ) + Zo(H/r(X) A u(x))[ 2

4~f

(24)

~(x) A (E/r(X) A u(x)). H/r(X) A u(x).

J OB,.

Let us recall the integration by part formula

s

(V A U . V - V A V . U) - ~o (u A (U/oo A u)) . (V/oe A u)da,

(25)

o

for all bounded regular domain O. Equations (16) imply that

0 = s (V A E -

ikZoH).-H-

(V A H

+ ikZolE)

9E,

r

that is

~o (u A (E A u)) .-~ A ~ : ik L Br

r

(Z~

So ~e jf

m (u A (E A u)). (H A u)

Br

+ Z~

-

0,

what implies

o

fo and /R zR 1

I - (. A (E A .)) + Zo(H A .)12 I+ (u A (E A u)) + Zo(H A

g,r.

(/o

B~

s R 1 (/0 "R

=

~,)12

-(uA(EAu))+Zo(HAu)[

2) d r -

[ + ( u A ( E A u ) ) + Z 0 ( H A u ' ) ' 2) dr. B.,-

We then let R goes to infinity. With the help of Hypothesis (17), we get

F. Collino and B. Despres

1/R~R~

lim

R---, cx~ - R

Br

I - (~ i (E

A

199

.)) + Zo(H A v)12dr = (26)

= / r ]Grn(x)]2dF(x) + 4~2 ]ain(d))]2dcr(d)" and, in a same way,

1//Rfo

lim

R---, oc -'R

B ,~

I+ (-

A

(E A .)) + Zo(H A . ) 1 2 d r

=

(27)

= / r [G~'t(x)12dF(x) + 4~2 la~ It proves (23). Once the isometry Lemma has been obtained, the minimization of I(E, becomes obvious. Indeed, definition (21) and equation (23) gives

I(E,H)

H)

=

1 ~n(E,H ) ll~c + 2 Ila~ in(E,H)l]2z - ~}~e(G~n(E,H), G in) z ~llGr

_-

1 G i n II2 1 ~(E,H) - G ~ II~: +211ao~m(E,H)I]~ - ~[[ ~llGr x.

(2s) It is then clear that the minimum is reached exactly for the electromagnetic field such that both the condition at infinity and the boundary condition are satisfied a ~ n (E, H) - - G ~ . (29) c oi on (E, H) - O, Now, all the remaining difficulty is to choose an appropriate -and useful for practical computations- parameterization of our set of in and out-coming electromagnetic fields to derive the expressions of the related quantities G~~, G~ut , a~in and u~-~ For doing that, we consider (/),_/:/)_the extension by zero of (E, H) in D - . It is classical to show that (E, H) satisfies in the sense of distributions of R 3

k2E § A E - -ikZo (JSr + ~---~VV. (JSr)) § V z,, (MSr) Zo MSr + V./~=~--'V-(JSr),

VV.

(MSr))

V-H=+:ikZ0

V.

- VA (JSr)

(MSr),

(30)

200

Integral equations and saddle point formulations for scattering...

with J - n A H/r,

M = - n A E/r.

(31)

The general solution of system (30) is

{

$(x)

=

iZoT~J(x) + Bi~M(x) + EH~r(x)

=

-KJ(x)

(32) + iZolT~M(x) + Hg~(x),

where we choose T~, /(r identical to T, /( given in (4) and (5) except that G ( x , y ) is replaced by ~ e G ( x , y ) while ( E H ~ r ( x ) , H H ~ r ( x ) ) i s some free electromagnetic field k2E Her + ~ E Her = O,

k2H Her + A H Her - - 0

(33) ~ . E Her = 0 ,

~ . H Her = 0 .

Note that the choice of the kernel ~eG(x, y) for the particular solution of system (30) is completely arbitrary at this stage, but it will be convenient for our purpose. Now, we use Theorem 6.30 of [5] which asserts that every entire solution of the Maxwell's system with the growth property (19) is an Herglotz pair. It means that there exists some tangential field ~ in L2(S 2) such that E H~(x) -

47r

2 ?(d)eikJXda(d)

(34)

ik

H H~'(x) - 4 7 r v / ~~

fs 2(id A (dl)d aXd (d).

Once more time, the normalization constant ~knV~~is here for convenience. Thus, a possible parameterization of in and out-coming fields might be (z, 7) with [ J ]1 J

M [ /V/~~ . _1

M J(x) (x)

(35)

Reciprocally, every (x, h') is associated to an electromagnetic field through (32)-(34)-(35). But it remains to verify that the fields vanish in D - , the open complement of D + (if not, equality (31) would not hold anymore). For that, it is enough to ensure that both the tangential interior traces on F are zero. Using once more time the classical jump conditions for the potentials, we get o -

A

(x) A

1 = iZo(TrJ)(x) + ( K r M ) ( x ) - -~n(x) A M ( x ) + i eHer(x),

(36)

F. Collino and B. Despres

o

=

~(x) A (HT~(x) A ~(x))

201

-

1 -(K~J)(x) + iZo](T~M)(x) + -~n(x) A g(x) + ihHe ~(x),

(37)

with

( k igT~o/s "Y(d)dkd'd~r(d)A n(z)) hHe~(x)=n(x)A ( 47r kix/-~oJfs2(idA~(d))eikdXda(d)An(x)). Let us normalize (36) by/x~~0 - ] and (37) by - ~ . (14) gives us immediately after transposition

(38)

A look at (11) and

(39)

T x + i(A~176 = 0.

Equation (39) defines a closed linear sub-manifold A4 of X • Z in which lives the pair (x, 7)D e f i n i t i o n 2 - Let A4 be the closed linear sub-manifold of X x Z defined

by A 4 - { ( x , 7) c X x Z ,

Wx+i(A~)*ff-0}cXxZ.

To every (x, "7) is associated an electromagnetic field through (32)-(34)-(35), vanishing in D - and with tangential traces satisfying to (31). Once this parameterization has been constructed, it only remains to rewrite the functional I(E, H). At first, we have

1

in

1 G~ut Z0

1

_~ Zo

= -~- (liJlll~ + IIM~II~) = ~-IIx]l~,

(GF(E,H),

ain)x -- Zo ~F(J1

- in A M1)- i v~in /~

= Zo (x, g)x,

(40) (41)

where g is defined in (79). To get the expressions of the far fields, we let x goes to infinity. For the

202

Integral equations and saddle point formulations for scattering...

potential, the calculations are classical (see [5]) page 157). From, eikix' ( v~ A O(~,y)a(y) = ~ 7 ] +~k~ A a(y)~ - ~

+ 0

(ia[z)) ~

,

Vx A (V~ A G(x, y)a(y)) =

(42)

~ A (a(y) A ~ ) ~ - ~ + 0 {\ lal~ I~i ]

4~lxl

'

as Ix[ goes to infinity, uniformly for all y in F, we get

=

E ( x ) - Egc~(x)

-

9~ e~iklxl

~

r 2 Ixl 1 ~ ( j 1 , M1; _ ~ ) e -ikixl (~) +~n lxl ' + 0 , 1 (-i~: A A ~ ( J i~o = -~

( H ( x ) - HU~"(x))

1(+~ A h~(a

~)) e-ikixl

M,

Ixl

1,

M1 ; X:))

eikixl

(43)

Ixl

(1~[2) + o

The asymptotic behavior of the Herglotz wave is obtained thanks to the stationary phase theorem for the sphere. If a E C I ( S 2) we have J(s a(d)e~k&da(d ) --, 27r ( a(2) ~eikl~l - a ( - 2 ) e -ikl~l) ~ kl~l kixl '

(44)

as Ix[ goes to infinity. It is also true in our context for a E L2(S 2) with convergence in the sense (18). We deduce E Her (x)

vqz0 Hgc~(x)

1

e ikixl

-~ ~ 7(~)

1

kl~l

2

e -ikixi

r(-~)

e/klxl ix/77~176 ---+ 5l(_i~A "Y(~)) kJx[

kl~l '

1 2 (i2

Ix] -~ e-iklz[

(45)

A ",/(-~))

Finally, gathering the expressions, we recover the asymptotic behavior (17) with a~

(AC~(J1,M1;~)) + ~,(~:))

H) -

' a ~in(E, H) - - - ~

2

(46) ( A ~ ( J 1 , M I ",- ~ ) ) - 7 ( - ~ ) ) ,

203

F. Collino and B. Despres

and, in particular, Z0 Z0 A ~ (J1, Mx)I]~ . + IIa~(E,H)I[2z = --~-I[TIl~ + -~-II

liar" ~

(47)

After all those tedious calculations, we get the expression for the functional I ( E , H ) = ZoJ(x,~y) = Zo

~[[xl[ ~ + ~1171[~ + ~I[

xll ~ - ~ e ( x g)x

9

(48) Let us adopt the definition D e f i n i t i o n 3 - For all (x,'y) C X x Z we define 1

1

1

J(x, 7) = -~tlx[[2x + ~l[Tll~ + ~l[A~xll~ - ~ e (x, g ) x -

T h e o r e m 2 - Let X = (L2(TF)) 2. The m i n i m u m of the functional .l(x, 7) given in (~8), over all the pairs (x, ~/) CA/I

I x + i(A~)*~/= 0,

(49)

is reached at

x:

[

__ iv/~o- n A E ~ r

],

7:

H+),

ace / v / ~0'

(50)

where (E +, H +) is the solution of the Maxwell system (22).

D e f i n i t i o n 4 - We define the Hilbertian space Y = {y e X, W*y e X } ,

(51)

equipped with the following norm,

Ily]IY--Iiyl[x + llW*y[Ix .

(52)

If y t is the dual space of Y, the constraint (49) can be viewed as an equality in yr Vy e Y, (x, T * y ) x + i(~, ( A ~ ) * y ) z = 0. (53) At this point it is classical to dualize the constraint, introducing the Lagrangian D e f i n i t i o n 5 - For all (x, ~/, y) C X x Z x Y we define s

7, Y) = .l(x, ~) - ~e ((x, W*y)x - i(% ( A ~ ) y ) z ) .

(54)

204

Integral equations and saddle point formulations for scattering...

It is well known (cf. [2]) t h a t if the Lagrangian admits a saddle point, then its first a r g u m e n t is the m i n i m u m argument of aT in j~4 s

~/, y) =

inf sup s (x*,~*)~x• y*~w

y*) =

min Y(x*, 7"), (x*,-~*)~A4

(55)

Furthermore, as J is quadratic, such a saddle point exists if and only if D x s (x, ~/, y) = O, D.~f__.(x,~/, y) = O, Dys ~,, y) = O, i.e. x + (AOr * A ~ x -

T * y -- g

+ i A~y - 0 -Tx-

(56)

i ( A ~ ) * ~ = 0.

Discarding -~, it gives x+(A~)

* A~x-T*y=g

(57) -Tx-

( A ~ ) * A ~ y -- 0.

This is the basic integral system we discuss in this work. The interesting and new feature of this system is t h a t it is is clearly derived from the minimization of a natural quadratic functional under constraint, and t h a t the functional is clearly related to the isometry L e m m a (equivalent to the unitarity of the scattering matrix). The additional unknown is interpreted as the Lagrange multiplier of our constrained minimization problem. It is possible to re-interpret the multiplier and prove t h a t y = ix (up to a term in K e r T * actually equal to K e r A 0r see Appendix ). To our knowledge such a system has not been discovered beforehand, even if m a n y researchers have stressed the interest of these kind of structure for practical computations. Nevertheless it is in some sense natural, since based on the isometry result which is equivalent to the unitarity of the scattering matrix, what has been well known for a very long time, [11]. To every unitary operator U U * = I, is associated at least formally a hermitian operator H -- II+- ii Uv with inverse formula U - II~-iH So in some sense this -ill" mixed integral system corresponds to a "scattering" like integral theory.

4. V a r i a t i o n a l

formulation

and well-posedness.

The penalized

systems Variational formulation are useful for minimization problems. It provides a good framework for the study of uniqueness and existence, and also for discretization and convergence of the discrete solution. The natural variational

F. Collino and B. Despres

205

formulation of our problem is as follows

XCVx, ~cv~, (x, 2) + ( A ~ x , A ~ s (x, T*~) + ( A ~ y , A ~ )

(T'y, s = 0,

- (g, ~),

v :~ e vx,

v~ ~v~.

(58)

It remains to define the functional spaces Vx and Vy. Let us remark that both smooth analytic operators A ~ y , A ~ are not involved in the coerciveness of the system. Due to the L 2 coerciveness of the formulation it is clear that the only choice for Vx is

v~ = X - L2(TF) • L2(TF). It may seem at first sight impossible to define a compatible space Vy such that the inf-sup condition of Babuska-Brezzi holds, [2], max

-----~T*y) (x,]lx[it

_>

kllyllw~,

k > 0,

w i t h Wy =

v~

KerT*

,

(59)

just because standard functional spaces in which T is continuous are known to be based on H -1 (div, F) and H- 89(curl, F). Nevertheless it is at least possible to provide an abstract framework in which the inf-sup condition holds. Let us take Vx = X, Vy = Y, defined in (51). Definition 6 -

We define the quotient Hilbertian space W

___

Y KerT*

equipped with the norm inf

yo E K e r T *

JlY- YoJIY --IIT*yJIx +

inf

yo E KerT*

IIY- yolJx-

System (57) is well-posed as soon as the inf-sup condition sup ~x

(x, T * y ) x

> Cllyl]w

]l~ljx

(60)

-

holds for some strictly positive constant C > 0. This inequality can be derived as follows. Picking x - T * y in (60), we find sup

x~x

(x, T * y ) x

II~llx

> IIW*yllx.

-

(61)

Integral equations and saddle point formulations for scattering...

206

Let y E Y. In appendix it is proved t h a t - I I T a projector (it is a Calderon Projector)"

with I I defined in (12), is

TIIT = -T.

(62)

Let yl = - Y I * T * y . We have t h a t T * ( y - Yl) = T * y + T*YI*T*y = 0. So yo = y - yl C K e r T * hence inf

Yo E K e r T *

IlY- Yollx < IlYllIX = IIII*T*yl]x = IIT*ylIx,

(63)

where we have used the isometric property YI*II - I. So we have sup ( x , T * y ) x Ilxllx

xeX

> 1 T* 1 inf IlY-Yollx = - ~11 YlIx -[- ~ yoEKerW*

1

IlYllw.

(64)

Thus it gives Lemma

C=!

2 -

The inf-sup condition (60) in spaces = X x W is true with

2"

Since the natural continuity T* 9Y -+ X holds and one has the bound (98) I I A ~ A ~ y l I x <_ C ' l l y l l w for some C' > 0, it proves following [2] T h e o r e m 3 - The variational system (58) is well posed, that is, for every pair (g, gy) C ( V'x, Vy) - (Vx, Vy) there exists a unique (x, y) c Vx x W such that (x, 2) + ( A ~ x , A ~ ) - ( T ' y , ~) = (g, ~), (x, W*~) + ( A ~ y , A ~ ) = w ' (gy, Y ) w ,

V2 e Vx, V~ e Vy.

(65)

However when discretization is considered a difficulty arises with the use of (X, Y) = (Vx, V~) spaces. The reason is t h a t we want to avoid the construction of some discrete space compatible with the L 2 based space (X, Y). We would like to take a classical integral code based on the duality H- 89(div, F) and H- 89(curl, F), rearrange all the routines, and use the iterative algorithms described further in order to solve our new discrete integral system. Then the question of the convergence of the discrete solution to the exact solution arises. All our efforts in order to prove the convergence using this strategy failed. The reason seems to be t h a t the classical discretization of integral operator is based on H- 89(div, r) and H - 1 ( c u r l , r ) , and not on (Z, Y), [4]. Moreover some numerical results in 2D for the Helmholtz equation, [1], show t h a t this problem may be a real one. It seems that there are cases where the discrete solution obtained through the strategy described above does not converge to the exact solution, even in some very simple and regular cases. Of course this conclusion has to be re-evaluated if the

F. Collino and B. Despres

207

discretization of the integral operators are compatible with (X, Y). Note also that the kernel KerT* is a large space of infinite dimension. Some algorithms are very sensitive to the dimension of this kernel. It is our purpose now to modify the system and to present what we will call the penalized problem, with much stronger coercivity. Let/3 some positive penalisation parameter (for instance/3 = 1). Remembering that y = ix, we modify system (15) to obtain the penalized system (1 +/3)x + (Ar162* A ~ x -

T * y + i~y = g (66)

+Tx-

i3x + fly + (A r162 A ~ y = 0.

which appears to be a system of the form X

A simple calculation shows that X

X

[ y 1, [ y ])xxx -Ilxll~ + 91Ix + :> m i n ( ~ ,

iyli2x

2

+ tIA~yl]~ + [tA~x][z

(68)

1

and the system appears now as coercive in the x variable and also in the y variable. Another possibility is to modify system (15) according to (1 - 3)x + ( A ~ ) * A C ~ x - (T*y + if~y) = g (69) (Tx-

i/3x) + 3 y + (Ar162* ACCy = O,

where/3 is now some positive number less than 1 (let/3 = 89 The interest of (69) is that it corresponds to a saddle point for the Lagrangian 1 1

1

1

+xllTIl~ + xl]AC~xI[~ - Re ((x, T * y + ifly)x - i(v, ( A ~ ) y ) z ) . L

The problem appears as a penalized saddle point problem.

(70)

208

Integral equations and saddle point formulations for scattering...

5. System for general impedance boundary conditions We turn now to the case of a general boundary condition. We assume that the electro-magnetic field is determined by some impedance boundary condition of the type

n(x) A (E~r(x) A n(x)) + ZoZ,.(H?r(X ) A n(x)) = F,

(71)

where Z~ is some impedance operator that we assume symmetric with a positive real part, i.e.

(NeZrJ, J) > 0,

VJ C D(Z~).

(72)

We associate to it its reflection coefficient operator

R = ( I d - Zr)(Id + Zr) -1,

(73)

which, thanks to (72), satisfies

ilRII_< 1 r IIRJ[I~(Tr) <--IIJIl~(Tr).

(74)

We begin by rewriting the boundary condition with the only help of R. We have G in = - R G ~ + (Id + R)F, (75) where G in -- Tt(x) A (E?F(x) A Tt(x)) -~- Zo(g?F(x ) A Tt(x)) while

G TM - n(x) A (E?r (x) A n(x)) - Zo(H~r(x ) A n(x)).

(76)

Note that this notation is compatible with (3) with Zr = 1. Setting

Fo -

fand using

1 iv/7~o( I d + R ) F,

(77)

[ - n ( -~Fo ] x ) A Fo '

(78)

9 1 g-

- ~(x) A ~

in G~(x)

] '

(79)

for g with (75) for G in in system (57), we have x+(A~) -Tx-

*A~x-T*y-f-NRx ( A ~ ) * A ~ y - 0,

(80)

209

F. Collino and B. Despres

where dl (x)

NRx

NR

-iR - n ( x ) A R Go~t(z)

Ml(x)

or

- i R (~ A M1 - i:1) NR

x :

1

] .

- n A R (n A M1 - idl)

(81)

(82)

Let fl some positive parameter (for instance fl = 1), remembering that y = ix, we finally modify the system to obtain (l+fl)x+(A~)

*A~x-T*y+NRx+ifly=f

(83) Tx6. C a l d e r o n

iflx + fly + ( A ~ ) * ACCy = 0.

projectors

Our aim in this section is to make the link between the operators we have defined with the Calderon projectors, [4] page 87, [6] page 93. Let (all,/~1) two fields given on F, (not necessarily corresponding to the traces of some exterior electro-magnetic fields). We can associate to them the two fields in ~+

E (x) ,/,/~o - x / 7 ~ 0 H + (x)

--

Tffl (x) -J-/(M1 (x)

=

/ ( d l (x) + TM1 (X),

(84)

where T d , / ( M are given in (5). If x goes to some points on F, we have via the jump conditions 1

~(x) A ( i,/~o ET~(~) A ~(~))

q-TYl( x ) + / ( J ~ l

:

(85) (x) -~- l n ( x ) A J~l(X),

n(x) A ( - - , / ~ o H ? r ( x ) A n ( x ) ) :

Ii

TJ~/1 (x) + K J l (x) +

(86)

1 ~(~) A ]~ (x)

Now, we can proceed exactly as in the first section: from the exterior traces, we construct the fields (all, M1) and x by x --

M1

:

"/~0-1M

'

210

Integral equations and saddle point formulations for scattering...

a n d we have Sx-

S

(ss)

-0.

M1

B u t , it is easy to see t h a t (86)- (85) reads (see definition (12) and (11))

-II

[ 3"1

ffl

M u l t i p l y i n g by S I I and using b o t h - I I 2 = I d a n d (88) we get

0 = S

= (sIis

M1

--l- S)

J~l

1"

,90

In o t h e r words, -HS

= (-IIS) 2 ,

a p p e a r s as a projector" it is one of t h e C a l d e r o n projector. c o m p o s i t i o n in real and i m a g i n a r y p a r t S - T + i R , we get TIIT-

RIIR

(91) W i t h our de-

- -T

(92) TIIR + RIIT - -R. U p to now, w h a t we have o b t a i n e d c o r r e s p o n d s to the Green function (6), b u t we can o p e r a t e exactly in t h e s a m e way with

G(x, g) = exp

-ikl~-yl

4~-[x - y[ "

(93)

t h e only modification being t h e r a d i a t i o n condition at infinity. T h e Ix-yl 1 s i n g u l a r i t y of t h e kernel r e m a i n s , so we get the same relation (91) also for S - T - iR. It e x p a n d s into TIIT + RIIR-

-T (94)

TYIR + R I I T = - R . B y c o m p a r i s o n we get = -T

(95)

RHR

- 0.

(96)

RIIT*

- R.

(97)

TIlT and It r e m a i n s to prove t h a t

F. Collino and B. Despres

211

This is a simple consequence of the fact that free fields are in the kernel of "exterior" integral operators [5]. To see this let us consider the free field (that is a continuous field in the whole space, with continuous traces on F) R y for some arbitrary smooth y. Let (Y, M) = I I R y . By integration by parts we get 0 = iZ0(TY)(2) + (/(M)(2) 0

=

+

for 2 in D +. Passing to the limit ~ -~ F we get that (T + II + i R ) I I R y = 0.

It means that T I I R - R : 0 which implies by transposition the desired relation (97). A consequence of the boundedness of R I I is the continuity bound IIRyIIx _< IIRIIIIc(x)llT*Yllx _< C'llyllw.. (98) Finally note that the identity (95) also proves by a density argument that I I T y C Y, with the notations of Definition (4). In other words - K I T is a well-defined projector in Y. References

[1] N. Bartoli, F. Collino, and B. Despr~s, Integral Equations via Saddle Point Problems for Acoustic Problems. in preparation. [2] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Number 15 in Springer Series in Computational Mathematics. SpringerVerlag, 1991. [3] F. Collino and B. Despres, Integral equations via saddle point problems for time-harmonic maxwell's equations, in preparation. [4] M. Cessenat, Mathematical Methods in Electromagnetism, Number 41 in Series on advances in Mathematics for Applied Sciences. WorldScientific, 1996. [5] D. Colton and R. Kress, Inverse Acoustic and electromagnetic scattering theory, Applied Mathematical Sciences 93. Springer-Verlag, 1992. [6] G. Chen and J. Zhou, Boundary Element Methods, Academic Press, 1992. [7] B. Despres, Fonctionnelle quadratique et 6quations int6grales pour les ~quations de maxwell en domain ext6rieur, Comptes Rendus de l'Acad6mie des Sciences, Paris, S6rie I, 323(1996), 547-552. [8] B. Despres, Fonctionnelle quadratique et 6quations integrales pour les probl~mes d'onde harmonique en domaine ext~rieur, M2AN, 31(1997), 679-732.

212

Integral equations and saddle point formulations for scattering...

[9] B. Despres, Quadratic functional and wave equations, In Mathematical and agation, 56-66, 1998. [10] P. Lascaux and R. Theodor, Analyse l'art de l'ing~nieur, Masson, 1987. [11] M. Reed and B. Simon, Scattering York, 1979.

Francis Collino CERFACS 42 avenue Coriolis 31057 Toulouse France E-mail:~inria.fr Bruno Despres Laboratoire d'Analyse Num~rique Universit~ Paris V I - CNRS 175 rue du Chevaleret 75013 Paris France E-mail: [email protected] ussieu, fr

integral equations for harmonic numerical aspects of wave propnum~rique matricielle appliqu~e Theory, Academic-Press, New

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 11 EXISTENCE A N D UNIQUENESS OF A STRONG SOLUTION FOR N O N H O M O G E N E O U S M I C R O P O L A R FLUIDS

C. CONCA, R. GORMAZ, E. ORTEGA AND M. ROJAS

1. Introduction In this paper, we study the equations of a nonhomogeneous viscous incompressible micropolar fluid. These equations are studied in a bounded domain ~t c ]R 3, with boundary F, in a time interval [0, T]. Let u(x, t) E IR 3, w(x, t) C ]R 3, p(x, t) E lit and p(x, t) c IR, denote respectively, the velocity, the angular velocity of internal rotation, the density and the pressure at a point x E ~ and at time t E [0, T]. The governing equations for this fluid are 0U

p - ~ + p(u. V ) u -

(# + # , ) A u + V p -

2#~rot w + pf,

div u = 0,

Ow

p-'~

+ p ( u " ~7)W -- (C a -nt- Cd)Z~W -- (C 0 "Jr-Cd -- Ca)~ 7 div w

(1.1)

+4p~w = 2prrot u + pg,

Op 0~ + ( ~ V)p = 0, in QT :--f~ • (0, T), with the following boundary and initial conditions

~(~, t) = 0, w ( ~ , t) = 0 o~ r x (0, T), ~(~, 0) = ~o(~), ~(x, 0) = ~o(~) i~ ~, p(x, o) = po(x) in ft.

(1.2)

Here f ( x , t) and g(x, t) are densities of the external forces field and the external torque field respectively. The positive constants #, #r, co, Ca, Cd characterize isotropic properties of the fluid; # is the usual Newtonian dynamic viscosity; #r represent the dynamic microrotation viscosity; co, ca

214

Existence and uniqueness of a strong solution ...

and Cd are called coefficients of angular viscosities. These new viscosities are related to the asymmetry of the stress tensor and in consequence related to the appearance of the field of internal rotation w. Furthermore, these positive constants satisfy co + Cd > ca. The expressions V, A, div and rot denote, the gradient, Laplacian, divergence and rotational operators respectively. We denote Ou/Ot by ut and the i th component of (u. V)v in cartesian coordinates is given by 3

3

Ov~

V)vl - E J-5 zj; j--1

vp =

j~l

Op

Oxj

"--

For the derivation and physical discussion of equations (1.1)-(1.2), see D. Condiff & J. Dahler (1964), L. Petrosyan (1984) and G. Lukaszewicz (1999). We observe that this model of fluids includes as a particular case the classical Navier-Stokes equations, which has been thoroughly studied (see for instance the classical books of O. Ladyzhenskaya (1969), J.L. Lions (1969), R. Temam (1979) and the references there in). It also includes the reduced model of the nonhomogeneous Navier-Stokes equations, which has been less studied than the previous case (see for instance S. Antoncev et al. (1990), J. Simon (1990), J. Kim (1987), O. Ladyzhenskaya & V. Solonnikov (1975), R. Salvi (1991), and J.L. Boldrini & M. Rojas (1992), (1997)). Concerning the model of a micropolar fluid that we consider in this paper, G. Lukaszewicz (1990) established the existence of local weak solutions for (1.1)-(1.2), using linearization and an almost fixed point theorem. In this paper Lukaszewicz remarked the possibility of proving the existence of a strong solution, under the hypothesis that the initial density is separated from zero, by the techniques used in G. Lukaszewicz (1988), (1989), that is, linearization and fixed point theorems; notice that Lukaszewicz assume constant density. The first result on the existence and uniqueness of a strong solution (local and global) for problem (1.1)-(1.2) was proved in J.L. Boldrini & M. Rojas (1998) using the spectral semi-Galerkin method and compactness arguments. The convergence rate of this method was established in J.L. Boldrini &= M. Rojas (1996). In this work, we use a different approach in order to obtain the existence and uniqueness of a strong solution. We propose an iterative process, which solves at each iteration a linear problem. For each linear problem it is easy to prove the existence and uniqueness of a strong solution. We obtain a priori estimates for the sequence generated by the iterative process. In the next step we show that this sequence of solutions of linear problems is a Cauchy sequence in an appropriate Banach space, and consequently, converges. From this convergence, the strong solution for the full original

C. Conca, R. Gormaz, E. Ortega and M. Rojas

215

nonlinear problem is obtained. As by-product we obtain bounds for the convergence rate of the method. We think that the techniques developed here could be adapted to a numerical scheme that solves these equations. We remark that, from the technical point of view, the hyperbolic character of the transport equation in (1.1) and the extra nonlinearities of the problem make the technical arguments more elaborated than those used in the case of constant density (compare with E. Ortega & M. Rojas (1997)). This paper is organized as follows: In Section 2, we describe the iterative method. In Section 3, we derive some important a priori estimates. In Section 4, we prove the Cauchy character of the sequence of solutions of the linearized problems and we give an error bound between an element of the sequence and the unique solution of the problem. We give also an error estimate for the pressure. Finally, a word about the notation adopted in this work. The constants appearing in the various estimates depending exclusively on ft, are generically denoted by Ca, except for the smallest eigenvalue of the Laplace operator B = - A in ft with a Dirichlet boundary condition, which will be denoted by A.

2. P r e l i m i n a r i e s Let ft be a bounded domain in IR 3 with a smooth boundary 0ft and T > 0 a positive real number. Denote by D either ft or f~ x (0, T). For n E IN and 1 <_ q < oc, we consider the usual Sobolev space

wm'q(D) = { f e Lq(D) l O~ f c Lq(D), lat < m }, endowed with the usual wm,q-norm. If q = 2 we denote Hm(D) Wm'2(D) and H ~ ( D ) " = the closure of 7P(D) in Hm(D). We put 12(ft) -- {v E :D(ft)3]div v = 0

in

a}

H=

closureof

12(Ft) in

L 2 ( a ) 3,

V=

closureof

P(ft)

H l ( f t ) 3.

in

"=

Since f~ has a smooth boundary, it is well known t h a t V = {v c H01(Ft)3 ! divv = 0 in f~}. We recall the Helmholtz decomposition of vector fields L2(ft) = H | G, where G = {r r = Vp, p c H 1(ft)}. Throughout this paper P will denote the orthogonal projection from L2(ft) onto H. The Stokes operator A : D(A) ~ H ......, H is given by

Existence and uniqueness of a strong solution ...

216

A - - P A with domain D ( A ) - V cl H2(f~). It is well known that A is a positive definite, self-adjoint operator and it is characterized by the relation

(Aw, v) - (Vw, Vv),

Vw c D(A),

v e V.

If f~ is of class C 1'1, the norms IlUl]H2 and IIAu]l are equivalent in D ( A ) (see C. Amrouche &: V. Girault (1991)). Similarly, we introduce the Laplacian operator B - - A with domain D ( B ) = H l ( f t ) N g 2 ( f t ) . By means of P , we can reformulate problem (1.1)-(1.2) as follows: Find u, w, p (in spaces which will be defined later on), satisfying

P(put) + (# + #~)Au + P ( p u . Vu) - 2#~P(rot w ) + P ( p f ) ,

(2.1)

p w t + (Ca ~- c d ) B w + p u . V w -- (Co -~- Cd -- Ca)~7div w -~- 4#rw

= 2 # t r o t u + pg,

(2.2)

Pt + u - V p = 0,

(2.3)

u(x, O) - uo(x), w(x, O) -- wo(x), p(x, O) = po(x) in ft.

(2.4)

We propose the following iterative process for approximating the solution of problem (2.1)-(2.4). For n - 1, we define U 1 (t) = tO,

W 1 (t) -- WO, and fll (X, t) -- po(X).

Assuming t h a t u '~, w ~ and p~ have already been computed, we define u ~+1, w n+l and p~+l as the unique solution of the following system of linear equations:

p ( p n u t + l ) + (# + # r ) A u n+l + p ( p n u n . Vu n+l) = 2 # r P ( r o t w n) + p ( p n f ) , pnwt+l

(2.5)

+ (Ca -~- C d ) B W n + l + p n u n . ~ W n + l

-(Co + Cd -- ca)V div w ~+1 + 4#rw n+l = 2#~rot u ~ + pn g, (2.6) (2.7)

p t + l _~_ u n + l . V p n + l __ 0, ~tn+l(x,0)

-- U0(X),

w n + I ( x , O) -- Wo(X),

pn+l(x, 0) -- po(x) in ft.

(2.8)

For simplicity, from now on we consider t o ( x ) - 0 and wo(x) - O. In this case, it is clear t h a t the first iterate is (u 1, w 1, pl) _ (0, 0, p0). In this way, we have reduced a nonlinear coupled system into a sequence of linear, weakly coupled systems. Our goal is to prove t h a t the resulting solutions (u n, w ~ , p ~) define a C a t c h y sequence in a suitable Banach space, which converges to a strong solution of our problem. More precisely, in this paper we prove:

C. Conca, R. Gormaz, E. Ortega and M. Rojas

217

T h e o r e m 2.1 - Assuming that the hypotheses (3.1), (3.12) and (3.28) hold (see Section 3 below), then there exists a unique solution (u, w, p) of problem (2.1)-(2.~). Moreover, the sequences u n, w n and pn satisfy un W n

pn

>u

strongly in

L~(0, T; V ) n L2(O,T; H2(f~) n V),

w

strongly in

L ~ ( O , T ; H I ( ~ ) ) n L 2 ( O , T ; H 2 ( ~ ) n Hl(f~)) and

>p

strongly in

L~(O,T;L~(s

>

3. A priori estimates Throughout this paper the external fields f and g are assumed to be L2(QT) functions, small enough with respect to the viscosities coefficients of the model (a precise formulation of this hypothesis will be given later on; see Section 3.2) p, #~, Ca and Cd. Concerning the initial density p0, we assume that it is a continuously differentiable function (p0 E C1), and that there exist a,/3 such that 0
VxE~.

(3.1 i

The Galerkin method used, resolves at each step a linear system. As a result, J.L. Boldrini & M. Rojas (1998) proved that the solutions (u n, w n, pn) enjoy the following conditions concerning their regularity: u ~ E L ~ (0, T; V) cl L 2 (0, T; D(A)), u? E L 2 (0, T; H), Aun E L2(O,T;L2(t2)), w ~ E L~(O,T;HI(ft))c3L2(O,T;D(B)), w~ E L2(0, T; L2(~t)), B e n E L2(0, T; L2(~-t)). We are going to prove on the one hand that these sequences are furthermore uniformly bounded in the corresponding spaces. On the other hand, applying the method of characteristics to the continuity equation (2.7), it follows immediately that whenever p~ exists, it satisfies 0 < a <_ p~ _
(3.2)

Furthermore, the hypothesis on p0 make possible to apply the results of O. Ladyzhenskaya &: V. Solonnikov (1978) (see Lemma 1.3, p. 705). In our case, uniform bounds for Vp ~ and p~ in L~(0, T; L~(f~)) are also obtained.

218

E x i s t e n c e a n d uniqueness of a s t r o n g solution . . .

3.1. T h e L 2 e s t i m a t e s of U n and W n Multiplying the equation (2.7) by v, and taking scalar product of the resulting identity also by v we get (p~v, v) - - ( d i v ( p n u n ) v , V) = 2(pnu ~" Vv, v) and consequently, for all v E H01(gt) such that vt E L2(~t), ld n 2 dt ]l V / ~ v "112-- "2 (Pt V, V) Jr-(pnvt, V) --- (pn~tn " V V , V ) + ( p n v t , V).

Let us keep this identity in mind. Multiplying (2.5) by u n+l and (2.6) by w n+ 1 we obtain ld

2 dt

IIx/~u'+1112 + (~ + P~)llVun+lll2 - 2#~ (rot w n, it n+l ) + ( p ~ f , un+l),

(3.3)

ld IIv~w'+Xll 2 § (Ca + cd)llVw'+lll 2 + (co § ~d - c.)lldiv wn+l[12 2 dt

+ 4 ~ l l w ' + ' II2 - 2#~(rot un, w "+1) + ( p n g , w n + l ) .

(3.4)

It is well known that for all u E H01(gt) the following classical inequalities hold [[rot u[[ < [[Vu[[, [[U[]L4"~ V/2 []U[[1/n[[~Tull 3/4 [[It[]2 ~__ /~--l[[Vit[[2 -(3.5) where, recall that A denotes the smallest eigenvalue of the Laplace operator. Using these inequalities, classical estimates for the right hand side of (3.3) and (3.4) lead to the following differential inequalities

d

dt ][v~un+X[[2 + (#

29~ -1

+ #r)IIVltn+ll[2 < 8#---~r2][Wn[]2 ~- ~ l I f [ ] 2 -

# + #~

# + #~

'

d

d-7 IIv~w'+lll~ + (c a -j- cd)llVw~+Xlli + 2(co + Cd - ca)lldiv w"+~ll l <

4#~

--

Ca Jr- Cd

/32

[[u.i]2+

~r

i[g[i2 "

Therefore, adding up both inequalities and integrating from 0 to t, we have c~([lun+l(t)ll 2 + ]]wn+l(t)][ 2) q- (# -+- Pr) +(C a -~- Cd)

/o I[Vwn+l(7-)]]2dT

< 8,s fot

--

[[Vu"+l(r)ll2dr +

-~- 2(C 0 -~- Cd -- Ca)

4,s Cd /o ~

Ca +

/o

[Idiv

wn+l(w)[]2dT

C. Conca, R. Gormaz, E. Ortega and M. Rojas

232 )~-1

219

~2 Ilfll~(Q~) + ~llgll~(Q~)

(recall that u0 = w0 = 0). This means that we can choose constants M and C such that

Ilun+l (t) [i2 + ilwn+X(t)l[ 2 + P +a

.L t IIVu~+I(T)II2d~

#~

Ca Jr-Cd i t

Oz < c

Jo

(ll~n(-,-)ll 2 + [lwn(-,-)ll2)d-, - + M.

(3.6)

Effectively, the following choice provides the desired inequality C

=

M

--

max{a(#+pr),a(ca

+Cd) }'

2/T'x-~

(3.7)

Z2

L.2

.

(3.8)

Thus, setting

(~n (t)

-

-

Ilu n (t)112 + Ilw ~ (t)[L 2,

(3.6) reduces in particular to r Observing t h a t that for n > 2,

~_ M + C ~0t ~n(T)dT.

~ l ( t ) = 0, a straightforward induction argument shows n--2 ( C t ) k

~(t) <_ME

k!

< M e ct,

(3.9)

k=O and hence for all n we have sup (llu~(t)[[ 2 + Ilwn(t)[[ 2 )

tE[0,T]

<

sup M

e Ct

--

M e CT.

te[0,T]

(3.10)

Combining (3.6) and (3.9), we get the following result: Lemma

3.1-

The sequences u n and w n satisfy

[[u~[[2L2(O,T;y) < --

a M e CT # -Jr- P r

a M e CT and []wn][2L2(O,T;Hlo (f~)) < -- ~Ca -Jr- Cd ,

(3.11)

220

Existence and uniqueness of a strong solution ...

i.e., u n remains bounded in L 2 ( O , T ; V ) and w n in L2(O,T;H~(f~)), uniformly as n ; oo. 3.2.

The

L~

estimates

of u ~ and

w~

These bounds are harder to obtain. Notice that inequality (3.9) already gives L ~ ( O , T ; L 2 ( f t ) ) estimates. For bounds in L ~ c ( O , T ; H I ( f ~ ) ) i t is necessary to introduce two parameters, 5 and r/, and prove that there exist at least one choice of these parameters for which a certain key inequality (see (3.20)) holds with positive left hand side terms. On the other hand, we will need that the following technical condition

eCT) /~1/4

O~

Ol5

(3.12)

-< 160C~ min { ~--~, ~-ff}

holds, where C is defined by (3.7) and Ca is introduced in (3.17). Notice that this hypothesis is fulfilled if f and g are small enough with respect to the viscosities # and #~. Multiplying (2.5) firstly by 5Au ~+1 and secondly by ut +1, an integration in ft yields

5(~ + , ~ ) l l A ~ + l f l : __ __5 ( p ~ t +~ , A~ ~+1) + 2,~ 5 (rot ~ n Au n+l) (3.13)

+5 (pnf, Aun+l) - 5 ( p n u n . V u n + l , A u '~+1)

and

2 dtllV~+lll2

- 2#r(rotw

~

, Ut~ + 1 )

+(p~/, ~?+1) _ ( p ~ . v ~ n + l , ~?+,),

(3.14)

respectively. Thus, using the fact that c~ _< p'~ _< ~, we have

o~llu?+lll 2 -~ ~ +2 ~ dt d IlVu"+ 1 II 2 + 5(p + ,ur)llAu~+lll2 15 (pnut+ ', Aun+ l ) l + 12~ ~ (rot wn, Aun+ : )I

+ 12,r(rotwn, ut+l)l + ](p~f, ut+l)l + 15 (p~f, Aun+l)l + 15(P'~un.Vun+I,Au'~+:)]

+ ](p~u'.Vu~+',u?+l)l.

(3.15)

Using the classical estimates (3.5), HSlder and Young's inequalities, the sixth term in the right hand side of (3.15) can be estimated as follows 15(pnu n- v u n + l , A u n + l ) l ~ 5 ~llunllL~llVu~+lllL~llAu~+llI

C. Conca, R. Gormaz, E. Ortega and M. Rojas

221

<_5 /3 v~ Ilunll~/411Vu'~ll3/411Vun+lllL, lJAun+l[I <_5 ~ V~-I/811Vu'~]]IIVun+IlIL~I]Au~+IlI. (3.16) Since the embedding H2(gt) ~-~ Wl'4(~) is continuous, for all have

u C D(A)

[]Vttl]L4 ~ [[UI[W1,4~ Cgt[Itt]]H2 ~ Ca][Aul].

we

(3.17)

Thus, from (3.16) and (3.17), we obtain

Is(pntt n 9vltn+l, Attn+l )] ~ ~ /~ x/~ ),-1/scallVunllilAun+l

jl2.

(3.18)

Similarly, we can estimate the seventh term in the right hand side of (3.15) as follows

i(pnun.Vun+l,u'~+l)l

_<

<

llun llz~ llVu"+ l llL411U':+l ll 5 ~ V~ ~-l/SCallVu"llllAu"+lll2 _t_8 V/2 "~-1/8Ca tt~+l 2 IlWnllll

45

II.

(3.19)

Therefore, using classical estimates for the remaining terms of the right hand side of (3.15), and by virtue of Young's inequality we obtain, (~ +

d vun+l 2 ( ~)~11 II + 2 c~- 3 r / -

+2~((~+.~) 3~4~ _<

+ r/,

~v/2"~-1/8C~ 45

IlWnll

/3 x/g,x-1/SCallVu~ll IlVwnlI2+

2'7+~

)

Ilut+ll[2

) IlAu~+lll2

II/112

where 7/ > 0 is any positive parameter. Integrating from 0 to t this last inequality, we obtain

(# + #r)llVu'~+l(t)ll 2 +2

a-

+26

3r/-

(# + #~) <

-

< --

IIvun(~-)][

[J

(T)IIZdT

9 v/-8a-'/sca[IWn(~-)!

ilAun+l(~-)lledT-

(8~,7 2~)~ ~IlVwn(~-)l[2d'r+(2rl + ~/~ )11/112L=tQT) "82 + rl (4r/

~+

~) 2#~c~MecT Ca -Jr-Cd

+

(

/32)

2rl+ ~

Ilfll L2(QT). 2 (3.20)

Existence and uniqueness of a strong solution ...

222

Here, we have to choose carefully the values of the two parameters r/and 5 in order to guarantee positiveness of the left-hand-side terms. Choosing

~(~ + ~ ) 6=

q = -~; we

4/32

have

(~ -4- ~)llVu~+x(t) II2 .4.2 f0t(4

+

/~3a(# V/2)k+ 1/8CFt #,.) [[~7urt(T) ],) "u~ I1~

Ol(# + ].t,`) jot (].Z .+ #r

2 ~2

4

[3

()11

v/S)k-I/8cft[[Vun(T)[[)],Aun+l(7")l[2dT

~< 2(C~2 .4. 4132)~ 2 ( Ca#~-4- Cd ) MeeT

1

+ -~

(a2 + 432)1[f[[2L 2 (Q T )

<-5/32a ( [[f[[2L2(QT)+"91'2L2(QT))(1"4" Ca+Ca#,`eCT(1~ .4. -~4))

(3.21)

where we used in this latter expression the definition of M (see (3.8)) and other standard estimates. Let e be any positive number. Choosing carefully the size of the data f, g and the viscosities # and #,`, it is possible to obtain

(~ + #~)l[Vu~+l(t)l] 2 +

~ot( O~ 133V~/~- 1/8C~ +

2/32

) n+l(w)[12d~l(w)ll2dw

4

< E2

(3.22)

Setting n = 1 in (3.22), we get

c~11~(~-)[12d~- + (~ + ~)llVu2(t)ll 2 + 2 f0 t -~ .+Ct(#2"4"f12#r)f0 t ~ "4"4~

IIA~2(~-)ll=dw -< ~2,

for all t c [0, T] since u 1 = 0. Thus

IlVu2(t)ll

(~ .4. #r) 1/2"

(3.23)

223

C. Conca, R. Gormaz, E. Ortega and M. Rojas

Using classical induction on n, that is, assuming that

Ilwn(t)Fr <__

(It -~ Itr)l/2'

we prove that for ~ sufficiently small, a similar inequality holds for Un+l Effectively, it is clear that for s sufficiently small we have

a 3 3 V/2)~-1/8C~ E 4" a ( # + #,.)3/2 > 0

It -~- Itr 4

and

/~ V/~)~-l/8Cft (It -'1- Itr) 1/2

>0.

This condition, stated in terms of c 2, is written as

s < (it ~t_ Itr)3)~l/4 O~4 1 32~2C~ min{ ~-~, ~}.

(3.24)

Therefore, from (3.22) it follows the desired estimate for u n+l. Thus, identifying ~2 with the right hand side of inequality (3.21), we have proved: L e m m a 3.2 - Assume that hypothesis (3.12) is fulfilled. Then there exists which depends on the smallness of f and g with respect to the viscosities it and #r, such that s

sup Ilvu (t)tl <

te[0,T]

-- (~ -~ Pr

(3.25)

)1/2,

i.e., u n remains uniformly bounded in L ~ ( 0 , T; V) as n

~ c~.

Similar arguments leads us to an analogous result for w n. 3.3. T h e L 2 e s t i m a t e s of the t i m e derivatives

Throughout this section we assume that hypothesis (3.12) is fulfilled. By virtue of Lemma 3.2, inequality (3.22) reduces to

(# + #r)l]Vun+l(t)ll 2 + 2

2/32

4

4

o~(# + pr)3/2

(it -~- Itr) 1/2

[[Ut+I(T)t]2dT

[[Aun+l(r)l]2dr <- s2.

Therefore, we conclude that there exists a constant C~, independent of n, such that

[[vun+l(t)[[ 2 -}-

11

(7)]12d7 q-

]tAun+I(T)II2dT < Cs,

(3.26)

224

Existence and uniqueness of a strong solution ...

which implies up and A u n remain uniformly bounded in L 2 ( O , T ; H ) , as 12

~(X).

Similarly, for all 12, we can prove that there exists a second constant C~, independent of n such that

llVw~+l(t)ll 2 +

fo t llBwn+l(T)ll2dr + fo t llwp+~(T)ll2& - _

c',

(3.27)

and analogous bounds for w n hold. 3.4. T h e L 2 e s t i m a t e s of the t i m e derivatives In this section we assume the following additional hypotheses on the data f and g" Vf, Vg

E

L2(QT) 3 and

ft, gt

E

L2(QT).

(3.28)

Differentiating (2.5) with respect to t, we obtain n n+l 1 )Au~+l P(pt ut ) + P(p~u~t + ) + (# + #r = 2 #~ P(rot w2) + P ( p ? f ) + P((p'~ft) - P(p;~u '~" V un+l) _ p ( p n U nt 9 Vun+I ) - - P ( p ~ u n" Vut+l). (3.29)

Multiplication of (3.29) by ut +1 yields ld

1

2 dt IIv~~+1112 + (~ + ~ ) l l W t + l l l 2 = _ ~(pp~p+~, ~p+l) +2 #r (rot w~, u~ +a) + (Ptn f, u~ +1) + (Pnft, utn + l )) -- (pin ?.tn . ~ 7 u n + l , ~t?+l) __ (pn~t~.V~tn+l ,U tn+l ) V U p 4-1 ,/t~q-1))

--(pnll.n. __

_

l(div(pn~n)~?+~ ~ t + ~ ) + 2 . ~ ( ~

rot~

n+l

)

- ( d i v ( p n u n ) f , Ztp+l)) q- (pnft, ~tt +1) + ( d i v ( p ~ u ~ ) u ~ 9Vun+l, u~ +1) _ (p~u~~ . Vu~+l, ut+l)) __(pn,un

"

V,U,t+I, ,U.,t+ 1)

(3.30)

since from (2.3), p ' ~ - - d i v ( p n u n ) . Using classical estimates, H61der and Young's inequalities, each of the seven right hand side terms in (3.30) can be estimated. A straightforward

C. Conca, R. Gormaz, E. Ortega and M. Rojas

225

but tedious estimate yields ld

Crlll?~t+l [I2 nu Crl][W~[[ 2 -[- Cv]If[12H1 + Cv[[ftl[ 2 + C~IIA~II 2 +

+Cn[lAun+ll[2 + cvllurl[2[lAun+lll2 + C~llut+lll 2 + St/IlVut+lll 2. where r/ is any positive parameter, and C n is a constant independent of n. Choosing #+#r rl= 16 ' and denoting C the corresponding constant Cn we get d

d--~Ilv~uF+lll2 + (~ + ~)IiWF§

~ ~ c IluF+lli 2 + c llwFI! ~

+ C Ilfll~ + c IlAunll 2 + C []Aun+lll2 +C liftll 2 + C [[uTll2l]Au~+lll 2 + C.

(3.31)

Let us now estimate IIAunll 2 in terms of Ilupll2. In order to do this, we multiply (2.5) by Au n+l and obtain

- ( p n u ~ + l , A u n + l ) + 2#r ( r o t w n , A u n+l) +(pnf, Aun+l ) - (pnun . V u n + l , A u n + l ) . Since

[(p'~un.Vun+l,Aun+l)l

< /3 ]IunIIL~IIYun+IlIL~IIAu~+lll < C IlVun+llll/4llAu~+~llT/4 <_ CnllVun+i]12 + 5llAun+ill2 ,

where d is any positive parameter used in this last application of Young's inequality, and using classical estimations for the remaining terms, we get

(# + #r)t[Aun+lll 2

<

Csl[u~-[-lll 2 -[-CsI]Vwnll 2 -1-Csllfll 2 --t-

-]-CsIIVun-1-1][2 nt- 4(~ IIA~n-t-lll2. Then, choosing 5 > 0 sufficiently small, from the last inequality, we obtain as needed IIA~n+lll2 <_ C 1[~7+~ll2 + c. (3.32) Thus,

(3.31) can be reduced to d

dtll.~n,,n+l 2 C Itut+l It2 -[- C Ilw~l] 2 -t- c Ilfl]~/, + C Ilft[I 2

+ c ilu~l[211~+'ll 2 + c !!~112 + c.

Existence and uniqueness of a strong solution ...

226

Integrating this inequality from 0 to t, we have

a[luT+l(t)ll 2 + (# + #~)

_< c

j~ot IIVu'2+I(T)II2dT

/o t (11~+1(-,-)112+

Ilu?(-,--)ll:~ + IlwF('r)ll2)dr

+C

IIf(')ll~/, + IIf~(")ll:~) d~

+c

I1~;~(-,-)112 II~r+x(-,-)ll2d-, - § o~ll~F+l(o)ll = + c t .

From equation (3.14), we can easily bound the rightmost term, Ilu7+l(0)ll 2. This is due to the fact that (d/dt)[]Vun+l(t)[I 2 is non negative at t -- 0 since Vun+l(0) = 0. Applying the estimates for fo [lut+t(T)]] 2d~- and fo [[w~+1(7) [[2d~- and hypothesis (3.28), we get

IluF+l(t)ll 2 + Let us denote s

fo ~ II rut +1 (r)ll2dr _< C + C fo ~ Ilu?(~-)ll211~F+l(r)ll2d~ -. -[lu~+l(t)[[ 2. The above inequality can be written as

p(t) _< C + C

jfot [lu~(r)[[2p(t) d~

which, by Gronwall's lemma, implies

IluF+x(t)ll 2 - ,,,:,(t) _< Cexp(C

jr0t 11~?(~)112tiT) < Cexp(CCE),

where C~ comes from one more application of (3.26). We conclude finally t h a t there exists a constant C independent of n such that

Iluy+x(t)ll 2 +

~0t IIVuy+l(7-)ll2d~ - _< C,

(3.33)

which means that up c L~(O, T; H) n L2(O, T; V). Moreover, from (3.32) we have for all n sup []Aun+l(t)ll 2 < C,

t

t h a t is, u n c L~(O, T, D(A)).

(3.34)

C. Conca, R. Gormaz, E. Ortega and M. Rojas

227

Similarly, it can also be proven that there exists a constant C, independent of n, such that

IIW tn + l ( t )ll 2 + f0t I I w ~ + 1 (~)lt ~d~ < c

and

suPt ]lBwn+l(t)ll 2 <_ C.

(3.35)

3.5. A n L2(0, T; W 1'~) e s t i m a t e of u ~ Let us write (2.5) as ( # -t- # r ) A u

n + l --

P(F),

(3.36)

where F = 2#trot w n +

flnf

_ flnut+l

_ flnun . ~un+l.

Making use of all the previous estimates, it is possible to prove that F c L2(0, T; L6(f~)). As a consequence of the Amrouche-Girault's results (1991), we conclude that u n remains uniformly bounded in L2(0, T; W2,6(f~)). This, using the Sobolev embedding, implies that u n remains uniformly bounded in L2(0, T; W l , ~ ( f t ) ) . 3.6. T h r e e key e s t i m a t e s of t h e s e c o n d o r d e r d e r i v a t i v e s Without any extra regularity assumed for the solution at t = 0, we can still obtain bounds for V u [ ~ in L ~ ( a , T ; L 2 ( ~ t ) ) and for ut~t and Au~ in L2(a,T; L2(gt)) for any ~ > 0. Analogously for w'-. Let first multiply (3.29) by u~t+1. Using (3.2), (3.25), previously obtained estimates, and HSlder and Young's inequalities, we have

2 +CvlifJl 2 +

d~ [lvut+lll2 ~ c~Ilu~+ll]2 + c~JlVwtll2 +

C,711ftll2 + c,7]lAun+llI2 + Cvlivur~l]21lAun+l[12

+ Cr/IIYU~+1112 -]- 7~lllt~t+l ]l2, where r/is any positive parameter, and Cv is a constant independent of n. Choosing r] = ]-~ and taking into account (3.33)-(3.35), we have

d t9/ l]U~t+l t12 -Jl- (# Jr- /-tv) ~--~ IlVltt+l [i2

c IIv~rll ~ + c Ily[I2 + c tlA[i ~ + c llW~ll 2 + c Ilvltt -t-1]12 -1- C.

228

Existence and uniqueness of a strong solution ...

It is possible to overcome the lack of boundedness for 11~7U~+1[[2 at t = 0. In fact, multiplying this latter inequality by a(t) - min{1, t}, we get

~(t)llu~+lll = + ( , + , ~ ) ~ d (~(t)llw~+l II ) < (# + #~)~'(t)llWF+lll 2 + c ~ ( t ) ( l l W F I I 2 + IIv~FIi 2) + c ~(t)(llfll = + IIf~ll 2) + c ~ ( t ) ( l l w F + l l l 2 + 1). (3.37) Observe t h a t as a consequence of (3.33), it exists a sequence Sk

> O, such

that ~k IIWF+I(~k)II 2 <_ C. Whe~efore, since ~(t) _< 1 ~nd o'(t) _< 1 ~.e. in [0, T], applying (3.33)-(3.35) and integrating (3.37) from ~k to t, we obtain

~r(T)[[u~t+t(w)[[2dr + (# + #~)a(t)llVu~+l(t)l] 2 k

< ( , + #~)~(Ek)IlVuF+l(~k)ll 2 + C. Taking limit as ~k ~

0 reduces the previous inequality to

f0

t~

+ r

[ [ V u t + l (t) [[2 < C,

which provides, for any ~ > 0, uniform estimates of ut~ and Vu2 in the ~p~c~ L~(~, T; L ~ ( s as ~ , oo. Analogously, for all n, it can be proved that

/o

t ~ ( ~ ) 1 1 ~ + 1 ( ~ ) 112d~ + cr(t) []Vwt+ l (t) [[2

_< C

and similar uniform bounds hold for w '~. Finally, from (3.29) we obtain

(~ + ~ )

/o t

o'(~-)llAu'~+l(~-)lled7

"

<_

/0

~(~)llC'(~)ll2d~

-

where Gn

2p~rot w t +ptn f Jr- pn f t -- p t n u t + l _ I3^n~ttttn+l -- ptnun " r u n 4 - 1 _ f i n u t n . Vun+l _ f l n u n . V u t + l .

Gathering together all the estimates already derived, we conclude that a l / 2 ( t ) G n remains uniformly bounded in L2(0, T; L2(gt)) and therefore, for any s > 0, Au~ is uniformly bounded in L2(s,T; L2(s A similar conclusion hold for w n.

C. Conca, R. Gormaz, E. Ortega and M. R o j a s

229

Remark. Using classical compactness arguments, we conclude that, up to a subsequence, the approximate solutions (u ~, w '~, p") converges to a strong solution of problem (1.1)-(1.2). Alternatively, this can be proved using a different approach which we develop in section 4.

4. T h e c o n v e r g e n c e

of the sequence

This section is devoted to prove t h a t u ~, w n and p~ are Cauchy sequences. Let us introduce the following notation for the difference of two terms of a sequence. For n, s > 1, It n ' s ( t ) --- U n+s (t) -- It n ( t ) , W n's (t) -- W n + s ( t ) -- W n ( t ) ,

and

p',~(t)

=

p'+~ ( t )

-

p" (t).

It is clear t h a t u n,s, w n,s and pn,* satisfy the following equations p ( p n - l + s ~ a n,s t ) + ( # + # r ) A u n's -- 2 # r P ( r o t w n-1 ,s) + p ( p n - 1 _g(fln-l,sur~) _ g(pn-l+sun-l+s . ~ u n , s) -p(pn-l+sun-l's

p~-~+~'~

" V u n) - p ( p n - l ' s u n - 1 .

,s f )

~Tun)

(4.1)

+ (~a + ~ d ) B ~ ~'~ -- (~0 + ~d - ~a)V di~ ~ ' ~ + 4 p r w n'~ -- 2pr(rot u ~ - l ' s ) + p~-l,s g _ p n - l , s w ~ _pn--l+sun--l+s

. ~wn, s _ fln--l+sltn--l,s . VW n

(4.2)

_ p n - - 1,sun-- 1 . V W n

pt '~ + u "'~- Vp n+~ + u ~- Vp ~'s = 0.

(4.3)

The following lemma which can be easily proven, is fundamental in order to obtain error estimates. L e m m a 4.1 - Let 0 <_ r (t) ~_~ M f o r all t E [0, T] and assume that there exist C > 0 such that for all n > 2, we have 0 ~ ~ n ( t ) ~_ C

~ot ~ ) n _ l ( T ) d T .

Then (Ct) n-1

Cn(t) _< M ( n for all t E [0, T], and n >_ 2.

1)!

Existence and uniqueness of a strong solution ...

230

The next lemma, proven in W. Varhorn (1994, Lemma 3.10 p. 122) is a variant of Gronwall's lemma and it will be also needed. L e m m a 4.2 - Let "y c IR ('y > O) and let ~, f, g c C([0, T]) real functions (f _> 0, g >_ 0 on [0, T]) satisfying for all t C [0, T] the inequality

,~(t) +

j~Ot f ('r) dT <_ fOt g(T) dT + ~/ j~Ot ~(T) dT.

Then for all t E [0, T] we obtain the estimate ~(t) +

/0 f (T) d T < (/0 g(T) dT ) e"~t.

4.1. Bounding the error of the density sequence The density sequence can be bounded in term of the velocity sequence. For this purpose, let us multiply (4.3) by (pn,~)5 and integrate in Ft. We get

1dip

S dt

ns6

r '1 dx

=

<

Un,s. Vpn+S(pn,S)Sdx_ -~

9v(pn,s)6dx

/a [u~'~]lVpn+~]lpn'S]Sdx + -g ira

div u ~ (pn'S)6dx

_

< IIVP'~+~IIL~(O,T;L~(a))fo[un'~llP"~lSdz <_

c

(~

I~,~l~dx

)1/6(~

Ip~,~16dx

)5/6 ,

that is, 6dt[ [ l d p~,Sllg~ _< C

II~n'~IIL~IIPn'~IISL~

1 S = IIP'~,s IIL6(d/dt)ll Since on one hand -g(d/dt)l Ipn '~][L~ 5 pn '~]IL~, and on the other hand the embedding H i ( a ) ~-, LS(fi) is continuous, d

~2trt,s II.

d-7 IIp~'~ IIL~ --< C II

Integrating from 0 to t and applying the Cauchy-Schwarz inequality, we get

IIp"S(t)llL 6 <_ C fo IlVu~'S(~-)l]d~- -< C T

(/ot IlVu~,~(~)ll2d~)1j2

(4.4)

C. Conca, R. Gormaz, E. Ortega and M. R o j a s

231

The latter inequality means that p~ is a Cauchy sequence in L ~ ( 0 , T; L6(~)) whenever u ~ is a Cauchy sequence in L2(0, T; V). Furthermore, from (4.3) we have p t 's -Jr-U n . V p n's At- ~tn's " V p rent's

--

0

pn'S(O)

--

O.

which, using the characteristic method (see O. Ladyzhenskaya & V. Solonnikov (1978), p. 730), lead us to ][pn'~(t)l]L~ < ]]~7pn+sl]Loo(O,T;L~(gt) )

/o

]lun'~(~)]]L~dr.

(4.5)

4.2. C o n v e r g e n c e of u ~ a n d w n in L ~176 In this paragraph we prove that u n and w n are Cauchy sequences in the space L~176 T; Hl(ft)). Multiplying (4.1) by 5 A u n's, where 5 is any positive parameter, an integration over f~ and some straightforward estimations yield, 5(# + #r)HAun'S]] 2

<

r/[[u~'S[[2 -F ?][[Vwn--l's[[ 2 + rl[[pn-l's[[2L6[lf[[2L a

+,711p~-1,~t1~o llV~ll ~ + ~l[W',~ll~ 1

+~2CIIA~'~,~112.

(4.6)

where ~ is any positive parameter and C is a constant independent of n. Analogously, multiplying (4.1) by u t '~, we get a[[u~'~ []2 -t # +2 #r dtd ][~ u n , s [[2 < CnllVwn-X,~]] 2 + Cnl[pn-~'~ll2L~l[fllL2 ~ +Cnllpn-l,~llg~ +

6nllu]'~ll 2,

(4.7)

where C n is a constant independent of n. Adding up (4.6) and (4.7) we obtain a[[u~,S[]2 _$ # + #r d 2 dt ]]Vun'~ ]12 + 3(# + #~)][Au"'~ []2

_< c,7 + w) {[IVwn-X'Sll2 + I1,o'-1'~11~ [Ifll~ + Ilpn-l'~ll~o[IVu~lt2 +llV~n'~ll 2 + IIV~'-X'slt 2 + ]l,on-l'sll~o ) + 'rwlluY'~ll ~ C,5 2

+--~

llAu'~'sii 2 .

232

E x i s t e n c e and u n i q u e n e s s o f a s t r o n g s o l u t i o n . . .

oz

We choose ~ - ~ ,

C6 2

and 5 > 0 such that (# + #~)5

4r;

-C1

> 0 . This

reduces the previous inequality to

~11,.,~' II~ + (# + #,-)~llVu"'~ll ~ + c111 n s

d

CIIVwn-l'sll

II

Aun,~ 2

2 -~- C[[pn-l's[[2L6 [[f[]2L3 Jr- C[Ipn-l's[12L6 [IVu~[[2

-JcCllVun'sl[2 -t- CllVun--l'sl] 2 nt- Cllpn-l's[12L6

(4.8)

with positive constants C1, C independent of n. From (4.2), we have p n - l + ~ w t ' ~ + L w n,~ + 4 # r w n'~ =

2#~(rot u n-l'~)

+

--pn--l+sun--l's

pn-l,Sg_

pn--l,swr ~ _ pn--l+sun--l+s

" Vwn,S

(4.9)

" V W n -- pn--l'sun--1 " V w n

where L w n,~ = (c~ + c d ) B w n'~ -- (Co + Cd -- c~)V div w '~'~. Since L is a strongly elliptic operator (see O. Ladyzhenskaya et al. (1968) p. 70), there exists a positive constant No depending exclusively on c~ + Cd, Co + C d - C~ and F, such that ( L w n ' S , B w n's) >_ (Ca + Cd)l[Bwn'S[[ 2 -

No[[Vwn'Sll2.

(4.10)

Multiplying (4.9) by O B w n'~, where 0 is any positive parameter, the use of (4.10) and some classical estimations yield

O(ca + ed)llBwn'~[12 <_ OCr

~;llw2,Sll 2 + dllVu'~-l,~l]2 +~ll,on-l,sll~ollgll~3 + ~ll#~-l,Sll~o IlVw?ll 2 +<{IVw'~,'ll + <{IW'~-x,'I{ +

02C + -.~-~ IIBwn,~ll2,

+r

(4.11)

where ~ is any positive parameter and C is a constant independent of n. Multiplying (4.2) by wt '~, classical estimations lead us to al]wt'~]]2 +

c~ + cd d

2

Vw~, ~ 2

d--t[]

[[ +

d + 2 # ~ l l w ~ ' ~ [ [ 2 < Cr _

Co + Cd -- C~ d

2

d--t[Idiv w~'~[12

2 + Cr

23 L

+c
+6r

(4.12)

i-.,o

0

9

B

c-.t--

c--'-l'-

.

=-.

~ ,--,.

Oq

0

+

+

+

e

D,..,

M

+

._.__.

--

o~...,.., o~......~ o~......,o~...~

+

~

~

~

IA

+

~

-F

p

~

"'-/

~

~

~.

~.~ "~

n~ ~< ~

~ ~

+

~

~

i:::s

I-'t

~

~

o

~'o~ o'q ,.,,...,.

~ .

~..~.

v

~

~

_

~

_.'-~

---

+ ,-.,

O-q

0

i.-,,,

1-,.~ ~

0 0

~

~

9

co

~o.

a~

9

IA

I

~"

~

o

e

~

<1

-%

~ ~

,_.,

o

~

<..l

~

IA

---'-

4--

_ _ . _ .

4-

<1 ~

p

On.

tO

h-

I

t9

~

~

=

~

~-~

IA

+

+

+

~

.~

~

o~-,

+

~

~

~

+

~

l:::r'

9

~

r~

+

~:

l:::r

~

~

~

~]

~

<:1

e I

~ .

<1

- -

~

~

r

<1

+

I

~

"~

- -

~

~

r

<1

+

to

~

I

~

"-I

+

<1

~

~

~

.,..}

-~

+

~

~"

+

+

~

~"

+

o

O

r~

O

,.Q

tO r

C. Conca, R. Gormaz, E. Ortega and M. Rojas

235

Since the application of (3.26) and (3.27) gives IlVul,s(t)ll 2 + IIVwl,~(t)ll 2 _< M~, Y s and t E [0, T] for some positive constant M2, using the Lemma 4.1, we obtain (Mlt) n-1 ( M T ) n-1 IIVun's(t)ll2 + IIVw~'~(t)ll 2 _< M2 ( n - 1)! -< ( n - 1)! '

(4.16)

where M is again a constant independent of n. This inequality proves that u ~ and w n are Catchy sequences in L~(O,T;HI(f~)), and then pn is a Catchy sequence in L ~ ( O , T ; L ~ ( f ~ ) ) in view of (4.4)-(4.5). Finally, since

~ (t[v~n--l's(7.)[] 2 -t-[[Vwn-l's(7.)ll2)dT. t

M1

_< M1

jr0 t (MT")n-2 (Mt) n-1 ( n - 2)! dT" _< M1 ( n - 1)!

(4.17)

from (4.15) we have

j['ot (llut 's (7")112 -1- llwt 's (T)]12)dT. < M -

~o

(Mlt) n-1 <_ M ( M I T ) n - 1 (n-

t(llAun,~(~-)lle+liBw~,~(~)ll2)d~ < M ( M l t ) n-1 -

(n-

( n - 1)! "

I)!

1)!

(4.18)

(M1T) n-1 _<M ( n - l ) ! " (4.19)

Let now summarize the results already obtained in the following lemma: L e m m a 4.3 - We assume that hypotheses (3.1), (3.12) and (3.28) are fulfilled. Then there exist u c L ~ ( O , T ; V ) C l L2(O,T;H2(f~)n V), w C L~(O,T;HI(f~))AL2(O,T;H2(f~)AHI(f~)) and p C L ~ ( O , T ; L ~ ( f ~ ) ) such that un W n

; u strongly in L~(0, T; V) AL2(O,T;H2(ft) A V), (4.20) .... ~W strongly in L~(O,T;HI(f~))NL2(O,T;HU(f~)) and strongly in L2(O,T;HI(f~))

p~ .....>p strongly in L~(O,T; L~(f~)).

(4.21) (4.22)

Following a similar pattern to the proof of Lemma 4.3 and multiplying the resulting inequalities by or(t) = min {l,t} (as in Paragraph 3.6), the additional convergences are proven in Conca et al. (2002)"

Existence and uniqueness of a strong solution ...

236

4 . 4 Under the same hypotheses of L e m m a ~.3, we have the following convergences:

Lemma

n Ut

". ut strongly in L ~ ( e , T ; H )

n Utt

n L 2 ( e , T ; V),

utt strongly in L2(e,T; H), W t

strongly in L ~ ( e , T ; L 2 ( ~ ) ) n

L2(e,T;HI(~)),

wtt strongly in L2(e, T; L2(~t)), Un

u strongly in L ~ ( e , T ; H 2 ( ~ )

W n

N V) N L 2 ( e , T ; W I ' ~ ( ~ ) N

w strongly in L ~ ( e , T ; H 2 ( ~ ) n

Y),

H I ( ~ ) ) N L2(e,T; w l ' ~ ( g t ) )

and strongly in L 2 ( e, T; H lo( ~ ) ) , where ~ is any small positive constant.

4.3. Passage to the l i m i t A p p l y i n g L e m m a 4.3, as a s t a n d a r d p r o c e d u r e we o b t a i n

lo

T ( p u t + p u . V u -- p f -- 2 # ~ r o t w -- (# + # ~ ) A u , v ) r

dt -- 0,

T

o (pwt +pu'Vw-pg-2#rrotu+4#~w-(Ca

+Cd)AW

- ( c o + Cd -- c a ) V d i v w, z )~(t) dt = O, for all z, v E L2(~t) and

r

(put + pu. Vu-

r E L ~ ( 0 , T). This clearly implies

p f-

2#,.rotw-

(# + # ~ ) A u , v) = 0,

( p w t + p u . V w - p g - 2 # ~ r o t u + 4 #~ w - (ca + Cd)AW - - ( C 0 --~ C d - -

Ca)~Tdiv w, z ) = 0

a.e. in [0, T], for every v E H, z E L2(~t). F r o m these equalities we have

P( p u t + p u . V u -

p f-

2 # ~ r o t w - (# + # ~ ) A u ) = 0

and

p w t + p u - ~Tw - p g - 2 # ~ r o t u + 4#~ w - - ( C a -~- C d ) / ~ W

- - ( C O -~- C d - -

c a ) V d i v w = 0.

In the case of t h e density, we observe t h a t un

~ u strongly in L2(O,T;L2(gt)),

p~

~ pt,

and

~Tpn

, V p weakly in L2(0, T;L2(gt)),

237

C. Conca, R. Gormaz, E. Ortega and M. Rojas

Thus, when n -~ c~, the approximated continuity equation gives Pt + u . ~Tp - 0 in the L2(0, T; L2(~t)) - sense.

Let us prove the continuity of the solution (u, w, p) at t = 0. Since u E L ~ ( 0 , T; D ( A ) ) and ut E L2(s, T; D ( A ) ) , then by classical interpolation theory (see R. Temam (1979), p. 260) u is a.e. equal to a continuous function from [6, T] into D(A), i.e., u E C([s,T];D(A))

Ve > O.

On the other hand, since ut E L e ( s , T ; D ( A ) ) , utt E L 2 ( r again interpolation theory we have

and using

~, E C([~, T]; V) V ~ > 0. Therefore, u E CI([s,T]; V) N C ( [ e , T ] ; D ( A ) )

Ve > O.

Analogously, we prove t h a t w C CI([e,T];Hlo(~))NC([e,T];D(B))

Ve > O.

To prove continuity at t - 0 , we already have lim

t--~0+

liu(t)

-

lim [[Vu(t) - Vu(0)l i -- 0.

~(0)ll = 0,

t-+0+

Also, we can prove indeed t h a t lim []Au(t) - Au(0)[[ = 0.

t-+0+

In fact, it is sufficient to show that

lim sup [IAu(t)ll ~ I]Auol]

t---+0+

since we already know that u(t) ---. uo in Hl(~t). Multiplying (2.5) by Au~ +] and integrating in ~, we obtain

# + #r d Aun+ 1 2

2

d--/ll

/"Z-'~V'Vun-t1 2

II + l l v p ' ~

-- - - ( f l n t t n " v u n +

II =

l, A u ~ + l ) -4- 2 # r ( r o t w n , A t t t ~+1)

+(pn f , A u t +1) - ( V p n . ~Tut4-1 , ttt4-1),

Existence and uniqueness of a strong solution ...

238

which, integrating from 0 to t, implies

IlAu~+l(t)[! 2

2

<__ IlA~ol{ 2 +

[(--pn(t)un(t) " Vun+l(t)

-~- 2#rrotwn(t)

+p~(t)f(t),Aun+l(t))-

(-p~u~(O) 9Vu~ +1 + 2#~rot w~ 2 +p~f(O),Au~+l)] + ~ N ( t )

where

N ( t ) = ~o t {(p'~u n . VU n-t-1 + print?" V?.tn+l + pnun " VU? +1 - 2 # ~ r o t w~ - p'~f - pnft, A u ' + l ) l d T +

<_ c

/o

({{vun+l{I + IlW?l}

+

/o

}(Vp ~. VU~+I,u'~+I)IdT

}}Vu~+lll + }IV~FI{

+l{.fll + {IAII)d~ ~_ ct ~/~,

by virtue of H61der's inequality and the estimates given in Lemmata 4.3-4.4. From this, we conclude {{Au(t){{2 _< {{Auol{ 2 + c [ ( - p ( t ) u ( t ) . V u ( t ) + 2#~rot w(t)

+p(t)f(t),Au(t)) -(-pou(O). +pof(O),Auo)] + ct 1/2.

Vuo + 2#~rot wo

Since p(t)u(t). Vu(t) --~ pouo" Vu0, p ( t ) f ( t ) --~ pof(O), rot w(t) --~ rot w0 in L2(gt) and Au(t) --~ Auo weakly in L2(f~) as t -~ 0 +, we obtain the desired result. It is now clear that lim {{ut(t) -

t--,0+

ut(O){{-

0.

The continuity of u(t) is therefore proved. The results for w are proved in a similar way. This completes the proof of existence in Theorem 2.1.

4.4. Uniqueness Suppose that there is another solution (Ul,Wl,Pl) of (2.1)-(2.4). Define U-ul-u

,

W--wl-w

and R - p l - p .

These auxiliary functions satisfy a set of equations similar to (4.1)-(4.3). If we multiply the first equation by U, the second by W and the third by

C. Conca, R. Gormaz, E. Ortega and M. Rojas

239

R and repeat arguments similar to those given in Paragraph 4.2, we obtain for q~(t) = IIU(t)ll 2 + IIW(t)ll 2 + ilR(t)ll 2 an inequality of the type: <_ c which, by Gronwall's inequality, is equivalent to assert U = 0, W = 0 and R=0. 4.5. Results on the p r e s s u r e To conclude, let us refer the interested reader to Conca et al. (2002), where the following result concerning the pressure is proven. L e m m a 4.5 - Assume that the hypotheses of Lemma ~. 3 are fulfilled. Then, for each n, there exists p'~ c L2(O,T;HI(f~)/1R) such that (un,wn, pn,p n) satisfy the first equation in (1.1). The approximate pressure pn converges to a limiting element p in L2(O,T;HI(f~)/1R) and (u,w,p,p) is the unique solution of (1.1)-(1.2), where (u,w,p) is the unique solution of problem (2.1)-(2.~). Moreover, we have the error estimate f0 T Ilpn(7") -- p(T)[I 2 gl(f~)/

/RdT < (MT) n-1 -( n - 1)l '

where again M is a positive constant. Also, pn converge to p in L~(~,T;HI(f~)//1R), for all ~ > 0 and we have the following error estimate:

supcr(t)lpn(t)t

(MT) n-2 - P(t)IIHI(~)/1R <- (n - 2)v. "

Acknowledgments. This work was supported by the Fondap Programme in Mathematical Mechanics. E1va Ortega was supported by CNPq-Brazil and Marko Rojas was partially supported by CNPq-Brazil, Grant 300116/934(RN) and FAPESP-Brazil, Grant 1997/3711-0.

References [1] Amrouche C. and Girault V., On the existence and regularity of the solutions of the Stokes problem in arbitrary dimensions. Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 171-175. [2] Antoncev S.N., Kazhikov A.V. and Monakhov V.N., Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam 1990.

240

[3]

[4] [5]

[61 [7]

[81 [91 [10] [11] [121

[13] [14] [15] [16] [17]

Existence and uniqueness of a strong solution ...

Boldrini J.L. and Rojas-Medar M.A., Global solutions to the equations for the motion of stratified incompressible fluids, Mat. Contemp. 3 (1992), 1-8. Boldrini J.L. and Rojas-Medar M.A., On the convergence rate of spectral approximation for the equations for nonhomogeneous asymmetric fluids, Math. Model. Numer. Anal. 30(2) (1996), 123-155. Boldrini J.L. and Rojas-Medar M.A., Global strong solutions of the equations for the motion of nonhomogeneous incompressible fluids, in Numerical Methods in Mechanics, C. Conca et al. eds., 35-45, Pitman Research Notes in Mathematics Series 371, Longman, Harlow (1997). Boldrini J.L. and Rojas-Medar M.A., Strong solutions of the equations for nonhomogeneous asymmetric fluids, Technical Report N~ 21 (1998), IMECC-UNICAMP, Brazil. Conca C., Gormaz R., Ortega-Torres E.E. and Rojas-Medar M.A., The equations of nonhomogeneous asymmetric fluids: An iterative approach, Math. Methods Appl. Sci. (2002), in press. Condiff D.W. and Dahler J.S., Fluid mechanical aspects of antisymmetric stress, Phys. Fluids 7(6) (1964), 842-854. Kim J.U., Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, Siam J. Math. Anal. 18(1987), 89-96. Ladyzhenskaya O.A., Solonnikov V.A. and Uraltseva N.N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society 23, revised edition, (1988). Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1969). Ladyzhenskaya O.A. and Solonnikov V.A., Unique solvability of an initial and boundary value problem for viscous nonhomogeneous incompressible fluids, Zap. Nauch. Sem. Leningrad Otdel Math. Inst. Steklov 52 (1975), 52-109; Engl. Transl. (1978), J. Soviet Math. 9, 697-749. Lions J.L., Quelques M~thodes de R~solution des Probl~mes aux Limites non Lin~aires, Dunod/Gauthier-Villars, Paris (1969). Lukaszewicz G., On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. XII (1988), 35-44. Lukaszewicz G., On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. XIII (1989), 105-120. Lukaszewicz G., On nonstationary flows of incompressible asymmetric fluids, Math. Methods Appl. Sci. 13(3) (1990), 219-232. Lukaszewicz G., Micropolar Fluids: Theory and Applications, Birkh/iuset, Berlin (1999).

C. Conca, R. Gormaz, E. Ortega and M. Rojas

[181 [19] [20] [21] [22] [23]

241

Ortega-Torres E.E. and Rojas-Medar M.A., The equations of a viscous asymmetric fluid: An iterational approach, Technical Report N~ 42 (1997), IMECC-UNICAMP, Brazil. Petrosyan L.G., Some Problems of Mechanics of Fluids with Antisymmetric Stress Tensor, Erevan (in Russian) 1984. Salvi R., The equations of viscous incompressible nonhomogeneous fluids: On the existence and regularity, J. Austral. Math. Soc. Set. B, 33(1) (1991), 94-110. Simon J., Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure, Siam J. Math. Anal. 21 (1990), 10931117. Temam R., Navier-Stokes Equations, revised ed., North-Holland, Amsterdam 1979. Varnhorn W., The Stokes Equations, Mathematical Research 76, Akademie-Verlag, Berlin (1994).

Carlos Conca, Radl Gormaz Departamento de Ingenieria Matems and CMM, UMR 2071 CNRS-UCHILE Universidad de Chile Casilla 170/3-Correo 3 Santiago CHILE

E-mail: cconca~dim.uchile.cl,

E-mail: rgormaz~dim.uchile.cl

Elva Ortega Departamento de Matems Universidad de Antofagasta Casilla 170 Antofagasta CHILE

E-mail: [email protected] M a r b Rojas Departamento de Matem~tica Aplicada IMECC-UNICAMP C.P. 6065

13081-970 Campinas-SP BRAZIL E-mail: [email protected]

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Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 12

H O M O G E N I Z A T I O N OF DIRICHLET M I N I M U M PROBLEMS W I T H C O N D U C T O R T Y P E PERIODICALLY D I S T R I B U T E D C O N S T R A I N T S

R. DE ARCANGELIS

O. Introduction The study of the homogenization of variational problems subject to pointwise oscillating constraints on the gradient was proposed in the 1978 book [31 by a. Bensoussan, J.L. Lions and G. Papanicolaou (cf. also [28], [1], [2], [14], [24], [4], [10] besides [31, fo~ general references on homogenization theory), where also a general conjecture on the limit problems was formulated (el. w of Chapter 1 in [3]). It originated from some problems from elasticplastic torsion theory for media with periodic structures (cf. for example [18], [6], [29], [22], [25], [19]) that involved integral energies defined on sets of functions u subject to constraints of the type IVu(x)[ ~_ p(hx) for a.e. x,

(0.1)

where ~a is an essentially bounded and ]0, 1In-periodic function, and h (c IN) is the homogenization parameter (cf. [10] for a bibliography on the subject). Analogously, some models in electrostatics and elasticity (el. [27]) suggested the treatment of homogenization problems in presence of periodically distributed conductors, or of rigid inclusions in the case of elasticity. Such models lead again to the consideration of gradient constraints as in (0.1), but with ~a taking only the values 0 and +oc (cf. [13], [30], [5], [15], [7]), or, in order to consider also "thin" (i.e. with null Lebesgue measure) conductors, to constraints as (of. [11])), u is constant in each element of a ]0, 1/h['~-periodic family of subsets of IRn.

(0.2)

Homogenization of Dirichlet minimum problems...

244

In all these papers the homogenization of some classes of Dirichlet minimum problems with null boundary data, or sometimes of Neumann minimum problems, for variational integrals has been performed. For example, by using the results of [11], the following homogenization result in BV spaces can be proved (cf. [11], [10], and Theorem 5.2 below). Let Y = ]0, 1[~, Cy be a collection of subsets of Y, and let C be the set of the periodically distributed conductors defined as C = { ( i l , . . . ,in)4" C: ( i l , . . . ,in) C 7zn, C E Cy}. Let f be

a

(0.3)

function satisfying

f: (x,z) e ] p n x ] R n ~-+ f (x, z) e [0, +c~[ f ( . , z ) measurable, Y-periodic and in LI(y) for every z C ]R~ f(x, .) convex for a.e. x C ]Rn,

(0.4)

and let fhom be defined by fhom" Z E ]Rn ~ inf { ;

(0.5)

f(y,z + V v ) d y ' v C Wllor(IRn),

v Y-periodic, uz + v constant in S for every S E C}, where, for every z C IR~, we denote by Uz the linear function having the gradient z. Then, by assuming that ]z I ~ f(x,z) for a.e. x E IR~ and every z c IRn,

(0.6)

int({z C ] a n : fhom(Z) < -[-OQ}) r 0,

(0.7)

S is connected for every S c C,

(o.8)

and it results that fhom is convex and finite in the whole ]Rn, and that IZ[ ~ fhom(Z)

for every

(0.9)

z e ]R~.

Moreover, for every bounded open subset ~ of IR" with Lipschitz boundary, A c ]0, + ~ [ , p C ]1, h-~---i-l[,/3e L ~ ( ~ ) , the values ih(ft, 0, ~) = inf { ~ f (hz, Vu)dx + ~ /3udz + A

1

ffa

[ulPdx

u E W 1'~176 (f~), u constant in ft N ~ S for every S E C

"

}

(0.10)

R. De Arcangelis

245

converge, as h tends to +oo, to the finite value ioo (ft, 0,/3) = min

fhom(Vu)dx +

fhC~m dlDsul

-FJfo~fh~m(--un~)d']-{n-l na Jf~ ~udx-F,~ jf ]ulPdx " u C BV(~)}, where f~om is the recession function of fhom, n a is the outward unit vector normal to 0f~, ?_/n-1 is the ( n - 1 ) - d i m e n s i o n a l Hausdorff measure, BV(f~) is the space of the functions in Ll(f~) whose distributional partial derivatives are Borel measures with finite total variation in f~, and DSu, IDSul' dlD~ul nDsu are defined in w Finally, convergence results in Ll(gt) for minimizing sequences of the problems in (0.10) to minimizers of ioo(~, 0,/3) too hold. In the above results the choice of the b o u n d a r y d a t u m ( = 0) has been forced by the shape of the constraints, and in [12] it has been observed by means of an example in one space dimension that, at least under constraints as in (0.1) with ~ in L ~176 (Y), homogenization result of the type just described can be no more true if non homogeneous b o u n d a r y d a t a are taken into account. Analogously, in the present paper and in the context of homogenization under constraints as in (0.2), we produce an example showing that, for n = 2, f(x, z) = Iz] 2 for a.e. x E IR 2 and every z E IR 2, for a suitable Cy verifying (0.7) and (0.8), Ft = Y, /3 C L ~ ( ~ ) , )~ E JR, p c ]1,3/2[, and a nice b o u n d a r y d a t u m u0, infima in (0.10), but taken in u0 + W01'~(Ft) in place of WI'~ can be +oo for every h C IN, whilst the m i n i m u m

min{/~tfhom(Vu)dx-~-/~udx+/~~'u]Pdx'uCuo+Wl'2(~)

}

can be finite (cf. Example 6.1). These features are essentially due to the fact that in general, for non zero b o u n d a r y data, for every h c IN there can be no admissible function verifying the corresponding constancy constraint. Therefore it seems more natural to consider, in place of a fixed b o u n d a r y datum, a sequence of boundary data, each one fulfilling the constraint, and converging to a limit boundary datum. This has already been done in [12] in the setting of constraints as in (0.1) with ~ in L~(Y). In the present paper we want to treat the same homogenization problem, but in the setting of constraints as in (0.2).

Homogenization of Dirichlet minimum problems...

246

To do this, we consider a collection Cy of subsets of Y, C given by (0.3), f by (0.4), fhom by (0.5), a bounded open set gt with Lipschitz boundary, a sequence of b o u n d a r y data {U0,h} C_ Wllo'~(IRn), and prove, for A E [0, +c~[, p E [1 , ~ n [, ~ C L ~ (t2) , some homogenization results for the problems

ih(~t, Uo,h,fl) =inf { ~ f(hx, Vu)dx+ / f l u d x + A / 'u'Pdx "

1

(0.11)

)

u C Uo,h + W 1'~ (~), u constant in ~ M ~ S for every S E C , under different sets of assumptions depending on the coerciveness properties of f. A first results deals with the case of superlinear coerciveness, and is the following. Let r ]Rn ----+ [0,-~-(:X:)[ be convex and superlinear, in the sense that lim r = +oc. z--.c~ Iz]

(0.12)

Let us assume t h a t r

_< f(x,

z) for a.e. x C ]Rn and every z C IRn,

(0.13)

and that (0.7) and the following conditions hold

1

to, h is constant in ~ N ~ S for every h E IN and S c C, the integrals

f

] f ( h x , Vuo,h)dx are equi-absolutely

(0.14)

(0.15)

continuous in gt when h E IN, there exist a > 1, M :> 0 such that

faf(hx, o'Vuo,h)dx < M

(0.16)

for every h C IN.

Moreover, let us assume that there exists u0 c Wllo'~(JR n) such that

UO,h ~ Uo in LI(~t),

(0.17)

fhom(CrVUo) C LI(~~) for every cr E IR.

(0.18)

R. De Arcangelis

247

Then fhom is convex and finite in the whole IRn, r

_~ fhom(Z) for every z C IRn,

(0.19)

and for every A e [0, +oc[, p e [1, ~-1 [, fl e L ~ (f~) the values ih (f~, uO,h, fl) in (0.11) converge, as h tends to +oc, to the finite value

i~(~,uo, fl) =min

{J;fhom(Vu)dxq- ~,udx+)~~luiPdx 9

(0.20)

(cf. Theorem 5.1). We emphasize that in this case we do not need to assume

(0.8). When (0.13) is replaced by the linear coerciveness assumption (0.6), the homogenization process must be carried out in the context of B V spaces, but to do this some of the above conditions must be strengthened. Thus we assume that (0.6)+(0.8) and (0.16) are fulfilled, and tat, if f~' is an open set such that f~ C f~P, the following conditions hold 1

uo,h is constant in f~t Cl ~ S for every h E IN and S c C,

(0.21)

the integrals l S ( h x , VUO,h)dx are equi-absolutely

(0.22)

continuous in f~' when h C IN,

uo,h ~ Uo in LI(fY),

(0.23)

fhom(aVu0) C Ll(f~ ') for every cr E ]R.

(0.24)

Then, we prove that fhor, is convex and finite in the whole ]Rn, that (0.9) holds, and that for every )~ e ]0, +oc[, p e ]1, ~n--~_l[, fl e L~176 the values ih(f~, U0,h, fl) in (0.11) converge, as h tends to +cx~, to the finite value ioo(f~, u0, fl) = rain

fhom(Vu)dx +

/ho~m dlD ul

'~-LO fh~176176176 - ")n')d~n-l + Y[ofl'dx+ ~ /o i"Pdx "" ~ BV(')} (cf. Theorem 5.2).

Of course, problem in (0.25) reduces to the one in (0.20) when (0.13) is fulfilled. In particular, when uo,h --0 for every h E ~ , the above described convergence result for the problems in (0.10) follows as corollary.

248

Homogenization of Dirichlet minimum problems...

In both cases, convergence results in Ll(~t) for the minimizing sequences of the problems in (0.11) to minimizers of io~(~, u0, t9) are also proved. In particular, the above results continue to hold if a E L~oc(lRn) is Yperiodic, q E [1, +oc[, f satisfies

f(x, z) < a(x)+ Izlq for a.e. x e

]R n

and every z e IRn,

(0.26)

and if assumptions (0.15), (0.16), (0.18), respectively (0.22), (0.16), (0.24), are replaced by the integrals

/IVuo,hiqdx are equi-absolutely

(0.27)

continuous in ~t when h E IN, respectively by the integrals / I V u o , h

]qdx are equi-absolutely

(0.28)

continuous in f~'when h E IN (cf. corollaries 4.3 and 4.4). Our results are obtained by exploiting De Giorgi's F-convergence theory, together with some recent results and techniques introduced in [11], [9], and

[16]. 1. Notations and p r e l i m i n a r y

results

We first recall some properties of BV spaces. We refer to [21] and [31] for a complete exposition on the matter. Let ~ be an open set. For every u E BV(~) we denote by IDuI the total variation of the lRn-valued measure Du. Moreover, according to Lebesgue Decomposition Theorem, we have

Du(E) - / E Vudx + DSu(E) for every Borel subset E of where we have denoted by Vu the density of the absolutely continuous part of Du, and with DSu the singular part of Du, both with respect to Lebesgue measure. We recall that BV(~) is a Banach space with norm

[L"llBv(a)'ue B V ( 9 ) ~ f luLdx+ ]Dul(~). Jn

249

R. De Arcangelis

If f~ has Lipschitz boundary, B V ( F I ) continuously embeds in La-~-l(f~) and compactly in L p(f~) for every p e [1, ~ [. If f~ has Lipschitz boundary and u c BV(f~), then the null extension u0 of u to ]Rn is in BV(]Rn), and there exists a function in Ll(0f~) (endowed with the ?_/n-1 measure), called the trace of u and again denoted by u, such that D u o = - u n ~ "-1 in Oft. As consequence we have t h a t if f~' is an open set such t h a t -~ c f~', u C B Y ( f ~ ) , and v C B Y ( f ~ ' \ -~), then the function

W --

u v

inf~ ~t \f~ in

is in B V ( f ~ ' ) , v - u e Ll(0f~), and D w = (v - u ) n a ~ n-1 in 0f~. For every Lebesgue measurable set E we denote by IE! its measure. For every f" ]Rn ~ [0, Ac-CX:)[convex, we define the recession function fc~ o f f by .t

f cX~ " z

E ] R n )-->

lim

l=f (t z ) .

t -+ 4- c ~ t

It is well known t h a t f ~ is convex, lower semicontinuous, and positively l-homogeneous. We now introduce the r-convergence theory. We refer to [17] and [14] for a complete exposition on the subject. Let (U, T) be a topological space satisfying the first countability axiom. D e f i n i t i o n 1.1. - For every h C IN let Fh" U --, [-oo, +ec]. We define the r - ( T ) - l o w e r limit and the F - ( T ) - u p p e r limit of {Fh} as F - ( T ) l i m i n f Fh: u C V ~ inf r~ liminf Fh(Uh) " Uh ~ U in T~ h--* + o o

~ h--+ + o o

)

and

F-(T) limsup F h ' u C U ~-~ inf { lim sup Fh(Uh) " Uh --* U in T}. h-++oo

h--++cx~

I f in u one has

F - (T) lim inf Fh (u) = F - (T) lim sup Fh (u), h--,+c~

h--~+c~

we say t h a t in u there exists the F - ( T ) - l i m i t of { F h } , and we define it as

F - (T) lim Fh (u) -- F - (T) lim inf Fh (u) = F - (T) lim sup Fh (u). h--*+c<~

h--++cx~

h--*+c~

I-Iomogenization of Dirichlet minimum problems...

250

We recall t h a t F - (T) lim inf Fh (u) <_ F - (~-) lim inf Fhk (u) <_ h--*+c~

k---*+c~

(1.1)

_< F - (~-) lim sup Fhk (u) _< F - (T) lim sup Fh (u) k---~+c~

h---~+ c ~

for every {hk} C_ IN strictly increasing, u E U,

F - ( r ) l i m i n f F h and F - ( r ) l i m s u p F h h--,+~

(1.2)

h--*+e~

are r-lower semicontinuous in U. For every h C IN, let Fh: U ~ I-co, +oo]. We say t h a t Fh are equicoercive if for every c C IR there exists a compact set K~ C_ U such that UheN{u E U ' Fh( ) <_ c) c_ 1.2. - For every h c IN let Fh" U ~ [-c~, +oo]. Assume that the functionals Fh are equicoercive, and that for every u E U the limit F - (T) limh-_,+~ Fh (u) exists. Then F - (r) limh__,+~ Fh attains its minimum in U, and

Theorem

min~F-(r) k

lim

h---~+cx~

Fh(u)'uCU~)

lim

h--~+c~

inf{Fh(u)'uCU}.

Moreover, if limh_,+~inf{Fh(U) 9 u E U} < +oo, and if {Uh} C_ U is such that l i m h ~ + ~ ( F h ( U h ) - - i n f { F h ( u ) 9 u E U}) - 0, then {Uh} is r-compact, and its converging subsequences converge to solutions of m i n { r - (r) l i m h ~ + ~ Fh (u) " u C U}. We now come to the homogenization of integral functionals. Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), {u0,h} c_ Wllo'~(IRn), and let f~ be a bounded open set. Then, by (0.4), it soon follows that f~ f(hx, Vu)dx < +co for every h c IN, u E Wlloc~(~n), and let us define iv(x)

Fh (f t,.)" u E L' ( f t) ~-~

n

f f l f ( h x , Vu)dx i f u E W l o c ( I R ) , u constant in f t n 88S for every S E C (1.3) +oo

otherwise,

and

fa f(hx, Vu)dx

if u c uo,h + wl'~(f~), u constant in ft n 1 S for every S C C (1.4)

F0,h(f~, "): u C LI(Ft) ~-~ +oo

otherwise.

251

R. De Arcangelis

It is clear that F-(Ll(f~))liminfFh(f~,u) <_F-(Ll(f~))liminfFoh(gt u)

(1.5)

F-(L1 (f~))lim sup Fh(f~, u) <_ F-(L1 (f~))lira sup FO,h(i], u) h --. + c ~

h ~"o + oo

for every bounded open set ~t, u E Ll(f~). The following results are proved in [11]. L e m m a 1.3. - Let f be as in (0.4), and fhom be defined by (0.5). Assume that (0.7) holds. Then there exists L E [0, +cx~[, and, for every bounded open set f~ and each compact subset I4 of f~, there exist {~h} C_ Wol'C~ and h~,K E IN such that 1

r

is constant in -~S for every S E C and h > h~,K,

and O ~ ~Ph ~ l i n ~ ,

Ch = 1 in K,

L ,,,illV~hlllL~(U) _< dist(K, 0Q) ,,,v~.,.

for every h E IN.

L e m m a 1.4. - Let f be as (0.4), E be a measurable set with ]E i < +oo, r ~> O, and (mh} C (LI(E)) n be such that I[mh]](L~(E))~ < r for every h E IN, and mh ~ 0 in (L~(E)) ~. Then lim suPh_.+oo/E f ( h x , mh)dx ~ 2n]EI /'y f(y, O)dy.

T h e o r e m 1.5. - Let Cr be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), and let, for every h E IN, Fh be defined by (1.3). Assume that (0.7) holds. Then fhom is convex and finite in ]Rn, moreover for every bounded open set f] with Lipschitz boundary, u E BV(f~) the limit F - ( L I ( U ) ) l i m h ~ + ~ Fh(f~,u) exists and F-(LI(f~))

lim Fh(f~ u ) =

h - * + oo

fhom(Vu)dx+

f~omkd, Dsu, dIDSul.

The following approximation result easily comes from Proposition 2.6 in [20].

Homogenization of Dirichlet minimum problems...

252

1.6. - Let g : ] R n ~ [0,-4-cx3[ be convex. Then for every bounded open set f~ with Lipschitz boundary, and v E W01'1(f~) there exists {vk} C_ C~(FI) such that vk -~ v in Ll(ft), Vvk --+ Vv in ( g l ( ~ ) ) n, and

Proposition

k--,+~

Finally, we recall from [16] the following relaxation result in B V spaces. 1.7. - Let g: lit n --~ [0, +c~[ be convex, u0 E WlXo'cl(lRn), ft be a bounded open set with Lipschitz boundary, and fY an open set such that fl c f Y . Assume that

Theorem

g(aVUo) E L 1(~t) for every (7 E JR. Then inf { liminfh_~+~/~g(Vuh)dx " {Uh} C_ uo + Clo(f~), Uh --~ u in L l ( f t ) } =

_/ -

g(Vu)dx+ /

g ~ ( dDSu diDsui)d[DSu[ + j f o g C ~ ( ( u o - u)n~t)d~-Ln-1 for every u C B V ( ~ ) .

R e m a r k 1.8. We observe that in [16] a slightly different version of Theorem 1.7 above is proved. In fact in [16], without introducing any open set ~t, it is assumed that

g(aVuo) E L~oc(IRn ) for every (7 E JR, and the same representation formula of Theorem 1.7 is deduced not only for u E BV(ft), but for every bounded open set ft with Lipschitz boundary, and u E B V ( ~ ) . On the contrary, when the open set ~t is fixed a priori, an analysis of the results of [16] proves that the assumptions of Theorem 1.7 are sufficient in order to obtain the representation formula only for every u E BV(ft).

2. E s t i m a t e s

from above

L e m m a 2.1. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), {uo,h} C - W~loc 1'~ (]Rn), and u0 E

R. De Arcangelis

253

m l loc ' l ( ] R n) " F o r e v e r y h C IN let I?h and F 0 ,h be de/~ned by (1.3) and (1.4) " Let ~ be a bounded open set with Lipschitz boundary, and assume that (0.7), and (0.14)+ (0.17) hold. Then

F-(Ll(ft)) lim sup F0,h (ft, u) _< F-(LI(ft))lim sup Fh (~t, u) h--~+c<~

h--*+c~

for every u C Ll(f~) such that spt(u - u0) C_ ft. Proof. Let u c Ll(ft) be such that s p t ( u - u0) C ft. Clearly we can assume that r-(Ll(~))limsuPh~+ ~ Fh(f~, u) < +oo, so that there exists {Uh} C Wllo'c~176n) such that Uh --* u in LI(Ft), Uh is constant in f~ A 88 for every S E C and h c IN, and lim s u p / ~ f ( h x , V u h ) d x h--*+oo

= r-(L~(f~))lim sup Fh(f~, u).

(2.1)

h--,+oo

For every k c IN, let Xk" ]R -~ ]R be a smooth function such that xk(t) -- ( k + 1) i f t < - ( k + 2 ) , xk(t) = t i f - k < t < k, xk(t) = k + 1 i f t > k + 2 , 0 _< X~ _< 1, and set ~ , h = UO,~ + Xk(Uh -- Uo,h),

e k = UO + Xk(U -- UO).

Let us fix ~ > 0, then, since for every k c IN (0.17) yields limsupi{x E f t ' i u h ( x ) - Uo,h(x)l > k}l < h---,+cx~

sup / ~ luh -- UO,h[dX = ~1 / ~ l~- ~01d~, -< kl lim h-~+~ by (0.15) we deduce the existence of kE C ]hi such that

f ( h x , Vuo,h)dx < ~ for every k _> ks. (2.2) lira sup f{ h--,+c~ xEf~:Iuh(x)--UO,h(x)l>k} Moreover, again by (0.15), let A be an open set such that A _C f~, s p t ( u u0) C A, and limsup]

f ( h x , Vuo,h)dx <_ ~,

(2.3)

h--*+oo J F t \ A

and let {~h} C W01'~176 (f t), ha,A E IN be given by Lemma 1.3 with Ch = 1 in A.

254

Homogenization of Dirichlet minimum problems...

For every h, k C IN, t C ]0, 1[ let us now define ~ , , k , h = t 2 (2 -- t ) ( r Wt,k

--

+ (1 -- ~ h ) U o , h ) + (1 -- t ) ( 1 + t -- t2)~O,~,

t2(2

~

t)~k + (1

~

t2)uo,

t)(l+t

wt = t2(2 - t)u + (1 - t)(1 + t - t2)u0. Then, by using also (0.14), it follows that, for every h, k E IN, t c ]0, 1[, wt,k,h C UO,h+ wl'l(fl), and t h a t wt,k,h is constant in 12 N ~ S for every S E C provided h >_ h~,A. By the convexity properties of f we get

/ f(hx, Vwt,k,h)dx <_

(2.4)

<_t / ~ f(hx, t(2 - t)(~hV~tk,h + (1 -- ~2h)VUO,h + (~k,h -- uo,h)VCh))dx+ +(1 - t)/~ f(hx, (1 + t - t2)VUo,h)dx <_ < t2(2 - t)

+ (1 -

a +t(1 - t(2 - t))

~h)VUo,h)dx+

t(2 - t)

f\hX'l( - t(2 - t ) ( U k , h -- uo,h)V~Zh/dx+)

+(1 -- t)/~ f(hx, (1 + t - t2)Vuo,h)dx for every h, k E IN, t E ]0, 1[. We estimate the first term in the right-hand side of (2.4). Again by the convexity of f we have successively

a f(hx, ~2hVSk,h + (1 -- Ch)VUO,h)dx = f f(hx, V~tk,h)dx + f f(hx, ChV~tk,h + ( 1 - r J~ \A JA _<

J~ \A

JA + f

Ja \ A

(1 - r

Vuo,h)dx <<

< f f(hx, Vttk,h)dx + f f(hx, Vuo,h)dx J~ \ A 3~

(2.5)

<_

R. De Arcangelis

255

f (hx, VUo,h)dx+

+ fxea:luh (x)-uo,h (x)l
+f

f(hz, Vuh)dx+

f(hx, X~(Uh--UO,h)VUh+(1--Xlk(Uh--UO,h))VUO, h ) d X + xE~:k
J gt\A

f(hx, Vuo,h)dx < f (hx, ~Tuo,h)dxq-

--< f{xea:l~h (x)-uo,h (x)l___k+2} + f~ea-I~h (~)-~o,h (x)[
f(hx, Vuh)dx+ -

o,h)f(hx,

Wh)dx+

(1 - X~(Uh -- uo,h))f(hx, Vuo,h)dx+

f(hx, Vuo,h)dx < /a f(hx, Vuh)dx+ f

+ fxea:l~h (x)-~,o,h(x)l>k}

f (hx, Vuo,h )dx + [

J gt\A

f(hx, Vuo,h)dx

for every h, k C IN, t E ]0, 1[. Therefore, by (2.5), (2.1), (2.2), and (2.3) we obtain f lim sup I f(hx, ~hV~tk,h + (1 -- Ch)Vu0 h)dx < h--~+cx~ Jgt

'

--

(2.6)

__ kE, t e ]0, 1[. h--,+cx~

We now fix k >_ kE, t C ]0, 1[, and observe that tl~tk,h-- Uo,hliL~(a) <_k + 2 for every h E IN. Then, since s p t ( u - u0) C_ A, by Lemma 1.3 we obtain limSUPh__.+ff~ ~ i(Uk'h--uo'h)V~hldx <-

lim sup dist(A, 0~) h-~+~

\A

l~k,h--UO,hidX=

Homogenization of Dirichlet minimum problems...

256

L

dist (A, 0ft)

ja

\A

I~k-uo[dx =

L

[

dist(A, 0~t) Ja\A

[xk(u-

u o ) l d x = o.

Therefore, by Lemma 1.4 applied with E

- - ~'-~ r - -

t(2--t) (k+2) {mh}={ t(2--t) 1 - t(2 - t) ' 1 - t(2 - t) (ftk,h -

} uO,h)V~)h

,

we get

f hx,

limsup h--~+~

< 2nlgtl ;

1 -- t(2 -- t)

(~k,h -- UO,h)V~2h dx <

(2.7)

f(y, O)dy for every k > k~, t e ]0, 1[.

Finally, again by the convexity of f, and (0.16) we infer

lim sup/a f (hx' (1 + t - t2)Vuo,h)dx <_

(2.8)

h--*+(x)

< limsup r + t - t 2 fJa [ f(hx, (TVUo,h)dx-ih - - , + cx~

(

+ 1l + t -t2

_<

(T

(7

l+t-t2)~

}

f(hx, O)dx <_

(7

M +

(l+t-t2) 1-

(7

~ [gt[ f(y,O)dy

for every t C ]0, 1[ such that 1 + t - t 2 __ (7. So, once we observe that for every k C IN and t E ]0, 1[, wt,k,h -+ Wt,k in Li(~) as h diverges, by (2.4), (2.6), (2.7), and (2.8), we obtain

F- (Ll(ft)) lim sup Fo,h(f~,wt,k) <_lim sup ~ f(hx, Vwt,k,h)dx <_ h---*+cx~

h--,+cx~

< t2(2- t){F-(Ll(f~))limsupFh(~,u)+

26}+

h--,+ec

+t(1 - t(2 - t))2n]ft[ Iv f(y' O)dy+

+(l_t){

1+t-t2 (7

M+(1-

l + t - t2)[ft[ /y

for every k >_ k~, t E ]0, 1[ such that 1 + t - t 2 < (7.

(2.9)

R. De Arcangelis

257

We now remark that for every t C ]0, 1[, wt,k ~ wt in L I ( ~ ) as k tends to +c~, and that wt ~ u in L l ( ~ ) as t increases to 1. Therefore by (2.9), (1.2), and (0.4) the lemma follows letting first k go to +oo, then t go to 1, and finally e decrease to 0. I By using Lemma 2.1 we can prove the estimate from above for the F - ( L 1 ) - u p p e r limit of {F0,h}. We start with the case dealing with Sobolev functions. P r o p o s i t i o n 2.2. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as ii1 (0.4), fhom be given by (0.5), {U0,h} c_ Wllo'cC~(]Rn), s u0 C Wllo'I(IR~). For every h E IN let Fo,h be defined by (1.4). Let f~ be a bounded open set with Lipschitz boundary, and assume that (0.7), and (0.14)+ (0.18) hold. Then

F - ( f l ( a ) ) l i m s u p F o , h ( a , u) <- f fhom(Vu)dx for every u E//,0 -~- wd'l(~'~). d~ h--,+oo Proof. Let u c u0 + w ~ ' l ( f t ) , and let us assume that ffl fhom(Vu)dx < +oo. Let us set u = u0 + v with v E W1'1(~), and take t E [0, 1[. Then, by the convexity of fhom, and Proposition 1.6, there exists {vt,k} C_ C ~ ( f t ) such that vt,k -* tv in Ll(ft), Vvt,k ~ t V v in (LI(f~)) ~, and lim f Shorn(W, k)dX k-++oo Jgt

-

-

i Shom(tVv) dx" Ja

(2.10)

Let us observe now that, by (0.18) and the convexity of fhom, it follows that

fhom (tVv)dx <__t

fhom ( V u ) d x + (1 - t)

fhom t -- i VUO dx ,( +cx3,

from which, together with (2.10), it is not difficult to deduce t h a t (2.11)

for every t E [0, 1[ the integrals f f h o m ( V V t , k ) d x J..

are equi-absolutely continuous in ~ when k E IN. For every k E ]hi we now set ut,k = uo + tvt,k. Then ut,k C uo + C~(~t) for every k E IN, and ut,k --~ uo + t2v. Moreover, the convexity of fhom yields

/ 1 fhom(VUt,k(X)) ~_ tfhom(VVt,k(X)) q- (1 -- t)fhom t I t Vu~ for a.e. x Efl, and every k c IN,

\

258

Homogenization of Dirichlet minimum problems...

by (2.11), Vitali-Lebesgue Theorem, and (0.18) we conclude that lim ~ fhom(VUt k)dx = ~ fhom(Vuo + t2Vv)dx k--*-I-oc

(2.12)

for every t 6 ]0, 1[. Finally, by the convexity inequality

fhom(VUO -4-t2Vv)dx ~ t 2 ~ fhom(Vu)dx -~-(1 - t 2) s fhom(V~to)d2 for every t e ]0, 1[, (0.18), and (2.12) we conclude that there exists {uk} C_ uo + C~(a) such that uk ---, u in Ll(ft), Vuk ~ Vu in (Ll(ft)) n, and lim ~ fhom(Vuk)dx= / k--,+oe

fhom(Vu)dx.

(2.13)

In conclusion, by (1.2), Lemma 2.1, Theorem 1.5, and (2.13) we obtain F - ( L l ( f t ) ) lim sup Fo,h(ft, u) <_lim inf F - ( L l ( f t ) ) l i m sup Fo,h(a, uk) <_ h--~+oe k--~+oe h--~+oe

<_/~ fhom(Vu)dx, that is the thesis.

II

Finally, we treat the case of B V functions. P r o p o s i t i o n 2.3. - Let Cv be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), {U0,h} C_ w'l'~ (]Rn), and u0 r Wllo'1(IRn). For every h c IN let FO,h be defined by (1.4). Let f~ be a bounded open set with Lipschitz boundary, f~' be an open set such that ft c_ ft', and assume that (0.7), (0.14)+(0.17), and (0.24) hold. Then

F-(Ll(a)) limsup Fo,h(f~, u) <_ h--~+o~

<

fhom(Vu)dx+

fhC~m dlD~ul dlD'ul + for every u

6 BV(f~).

a fl~~176

(2.14)

u)na)dT-l"-I

259

R. De Arcangelis

Proof. Let u E BV(~)), and assume that the right-hand side of (2.14) is finite. Then, by (0.24) and Theorem 1.7 there exists {Uh} C uo + Clo(~2) such that Uh --~ u in Li(fft), and

lim L fhom(Vu)dx = h---,+cr =

fhC~m dln*ui dlDSul+

fhom(Vu)dx+

(2.15)

a ff'~m((U~

Consequently, by (1.2), Lemma 2.1, Theorem 1.5, and (2.15), the proposition follows as in Proposition 2.2. I

3. Estimates from below L e m m a 3.1. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), and {U0,h} C_ Wl~o'c~(lRn). For every h c IN let Fo,h be defined by (1.4). Let ~ be a bounded open set with Lipschitz boundary, and assume that (0. 7) holds. Then F-(Ll(~))liminfF~ '

_

fhom(Vu)dx+

fhC~m dID~ui dlD~ul

for every u 6 B V ( f t ) . Proof. Follows from (1.5), and Theorem 1.5.

I

It is clear that Lemma 3.1 does not fit well with the corresponding estimate from above given by Proposition 2.3. To do this, we need to assume (0.8) and, for a given bounded open set ~t, to consider another open set ~t' such that ~ c_ ~t', for which the following conditions hold 1 uo, h is constant in ( f t ' \ ~) N ~ S for every h e IN and S e C, the i n t e g r a l s / s ( h ~ ,

Vu0, )d

are equi-absolutely

(3.1) (3.2)

continuous in Ft' \ ~ when h c IN, uo,h --~ Uo in L I ( ~ ' \ - ~ ) .

(3.3)

P r o p o s i t i o n 3.2. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be giveli by (0.5), {uo,h} C_ Vlflllo'T(]Rn), and

Homogenization of Dirichlet minimum problems...

260

Wllo': ( ] R n ) . For e v e r y h E IN let 120,h be det]Iled by (1.4). Let a be a bounded open set with Lipschitz boundary, f~t be an open set such that f~ c f~', and assume that (0.7), (0.8), and (3.1)+(3.3) hold. Then

U0 e

r-(Ll(a)) liminf F0 h(f~ u) h---,+oo ' ~ >

>

--

d D S U ) d l D S u l n t - f o a fhC~m((Uo--u)nf~)d~-~n-1 f h o m ( V u ) d x + ~ f h ~ m ( dlD~u]

for every u e BV(f~). Proof. Let u c BV(f~), and assume that F-(LI(f~))liminfh--,+oo FO,h(a, u) < +oc, then there exist {Uh} C_ Wllo'~(IR~), and {hk} C_ IN strictly increasing such that Uh E uO,h + wl'~176 for every h C IN, Uh --* u in Ll(f~), Uhk is constant in f~ O ~ S for every S c (J and k E IN, and

r-(L~(a)) lim inf F0 h(a, h--,+oe '

u) =

lim ~ f ( hkx, Vuhk )dx.

k ~+oc

(3.4)

Let e > 0, and let, by (3.2), A c_ f~' be a bounded open set with Lipschitz boundary such that f~ C_ A, and f lim sup I

h--.+oo J A\-~

f ( hx, Vuo,h )dx < ~.

(3.5)

Then, by (3.3), (3.1), and (1.1), it is clear that Uh---*W--

u u0

in f~ inAkf~

L1 in

(A),

and that Uhk is constant in A O h@S for every S r C and k E IN. In fact, given S E C and k E IN, this comes trivially if ~ S C f~, whilst, if h-LkSN (]Rn \ f~) =/= 0, it follows once we observe that (0.8) implies the coincidence of the constant value taken by Uhk in f~ C3 h![S with the one taken by UO,hk in 1 S. Moreover, f

lim inf I

k --, + oo J A

f (h k x, V u hk )dx > F- (L 1(A)) lim inf Fhk (A w) > --

k ---++ oo

>_F-(LI(A))lira

inf Fh(A, w).

h--,+oe

'

--

(3.6)

R. De Arcangelis

261

By combining (3.4) with (3.6), (3.5), and Lemma 3.1, it results

F-(LI(~)) lim inf F0 h(ft, u) = h--*+c~ '

(3.7)

f f = lim infk_~+~ L f(hkx, VUhk)dx - l isuPk__,+~ m ]A\a f(hkx, VUo,hk)dx _> _>

i.

fhom(Vw)dx +

S.

fho~m d[D,wl dID'w I - e >

> dD'u - ~ fhom(VU)dx_V ~ fh~om( d]Dsul)dID~ui+ +

~o fhom ~ (\diD~w[/dl dD'w D~w] a

e.

By (3.7) the proposition follows as e tends to 0, once we recall that m

Dw = (u0 - U)~-~n--1 in Oft.

4. Representation results for homogenized functionals In this section we collect the previously obtained estimates to get some integral representation results, in Sobolev and B V spaces, for the F-(L1)limit of the functionals in (1.4). T h e o r e m 4.1. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), {U0,h} C Wllo'~(]Rn), and uo E Wllo'~(]Rn). For every h e IN let Fo,h be defined by (1.4). Let ft be a bounded open set with Lipschitz boundary, and assume that (0.7), and (0.14)+(0.18) hold. Then fhom is convex and finite in IRn, for every u e uo + w l ' l ( f t ) the limit F-(Ll(ft))limh_~+~Fo,h(ft, u) exists and

F-(L~(~)) h-~+~limF0,h(a, u) =

~ fhom(Vu)dx.

Proof. The properties of fhom follow from Theorem 1.5, whilst the remaining part of the theorem by Proposition 2.2, and Lemma 3.1. m T h e o r e m 4.2. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), {U0,h} C Wllo'F(IRn), and U0 E w~l'l(]Rn).loc For every h E IN let Fo,h be defined by (1 .4). Let ft be a bounded open set with Lipschitz boundary, fY be an open set such that ft C ft', and assume that (0.7), (0.8), (0.16), and (0.21)+(0.24) hold.

Homogenization of Dirichlet minimum problems...

262

Then fhom is convex and finite in ]R~, for every u C BV(~t) the limit F - ( L I ( f t ) ) limh~+cr Fo,h(~t, u) exists and

r-(Ll(f~)) h-*+c~ lim Fo,h(f~, u) =

=

/f~ f h o m ( V u ) d x A - /Ftfh~m ( dlD~ul dDSu )diDSul~-~oa f~c~~

"

Proof. The properties of fhom follow from Theorem 1.5, whilst the remaining part of the theorem from Proposition 2.3, and Proposition 3.2. I In particular, by the above results we deduce the following corollaries. C o r o l l a r y 4.3. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in the first two lines of (0.4), fhom be given by (0.5), {uo,h } C_

Wllo'cC~(]Rn), u0 E Wllo':(]Rn), a e L~oc(]R n) be Y - p e r i o d i c , and q e [1, +c~[. For every h E IN let Fo,h be defined by (1.4). Let ~t be a bounded open set with Lipschitz boundary, and assume that (0.7), (0.14), (0.17), (0.26)1, and

(0.27) hold.

e 0+W0 ,l(a)

/horn

the limit F-(LI(ft))limh_~+~ Fo,h(gt, u) exists and

r-(Ll(n))

lim Fo h(~) u) = ] fhom(Vu)dx. ' ' Ja

h---,+cx~

Proof. We prove that the assumptions of Theorem 4.1 are fulfilled. It is clear that (0.15) and (0.16) follow trivially from (0.26) and (0.27). Let us observe now that (0.17), (0.27), and the Ll(~t)-lower semicontinuity of the functional v E Wl'q(~t) ~-~ fa ]Vviqdx imply that u0 E wl'q(ft). By virtue of this, since by (0.26) and Jensen's inequality it results fhom(Z) _<

ady + inf

Iz + Vviqdy " v E

loc (IR'~),

f

v Y-periodic, Uz + v constant in S for every S c C~ - Jy ady +

IzIq

for every z C IRn, condition (0.18) too follows, in fact fhom(O'~7?.t0) ~

/ y ady + .lXT 01 . c LI(~) for every a c ]R.

In conclusion, the corollary follows from Theorem 4.1.

I

R. De Arcangelis

263

C o r o l l a r y 4.4. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in the first two lines of (0.4), fhom be given by (0.5), {/20,h } C Wllo,C(]Rn), ?-to E Wllo':(]Rn), a E L~o~(IRn) be Y-periodic, and q c [1, +oo[. For every h c IN let Fo,h be defined by (1.4). Let f~ be a bounded open set With Lipschitz boundary, f~' be an open set such that -~ C_ f~', and assume that (0.7), (0.8), (0.21), (0.23), (0.26), and (0.28) hold. Then fhom is convex and finite in lit ~, for every u E B V ( ~ ) the limit F - ( L l ( f t ) ) limh-.+~ Fo,h(f~, u) exists and F-(LI(f~))

--- s fh~

lim Fo,h(f~, u) =

h--++oo

fhc~m( dID*ui dDSu ) d l D * u l + ~oa fh~176176176

Proof. Follows the outlines of the proof of Corollary 4.3, but by exploiting Theorem 4.2 in place of Theorem 4.1. I 5. T h e c o n v e r g e n c e o f i n f i m a a n d o f m i n i m i z i n g s e q u e n c e s In the present section we derive, from the results of w the statements on the convergence of infima and of minimizing sequences. T h e o r e m 5.1. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), ~: IRn --+ [0,-[-(20[ be convex and verifying (0.12), {uo,h} C_ Wlo1coo (IR), n and u0 c wl'l" loc (IR n). Let f~ be a bounded open set with Lipschitz boundary, and assume that (0.7), and (0.13)+ (0.18) hold. Then fhom is convex and finite in IR ~, and (0.19) holds.

a

[0, +oo[, p e [1,

[,

e L

(a)

given by (0.11), and i~(f~, uo,/3) by (0.20). Then the following facts hold: a) ioo (f~, u0,/3) has a t / e a s t one solution, and i ~ (f~, u0,/3) =

lim ih (f~, uO,h,/5) < + ~ ; h---++oo

W if for every h C ]N Uh C uO,h + wl'~176 is constant in ft Cq1 S for every S CC, and lim

h--++oo

{f~t f ( h x , V u h ) d x + / a / 3 u h d x + &

f~q [uhiPdx -

ih(f~,Uo,h,/3)}=O,

then {Uh} is compact in Ll(f~), and its converging subsequences converge to solutions of (0.20). Proof. The properties of fhom come by Theorem 1.5.

264

Homogenization of Dirichlet minimum problems...

For every h C IN let FO,h(f~, ") be given by (1.4), and let A, p and ~ be as above. Then, by (0.13), and Theorem 4.1, it soon follows that for every u C L l ( a ) the limit r-(Ll(a))limh-.+oo{F0,h(a, u ) + f a ~udx+A fa lul pdz} exists and F - (Ll(f~)) h-.+oolim{F0,h(a, u)+ ~ ~udx + X fa lulpdx} -

= { -+-oo ff~ fhom( V u ) d x

-Jr-f ~ / 3 u d x

+ ,~ fa lulPdx

(5.1)

if u E uo A- W 1'1 (~"~) if u e Ll(f~) \ (uo + wl'l(a)).

In fact, if u E uo + Wol'l(f~) is such that fa fhom(Vu)dx A- fFt/3udx + ,k fa lul pdx < +oc, then, Theorem 4.1 provides {Uh} such that {Uh--Uo,h} C_ Wol'Cc(f~), Uh ~ u in Ll(gt), and

;~fhom(Vu)dx

= lim sup ~ f(hx, Vuh)dx. h---*+ or

Now, by making use of (0.13), {Uh} turns out to be bounded in WI,I(~). Consequently, the Rellich-Kondrachov Compactness Theorem implies that Uh --* u in LP(f~). Therefore

F-(Ll(f~)) limsup {F~

+ fafludx + 1 f a lulPdx}

(5.2)

<_limsup { ffaf(hx, Vuh)dx + fa~uhdx + A ~ luhlPdx} = h--. + oo

=~fhom(Vu)dx-t-/

~udx-+.k~'ulPdxforeveryucuo-f-Wlo'l(ft)

.

Conversely, if u c Ll(f~) satisfies r-(nl(~))liminfh--.+oo{Fo,h(f~,U) + < +oc, there exists {Uh} C Ll(f~) and {hk} _C 1N strictly increasing, such that uh --* u in Ll(f~), {Uhk --Uo,hk} C_ Wlo'~176 and

fa/%dx + fa lulpdx}

F - ( L l (~ ) ) li~m +in~f { F~ h ( ~t ' u ) + gf~ ~ u d x + )~gf~ lu lPd X }

(5.3)

Then, (0.13) and the de la Vall~e Poussin Compactness Theorem provide that u c WI'I(~) and that Uhk ~ u in weak-Wl'l(~). Moreover the weakW 1'1 (~t)-closedness of W01'1(~) also implies that actually u c u0 + W 1'1(~).

R. De Arcangelis

265

Consequently, (5.3), again the Rellich-Kondrachov Compactness Theorem, and Theorem 4.1 imply that

F-(Ll(~)) liminf {F~'

+ s fludx + A s u[Pdx}

(5.4)

> liminf /nf(hkx'Vuhk)dx + s fludx + A s

> { f~ fhom(Vu)dx + f~ fludx + A fn [ulPdx --

if u E uo + W1'1 (a) if U e L l ( a ) \ (u0 + W 1'1(a)).

+OC

By (5.2) and (5.4), equality (5.1) follows. Finally, again by (0.13), it soon follows that the functionals u E LI(~) H Fo,h(~, u) + fn fludx + A fn lulpdx are equicoercive in LI(~). In conclusion, the theorem follows from (5.1), and Theorem 1.2, once we observe that, by (0.22), (0.23), (0.13), and the Sobolev Imbedding Theorem lim i h (~, uO,h, fl ) <_

h--++oc

51imsup ]~ f(hx' Vu~

+ ]~ flU~

+ A /~ [u~

< m

T h e o r e m 5.2. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), {U0,h} C_ Wllo'cC~(]Rn), and l'l (IRn). Let ~ be a bounded open set with Lipschitz boundary, ~' UO C W~loc 6~ ~ ow~ ~ t ~uch that ~ c ~,, ~nd ~ u m ~ that (0.7), (0.8), (0.6), (0.16), and (0.21)-(0.24) hold. Then fhom is convex and finite in IRn, and (0.9)

holds. For every ), e ]0, +co[, p e ]1, h-~-I[, and fl e L~176 let ih(~, uO,h,fl) be given by (0.1~), ~ d ~oo(~, ~o, ~) by (0.25). T h ~ the e o l l o ~ g e~ct~ hold: a) ioo(~2,uo, fl) has a t / e a s t one solution, and ioo(~,uo, fl) =

lim ih(~,uo,h,fl) < +oo;

h-++oo

b) if for every h E IN Uh C U0,h § W l'c~ (~) {Uh } is constant in ~ N ~ S for every S C C, and

266

Homogenization of Dirichlet minimum problems...

then {Uh} is compact in Ll(f~), and its converging subsequences converge to solutions of (0.25). Proof. The proof is similar to the one of Theorem 5.1. We develop it by emphasizing the main differences. The properties of fhom come by Theorem 1.5. For every h E IN let Fo,h(~, .) be given by (1.4), and let A, p and ~ be as above. Then, by (0.6), and Theorem 4.2, it soon follows that for every u e L1 (~t) the limit F-(Ll(~t))limh--,+o~{Fo,h(~, u)+fn ~udx+A fa ]ulPdx} exists and

F-(LI(~))

hl_iI~cx~{Fo,h(~-~,U) -Jr-/~~udx -~-~ /~ ,u,Pdx} "--

(5.5)

I f~ fhom(Vu)dx + ffl fhC~om dDSu dlD~ul+ foa fh~m(( uO -- u)nfl) d ~ n - 1 + f~ ~udx + A f~ ]ulPdx if u e BV(a) +oc if u e Ll(~) \ BV(~). In fact, if u C BV(~) satisfies f~ fhom(V?.t)dx-~ f~

)dlD~ul+

fhC~m( atD~ulaD~u

fon fh~m ((to -- u)na)dTl n-1 + fa ~udx + )~fa lulPdx < +co, then, Theorem 4.2 provides {Uh} such that {Uh- tO,h} C_WI'~(~), uh ~ u in LI(D), and fhom(Vu)dx +

fh~m

dlD~ul dlDSul +

fl fhC~m((Uo-- u)n~) dT/n-1 =

f h---~+c~ J~

= lim sup ] f (hx, Vuh)dx. Now, since A > 0 and p > 1, by (0.6) it follows that {uh} is bounded in BV(~t). Consequently, the compact embedding of BV(~) in LP(~), implies that Uh ~ u in LP(~t). Therefore

F-(Ll(~)) limsup {F~

=

fhom(Vu)dx-t-

+ j f ~ u d x + A jf~ [u[Pdx}

fhC~m diDst[ dlDSul+

(5.6)

f~C~m((Uo-u)n~)d~n-l+

R. De Arcangelis

267

Conversely, if u e LI(~) satisfies F-(Ll(f~))liminfh__,+~{Fo,h(f~,u)+ fa~udx + ~ fa lulpdx} < +oo, there exists {Uh} C Ll(f~) and {hk} C_ IN strictly increasing, such that uh ~ u in LI(Ft), {Uhk -- UO,hk} C_ W~'~(~), and F-(L1 (Ft))lim infh..~+oo {F0,h(~~, u)+

9fa/3udx + )~~ lu lPdx } =

(5.7)

:liminf { s f(hkx'Vuh)dx+ Zuh dz+ ]i Then, since )~ > 0 and p > 1, (0.6) and again the compact embedding of BY(ft) in LP(~) provide that Uhk ~ u in nP(f~), and that u e BV(~). Consequently, (5.7) and Theorem 4.2 imply that F-(L1 (f~))lim infh~+~

{Foh(~,u),

+ Jf~t~udx + )~~ lu,Pdx} >_

(5.8)

>- liminf ~ f(hkx'Vuhk)dx+ jfa~udx+)~jf~ lulPdx I r a fhom (Vu)dx

:>

fogt fh~ +oo

+ fa fho~m dDSu dlD~ul + u0 -- u)na) d~'~n-1 + f~ /~udx + )~f~ lulPdx if u C BV(a) if u e n 1(ft) \ BV(f~).

By (5.6) and (5.8), equality (5.5) follows. Finally, since A > 0 and p > 1, again by (0.6) it soon follows that the functionals u e L1 (ft) H F0,h(~, u ) + fa Zudx+ A f~ lulPdx are equicoercive in L 1(ft). In conclusion, the theorem follows from (5.5), and Theorem 1.2, once we observe that, by (0.22), (0.23), (0.6), and Sobolev Imbedding Theorem lim

h--~ + oo

<--limsup /~ f(hx'Vu~

i h (ft, tO,h, ~ ) <_ + ff~ /~U~

T /~/gt lU~

< m

6. A n e x a m p l e This section provides an example proving that the previous homogenization results can be no more true when a single Dirichlet boundary datum is taken into account.

268

Homogenization of Dirichlet minimum problems...

E x a m p l e 6.1. Let n = 2, Cy = {]0, 1/212}, f: (x, z) C ]Rn and fhom be given by (0.5). Let us first prove that

4 5[Z

fhom(Z)

21Z[2 for

every

•

]R n

~

Iz[ 2

(6.1)

z e IF[ n,

from which (0.7) follows trivially. To do this, we observe that for every v c wllo'c (IRn) such that v is Yperiodic and Uz + v is constant in S for every S E C, by Jensen's inequality, Gauss-Green Theorem, and the periodicity of v, it results

'y [z + Vv[2dy = / y

[z + Vv[2dy >_ \10,1/2[ 2

41/

> -

3

(z + Vv)dy

\]o,1/212

I 41/o -- ~

2

(Y\10,1/212

(Uz q- v)nY\lO,1/2[2d'][ n - 1

--

4 -- 3 [z[2 for every z E IRn, from which the left-hand side of (6.1) follows. In order to prove the remaining inequality, we take z c IRn and define v (z) as the Y-periodic extension of --XlZ

v (z)" (xl,x2) c Y ~-~

1 --

X2Z 2

- ( 1 - Xl)Zl - ( 1 -- Xl)Z 1

--

X2Z2

if (Xl, x2) e ]0, 1/2[• if (xl,x2) E ]1/2, 1[•

1/2[ 1/2[

(1 - x2)z2 if (Xl,X2) e ]1/2, 1[• -XlZl - (1 - x2)z2 if (Xl,X2) e ]0, 1/2[• --

1[ 1[,

then v (z) C VVllo'c~(IRn), Uz + v (z) is constant in S for every S c C, and

fhom(z) ~ / y [z + Vv(Z)]2dy = 2]zl 2 Let us now take ~ = Y, and u0: x E ] R n ~ X l , then it is clear that all the assumptions of Theorem 5.1, except (0.14), are fulfilled with u0,h = u0 for every h E ]N, and that for every h E ]hi there cannot be functions in u0 + W0~' (Y) t h a t are constant in each set of the type (~, i or (0, ~)+]~h,0[ 2 for some i E { 0 , . . . , h 1}. Consequently, for every t3 E L ~ ( Y ) , and A C IR, it results

inf { / y f (hx, Vu)dx + / y /3udx + A / y ]u[ dx :

R. De Arcangelis u E uo + W0t'~ (s

1

269

}

u constant in f~ N ~ S for every S E C -- +cx~

whilst, by (6.1), min

{ j f fhom(Vu)dx+jf~ ~udx + A 9fn luldx . u c uo + w l ' l ( ~ ) } <

-~-(:X::).

References [1] Attouch H., Variational Convergence for Functions and Operators, Pitman (1984). [2] Bakhvalov N.S., Panasenko G.P., Homogenization: Averaging Processes in Periodic Media. Math. Appl. (Soviet Ser.) 36, Kluwer Academic Publishers (1989). [3] Bensoussan A., Lions J.L., Papanicolaou G., Asymptotic Analysis for Periodic Structures. Stud. Math. Appl. 5, North Holland (1978). [4] Braides A., Defranceschi A., Homogenization of Multiple Integrals. Oxford Lecture Ser. Math. Appl. 12, Oxford University Press (1998). [5] Braides A., Garroni A., Homogenization of periodic nonlinear media with stiff and soft inclusions, Math. Models Methods Appl. Sci. 5, (1995), 543-564. [6] Brezis H., Sibony M., Equivalence de deux in~quations variationnelles et applications, Arch. Rational Mech. Anal. 41, (1971), 254-265. [7] Briane M., Homogenization in some weakly connected domains, Ricerche Mat. 47, (1998), 51-94. [8] Buttazzo G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes Math. Ser. 207, Longman Scientific & Technical (1989). [9] Carbone L., De Arcangelis R., On the relaxation of Dirichlet minimum problems for some classes of unbounded integral functionals, Ricerche Mat. 48-Suppl., (1999), 347-372; special issue in memory of Ennio De Giorgi. [10] Carbone L., De Arcangelis R., Unbounded Functionals in the Calculus of Variations. Chapman s Hall/CRC Monogr. Surv. Pure Appl. Math. 125, Chapman &=Hall/CRC (2001). [11] Carbone L., De Arcangelis R., De Maio U., Homogenization of media with periodically distributed conductors, Asymptotic Anal. 23, (2000), 157-194. [12] Carbone L., Salerno S., Some remarks on a problem of homogenization with fixed traces, Appl. Anal. 22, (1986), 71-86.

270

Homogenization of Dirichlet minimum problems...

[13]

Cioranescu D., Saint Jean Paulin J., Homogenization in open sets with holes, J. Math. Anal. Appl. 71, (1979), 590-607. Dal Maso G., An Introduction to F-Convergence. Progr. Nonlinear Differential Equations Appl. 8, Birkh~iuser-Verlag (1993). De Arcangelis R., Gaudiello A., Paderni G., Some cases of homogenization of linearly coercive gradient constrained variational problems, Math. Models Methods Appl. Sci 6, (1996), 901-940. De Arcangelis R., Trombetti C., On the relaxation of some classes of Dirichlet minimum problems, Comm. Partial Differential Equations 24, (1999), 975-1006. De Giorgi E., Franzoni T., Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur. (8), 58, (1975), 842-850. Duvaut G., Lanchon H., Sur la solution du probl~me de torsion ~lastoplastique d'une barre cylindrique de section quelconque, C. R. Acad. Sci. Paris Sdr. I Math. 264, (1967), 520-523. Duvaut G., Lions J.L., Inequalities in Mechanics and Physics. Grundlehren Math. Wiss. 219, Springer-Verlag (1976). Ekeland I., Temam R., Convex Analysis and Variational Problems. Stud. Math. Appl. 1, North-Holland American Elsevier (1976). Giusti E., Minimal Surfaces and Functions of Bounded Variations. Monogr. Math. 80, Birkh~iuser-Verlag (1984). Glowinski R., Lanchon H., Torsion ~lastoplastique d'une barre cylindrique de section multiconnexe, J. Mdcanique 12, (1973), 151-171. Goffman C., Serrin J., Sublinear functions of measures and variational integrals, Duke Math. J. 31, (1964), 159-178. Jikov V.V., Kozlov S.M., Oleinik O.A., Homogenization of Differential Operators and Integral Functionals. Springer-Verlag (1994). Lanchon H., Torsion ~lastoplastique d'une barre cylindrique de section simplement ou multiplement connexe, J. M~canique 13, (1974), 267320. Murat F., Tartar L., H-convergence, in "Topics in the Mathematical Modelling of Composite Materials", A. Cherkaev and R. Kohn editors, Progr. Nonlinear Differential Equations Appl. 31, Birk~iuser-Verlag (1997), 21-44. Rauch J., Taylor M., Electrostatic screening, J. Math. Phys. 16, (1975), 284-288. Sanchez-Palencia E., Nonhomogeneous Media and Vibration Theory. Lecture Notes in Phys. 127, Springer-Verlag (1980). Ting T.W., Elastic-plastic torsion of simply connected cylindrical bars, Indiana Univ. Math. J. 20, (1971), 1047-1076.

[14] [15] [16] [17] [18] [19] [201 [21] [22] [23] [24]

[26]

[27] [28] [29]

R. De Arcangelis

271

[30] Zhikov V.V., Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29, (1987), 33-66. [31] Ziemer W.P., Weakly Differentiable Functions. Grad. Texts in Math. 120, Springer-Verlag, (1989).

Riccardo De Arcangelis Universit~ di Napoli "Federico II" Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" via Cintia, Complesso Monte S. Angelo 80126 Napoli Italy E-mail: [email protected]

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 13 T R A N S P O R T OF T R A P P E D PARTICLES IN A SURFACE POTENTIAL

P. DEGOND

1. Introduction The purpose of this paper is the rigorous derivation of a diffusion model for trapped particles in a surface potential subject to collisions with the surface. This physical situation frequently occurs in plasma as well as semiconductor physics. For instance, surface discharges on satellite solar generators induce severe damage when metallic parts are biased (by the plasma environment) at a lower potential than the neighbouring dielectric parts. In this configuration, the electrons emitted from the metallic parts are confined along the surface of the dielectric parts by the surrounding potential and undergo many collisions with the dielectric surface (see e.g. [12] for more details and references). A similar situation occurs for electrons moving inside the inversion layer of an MOS transistor: they are confined in the vicinity of the insulator surface by the potential profile of the MOS structure and undergo many collisions with the insulator surface in their way from the source to the drain. It has been estimated t h a t the electron mobility in an MOS inversion layer is strongly reduced (by a factor 3 at least) compared with that of a bulk semiconductor because of collisions with the oxide surface (see e.g. [10]). Particles colliding with a surface may suffer a large number of physical processes, like specular reflection, attachment to the wall, secondary emission, etc (see e.g. [25]). Our assumption is that particles are reemitted with their incident energy and random directions (in other words, the interaction with the surface is supposed elastic). The resulting macroscopic dynamics is a diffusion process in an extended space consisting of the position and energy coordinates of the particles. The associated model is often referred to in the literature as the "SHE" model (for Spherical Harmonics Expansion, a terminology arising from its early derivation by physicists [26]). It consists of a diffusion equation for the energy distribution function F(x, ~) (i.e. the number density of the particles at position x and energy ~). Although

274

Transport of trapped particles in a surface potential

seemingly too simple, the assumption of elastic collisions often leads to a satisfactory description of the macroscopic dynamics. For instance, the diffusion of particles between two plates subject to a large magnetic field has been correctly described by such an approach [14], [15], both qualitatively and quantitatively. The diffusion dynamics induced by particle-surface interaction has first been investigated in the case of inelastic interactions in [1], [2] (see also [11] for references) where the collisionless motion of particles in between two plates is considered. In the case of inelastic collisions, the resulting diffusion process takes place in the usual position space. However, for this problem, [8] shows that the diffusivity is actually infinite and that a time rescaling is necessary to recover a finite diffusivity. This is the so-cMled 'anomalous diffusion' phenomenon. An alternate proof is given in [21]. In our framework of elastic collisions and confinement by a surface potential, anomalous diffusion can occur if the potential is too flat in the vicinity of the surface. An example of such a situation will be given. The SHE model naturally appears in semiconductor modeling [26], [20] and gas discharge modeling [25], when the collisions of the particles against the medium (phonons in semiconductors, atoms in gas discharges) are assumed elastic. In the physics literature, the SHE model is heuristically derived from a truncated expansion of the Boltzmann equation in Spherical Harmonics [26], [20], [25]. In [18] (in the case of relaxation operators) and in [5] (in the general case), the SHE model is shown to derive from a diffusion approximation of the Boltzmann equation (see also [13] for a review and references). The derivation of the SHE model in connection with surface collision mechanisms is reported in [14], [15] to describe the confinement of particles between two plates subject to a large transverse magnetic field. The SHE model also appears in [6] and [17] in the context of scattering by plane interfaces (a situation which bears some similarities with surface scattering), with applications to the modeling of semiconductor superlattices. The diffusion approximation (or diffusion limit) is the singular limit of the Boltzmann equation when the ratio a of the collision mean free path to the characteristic length scale is small, and when simultaneously, the ratio of the mean time between collisions to the characteristic time scale is of order c~2. This limit has been extensively studied in the literature, in various contexts (see e.g. [23], [7], [4] for neutron transport, [3] for radiative transfer and in [24], [22] for semiconductors). In the present work, we investigate the diffusion limit for the collisionless Boltzmann equation describing trapped particles in a surface potential subject to elastic collisions with the surface. We shall focus on a formal result, leaving the rigorous proof of convergence to future work. We refer to [16] and [6] for rigourous convergence results in related frameworks.

275

P. D e g o n d

The summary of the paper is as follows. In Section 2, the starting point of our analysis i.e. the kinetic equation describing trapped particles in the vicinity of a solid wall is established. The main result of the paper i.e. the formal convergence of the kinetic model to the SHE model as the small parameter a tends to 0 is also stated. Section 3 is devoted to important properties of the boundary collision operator. Then, the main theorem is proved in Section 4. Explicit computations of the diffusion coefficients are possible in a certain number of examples listed in Section 5. Finally, in Section 6, we investigate various extensions of the model.

2. A kinetic model for the transport of trapped particles in a surface potential We consider the domain Ft = IR2 x (-cx~, 0] and denote the position vector by x = ( X l , X 2 , Z ) E ft. A solid wall is supposed to be located along the plane F = 0ft = {z - 0} ~ 1R2. Our aim is to describe the motion of the particles subject to a force field in ft and to collisions with the solid boundary at F. We decompose x into its perpendicular and parallel components _z and z relative to the boundary F: z__= (Xl,X2) C IR2 and z E ( - c o , 0]. Similarly, the velocity vector of the particles v = (Vl, v2, Vz) c ]Ra is decomposed into v = (V, Vz) with v = (Vl,V2) C IR2 and vz E IR. We suppose that the particles are subject to a potential force field F ( x ) - (F1, F2, Fz), where f(x) = -Vxr and r is the potential. Again, denoting by __F= (F1, F2), we have __F = - V x r where Vx denotes the 2-dimensional gradient with respect to _z. The set of particles is described by its number density in phase space (x, v) (or distribution function). It is assumed to evolve without collisions in the domain ft. Therefore, the distribution function f ( x , v, t) satisfies the collisionless Boltzmann equation for (x, v) E ft x IR3" m

m

Otf q- v . V x f

-t- F . V v f

= O,

(2.1)

where the mass is set equal to 1. Expression (2.1) simply expresses the conservation of the number density in phase space along the classical trajectories of the particles. We now need to specify the boundary condition at the boundary F x IRa of the phase space. We introduce the traces (or boundary values) of f on F x 1R3 according to:

7(f)=fl{z=o}, 7+(f)--fi{z=O,+vz>O}. 7 + (f) is the distribution function of the particles exiting the domain ft at the boundary P, while 7 - ( f ) is that of the incotning particles. A well-posed

276

Transport of trapped particles in a surface potential

kinetic problem requires the prescription of the incoming trace. Here, we suppose t h a t it is a function of the outgoing trace through an operator B which expresses the interaction of the particles with the solid boundary:

~'-(f) =

B(~/+ ( f ) ) .

(2.2)

We shall consider an expression of B as follows:

J~

K(x, [vl2/2; w' --, w)r

,

' ES2+ VV C ] R 3 s . t .

(2.3)

v z < O,

where v - ]vIw, Iwl - 1 is the decomposition of v into spherical coordinates, $2 _ {a; C IR3, [w[ = 1} is the unit shere and $~: = {w e S 2,+wz > 0} are the two hemispheres. Expression (2.3) models an elastic bounce against the solid wall, with a random deflection of the velocity direction. K is an integral kernel which describes the reflection law of the particles. Indeed, K ( x , Iv]2/2, w ' ~ w)[wzldw is the probability for a particle hitting the wall at point x with velocity v' - [vlw' to be reflected with the same Iv[ and velocity direction w in the solid angle dw. We shall comment on the physical relevence of this model in Section3. Note t h a t B = B(x, [v[2/2) operates on the angular variable w only while x and Iv[ are mere parameters. We are interested in the situation where the potential r confines the particles close to the boundary F. Therefore, we assume H y p o t h e s i s 2.1 - For any given fixed x E F, the function z E ( - o o , 0] -~ r z) is decreasing and lim r

Z--~ --(:X3

z) - +oc.

This hypothesis could somehow be relaxed but we shall leave this point to future work. We define: -

r

z -

0),

z)

-

r

z) -

9

(2.4)

We note that r 0 ) - 0. Under hypothesis 2.1, the motion of the particles can easily be pictured. After some excursion in the domain of negative z, the particles are attracted back to the wall by the potential field. As they hit the wall, they are reflected elastically into the domain and a new ballistic loop begins. As a result of this succession of bounces, the large scale dynamics in the direction parallel

P. Degond

277

to the boundary should resemble a diffusion process. Therefore, we are led to rescale the longitudinal space coordinate x and the time t according to x' = c~x_,t' - a2t, where a << 1 is the (supposedly small) ratio of the size of a typical ballistic loop to the longitudinal length scale under consideration (e.g. the size of the device). The time rescaling like a2 is typical of a diffusion process. After rescaling, the problem reads (dropping the primes for clarity)"

c~Otf~ + (v_. V s

Vs162

+ - 1- ( v~ O

f~

a)f~

ar Oz OVz

Oz

--0,

(2.5) (2.6)

~,-(f~) = U(-),+ ( f ~ ) ) .

In (2.5), it is implicitely assumed that the potential has only macroscopic scale variations in the direction parallel to the boundary F. We supplement the problem with an initial condition

f~(x, v, t = O) = fi(x, v),

V(x, v) e a x ]R3 .

(2.7)

The goal of this work is to investigate the limit a ~ 0 in problem (2.5), (2.6), (2.7). Defining the measure

5~(x, ~) = 5 ( 1~l~ [2 + r

z) -

),

(2.8)

where 5 denotes the delta measure, we aim at the following formal result: T h e o r e m 2.1 - Under hypotheses 2.1 and 3.2, the solution f~ of problem (2.5), (2.6), (2.7.) f o ~ , ~ a @ c o n v e r g e s ~ ~ ~ 0 to ~ f . . c t i o , 1 F ( z , -~ ivl ~ +

~(z, z), t),

where the function F(x,~, t) satisfies the following diffusion ('SHE') problem: (2.9)

N(~_, ~)O~F + (V~ - V~Oo a~)._J = 0, _Y(z, ~, t) - - D ( z , ~) (V~_ - V~Oo a~) F ,

(2.10)

F(x, ~, t - 0) = Fi(x, e) =

(2.11)

1 j. N(x, e) f1 5~(x, v) dvdz,

(2.12)

J_(z, ~ = o, t) = o .

The coefficients N (the 'density-of-states') and D (the 'diffusivity tensor') are g i v e n by:

N(x_,~) = f S~(x,v) dvdz,

D(x,s) - /X_|

(2.13)

Transport of trapped particles in a surface potential

278

The vector function X_. ---- X(x, v) 'auxiliary problem'

( -Vz-ff~z 0 0 , 0+) 0---~Ov---~

--

(Xl,

~- = v,

~2) is the unique solution of the

~+(_~) = 13*(~y (_~)),

(2.14)

tis ying f v)d dz = O, [or all (x_,E), B* is the operator djoint of 13. Equation (2.14) must be understood componentwise (i.e. Xi is the solution with v~ at the right-hand-side of (2.14) for i = 1, 2). This theorem describes the large scale dynamics of the particles in terms of a distribution function which only depends on time, on the position coordinate along the surface _x and on the 'energy variable' s = 1[vl2 + r z). r is the kinetic energy of a particle when it hits the wall (since then, z = 0 and ~b(x_,0) - 0). When z ~ 0, s is not the kinetic energy of the particle, but rather, its total energy in the potential ~. Note that it is not the actual total energy in the 'true' potential r The quantity N(x_, r r t)dx_dr is the number of particles in a certain elementary volume of phase space. This volume is the cartesian product of the interval of width de about the energy s and the infinite cylindrical volume consisting of all half lines parallel to z originating from the surface element dx_. The quantity N(x_, r162 is the geometrical volume of this region of phase space. It bear similarities with the 'density-of-states' in solid-state physics (hence justifying our terminology). Similarly, _J(_x,r t) is the parallel component to the wall of the total flux carried by the particles belonging to the same elementary volume. In view of this interpretation, the prescription of the initial datum by (2.11) is natural. System (2.9), (2.10) is a diffusion equation in the extended space (_x,r C IR2 x [0, oo) known as the 'SHE' model (see bibliography in Section 1). in the present case, the drift is due to the gradient of the potential r at the wall, which results in an average force F0 = V~r in the direction parallel to the wall. At variance, both the density-of-states and the diffusivity only depend on the transverse profile of the potential described by r Therefore, the longitudinal and transverse variations of the confining potential r play different roles in the limit model. This decoupling is due to our assumption on the different scales of the longitudinal and transverse motions. Formula (2.13) can be understood by means of the coarea formula [19]. For a function ~(z, v), we have, omitting the dependence with respect to x_"

f ~(z, v) 5~(x, v) dv dz =

/o

z(~)

p(z, v/2(E - ~2(z))w)dw 2

)

V/2(r - ~(z))dz,(2.15)

279

P. Degond

where Z(s) = Z(x,e) is the unique root (by virtue of hypothesis 2.1) of the equation ~(z) = s. Z(e) is also the turning point of a particle starting from 1 [2 = ~. In particular, formula (2.15) the wall with directional energy -~]Vz applied to (2.9) gives N(x, e) = 47r

~

0

V/2(e- r

< oo.

z(E)

(2.16)

The auxiliary function X provides the response of the microscopic scale to gradients of the macroscopic variables. System (2.14) is constructed by means of the operator adjoint of the leading order term (in factor of l / a ) of the original equation (2.5). This system has multiple solutions and the integral constraint is used to single out one particular solution. It is easy to see (cf (2.13)) that the limit model does not depend on this choice.

3. The b o u n d a r y collision o p e r a t o r We denote by L2($2~) the space of square integrable functions ~ from S~= into ]R for the measure Iwz[dw. We suppose that B(x,e) is a bounded 2 operator from L2(S+) onto L 2 (52_). Furthermore, we suppose, for all x c ]R2, e > O: H y p o t h e s i s 3.2 - (i) flux conservation: f~ es 2 K ( x , z ; w ' - - + w ) l w z l d w = l ,

V w ' C S +2l

(3.1)

W c S 2 ~' e S+2

(3.2)

0i) Reciprocity:

K(z_,s; w ' - + w) = K(x, ~;-w -+ - w ' ) ,

~

(iii) Positivity: K(x, e; w' --+ w) > 0, for all w c S 2 w' c $2+ _y

(iv) B(~,~) i~ ~ compact o p ~ t o ~ e~om L~(S~) onto L~(S~). Relation (3.1) expresses the conservation of the normal flux of particles of given energy e at the point x of the boundary. Indeed, the magnitude of the incoming normal flux J~- is "

J z (x_, ~, t) = ]~

ES 2

f(x, z = 0, t~!~)I~ll~zl d~.

Transport of trapped particles in a surface potential

280

Using boundary conditions (2.6), we have:

--

[ f(Ivlw')lv]lw'~ldw' - J+, dw'ES~_

(3.3)

where the integrations with respect to w and w t have been exchanged, hypothesis (3.1) has been used and d + denotes the outgoing normal flux. This hypothesis is crucial for the validity of theorem 2.1. In practice, this assumption is not rigorously satisfied because surface interactions, like secondary emission [25] can induce jumps of the electron energy. However, over a wide energy range, these jumps are small and the interaction process can be approximated by the elastic process (2.3) with good accuracy. Even if the discrepancy between the two processes is large, the limit SHE model (2.9)-(2.10) can still give a good qualitative picture of the phenomena (see an example in a slightly different context in [15]). The reciprocity relation (3.2) is a macroscopic effect of the time reversibility of elementary particle-surface interactions. It may not always be true (see [11] for references about the validity of the reciprocity relation), but it considerably simplifies the analysis and we shall take it for granted. By the change a; ito - w in (3.1) and the use of (3.2), we easily deduce the following 'normalization' relation:

K(x_,r

V~E$2

(3.4)

The normalization relation, together with the positivity of K has important consequences. First, from (2.3), 139~(w) for w c S 2_ appears as a convex 2 From Jensen's inequality, we deduce the folmean value of 9~(cv~) over S+. lowing inequality, which bears some similarities with the Darroz6s-Guiraud inequality in gas dynamics [11]:

/s

I zld <_fs

(3.5)

Therefore, the operator norm of B in the space s L2($2_)) of 2 to L 2 ( $2 bounded operators from L2(S+) _ ) is less or equal one" I1~11 _< 1. 2 Furthermore, from (3.4), if p is constant over $+,/39~ is constant over $2_. Therefore, 11/311 = 1. Hypothesis (iv) is a regularity hypothesis on the integral k e r n e l / ( . Its consequences are best expressed if we introduce the specular reflection operator ,.7 which operates from L2(S]_) to L2(S2_) according to ffg~(cz) - ~(w*),

P. Degond

281

where co - (~, coz) and co* - (w_.,-coz). Its adjoint fl* is the specular reflection operator from L2($2_) to L2($]_). Then, the operator Bff* and its adjoint fiB* operate on L2($2_) while B*ff and ff*B operate on L2(S~). By hypothesis (iv), the operators I - BSr*, I - B*fl, etc, are Fredholm operators [9]. By using Fredholm theory and the Krein-Rutman theorem [9], it is possible to prove the following (see [16], [6J)L e m m a 3 . 2 - O) The Null-Spaces N ( I - B * f l ) and N( I - i f * B ) are spanned by the constant functions on $2_ and $2+ respectively. (ii) The range R( I - B*ff) is such that R( I - B*,7) - N ( I - Y ' B ) • Equivalently, the equation ( I - B* ff) f = g has a solution f if and only if fs~+ g(co)[wz[dw = O. Then, the solution f is unique under the additional

constraint fs~+ f (co)lwzldw = O.

4. T h e m a c r o s c o p i c

limit

In this section, we give a proof of theorem 2.1. The proof is divided in four a [2 + r steps. First, we show that the limit is of the form F(x, ~[v z), t). Second, we prove the continuity equation (2.9). Third, we show the existence of the auxiliary function X- Finally, we prove the current equation (2.10).

4.1. The limit is a function of the e n e r g y We suppose that f~ --~ f as c~ --~ 0 in adequate functional spaces which will not be precised any further in the present work. Formally, from (2.5), (2.6), f is a solution of the problem

0 0r

vz Oz

Oz OVz

f = 0,

7 - ( f ) = B(~/+(f)).

(4.1)

We show the L e m m a 4.3 - The solutions of (4.1) are of the form

f (x, v, t) = F(x_, -~

+r

z), t) .

(4.2)

Proof. Obviously, x and t are mere parameters in problem (4.1) and will be omitted in the remainder of the discussion. For (z, v) given, we introduce the change of variables uz -- sgn(vz)V/Vz2 + 2 r

Vz = sgn(uz)V/U~ - 2~b(z),

(4.3)

Transport of trapped particles in a surface potential

282

between Vz E ]R and u~ E D~ = (-ce,-v/2r

u [V/2~(z), oc).

We note u = (v, uz) and f(z, v) - f(z, u). Notice that uz = Vz for z = 0. Then, f satisfies the problem"

Vz(Z, Uz)--~zf = O,

~-(f)

=

B(',/+(f)).

(4.4)

Let u e ]R3 be given and z be such that Uz e Dz, i.e. z e [Z(u2/2),O], where Z(r is the turning point as defined in section 2.1. We shall denote it simply by Z when the context is clear. Integrating the first equation (4.4) with respect to z between Z and 0, we obtain for Uz > 0 that f ( 0 , v , uz) = f ( Z , v , u ~ ) and f ( O , v , - U z ) = f ( Z , v , - U z ) . But f ( Z , v , u z ) = f ( Z , v , - u z ) = f ( Z , v , 0) since Z is the turning point. So, f(O,v, Uz) = f(O,v,-Uz), or equivalently: - y + ( f ) = J*',/-(f). Inserting this equation into the second equation (4.1), we deduce that ( I - B f l * ) ( 7 - ( f ) ) = 0, which, by virtue of lemma 3.2, implies that 7 - ( f ) = F(]vl2/2). Then, going back to (4.4) and integrating it in z on the interval [z, 0] easily leads to (4.2) and concludes the proof. I

4.2. The continuity equation For finite a > 0, we define the average density F ~ and current J~ by F~(x, ~, t)

=

1 f f~ hE(x, v)dvdz, N(x, e)

J~(x,e,t)

=

la / vf~ 5E(x,v) dvdz.

(4.5)

We prove: L e m m a 4.4 - F ~ and J~ satisfy the continuity equation (2.9).

Proof. We multply (2.5) by 5~(x, v) and integrate it with respect to v and z. First, using Green's formula, we have, recalling (2.8) and omitting the arguments in the delta measures"

/(Vz

Oz

~ Iv12 + ~(x, z) - ~

Oz Ov=

= =

f /

Vzf~Sdv[z=O 1

~z/~l~=o~(~l~l

dzdv -

f Vzf~5'-5"~zdVdz o~/ + / Vzf~5' ~dvdz 2 - e)d~ = 0,

(4.6)

P. Degond

because

0~

0r

Oz

Oz

283

and by virtue of the flux conservation relation (3.3). 5 t denotes the derivative of the delta measure. Then, we compute 9

:

= ave-Y~

+ V~_r

= c~ V ~ . J~ - V~_r

,

(4.7)

because f ~f~'e~ez = -(o/o~)(f j . ~ e v e z ) . Finally, collecting (4.6), (4.7) and (4.5) leads to the continuity equation (2.9) for F ~ and Y~. I Obviously, F ~ --~ F as c~ ~ 0. Thus, the continuity equation (2.9) for F and Y will be proved as soon as we know t h a t g~ ~ Y. This is the aim of the next two sections. We start with proving t h a t the auxiliary equation has a non e m p t y set of solutions.

4.3. The auxiliary equation In this section, we consider the most general problem of which (2.14) is a particular case. Indeed, let g(z, v) be given and let us consider the problem of finding X such that:

-~zN

+ O z 0%-7 ~ - g'

(~) =

(~-(~1).

(4s)

Again, we omit the dependence upon the x variable in the forthcoming discussion. We prove: L e m m a 4.5 - Problem (4.8) has a solution if and only if g satisfies

gSE(x, v)dzdv = O. Furthermore, if this condition is satisfied, the solution X is unique under the condition f XS~(x, v)dzdv - 0 and the set of solutions is the one-dimensional linear manifold {X + F(Iv[2/2 + ~) with F(e) arbitrary.

Transport of trapped particles in a surface potential

284

Proof. We use the change of coordinates (4.3). With the same notations as in section 4.1 the problem is then written: 0

-Vz(Z, Uz)-~z ~ = O,

-),+(~) = B*(-y-(~)).

(4.9)

Let u be fixed. We integrate the first equation (4.9) with respect to z 2 between the turning point Z(uz/2 ) and 0 and obtain for u~ > O:

~(0,V,~z)

~(0,~,-~z)

-

~(z,_~,~)=-

0(z,~,~z)l~(~,~)l-~e~,

(4.10)

-

)?(z,_~,-~)=

/; o(z,~,-~)l~z(~, ~)l-~ez,

(4.~1)

Since ~(Z, v, Uz) = 2 ( Z , v , - u ~ ) = x ( Z , v , 0), we deduce that 2(O,v_,-Uz)2(O, v, Uz) - G(v, u~) with

G(v_, uz) =

/;

(O(z, v_, uz) + O ( z , v , - u ~ ) ) IVz(Z, U~)l-ldz .

(4.12)

This is equivalently written ~'-(2) = J ~ ' + (~) + a .

(4.13)

We insert this relation into the last equation (4.9) and obtain

( I - B*J)~/+ (2) - B*G.

(4.14)

By lemma 3.2 (ii), a solution 3`+ ()~) exists if and only if s (13*a)(lulw)lwzldw - O,

which, in view of the flux conservation relation for B* (which results from (3.4)), is equivalent to s~ a ( l ~ l ~ ) l ~ z l d ~

- o.

This relation can equivalently be written, for all s -

0

~1~1~ - ~ =

(~2/2)

) I~zld~

O(z,V, Uz)lVz(Z, U z ) l - l d z

O(z,V, Uz)5 z

lul2/2:

-~lul2 - ~

)(1 ~

-~1~12 -

IVz(Z, Uz)l-lluzldu

~) [uzldu dz. (4.15)

P. Degond

285

Using the change of variables (4.3) to change to the variable v in the integral (4.15), we find" o =

/~J(/R O0

g(z,

1'2 + r ~,~z)5 ( ~[~

- ~) dv

)

(4.16)

dz,

3

which is the condition of lemma 4.5. Therefore, under this condition, there exists a solution 7+()~) of (4.14), which determines the outgoing trace of )~. Then, the incoming trace 7-()~) is determined by (4.13). Once the traces are known, 2(z, u) for all possible values of z are obtained by integrating (4.9) between z and 0. Two solutions of (4.14) for 7 + (2) differ by a constant function of w, i.e. a function of lu] 2 only. Then, the associated 7 - ( 2 ) differ by the same function of I~12 and so do the associated ~. Back to the v variable, we deduce that two solutions of (4.8) differ by a function of Ivl 2 + @(z). A unique solution can clearly be singled out by imposing that

/ xSe(x, v)dzdv = O. This ends the proof. 1 The function g(z, v) = v obviously satisfies the solvability condition of lemma 4.5 by oddness. Therefore, the auxiliary function X defined by (2.14) exists and is unique under the constraint f )iS~(x, v)dzdv = 0. Vie are now ready to derive the current equation. 4.4. T h e c u r r e n t e q u a t i o n L e m m a 4.6 - The current equation (2.10) is satisfied.

Proof.

We multiply equation (2.5) by XSe(x, v). Using Green's formula and the fact that

/("'Oz

or

0~

Oz

Oz

(1..

we compute, omitting the arguments in the delta measures:

K-

-=

=

.

0.0..

( /v,f'x__ddv )

~~

+

) ("

Ivl + ~ ( ~ _ , z ) - .

' 0 0)

)

d.dz

Vz-~z + 0--~cgv-~ X__5 dvdz.(4.17)

z---O

Now, we have, using (2.6) and the second equation (2.14)"

286

T r a n s p o r t o f t r a p p e d particles in a surface p o t e n t i a l

Ivl 2 [ ~_ 7+(f) Js

=

(7+(x) - u* (7- (x)) I~: ld,.,.,= o, -

-

(4.18)

_

and with the first equation (2.14)"

S ( ' '+0) f"

- V z - ~ z 4- 0--~ Ov---~ X_ 5 d v d z =

J,-

v 5 dvdz = aJ" .

(4.19)

Therefore, inserting (4.18) and (4.19) into (4.17), we deduce that K " = a g ~. This a posteriori justifies the definition of XLFrom (2.5) and (4.17), we deduce:

"': - ~

i

"s"-"("v) 'v'z-i (v

~ _ - ~_+ . . ~ ) s. ,_,.(.,v) , v , z .

Taking the limit a --~ 0 and using (4.2), we obtain: J = - S ( v . V~_ - V~_r V,__) F ~ 5~(x, v) d v d z .

But

(.. v . - v.r v~) Y - . .

(V . - V.r ") Y.

So, we get J - -

(i

X_ | v 5,(x, v) d v d z

)(

V a - V~_r

')

which leads to equ. (2.10) and (2.13). This ends the proof.

F, m

5. E x a m p l e s In this section, we consider particular examples of b o u n d a r y collision operators B. A rotationally invariant B means that K only depends on Wz, / ~z' and the angle between _~ and _w', i.e. K ( ~ ' ~ w) - K ( ~ z , ~ z , u~ . u~,), where u~ - w/iw_ I C S 1 and S1 is the unit circle. If in addition, K does not depend on u~ .u~,, B is said to be isotropic. We show t h a t in the case of an

P. Degond

287

isotropic B, the density-of-states and diffusivity can be expressed in terms of the bounce period of a particle, given by

T(x, U2z/2) =

IVz(Z, Uz)l-ldz,

2

(5.1)

(_~,~,~/2)

where u~ is the transverse velocity at the origin z - 0. L e m m a 5.7 - ff B is isotropic, we have

27r~ jfo 1 T(x, ep)dp,

N(z_.,s)

=

D(z__,e)

-- "fie2

/01

T 2 (z_, ep)(1 - p)dp I ,

(5.2)

where I is the 2 x 2 identity matrix. Proof. We first begin with N. Using the change of variables (4.3), and the coarea formula, we have, ignoring the _z-dependence: N(s

--

]r

(~

3

( 1

)

~]ul2-5

"

]Vz(Z, Uz)l-lluzld~tdz

(u~/2)

1]'n ~ T ( u ~ / 2 ) 5 ( 1 -~[~1 ~ - ~ ) I ~ z l d ~ = ~ = -~

~ T(~l~z[~)t~ld~

Using w E B(0, 1) as parametrization of 5;~, where B(0, 1) is the unit ball in N2 and noting that Iwz [dw = &o, we obtain: N(s) = 2~ ]B(o,n T(s(1 -Iw__}2))dw. Changing to cylindrical coordinates in B(0, 1),-we finally get:

/01

N(e) = 47r~

f01

T(e(1 - p2))pdp = 27re

T(sp)dp,

which is the first formula (5.2). Now, we turn to D. We use the notations of section 4.3. First, we note that with g - vi, (i = 1,2), equ. (4.12) gives G(v, uz) = viT(u~/2). Therefore, G is an odd function of vi and since B is isotropic, BG = 0. It

Transport of trapped particles in a surface potential

288

follows from (4.14) and lemma 3.2 (i) that 7+(X,) does not depend on the angular variable w. In fact, we can take "y+ (Xi) = 0 since any other choice will simply add a function of 1vl2/2 + ~ to x~, which will not modify the value of D. From (4.13), it follows that ~/-(Xi) = G. Then, integrating (4.9) between z and 0, we deduce, for uz > 0: x~(z, v, ~z)

=

2 v~ ~-(z, ~z/2),

Y(i(z, v, -Uz)

=

2 v, (T(u2z/2) - T(Z, Uz/2)) ,

where

T( Z, ?.t2/2) z

(5.3)

Z0

[Vz(~,Uz)[-ld~.

Now, it is clear that the vector X is proportional to v, so that the diffusivity D - d I is a scalar with / / 1 ) d XlVl~z(X, v)dzdv - )(lVl(~( ~lu[ 2 -- ~ [Vz(Z, Uz)[-l[uzldudz. Additionally, the term proportional to 7 in X1 is odd with respect to Uz and has a vanishing contribution to d. Thus, using the coarea formula and with the same computations as for N, we get:

-

j/

z
v~T(u~/2)3

(1 ) ~1u12-r [v~(Z, Uz)l-Xluzldudz

=

-~ I J" v~T(u2z/2) ( / z

=

1 / 2 2VlT 2 -~

(Uz/2)(~

=

~2

1~12r2(e(1 _ ]~[2))d~ = 2 ~ 2

(0,1)

(u~/2)

IVz(Z, Uz)l-ldz ) ~ ( ~ 1[ u l 2) - ~

luz]du

( 1~[U[2--s ) Juzldu--~12~s[~,2T2(~W2z),wz,dw 2 p2T2(e(1 _ p2))pd p

71-s 2 jfO 1 (1 - p)T2(ep)dp,

which was the formula to be proved. We now review a certain number of transverse potential profiles ~" (i) Linear potential" ~(z) - - F z , where F is a constant force: T - 2v/-2-~ ~ ,

N = 8V/27rs 3 F

'

D-

47r e 3 3 F2

m

(5.4)

(ii) Harmonic potential" ~p(z)- 7w 1 2z 2 where c~ is the angular frequency of the harmonic oscillator" s T = -,Tr N - 27r2 -~, D - 7ra . (5.5) co w 2w 2

P. D e g o n d

289

(iii) Infinite wall at distance L" ~b(z) - 0 for z 6 [ - L , 0]; r z<-L: Then" T-

1 v/2L---~,

N-

4v~TrLv~,

- +cx~ for

D = +oo.

(5.6)

T h e divergence of the diffusivity is related to anomalous diffusion (see introduction). (iv) E x p o n e n t i a l potential: ~b(z) = r exp(z/A)). This the most realistic potential profile for plasmas or s e m i c o n d u c t o r modeling. ~boo is the potential at infinity and A is the screening (or Debye) length. It does not satisfy hypothesis 2.1 because ~b has a finite limit as z --+ - o c , b u t the t h e o r y is applicable for energies e _< ~boo. After some c o m p u t a t i o n s , we get, denoting e = (g/~)cxD)l/2: 2~ T

A arcsin e

___

N-

D = 16rrA2~booa(e) (5.7)

with t~(e) --

J•o

5(e) =

e arcsin u ~--------~ ~/iu -du u '

~0 e arcsin 2 u e2 -- it2

1 - u 2 udlt.

t, can also be expressed according to t,(e) - e - x / 1 - e 2 arcsin e. As expected, N and D are only defined for s _< ~boo. For s ~ 0, formulae (5.7) reduce to those of the linear potential case (5.4) with F - ~bo~/A. For s -+ ~boo, we have 3 7r 2

N --+ 3Nlin(~b~), where Nlin(~c~) and Dlin(r linear potential case. Since

D --+ ~(-~- - 1)Dlin(~boo), are the value of N and D at s = ~b~ in the

3 71.2

~(-~

1)~6,

we see t h a t the linear and exponential cases give rise to densities-of-state and diffusivities of the same order of m a g n i t u d e t h r o u g h the entire range of admissible energies s E [0, ~b~]. Therefore, it is reasonable to think t h a t , in most physical situations, the linear p o t e n t a l case gives acceptable results, within the error b o u n d s on the physical data. We also notice t h a t the diffusivity in the exponential case is finite even at the b o u n d a r y of the d o m a i n of admissible energies. However, its derivative with respect to the energy is infinite at this point, as it should be.

290

Transport of trapped particles in a surface potential

6. Extensions In this section, we review a certain number of extensions of the model. So far, only elastic surface interactions have been considered in the model. However, as already mentioned, particle-surface interactions can be quite complex. In particular, they certainly involve inelastic mechanisms such as attachment, secondary emission, etc. Aditionally, the medium corresponding to gt may be something else than pure vacuum (like, for instance, a gas, a plasma or a crystal). In this case, the particles can suffer collisions with the medium during their excursion into ft (e.g. collisions with neutral molecules or ions in a gas or a plasma, collisions with phonons and impurities in a crystal). These so-called 'bulk' collisions can be elastic or inelastic. In this section, we shall first discuss bulk and surface inelastic collisions. Then, we shall consider bulk elastic collisions. 6.1. T h e c u r r e n t e q u a t i o n To account for inelastic interactions, we need to suppose that they are a perturbation of order c~2 of the elastic interactions. More precisely, we start from the following kinetic model

c~Otf ~ + (v. V~ - V ~ r

1(0

V,__) f~ + -

Vz

0r O ) f ~ Oz Ovz

= c~Q(f~), ")'- (f~) - (1 - c ~ 2 ) B ( ' ) ' + ( f ~ ) ) + c~2Z(')'+(/~)).

(6.1) (6.2)

where Q ( f ) is the bulk inelastic collision operator while Z(~,+(f)) is the inelastic surface interaction operator. We shall leave the expressions of Q and 27 unspecified since they strongly depend on the precise physics involved. We see that the bulk inelastic operator in (6.1) appears mutiplied by a factor c~, to be compared with the factor c~-1 of the transport operator modeling the confinment in the z direction. Similarly, the boundary condition (6.2) expresses that elastic collisions appear with probability of order (1 - c ~ 2) while inelastic ones appear only with probability c~z. For this model, the following modification of Theorem 2.1 can be proved: T h e o r e m 6.8 - Under the same hypotheses as in theorem 2.1, the same

conclusion holds true for system (6.1), (6.2), except for a modification of the continuity equation (2.9), now written: N(~, e)OtF + ( V ~ - V~r

0~). _J - I ( F ) ,

(6.3)

1). Degond

291

with I(F) given by I(F) = Z(F) + Q(F) and Q(F) dzdv

Q(F)

-F ~

Z(F)

(6.4)

5 (11vl2 - ~) lv~[dv ~<0

+

Z(F) 6

( 1 ) -~lvl 2 - ~

Ivzldv.

(6.5)

z
The "collision operator" I(F) of the SHE model collects the contributions of the bulk inelastic collision operator Q and of the surface inelastic operator f . The structure of :~ shows a loss term (with a minus sign) corresponding to the absorption of particles by the boundary and a gain term (with a plus sign) corresponding to the inelastic reemission. The loss term can be written I - ( F ) = -r, bNF, where Ub = / 2 b ( s

is a collision frequency with the surface, given by

Ub -- N -1 f~.
-

-

g)Ivzldv = 27raN -1

Proof. Only the proof of the continuity equ. (2.9) is modified, and more precisely, the computation of (4.6). Indeed, we get, using (6.2) and the flux conservation relation (3.3)" --2

O~

+r f(~ OzO O()OZOVzO )f'5(~llv`'

)

z) - ~ d~dv =

(J(~/+(f~)) - ~/-(f~)) 5(~lv 1 iI - ~)lvzldv

= c~-2 f~ z<0

=

(j(~/+(fa)) _ ff(.),+(fa))) 5(~[v12 _ e)]vz]dv, z<0

which, in the limit a --~ 0 leads to (6.5). The obtention of (6.4) is straightforward. I1 6.2. Bulk elastic collisions Contrary to inelastic collisions, bulk elastic collisions must be of order 1 (i.e. of the same order as surface elastic collisions) to yield a modification

Transport of trapped particles in a surface potential

292

of the limit SHE model. Thus, we consider the following kinetic model

+ (v. + - 1- ( v~ 0 a Oz

0r 0 ) f ~ Oz OVz

---15(f~), a

~/-(f~) -- B(~/+ ( f " ) ) .

(6.6) (6.7)

where $ ( f ) is the bulk elastic collision operator. We shall restrict to relaxation operators of the form

E ( f ) -- u~(x,e) ( - f + -~1 ~s 2 f(v(e)w)dw ) ,

(6.8)

where e = [vl2/2 is the kinetic energy of the particle, v(e) - (2e) 1/2 and u~ is the elastic collision frequency. The treatment of more general elastic operators is left to future work. The presence of the elastic operator at the right-hand-side of (6.6) results in a modification of the diffusivity. More precisely: T h e o r e m 6.9 - Under the same hypotheses as in theorem 2.1 and the additional isotropy assumption on 13 (see Section 5), the same conclusion holds true for system (6.6), (6.7), except for a modified auxiliary function, now a solution of:

(

~

-~ O~b OVz

E) X

"Y+(X) --

_

(6.9)

Proof. The proof of theorem 6.9 consists of the same four points as for theorem 2.1 (see Section 4). However, we shall solely concentrate on the existence of a solution of the auxiliary problem (6.9) since the other steps can be proved by using the elementary properties of $ (such as particle conservation, entropy, etc, see e.g. [5], [13]). To solve (6.9), we first solve:

o

o

-Vz-~z ~ Oz Ov~ t-ur

) X~ - v~,

V+(__X)- g*(V-(X)).

(6.10)

for i -- 1, 2, and prove that Xi is odd with respect to v~. This implies that

x (z,

- 0,

for all z _~ 0 which shows that Xi is also a solution of (6.9). Of course, this method does not gives the uniqueness of the solution; this question is left to future work. As usual, we forget x_ in the following discusssion.

P. Degond

293

Using the same method and notations as in section 4.3, we obtain, by changing to the variable u:

02i

-Vz(Z,~z)-~z

(6.11)

+ ~,~x~ = v~,

with

a(z, I~J~/2) = .~(z, l~l~/2 - r Integrating the first equation (6.11) between Z(u2z/2) and 0, we obtain

.y- ( ~ ) = ~(o, z ) 2 j ( ~ + ( ~ ) ) + a, with OZ(Zl, Z2)

-

-

F'e(~, [ul2/2)lVz(~, Uz)[ -1 d~

exp 1

G = rite,

T~ - a(O, Z)

/;

,

(c~(Z, z) + a(z, Z))[Vz(Z, u z ) [ - l d z .

Inserting it into the second equation (6.11), we obtain ( I - B ' a ( 0 , Z ) 2 J ) ( 7 + ()~i)) = B*G. Since G is odd with respect to vi, the isotropy assumption on B implies t h a t B*G = 0. Now, because Z < 0, we have a(0, Z) < 1 and the operator norm of B*a(O,Z)2,] is strictly less t h a n 1. It tollows that ( I - B * a ( O , Z ) 2 J ) is invertible and therefore 7 + (2~) = 0. T h e n 7-(2~) = G. Now, integrating (6.11) between z and 0, we deduce, for Uz > 0

~(z,_~, ~ z ) = ~ ~+(z), ~(~, ~_,-~z) = ~ ( ~ ( z , O)T~ - ~ - ( ~ ) ) , ~-+(z) =

/z ~

~(~,z)lVz(~,~)l-~d~,

~-- (z) =

/z ~

a(z, ~)]Vz(~, Uz)[-ld(.

(6.~2)

(6.13)

From (6.12), (6.13), it is clear t h a t 2~ is odd with respect to v~, proving t h a t it is also a solution of (6.9). This concludes the proof, m The computation can be pursued to obtain an almost explicit formula for the diffusivity: D = 2~rs 2

I(e, p)(1 - p)dp,

Transport of trapped particles in a surface potential

294

(

2

'l

l

-

-

+

The computation can be pursued even further in the case of a constant ~e, then leading to

where g(s) = u~T(e), T being the bounce period in the absence of collisions (5.1). The quantity g measures the number of collisions in a bounce period. Finally, in the case of a linear confining potential r = - F z , we obtain

D(e)

=

47re2u-j25(g(e))

5(~)

=

2 1 1 1---~- ~ -t ~2

(6.14) 6 e_~(2 6 6) ~4 t~ + ~ + ~

.

(6.15)

It is easy to check that in the limit u~ --~ 0, expression (6.14) yields (5.4). It is remarkable that, in spite of the complexity of the situation, an anMytic formula for the diffusivity can be obtained. Acknowledgements. The author would express his gratitude to J. P. Boeuf for having suggested this problem and provided encouragements and references. This work has been supported by the TMR network No. ERB FMBX CT97 0157 on "Asymptotic methods in kinetic theory" of the European Community.

References [1] H. Babovsky, On the Knudsen flows within thin tubes, J. Statist. Phys., 44(1986), 865-878. [2] H. Babovsky, C. Bardos and T. Platkowski, Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary, Asymptotic Analysis, 3(1991), 265-289. [3] C. Bardos, F. Golse and B. Perthame, The Rosseland approximation for the radiative transfer equations, Comm. Pure Appl. Math. 40(1987), 691-721 and 42(1989), 891-894. [4] C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. A. M. S., 284(1984), 617-649. [5] N. Ben Abdallah, P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys. 37(1996), 3306-3333. [6] N. Ben Abdallah, P. Degond, A. Mellet and F. Poupaud, Electron transport in semiconductor superlattices, preprint

P. Degond

295

[7] A. Bensoussan, J. L. Lions and G. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. RIMS Kyoto Univ. 15(1979), 53-157. [8] C. BSrgers, C. Greengard et E. Thomann, The diffusion limit of free molecular flow in thin plane channels, SIAM J. Appl. Math., Vol. 52, No. 4, (1992), 1057-1075. [9] H. Brezis, Analyse Fonctionnelle, th~orie et applications, Masson, Paris, 1983. [10] R. Castagn~, Dispositifs semiconducteurs et structures ~l~mentaires des circuits int~gr~s, Vol. 3, Cours de DEA, University Paris Sud. [11] C. Cercignani, The Boltzmann equation and its applications, Springer, New-York, 1998. [12] M. Cho and D. E. Hastings, Dielectric charging processes and arcing rates of high voltage solar arrays, J. Spacecraft and Rockets 28(1991), 698-706. [13] P. Degond, Mathematical modeling of microelectronics semiconductor devices Proceedings, Morningside Institute, Beijing, to appear. [14] P. Degond, A model of near-wall conductivity and its application to plasma thrusters, SIAM J. Appl. Math., 58(1998), 1138-1162. [15] P. Degond, V. Latocha, L. Garrigues, J. P. Boeuf, Electron transport in stationary plasma thrusters, Transp. Th. Star. Phys. 27(1998), 203-221. [16] P. Degond and S. Mancini, Diffusion driven by collisions with the boundary, preprint, 1999. [17] P. Degond, K. Zhang, Diffusion approximation of a scattering matrix model of semiconductor superlattices, preprint, 1999. [18] P. Dmitruk, A. Saul and L. Reyna, High electric field approximation in semiconductor devices, Appl. Math. Letters 5(1992), 99-102. [19] H. Federer,Geometric Measure Theory, Springer, Berlin, (1969). [20] N. Goldsman, L. Henrickson, J. Frey, A physics based analytical numerical solution to the Boltzmann transport equation for use in device simulation, Solid State Electron. 34(1991), 389-396. F. Golse, Anomalous diffusion limit for the Knudsen gas, Asymptotic Analysis, (1998). [22] F. Golse and F. Poupaud, Limite fluide des ~quations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6(1992), 135-160. [23] E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys. 15(1974), 75-81. [24] F. Poupaud, Diffusion approximation of the linear semiconductor equation: analysis of boundary layers, Asymptotic Analysis 4(1991), 293-317. [25] Yu. P. Raizer, Gas discharge Physics, Springer, Berlin, 1997.

296

Transport of trapped particles in a surface potential

[26] D. Ventura, A. Gnudi, G. Baccarani and F. Odeh, Multidimensional spherical harmonics expansion of Boltzmann equation for transport in semiconductors, Appl. Math. Letters 5 (1992), 85-90. P. Degond MIP, UMR CNRS 5640 Universit~ Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex France E-maih degond@mip, ups-tlse, fr

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved C h a p t e r 14

D I F F U S I V E E N E R G Y B A L A N C E M O D E L S IN CLIMATOLOGY

J.I. DfAZ

1. I n t r o d u c t i o n This paper contains an expanded and updated version of my lecture at the Coll~ge de France, on May 1997, where I collected several results on a diffusive energy balance model given by a nonlinear parabolic problem formulated in the following terms: ut - d i v ( I V u [ p - 2 V u )

(P)

e QS(x)/~(u) - G(u) + f (x, t)

~(~, 0) - ~o(x)

in (0, T) x j~4, in A/t,

where A4 is a C ~ two-dimensional compact connectedoriented Riemannian manifold without boundary. We assume T > 0 arbitrarily fixed, Q :> 0, S E L~(A4) and p > 2. The function G is increasing and /~ represents a bounded maximal monotone graph in IR 2 (of Heaviside type). We also consider the associate stationary problem

(PQ,~)

- d ~ ( l ~ l ' - 2 X T ~ ) + G(~) e Q s ( x ) Z ( ~ ) + f ~ ( ~ )

on M .

Through the paper we shall use the notation div(]Vu]p-2Vu) = Apu. Problem (P) arises in the modeling of some problems in Climatology: the so-called Energy Balance Models introduced independently, in 1969 by M.I. Budyko [15] and W.D. Sellers [64]. The models have a diagnostic character and intended to understand the evolution of the global climate on a long time scale. Their main characteristic is the high sensitivity to the variation ofsolar and terrestrial parameters. This kind of models has been used inthe study of the Milankovitch theory of the ice-ages (see, e.g. North, Mengel and Short [60]). The model is obtained from the thermodynamics equation of the atmosphere primitive equations via averaging process (see, e.g. Lions, Temam and Wang [53] for a mathematical study of those equations, Kiehl [50] for the application of averaging processes and Remark 1 for some nonlocal variants

298

Diffusive energy balance models in climatology

of (P)). More simply, the model can be formulated by using the energy balance on the Earth's surface: internal energy flux variation = Ra - Re + D, were Ra and Re represent the absorbed solar and the emitted terrestrial energy flux, respectively and D is the horizontal heat diffusion. Let us express the components of the above balance in mathematical terms. The distribution of temperature u(x, t) is expressed pointwise after standard average process, where the spatial variable x is in the Earth's surface which may be identified with a compact Riemannian manifold without boundary A/l (for instance, the two-sphere $2), and t is the time variable. The time scale is considered relatively long. Nevertheless, in the so called seasonal models a smaller scale of time is introduced in order to analyze the effect of the seasonal cycles in the climate and in particular in the ice caps formation (see Remark 2 for the connection with the associate time periodical problem). To simplify the presentation we assume t h a t the internal energy flux variation is simply given as the product of the heat capacity c (a given constant which can be assumed equal to one by rescaling) and the partial derivative of the temperature u with respect to the time. For a more general modeling see Remark 1. The absorbed energy R~ depends on the planetary coalbedo ~. The coalbedo function represents the fraction of the incoming radiation flux which is absorbed by the surface. In ice-covered zones, reflection is greater than over oceans, therefore, the coalbedo is smaller. One observes t h a t there is a sharp transition between zones of high and low coalbedo. In the energy balance climate models, a main change of the coalbedo occurs in a neighborhood of a critical temperature for which ice becomes white, usually taken as u = - 1 0 ~ The different coalbedo is modelled as a discontinuous function of the t e m p e r a t u r e in the Budyko model. Here it will be treated as a maximal monotone graph in IR 2

m ~(u)=

[m,M] M

u <-10 u--10 u >-10,

(1)

where m =/3i and M =/3w represent the coalbedo in the ice-covered zone and the free-ice zone, respectively and 0 < ~i < ~w < 1 (the value of these constants has been estimated by observation from satellites). In the Sellers model, t3 is assumed to be a more regular function (at least Lipschitz

J.I. Diaz

299

continuous), as for instance m

?.t ~ u i ,

m +( M

) ( M - m)

< < u >uw,

where ui and uw are fixed temperatures closed to - 1 0 ~ both models, the absorbed energy is given by Ra = QS(x)~(u)where S(x) is the insolation function and Q is the so-called solar constant. The Earth's surface and atmosphere, warmed by the Sun, reemit part of the absorbed solar flux as an infrared long-wave radiation. This energy Re is represented, in the Budyko model, according to the Newton cooling law, that is, R~ = B u + C. (2) Here, B and C are positive parameters, which are obtained by observation, and can depend on the greenhouse effect. However, in the Sellers model, Re is expressed according to the Stefan - Boltzman law

Re

--

(TU 4 ,

(3)

where cr is called emissivity constant and now u is in Kelvin degrees. The heat diffusion D is given by the divergence of the conduction heat flux Fc and the advection heat flux Fa. Fourier's law expresses Fc = k~Vu where k~ is the conduction coefficient. The advection heat flux is given by Fa = v.~Tu and it is known (see e.g. Ghil and Childress [35]) that, to the level of the planetary scale, it can be modeled in terms of ka~TUfor a suitable diffusion coefficient ]Ca. So, D = + ( k V u ) with k = kc-+-ka. In the pioneering models, the diffusion coefficient k was considered as a positive constant. Nevertheless, in 1972, P.H. Stone [68] proposed a coefficient k = i~Tul, in order to consider negative feedback in the eddy fluxes. So, in that case the heat diffusion is represented by the quasilinear operator D = div(IVul~Tu). Our formulation (P) takes into account such a case which corresponds to the speciM choice p = 3 (notice that the case p = 2 leads to the linear diffusion). These physical laws lead to problem (P) with Re(u) = G ( u ) - f. In Section 2 we start by presenting some results on the existence and uniqueness of solutions which generalize some previous results in the literature for a one-dimensional simplified formulation. Such simplification considers the averaged temperature over each parallel as the unknown. So, the two-dimensional model (P) is reduced in a one-dimensional model when A4 is the two dimensional sphere and considering the spherical coordinates. Therefore, the model becomes

(p1)

=

0(x)

in ( - 1 , 1) x (0, T), in ( - 1 , 1),

Diffusive energy balance models in climatology

300

with p(x) = (1 - x 2 ) ~ where x = sin0 and 0 is the latitude. Notice t h a t again there is no b o u n d a r y condition since the meridional heat flux ( 1 x 2) ~ luxlp-2ux vanishes at the poles x = J:l. We also include in this section some comments on the free boundaries associated to the Budyko type model (the curves separating the regions { x " u(x,t) < - 1 0 } and {x " u ( x , t ) > - 1 0 } ) . We end the section with a result on the stabilization of solutions as t -+ c~. Some references on the question of the approximate controllability for the transient model are given in Remark 4. Section 3 is devoted to the study of the number of stationary solutions according to the parameter Q, when 13 is not necessarily Lipschitz continuous and p >_ 2. We start by estimating an interval of values for Q where there exist at least three stationary solutions and other complementary intervals for Q where the stationary solution is unique. A more precise study of the bifurcation diagram of solutions for different positive values of Q is available once we specialize foo(x) - C with C a prescribed constant. Then problem (PQ,f ) becomes

(PQ,c)

- div(IVu]p-2Vu) + G(u) + C C QS(x)/3(u) on A/I.

We denote by E the set of pairs (Q, u) c IR + x V, where u satisfies the equation (PQ,c). We show that, under suitable conditions, E contains an unbounded connected component which is S - s h a p e d containing (0, ~ - 1 ( _ C ) ) with at least one turning point to the right (and so at least another one to the left). We end Section 3 with a remark on a simplified version of problem (PQ,c) for which it is still possible to find more precise answers: if Ol < Q < 02, for some suitable positive constants Q1 < 02, then we have infinitely many solutions. More precisely, there exists k0 c IN such that for every k c IN, k _> k0 E ]N there exists at least a solution uk which crosses the level uk = - 1 0 , exactly k times.

2. The transient model 2.1. O n the existence of solutions Motivated by the model background described in the Introduction, we introduce the following structure hypotheses: p _> 2, Q > 0, - (HM) A/I is a C ~ two-dimensional compact connected oriented Riemannian manifold of IR 3 without boundary, - (Hz)/3 is a bounded maximal monotone graph in IR 2 i.e m < z < M ,

~

- -

w

Vz e/3(s), Vs c IR, - (HG) (~ 9IR -~ IR is a continuous strictly increasing function such t h a t (}(0) - 0, and [(j(a)] _> Cla]" for some r _> 1,

,

301

d.I. Diaz

- (Hs) S : M ~ IR, S c L ~ 1 7 6 Sl ~ S(X) ~ SO ~>0 a.e.x C A/I, - (Hr f e L ~ 1 7 6 x (0, T)), ( r e s p . - (H~~ f e L ~ 1 7 6 x (0, oo))), -

(H0)

u0 e L ~ 1 7 6

The possible discontinuity in the coalbedo function causes that (P) does not have classical solutions in general, even if the data uo and f are smooth. Therefore, we must introduce the notion of weak solution. The natural "energy space" associated to (P) is the one given by V := {u: M ~ R, u c L 2 ( M ) , ~7~4u E L P ( T M ) } , which is a reflexive Banach space if 1 < p < oo. Here T M denotes the tangent bundle and any differential operator must be understood in terms of the R i e m a n n i a n metric g given on A4 (see, e.g. Aubin [8] and D~az and Wello [26]).

D e f i n i t i o n 1 - We say that u : j~4 --, IR is a bounded weak solution of (P) if i) u e C ( [ O , T ] ; L 2 ( A 4 ) ) r q L P ( O , T ; V ) A L ~ 1 7 6 x (0,T)) and ii) there exists z e L~(A// x (0, T)) with z ( x , t ) e f l ( u ( x , t ) ) a.e. ( x , t ) e A4x(0, T) such that

L

u(x,T)v(x,T)dA

+

=

-

1o

<

v t ( x , t ) , u ( x , t ) >V, xV dt+

< [~Tu]p-2Vu, V'v > dAdt +

Q S ( x ) z ( x , t) v d A d t +

Vv e LP(0, T; V) N L ~

f v dAdt +

~(u)v dAdt =

u o ( x ) v ( x , O) d A

such that vt e L p' (0,T; Y'),

where <, >V, x y denotes the duality product in V ~ x V.

We have T h e o r e m 1 - There exists at least a bounded weak solution of (P). Moreover, if T -- +co and f verifies (H}~), the solution u of (P) can be extended to [0, co) x A//in such a way that u C C([0, co), L2(]vl)) N L ~ ( A / / x (0, co)) r L~oc(O, oo; V). The above result can be proved in different ways. As in the case of the one-dimensional model (Diaz [19]) we can apply the techniques of Diaz and Vrabie [30] based on fixed point arguments which are useful for multivalued non monotone equations. We start by defining the operator A : D ( A ) C

302

Diffusive energy balance models in climatology

L2(M) ~ L 2 ( M ) , A ( u ) = - A p t + G ( u ) i f u e D(A) = {u c L2(M) : - A p t + G(u) C L2(AA)}. The Cauchy associated problem (Ph)

du --~(t) + A t ( t ) ~ h(t)

t e (0, T), in X = L2(AJ)

u(o) = to,

u0 c L2(A/I),

is well posed (it has a unique mild solution in C([O,T];L2(M)) for every h c L2(0, T; L2(A/t)) by the abstract results of Brezis [14]) since we have Proposition

1

-

Let

u e D(r

(4)

u r D(r

where G(u) =

jr0u G(a)da

with D ( r

{u C L2(M), Vu e L P ( T M ) and

f ~ G(u)dA < +c~}. Then i) r is proper, convex and lower semicontinuous in L2(AA). ii)A = 0r and D(A) = L2(M), and iii) A generates a compact semigroup of contractions S(t) on L2(A/~). Besides, from Brezis [14] we know that u, solution of(Ph), verifies that u E LP(O,T;V), v/tut C L2(O,T;L2(A/~)), u c WI'2(5, T;L2(AJ)), 0 < 5 < T. Let us prove the existence of solutions for the problem (P) via a fixed point for a certain operator s Let Y = LP(O,T; L2(A/~)) and define ~_." K ---, 2 Lp(~ by the following process: Let us define

K = {z e LP(O,T; L ~ ( M ) )

: [Iz(t)[In~(M) <_ Co a.e.t e (0, T)}

with Co = Q S 1 M + IIflIL~(O,T;L~(~)). Now, we fix u0 C L2(y~4) and define the solution operator (or generalized Green operator) I0 : K ~ C([0, T]; L2(A/~)) by Io(z) = v solution of (Ph) associated to h _-- z. Given f E LP((O,T);L2(y~H)), we define the superposition operator ,~:LP((O,T);L2(A/~)) ~ 2 Lp((~ by ~ ( v ( t ) ) = {h : h(x, t) e QS(x)~(v(x, t)) + f ( x , t) a.e. z e A/I}. Finally, we define

s

= {h E LB(O,T; L2(Aj)): h(t) E .P(Io(z)(t)) in L2(A//)a.e. t c (0, T)}.

It is not difficult to check (see Diaz and Tello [26]) that using the results by Vrabie [71] and Diaz and Vrabie [30], s verifies the assumptions required to apply a version of the Schauder-Tychonoff Theorem due to Arino et al. [7] and hence the existence of a local solution to (P) is proved.

J.I. Diaz

303

In order to complete the proof of Theorem 1, we are going to show t h a t the solution u can be continued up to t = oo, when (H~ ~ is fulfilled. Taking u as test function we get

/M utudA+ /M [VulPdA+ /" G(u)udA =/~aQSzudA+j'MfudA,

zcl3(u).

(5)

Then

2l ddtL

luledA§

L

IVulPdA + c L

lul2dA < C + ~ll~ll~(~)

and by Gronwall's inequality Ilu(t)llL2(Z4 ) _< k Vt > 0 with k independent of t. By a well known result (see e.g. Theorem 4.3.4 of Cazenave and Haraux [16] ) u can be extended to (0, oc) and so u E C([0, oc); L2(Ad)). To see t h a t u c L ~ ( A d x (0, oc)) we introduce g(x, t) and u_(x, t), given as the unique solutions of the problem

{ Et - ApE + ~(~) = MQS(x) + f + (x, t) ~(0, x) = u + (x) = max {0, uo(x)} and

o n M x (0, oc) on.A/[

{ ~_~_/x~_ + ~(~) - mQS(x) + f - ( x , t) u_(0, x) = u o (x) =

m i n {0,

uo(x)},

respectively. Since the operator - A p u + 6(u) is T-accretive in L2(A4), it is easy to see that u(x, t) < u(x, t) < E(x, t) which proves the assertion once that

ll~llL~((0,oo)•

~ max{ilu0+ltoo, ~-l(llMQSlloo + IIf+l[oo)}

II~_liL~((0,oo)•

_~ max{ll~olloo, G-l(llmQSiloo + ll/-I[oo)). m

Remark 1. More realistic energy balance models are formulated in terms of functional equations adding some non local terms to problem (P) . So, for instance, in linking the albedo to the surface temperature u alone, one neglects the very important long response times the cryosphere exhibits. E.g. the expansion or the retreat of the huge continental ices heets occurs with response times of thousands of years, a feature which Bhattacharya, Ghil and Vulis [13] proposed to incorporate by substituting u by a long term average of u, e.g. w(x, t) := f~ T V(x, s)u(x, t + s)ds with T ~ 10 4 years with

304

Diffusive energy balance models in climatology

~/(.,-T) - 0, ~(., s) > 0 for s E ( - T , 0] and f ~ T ~/(., s)ds - 1. Of course, one can refine this procedure by having independent ice- and snow-lines. In that case, one understands ice-lines as the boundaries of regions that are covered by continental ice-sheets or huge glaciers (slow response times in comparison with the ten-year mean), whereas snow-lines refer to boundaries of regions where the variations in ice- or snow cover occur on the time-scMe of u. This approach was chosen in Diaz and Hetzer [25] (see also Hetzer and Schmidt [48] and Hetzer [45],[46], [44]). On the other hand, if we disregard the latent energy stored in continental ice sheets and glaciers, the internal energy flux is given by e(x, t) = c(x)u(x, t) with c the heat capacity which varies considerably with x due to the land-water distribution. However, a more accurate modeling suggests to set e(x,t) = c ( x ) u ( x , t ) + h(w(x, t)) where c denotes the thermal inertia and h(w(x,t)) stands for the latent energy density due to huge ice accumulations. This approach is closely related to the one for the Stefan problem (cf., e.g., Meirmanov [55])with the obvious change that h should depend on the long-term temperature mean w rather than on u in view of the time scales relevant for the latent fluxes. Here h is a nonnegative bounded decreasing function with derivative ~ having compact support. Using that et = [cu + h(w)]t = cut + ~(w)wt and observing that wt - f ~ T ~/(s, .)ut(., t + s)ds in case that u is sufficiently smooth, one obtains cut + ~(w)~(., O)u - ~(w) f ~ T %(., s)u(., t + s)ds via integration by parts for et. Collecting all terms one is led to a non linear and nonlocal quasilinear parabolic problem. Some results on the existence of solutions for such a model can be found in Diaz and Hetzer [25]. We also mention the treatment made in Bermejo, Diaz and Tello [12] for the study of the general case c = c(x) (but without any nonlocal terms) and study of the multi-layer model made in Hetzer and Tello [49].

Remark 2. A more realistic description of the incoming solar radiation flux QS(x) is obtained by replacing it by a time depending function Q(x, t) under the general assumption Q(x, t) is T - p e r i o d i c a l in time and Q(x, t) >_ O. This last inequality allows to consider the polar night phenomena (time where Q(x,t) = 0 for some subsets of the manifold A4). The consideration of the periodicity of the forcing term is motivated by the seasonal variation of the incoming solar radiation flux during one natural year. The existence of periodical solutions for the associated model was the object of the paper Badii and Diaz [9]. The existence of periodic solutions for the Sellers type model was obtained by considering the Poincar~ map F associated to the C a t c h y problem (P), i.e. the operator assigning, to every initial data, the solution of (P) evaluated after T-period. We prove that F is a continuous, compact and pointwise increasing map and so, the Schauder fixed point

.l.I. Diaz

305

theorem can be applied. The existence of a periodic solution for the Budyko type model needs some different arguments. The Poincar~ map can not be well defined as a univalued operator and so we apply a variant of the Schauder-Tychonoff fixed point theorem for a suitable multivalued operator. Remark 3. For different purposes it is useful to get existence results via regularization of the multivalued term/3(u). See, e.g., Xu [72] and Feireisl and Norbury [33] for some special formulations when p = 2. In our case it can be obtained as an easy adaptation of the results of Section 3. We also mention some results on the numerical approach due to Lin and North [51], Hetzer, Jarausch and Mackens [42], Bermejo [11] and Diaz, Bermejo and Tello [12]. 2.2. O n t h e u n i q u e n e s s of solutions The question of uniqueness has different answers for the different coalbedo functions under consideration depending on whether the coalbedo is supposed to be discontinuous or not. For the Sellers model (/3 locally Lipschitz), the uniqueness is obtained by standard methods (see e.g. Diaz [19]). Nevertheless, in the Budyko model (/3 multivalued), there are cases of nonuniqueness (in spite of the parabolic nature of(P)). The first nonuniqueness result in this context seems to be the one given in Diaz [19] where infinitely many solutions are found for the one-dimensional model (p1) for any initial condition u0 satisfying u0 C C ~ ( I ) , uo(x) = uo(-x) Vx C [0, 1], ] u0(0) -- - 1 0 , u(0k) (0) = 0, k - 1, 2 u~(1) - 0 , U~o(X) < 0, x E (0, 1).

(6)

Notice that these initial data u0 are very "flat" at the level-10. A similar non uniqueness result for the Budyko model with a suitable initial d a t u m carries over to the two-dimensional model when A/I = S 2. Each solution ul(x,t) of (p1) generates a solution u2(x,y,t) of 2D model by rotation about the axis through the poles (notice that the initial datum u2(x, y, 0) is independent of the longitude), i.e. u2(x, y, t) = ~t 1 (sin0, t) where (x, y) E S 2 with latitude 0. It is not difficult to prove that u2 is asolution of (P) for the initial datum Ul sin 0, 0). Other non uniqueness results can be found by using selfsimilar special solutions as in Gianni and Hulshof [37]. Since non uniqueness of solutions may arise, it is useful to know (see Diaz and Tello [26]) that in any case problem (P) has a maximal solution u* and a minimal solution u. (i.e. u* and u. are solutions of (P) such that everysolution u of (P) verifies that u. < u _< u* in(0, T) • ~r In order to obtain a criterion for the uniqueness of solutions for Budyko type models

306

Diffusive energy balance models in climatology

we introduce the notion of nondegeneracy property for functions defined on M. 2 Let w C L ~ ( A d ) . We say that w satisfies the strong nondegeneracy property (resp. weak) if there exist C > 0 and co > 0 such that for any e e (0, e0), ]{x e A/t : ] w ( x ) + 10] _ e)] < Cs p-1 (resp. ]{x e ,~4 : 0 < ] w ( x ) + 10[ < e}] _ Cep-1), where [E[ denotes the Lebesgue measure on the manifold Ad for all E C A/t.

Definition

T h e o r e m 2 - i) Assume that there exists a solution u of (P) suchthat u(., t) verifies the strong nondegeneracy property for any t C [0, T]. Then u is the unique bounded weak solution of (P). ii) There exists at most one solution of (P) verifying the weak nondegeneracy property. The proof is based in the fact (adapted from Feireisl and Norbury [33]) that fi generates a continuous operator from L~(A/t) to Lq(M), Vq C [1, c~), although/3 is discontinuous, when the domain of such operator is the set of functions verifying the strong nondegeneracy property. More precisely, we have (see Diaz [19], Diaz and Tello [26]) L e m m a 1 - (i) Let w, ~v C L ~ ( M ) and assume that w satisfies the strong nondegeneracy property. Then for any q E [1, c~), there exists C > 0 such that for any z,~ C L ~ ( M ) with z(x) e Z(w(x)) and ~(x) e Z(~(x)) a.e. x E All, we have II z - ~ IIL~(M)_< (bw - bi) min{C II w - ~ II(p-1)/qL~(M) , I M I l / q } .

(7)

(ii) If w, ~v C L ~ (All) and satisfy the weak nondegeneracy property then (z(x) - ~(x))(w(x) - ~v(x))dA -< (b~ - b~)C II w - ~ IIpL = ( ~ )

/

9

(s)

/dec of the proof of Theorem 2. Assume that there exist two bounded weak solutions u and ~ of (P), where u verifies the strong nondegeneracy property, i.e. ut - ApU + G(u) = Q S z + f and ~t - Ap~ + ~(~) = QS~ + f , in (0, T) x A/t, for some z E ~(u) and ~ C/3(~). Taking ( u - ~) as the test function we get

2 dt /

lu(t) -

~(t)12dA +

< IVu(t)lP-2Vu(t)-

L

(G(u) - G ( f i ) ) ( u -

~)dA+

]V~(t)IP-2V~(t), V u ( t ) - V~(t) > d A -

= Qf S ( x ) ( z ( x , t) - ~(x, t))(u(x, t) - ~(x, t))dA. JM

(9)

307

d.I. Diaz

By using the embedding V ~ L + ( . M ) if p > 2 and V c L " ( . M ) for all a E [1, c~) if p = 2 (recall that Ad is a two-dimensional compact Riemannian manifold: see, e.g. Aubin [8]) we arrive at ld 2 dt 11u - 5 1]~2<~/I)

~

Co

(CiQ II S []L~(A4) +0o

IF ~ -

)[]u

~ Cl,p,~

-

fi P IIL=(M> +

~ [IL2(M), ~

(10)

in the case p > 2 and 1 d 2 dt 1] u

u II~<M)

--

(CzQ It s

}2~4]~ C1,2,c r

IIL~<M)

~

- ~ ,

(11)

for the case p = 2 where e and a = a(E) .Now, we distinguish two cases:

Co

Cl,p,~ -< 0 and p > 2, then

CASEI: ifCtQ]lS]]~ ld

and the result holds by Gronwall's Lemma. 1

If p = 2 and CzQ [] S []~

<0

C1,2,a -

II ~-~

I1~(~)

-< ~ II ~o- ~o II~(~)

2E

c1,2,a

(e2t_l)

_<

and since the above inequality is true for all c, we conclude the uniqueness. 1 > 0, we consider a suitable rescaling CASE 2- if CiQ ]l S [l~ cl,p,c~ (A//~-+ ,A~4a) given by the dilatation D of magnitude 6 > 0 on the manifold (A/I, g), D " A/t C IR 3 -+ IR a, D ( x ) = Yc = 6x. So, if u is any function defined on +9l, its local representation in the new coordinates is 5 " A//a -+ IR, u(~) = u(~) and we have 05 ~ (- 5 : )

1 0 u 5: = 5 Oxi (~)

i = 1,2, 3.

So problem (P) in the new coordinates becomes

(Pa)

-6PdivM~ ( ] V M ~ S I P - 2 V M ~ 5 ) - ~ ( 5 ) e Q S f l ( 5 ) + f (0, ~) - u0(-~ )

{Su

in (0, T ) • on Aria.

Diffusive energy balance models in climatology

308

Clearly, if 5 is a solution of (P~) then ~t(Sx, t) is a solution of (P). Moreover, the uniqueness of (P~) implies the uniqueness of (P), and conversely. Let us see that there exists 5 > 0 such that the solution of (P~) is unique. Let u~ and 5~ be two solutions of (P~) with u~ verifying the strong nondegeneracy property. Arguing as before we arrive at

1d f lu~(t)_fi~(t)12dA~ 2 dt J~ 5 < IVu~Ip-2Vu~ -IVfi~IP-2Vfi~, Vue - Vt~ > dA~

+5 p f J.~4 5

+[

-

-

=

Q [

-

-

J,.~[ 5

JiM 5

for some z~ E/~(u~) and ~ E / ~ ( ~ ) . Here, S~ is defined by S~ :Ad~ -~ ]R, S~(2) - S ( ; ) . (10) and (11) allow us toestimate u~ - ~ for u~ and ~ solutions of (P~) ld 2 dt

~__(C1,5 Q I] S5 + II ~ -

IIL~(.Ad~) --

C1,2,a,5

C~ ( M ~ )

(12)

~;~ II2

L2(JM~) nc C1,2,a,5"

A careful study of the dependence on 5 of the involved constants (see Diaz and Tello [26]) allows us to see that if we define the constant

K.,~ = G,~Q

II s~ IIL~<M~>- c1,2,o., 5

we have that lim Kp,~ = 0. This reduces the proof to Case 1 and the proof 5--.0

of (i) follows. For the proof of (ii) we use the second part of Lemma 1 and SO

ld

2 dt II u - ~ II~<_(CdQ II s IIL~(~)

Co

-Cl,p-----~q)II~ -

-

~

IlL +Co II ~ - ~ tIN

where Cd is the constant of the weak nondegeneracy property (Lemmal). The uniqueness follows as in (i), by studying the sign of the constant CdQ II 1 S I[L~(jk4) Cl,p,q and by rescaling when it is negative, m Remark 4. It is possible to give several sufficient criteria for the nondegeneracy property. For instance, in the one-dimensional case and p = 2,

J.I. Diaz

309

if u0 E C 1 ( ( - 1 , 1)) is such that there exists e0 > 0 such that the set {x E ( - 1 , 1) : luo(x)+ 101 _< e0} has a finite number of connected components Ij with j = 1, .., N and for any j there exists xj c Ij such that uo(xj) = - 1 0 , a n d luox(x)l >>50 for some 50 > 0 and any x C /j close to xj, then there exists a solution u(x, t) satisfying the strong nondegeneracy property on (0, T*) for some T* (see Diaz and Tello [26]). Some results on solutions with IVul ~- 0 on the level where /3 becomes multivalued for a similar bidimensional problem are given in Gianni [36]. 2.3. O n t h e free b o u n d a r y for B u d y k o t y p e models The discontinuity of the albedo function assumed in the Budyko model (/3 multivalued) generates a natural free boundary or interface ~(t) between the ice-covered area ({x e A/I : u(x,t) < - 1 0 } ) and the ice-free area ({x c A/l : u(x,t) > - 1 0 } ) . The free boundary is then given as r = {x e A// : u(x, t) = -10}. In Xu [72], the Budyko model for p = 2 is considered in the one-dimensional case. He shows that if the initial d a t u m u0 satisfies

t o ( x ) - - U o ( - x ) , uo e 6 3 ( [ - 1 , 1]), U'o(X) < 0 for any x e (0, 1) and there exists r e (0, 1) such that (to(x) + 1 0 ) ( x - r < 0 for any x e [0, r

(((0), 1],

then there exists a bounded weak solution u of (P) for which one has r = {r U {r with x = r a smooth curve, r = r and r e C ~ ( [ 0 , T * ) ) , where T* is characterized as the first time t for which 4+(t) = 1. He also gives an expression for the derivative r (t) (some related results for a model corresponding to p(x) = 1 can be found in Feireisl and Norbury [33], Gianni and Hulshof [37] and Stakgold [67]). We point out that the uniqueness result can be applied for such an initial datum. For the study of the free boundary in the bidimensional case see Diaz [22] and Gianni [36]. The interpretation of the size of the separating zone r for other models is in fact a controversial question. So, some satellite pictures (Image of the Weddell sea taken by the satellite Spot on December 10, 1987) show that the separating region between the ice-free and the ice-covered zones is not a simple line on the Earth but a narrow zone where ice and water are mixed. Mathematically it could correspond to say that the set

M(t) = {x E All: u(x, t ) = - 1 0 } is a positively measured set. In the following we shall denote this set as the mushy region (since it plays the same role than in changing phase problems, see e.g. D i a z , Fasano and Meirmanov [23]). Using the strong maximum principle, it is possible to show that if p = 2 the interior set of the mushy region M(t) is empty even if the interior of

Diffusive energy balance models in climatology

310

M(0) is a n o n e m p t y open set (see Gianni and Hulshof [37]). As we shall see, this is not the case when p > 2 (recall t h a t p - 3 in Stone [68]). A o

necessary condition for the Budyko model (with R~ - B u + C) for M(t)7~ is t h a t C - 10B E [~iQS(x), [3~QS(x)] for a.e. z e Ad. (13) It is possible to show t h a t if p > 2, this condition is also sufficient. Here we merely present a result for the one-dimensional case (see Diaz [22] for the bidimensional case)" T h e o r e m 3 - Let p > 2. Assume (13) and uo E L ~ ( I ) such that there exist xo E I and Ro > 0 satisfying

M(O) = {x e I ' u o ( x ) = - 1 0 } D B(xo, R 0 ) ( = {x e

I'lx

-

x01 <

R0}).

If u is the bounded weak solution of (P) satisfying the weak nondegeneracy property, then there exist T* c (0, T] and a nonincreasing function R(t) with R(O) = Ro such that M ( t ) = {x E I ' u ( x , t ) = - 1 0 }

D B(xo, R(t))

for any t C [0, T*). Proof. We shall use an energy method as developed in Diaz and Veron [29]. Given u bounded weak solution of (P), we define v = u + 10. As in L e m m a 3.1 of the above reference, by multiplying the equation by v we obtain that for a.e. R C (0, Ro) and t c (0, T), we have

1

2

I.

(xo,R)

Iv(x,t)12dx +

io'l.

+B (xo,R)

< (xo,R)

plv.I,-2v..

(xo,R)

p(z)lvxl'dxdr

I~(x, r)12dxdr <_

~vdsdT +

{ Q S z - C + lOB}vdxd~(xo,R)

= I1 + I 2 , where p(x) - (1 - x2)~, S(xo, R) - OB(xo, R) = {xo - R} t_J {xo + R} and z ( x , t ) c 13(u(x,t)) for a.e. x E B(xo, R) and t c (0, T]. We introduce the energy functions

E(R, t)

-

So l;

p(x)lv.l'dzd~

(xo,R)

b(R, t)

-

sup ess [ 0
JB(xo,R)

Iv(x, ~)12dx.

J.I. Diaz

311

Using Holder's inequality and the interpolation-trace Lemma of Diaz -Veron [29], we get

I 1 < (OE -~ (R, t) )(P--1)/P(]i t f S(xo,R)IvlPdxdr ) a/p < _
( OE

, t)) (p-1)/p

X (E(R, t) 1/p + R6tl/Pb(R, t)l/2) ~ b(R, t) (1-~ where

0 = p/(3p-

2) and 6 = - ( 3 p -

Using the assumption (13), we have ~(-) = [ ( C Then applying Lemma 3 we get

I2 < -- ( M - rn)Q II S

IIL~(I)

c, f0

2)/2p.

IOB)/QS(.)] e

I! v ( r ) i [ ~

~9(-10).

(B(xo,R) dT.

From Theorem 4 of Rakotoson and Simon [62], we have the estimate

1] V

ltL ~

(J)---~C1 II v~ IIL~(J:~) + c 2 II v IIs~(J:~),

vv e v,

(14)

for some positive constants independent on the interval J. Then we obtain I2 _< (M - m)Q

II S IIL~(I) C'(C1E(R, t) + tC3(R)b(R, t)),

where now

c~(R)

=

(L(~o,~) p(~)~d~)"-~

+

As in the proof of the uniqueness, we can a s s u m e C 1 small enough without loss of generality. Then, there exists T* E (0, T] and A c (0,1] such that

,~(E(R, t) + b(R, t)) < I1 which implies t h a t

,~E~ < t(l_O)/pOE -

OR

for some # C (0, 1) and for any t E [0, T*). The proof ends as in Diaz and Veron [29](see also and Antonsev and Diaz [4]). II

312

Diffusive energy balance models in climatology

Remark 5. The existence of the mushy region (for anyvalue of p E (1, oo)) can be proved for a different class of models by taking into account a discontinuous diffusivity (see Held, Linder and SuArez [40]). In that case the problem is a variant of the Stefan problem (see, e.g., Meirmanov [55]). It would be interesting to find sufficient conditions implying the persistence of a mushy region for any time t > 0. The fact that a mushy region may exist for the stationary problem can be found from the results of Diaz [?] (see Theorem 1.14). 2.4. Stabilization of solutions w h e n t --~ + c o

In order to analyze the stabilization of the solutions of (P), following Diaz, Herns and Tello [24], we assume the additional condition - (H~)] f E L~ oc) x A4) and there exists foo E V' such that '

f

t+l

Iif(7,.) -- f~(.)l]v, dT ~ 0

as t --- cx).

-1

We start by recalling a global regularity of the solutions on (0, c~). L e m m a 2 - A s s u m e we are given uo E VML~ f E L~176 NWllo'1((0, oc); LI(A4)) and f : + l lift(s, ")]]Ll(M)ds <_ Co, Vt > 0 where a time independent constant. Then there exists a weak solution of (P) that u E L~176 oo; V) and ut E L2(0, oo; L2(A/[)).

oc)) Cois such (15)

A key point in the proof is to check that ~(t) - F1 f M I ~ u ( x , t ) I P d d satisfies p(t + 1) < C [ ~ ( t ) - qp(t + 1)] + O(t) t :> 0, (16) where C is a positive constant and O(t) > 0 when t is large enough with O(t) = O(1) when t --~ c~. Then, thanks to a technical lemma due to Nakao [56], we conclude that ~(t) = O(1) which is equivalent to u E L~(0, oo; V). The following theorem proves the stabilization of the solutions u satislying (15). As usual, given u bounded weak solution of(P), we define the ca-limit set of u by ca(u)= { u ~ E V n L ~

3tn ~ +oo such t h a t u(tn, .) -~ u ~ in L2(A//)}.

T h e o r e m 4 - Let uo E L ~ ( M ) A V and let u be any bounded weak solution satisfying (15). Then, i) ca(u) ~ 0 and if u ~ E ca(u), 3tn ~ +(x~ such that u(.,t,~ + s) ---, u ~ in L 2 ( - 1 , 1; L2(Ad)) and uo~ E V is a weak solution of the stationary problem associated to fo~ ; ii) in fact, if u ~ E ca(u), then 3{t~ } --, +oo such that u(-, tn) ~ uo~ strongly in V.

J.l. Diaz

313

Proof. Let u ~ be an element of w(u). Then,

ll~(tn + s)- ~(tn)ll~.(~) < 2 Ilu~ll~,=( ( t ~ - l , t ~ -

+ l);L2(Ad))"

-

Since ut C L2(0, oc; L2(jt4)) , [lut]12L2((t_l,t_t_l);L2(./M)) --~ 0 when t~ -~ oc and by the Lebesgue convergence theorem we conclude that u(., tn + ") U~ in L 2 ( ( - 1 , 1); L2(Ad)). To prove that u ~ is a solution of (PQ), we consider the test functions v ( x , t ) = ( ( x ) g ) ( t - t~) with ( E V O L ~ ( A d ) and ~ c Z)(-1,1), ~__ 0, fll ~ --- 1. Then

fti~+l /Ad ut~q~(t -- tn) + fftlt~+l/Ad I V u F - 2 V u

]i i

J M O(u)(7)(t - t~)

f t +l

9 1

-- tn)

f

+ =

. V~(t

f(x, t)~(t

- t~)

z ~ ~ ( u ( x , t)).

Jtn-1

Changing variables, namely s = t - t~ and defining U~ (x, s) = u(x, t~ + s), we obtain that Un

__.x u o o

weakly in weakly in

'

IVUnlp-2VUn ~ Y

L ~ ( ( - 1 , 1); V) L ~ ( ( - 1 , 1);LP(TA/I))

Va > 1 Ycr > 1.

Applying Aubin's compactness result (see e.g. Simon [66]), a well known property of the maximal monotone graphs (see Brezis [14]) and Lebesgue's theorem, w e g e t t h a t z, - - zoo C/3(uoo) weakly in L~ (Ad x ( - 1 , 1 ) ) Vcr > 1 and ~(U~) --+ ~(uoo) in L I ( A d x ( - 1 , 1)). Letting n --, oc, we arrive to Y-V~p+

6(uoo)~=

QSzoo~+

foo~

V~ r V r q L ~ 1 7 6

1

Now, the main difficulty is to prove that ~ Y(s,-)~(s)

1

[ W ~ IP-2Wo~.

Due to the coercivity of the p-Laplacian operator we obtain the following inequality: lim n--+oo 1

(IVUnIP-2VUn-IVXI~-=VX)

9( W o o

-

VX)~(~)

gAds >

O,

(17)

314

Diffusive energy balance models in climatology

which holds for X E V. We arrive to the desired convergence by taking = u ~ + A~ and applying a Minty type argument to(17) as in Diaz and de Th~lin [28]. The proof of (ii) uses the coerciveness of the operator and the fact t h a t ( ] V U n l p - 2 ~ U ~ - [ V u o o [ P - 2 V U o o ) 9(VUn - Vuoo)~p(s) dAds --, O.

The inequality I ~ - r to obtain

-< (]~[p-2r

[VU~ - Vuoo[P~(s) d A d s = 0, Vp. n---,oo

1

This implies t h a t there exists a subsequence { S n } n ~ , where $n E (--1, 1) such t h a t lim ] [Vu(t~ + s~, .) - Vuoo[PdA - 0 J~

rt---, oo

and so we prove the assertion,

m

R e m a r k 6. If uoo is an isolated point of w(u), it is easy to see that in fact the above convergences hold as t --, oo (and not merely for a sequence tn -~ oc ). The proof of this convergence is an open problem in the remaining cases. In fact, in some cases the set of stationary points is a continuum (see Remark 11) and the convergence when t ~ oe is far from trivial (for some results in this direction see Feireisl and Simondon [34]). R e m a r k 7. A result on the convergence (in a suitable sense) of the free boundaries to the free boundary of the solution of the stationary problem is given in Diaz [22] (see also Gianni [36]). R e m a r k 8. The question of the approximate controllability was considered in Diaz [?] and [21]. To avoid technical difficulties, in these articles the manifold A//is replaced by an open regular bounded set ~ of IR 2 (here IR 2 can be also substituted by IR N with N C IN ) and p is taken as p = 2. As a boundary condition on (0, T) x/)f~, it is chosen the one of Neumann type since it leads to a set of test functions for the weak formulation very similar to the one corresponding to the case of a Riemannian manifold without boundary. The case of an internal control is considered by taking f ( x , t ) = v ( x , t ) x ~ with v the control to be searched, and X,~, the characteristic function of w, a given open bounded subset of ~. Thus, the new formulation is now the following: given Yo, Yd : ~ - - * IR and ~ :> 0, find v~ : w x (0, T) ~ IR such t h a t d ( y ( T : v~), Yd) ~ E where, in general, y ( T :v) represents the solution

d.I. Diaz

315

of problem

(P~)

l Yt -- Ay -t- ~(y) E QS(x)/~(y) ~- vxco Oy

in ft x (0, T )

~nn - 0

on Oft x (0, T)

y(0,-) = y0(')

on ft,

where n is the outer unit vector to 0f~. It is shown t h a t the answer to the approximate controllability question depends on the asymptotic behaviour of the nonlinearity G(y) of the equation. If, for instance ~(y) = lylP-2y, then the approximate controllability property holds when p E (0, 1] but if p > 1, an obstruction phenomenon appears, implying the impossibility of the controllability for general data. Some results concerning a special class of data for the superlinear case p > 1 are presented in Diaz [21]. We point out that in 1955, John von Neumann [57] proposed to control the climate by acting on the albedo and that this still remains a mathematical open question. Finally, we mention the "rain making" (see Dennis [17]) as a practical example of the application of control problems in environment.

3. O n t h e s t a t i o n a r y

problem

We consider the problem (PQ,f) obtained in the last subsection. Following Diaz, Herns and Tello [24], we made in this section the additional assumptions - ( ~H* ) ~ satisfies (g~) and liml~l_~~ I~(s)[ = +oc - ( H f ~ ) fc~ E L ~ ( A d ) and there exists Cf > 0 such that --[If~IIL~(.M) <_ <_ - c s 9c M - ( zH* ) there exist two real numbers 0 < m < M and e > 0 such t h a t / 3 ( r ) = {m} for any r E (--oc,--10--e) and/3(r) = {M} for any r E ( - 1 0 + e , +co). -(Hcf)

G ( - 1 0 - e) +

CI

> 0 and 6 ( - 1 0 + e) + [Ifo~llg~(~) < S0M

9

6(--10

-- C) -Jr- Cf

-- Sl?7"t"

A function u E V N L ~ ( A 4 ) is called a bounded weak solution of (PQ,I) if there exists z E L~ z(x) E/3(u(x)) a.e. x E $r such that

Vv dA + f .~ 6(u)v

dA-/

QS(x)zv dA + J " f~v dA,

for any v E V. 3.1. E x i s t e n c e of at least t h r e e s o l u t i o n s for s u i t a b l e Q V~re start with a multiplicity result given in Diaz, Herns

and Tello [24]

316

Diffusive energy balance models in climatology

Theorem

5 - Let urn,

UM

be the (unique) solutions of the problems

(Pro)

- Apu 4- ~(u) -- Q S ( x ) m 4- foo(x) on .M,

(PM)

-- Apu + 6(u) = Q S ( x ) M + fo~(x) on .M,

respectively. Then: i) for any Q > O, there is a minimal solution u_u_(rasp. a maximal solution ~) of problem(PQ,f). Moreover, any other solution u must satisfy Um < U ~ U ~ ~ ~ U M (18)

(19)

G-I(QSo m - I I f ~ l I i ~ ( M ) ) lequm

<

~-l(QSlrn-Cf),

<_

~ - I ( Q S , M - C.f ).

G-I(QSoM- III~IIL~(~)) lequM

ii) for any Q there is, at least, a solution u of (PQ,f) m i n i m u m of the functional

j(~) - ~1 ]'~ [Vw]PdA + / ~ G ( w ) d A - / ~ f ~ w d A -

(20)

which is a global

]" Q S ( x ) j ( w ) d A ,

on the set K - {w E V, G(w) C LI(A//)}, where ~ - aj. Moreover, if (Hc.~) holds, then: iii) if 0 < Q < Q1, then ( P Q , f ) has a unique solution u - urn, u < - 1 0 , u is the minimum of J on K , and

{}-l(-I]fo~[[L~(~)) < lim inf ]]UI[L~(M ) _< lim Q\O Q\0

< G-~ ( - c i ) , iv) if Q2 < Q < Q3, then (PQ,f ) has at least three solutions, ui, i = 1,2, 3 with Ul = UM, Ul > --10, u2 = urn, u2 < - l O a n d Ul > u3 >_ u2 on ,All. Moreover, Ul and u2 are local minima of J on K A L c~(A/[) and, if p > 2 , and v) if Q4 < Q, then (PQ,f ) has a unique solution u = UM, u > --10, u is the minimum of J on K and I]U][Loc(M) ~ 4-00 when Q --~ 4-00, where (~1

Q3

-

-

S1M

Q2 - { } ( - 1 0 + E) + Ilf~llL~<~> SoM

G(-lO - ~) + c~

Q4 - G(-lO + ~) + I]f~JiL~(~).

{ } ( - 1 0 - e) + Cf

S1 m

Sore

(21)

(22)

J.I. Diaz Proof. i) It

317

is a consequence of the fact that the comparison principle holds for problems (Pro), and then the method of sub and supersolutions can be applied (see e.g. Amann [1]). ii) The conclusion follows from the Weierstrass argument by using Lebesgue convergence theorem, iii) From assumption (Hcj) and since 0 < Q < Q1, we obtain that ~1 < --10--([ and < - 1 0 - e. The proof of v) is analogous to i). The proof of iv) is divided into several steps. First, we construct two constant subsolutions Vi and two constant supersolutions Ui such that

(PM)

overlineu2

V2 < U2 < - 1 0 - E

< -10+e

< V1 < U1,

(23)

proving the existence of, at least two solutions of (PQ,f). Later, we introduce the approximate model (P~) +

=

+ f

(x)

on M ,

where ~ is the Lipschitz function/3~ = ~ ( I - ( I - A/~)-I), A > 0 (the Yosida approximation of f3). Since 13 verifies (H}), we get that /3~ is a bounded and nondecreasing function VA > 0, /3a(s) = ~ ( s ) f o r any s [-10 - e , - 1 0 + e + AM],/3x(s) --*/3(s) in the sense of maximal monotone graphs when A ~ 0. If/~ is a Lipschitz function, we take simply/3~ =/3. The existence of a solution of (P~) is obtained by a topological fixed point argument. Let us show the convergence of the mentioned solution of (P~) to a third solution of (PQ,f). For A < A0 (a certain positive parameter) U1, U 2 a r e supersolutions of(P~) and V1, V2 are subsolutions of(P~). So, arguing as in ii), we obtain two families of solutions {u)} and {u2~'} of (P~) such that -lO+c+AoM

< V1

_<

u1

_<

U2 < - 1 0 - c .

Moreover, since/3X(Ul~) =/3(u~), we deduce that U~l' - - u I . Analogously, we conclude that u~ = u2 . In order to prove that (P~) has a third solution u3~', different from u) and u~, we show the applicability of a result due to Amann [1] to the function (-Ap + on the space E - L~(JVI). Finally, we get the a priori estimates

F(v)"-

G)-I(QS(.)t3x(v)+f~(.))

/~. lVu~,iPdA+f G(u~,)u~,dA=f QS(x)~,(u~,)u~,dA+f~.f~u~,dA" ~, QS(x)~),(u),)u),dA+/~ f~u:,,dA < C1

Diffusive energy balance models in climatology

318

which allows to conclude that ux ~ u strongly in L2(Ad), ~x(u~) -~ weakly in L2(jt4) and that ~ --~ /3 in the sense of maximal monotone graphs (so, z e / 3 ( u ) see, e.g.,Benilan, Crandall and Saks [10]). Moreover, we get t h a t limx__.o [[ V u x - V u [[Lp(T~) -- 0. Since u3~ --~ u3 uniformly and ul > - 1 0 + Co, there exists el such t h a t Ve < el, u~ > - 1 0 + eo, which is a contradiction ( u3 necessarily crosses the level-10), il C o r o l l a r y 1 - Let Re(u) = B u + C with ~ given by (1), - 1 0 B + C > 0 and s_x < M __ Then we have i) if O < Q < - ISO1BM- t - C ~ then (PQ , I) has a So -m" unique solution, ii) if -XOB+C SoM < Q < -10B+C S~m ' then (PQ,I) has at least

three solutions, iii) i f - x oSom B + c < Q, then (PQ,I) has a unique solution. R e m a r k 9. As pointed out in Hetzer [44], the uniqueness of solutions for Q small and Q large still holds if conditions (H~) and (Hg) are replaced by G c

C1 (JR),/~ C C I ( I R - { - 1 0 } ) , rn <_/3(r) _< M, Vr E I R - { - 1 0 } , inf{ G'(~)~-r~, r C [ U , - 1 0 - e l } > 0, where -U " - ~ - l ( - - [ [ f ~ l l L ~ ( ~ ) ) and inf{ ~-2s ~'(r), r C [ - 1 0 + e, +oc) } > 0. Indeed, if Q is small enough, we can construct a supersolution showing that any possible solution u satisfies that U < u _< - 1 0 - e on AA. Then, any solution u must satisfy - A p t + JZ'(x,u) - f ~ ( x ) with ~'(x, u) "- G(u) - QS(x)~(u). Since ~-(x, u)is a strictly increasing function on [ U , - 1 0 - el, for a.e. x c Ad we have the uniqueness of solutions. The assumption on ~ leads to a similar conclusion when Q is large enough. 3.2. S - s h a p e d bifurcation d i a g r a m

As a continuation of the previous results we can improve the answer for the special formulation

(PQ,c)

- div(IVulP-2Vu) + ~(u) + c

c QS(x)/3(u) on M .

Following Arcoya, Diaz and Tello [6], we shall describe more precisely the bifurcation diagram and in particular, we shall prove that the principal branch (emanating from (0, G - I ( - C ) ) c IR + x L~ is S-shaped, i.e. it contains at least one turning point to the left and another one to the right. By a turning point to the left (respectively, to the right), we understand a point (Q*,u*) in the principal branch such that in a neighborhood in IR + x L~ of it, the principal branch is contained in {(Q,u) c IR + x L ~ ( A J ) / Q <_ Q* } (respectively, {(Q,u) E IR + x L ~ ( A J ) / Q >_ Q*}). A previous result is due to Hetzer [43], for the special case of p = 2 and /3 a C 1 function. He proves that the principal branch of the bifurcation diagram has an even number(including zero) of turning points. Our main

J.I. Diaz

319

result already improves this information showing that indeed, this number of turning points is greater than or equal to two. Semilinear problems with discontinuous forcing terms on an open bounded set and with Dirichlet boundary conditions have been considered in Ambrosetti [2], Ambrosetti, Calahorrano and Dobarro [3], Arcoya and Calahorrano [5] (see also Drazin and Griffel [31], North [59] and Schmidt [65] in the context of energy balance models). We make the additional assumption G ( - 1 0 + e) + C < S2M -(He) G(-10-e)+C>0 and { 7 ( - 1 0 - e ) + 6 Slm" We start by considering the problem with/3 a Lipschitz function (as in the Sellers model). T h e o r e m 6 - Let/3 be a Lipschitz continuous function verifying (H~).

Then E contains an unbounded connected component which is S-shaped containing (0, G-~(-C)) with at least one turning point to the right contained in the region (Q1,Q2) x LC~(2vl), and another one to the left in (Q3, Q4) x L ~ ( A J ) . Proof. Step 1. E has an unbounded component containing (0, G - I ( - C ) ) 9 We claim that the following result, due to Rabinowitz [61], can be applied to our case: "Let E a Banach space. If F 9IR x E -~ E is compact and F(0, u) - 0 , then E contains a pair of unbounded components C + and C in IR + x E, IR- x E, respectively and C + N C - - {(0, 0)}". To do so, we consider the translation of u given by v := u - G -1 ( - C ) . Obviously, v is a solution of -Apv + r - QS(x)~(v) on AA (24) where G(a) = ~(G-[-~-I(-C))-t-C and r = / ~ ( G - t - ~ - I ( - C ) ) . We define in an analogous way to E. Let E = LC~(2vt) and define F(Q, v) = ( - A p + G)-l(QS(x)~(v)). Then F is the composition of a continuous operator and a compact one (recall that p >_ 2), so F is also compact. On the other hand, if Q = 0 problem (24) has a unique solution v = 0 , so F ( ~ 0) = 0. In conclusion, E contains two unbounded components C + a n d C - on IR + x L~176 and IR- x L~ respectively and C+ N C - = {(0, 0)}. Since E is a translation of E, E contains two unbounded components C + and C - on IR + x L~(Ad) and IR- x L~ respectively, and that C + n C - = { (0, g - 1 ( - C ) ) } . Since Q _> 0 in the studied model, we are interested inC +. In order to establish the behaviour of C +, we also recall that for every q > 0, there exists a constant L = L(q) such that, if 0 < Q _< q, then every solution UQ of (PQ,c) verifies IluQllL~(~) <_ L(q). Since the principal component is unbounded, its projection over the Q-axis is [0, oo). On the other hand, if Q is large enough, (PQ,c) has a unique solution uQ, and this solution

320

Diffusive energy balance models in climatology

is greater than G - I ( Q S o M - C). Since limlsl__,~ IG(s)l - +c~, then the unbounded branch containing (0, G - : ( - C ) ) C + should go to ( ~ , c~). Step 2. Bifurcation diagram for two auxiliary problems: We consider the auxiliary zero-dimensional models (P1)

~(u) + C = QS1/3(u)

u c IR,

(P2)

G(~) + C =

~ c ]R.

Qs23(~)

The number of solutions to these problems depends clearly on the values of Q. Let us call E: and E2 the bifurcation diagrams of (P1) and (P2), respectively. By assumptions (H a), (H~) and (Hc), the principal components of E: and E2 are S-shaped. We also remark that the sets K i := {(Q, ~Q) e ]R ~

-O<_Q__

~ ( - 1 0 - e) + C , ~o = G-:(Qs~.~ - c)}, Sire

K~ := {(Q, uQ) e IR 2" Q > G(-10 + e) + C S,M , uo - ~-I(QS~M - C)} are contained in Ei, i = 1, 2. Step 3. A comparison argument: If Q < Q3, there exists UQ solution of (PQ,c) such that uo < 1 0 - c. Thus UQ satisfies

QS2m ~ -ApU + ~(u) + C ~ Q S l m on M . Let u~ and u~ be the solutions of the problems

~(u) + C

-

QSlm OhM

g(u) + C

-

QS2rn onAJ,

respectively. That is, (Q,u~) and (Q, U2Q)live in E1 and E2, respectively. Now, if Q < Q3,

and so by the comparison principle for the monotone problem -/kpU + (}(u) - f E L2(AJ) on A4, we have that u~ leq uQ <_ u~. Therefore, the component of r starting in (0, ( } - I ( - C ) ) , lives between F~1 andE2 to arrive at (Q3, UQ3), where uQ3 is the minimal solution of (PQ3). Analogously, if we denote by tO2 the maximal solution of (PQ2), we can prove that the component ore which connects (Q2, u~) with (:xD, c~) lives between E1 and E2. From Q2 < Q3, the branch containing (0, G - I ( - C ) ) is un bounded and by the uniqueness of solution for (PQ,c) when Q > Q4, we get that this branch is necessarily S-shaped. m Our next result avoids the Lipschitz assumption made in Theorem 6.

J.I. Diaz

321

T h e o r e m 7 - Let/3 a general maximal monotone graph satisfying (H~)

and assume (He). Then E has an unbounded S-shaped component containing (0, 6 -1 ( - C ) ) with at least one turning point to the right contained in the region (Q1, Q2) x n~ (Ad) and another one to the left in (Q3, Q4) x L~ respectively. To prove Theorem 7, we approximate problem (PQ,c) when /3 is not Lipschitz continuous. We only need to show the convergence of the principal branches Cn of these approximating problems to a S-shaped unbounded connected set C of solutions of (PQ,c). For this reason, let us recall the notions of liminf and limsup of a sequence of subsets Cn of a metric space X: liminf Cn "= n--+(~

{p C

X : for any neighbourhood U(p) of p in X 3n0 C IN: U(p) N C n r 0 Vn _> n0},

limsup C~ "=

{p C

X : for any neighbourhood U(p) of p in X

n---~(x)

U(p) N Cn r 0 for infinitely many n}. A lemma due to W h y b u r n [70] shows that if i) lim~_.~ i n f Cn r 0 and ii) t2~__1C n is precompact, then limn_.~ sup Cn is a nonempty, precompact, closed and connected set. Proof of Theorem 7. The method of super and sub solutions proves that if Q > Q2, then there exists a solution of (PQ,c) greater than - 1 0 + e. Analogously, we know that if 0 _< Q < Q3, then (Po) has a solution smaller than - 1 0 - e. It is clear that these functions are not the unique solutions of (PQ,c) in those intervals and that the uniqueness holds at least in the Q-intervals [0, Q1) and (Q4, oc). Since we can not apply directly Rabinowitz theorem to our problem, we consider the family ]~n - - n ( I 1 --1 ), n C ]hi to approximate/3 in the sense of maximal - (I - n/3) monotone graphs when n --~ oc. Notice that since/3verifies (H}), then/?n is a Lipschitz bounded nondecreasing function (see Brezis [14]) and that fin(S) = fl(S) for any s r [--10 -- e,--10 + e + ~M] , Vn. Let u~ be the solutions of the approximated problem (P~))

- ApU~ + G ( u ~ ) + C = QS(x)/3~(un) on .Ad

and let En the bifurcation diagrams for (P~)). Let us denote by Sn the component of En containing (0, G - I ( - C ) ) . By Theorem 6, every Sn is an unbounded, connected and S-shaped set. First of all, we are going to prove that lira sup Sn is a connected and closed set of solutions to problem (PQ). In order to apply W h y b u r n ' result, we consider the sets C j (j > Q4) defined as S~ A ([0, j] x L~(f~)), Vn C IN containing (0,~-l(-C)). It is

322

Diffusive energy balance models in climatology

easy to see that these sets are connected and that i) is verified. Let us check (ii),Un~__lC~ is precompact. Since X is a Banach space, it suffices to prove that every sequence {(Ql, ut)}l~IN C Un~__IC~ contains a subsequence {(Qza,uta)} converging in X. From Qt c [o, j], there exists Q E [O, j] and a subsequence of {Qt}which we still call {Ql}, such that Qz --+ Q. On the other hand,ul is a solution of the problem

Taking ul as a test function in this equation, we obtain the estimate

I V u , ] ' d A -4-

-~lm[2dA <_

where [j~41 is the Hausdorf measure of Ad. Then uz is a bounded sequence in V. From the compact embedding V C L ~ ( A d ) when p > 2, there exist u e L~(;~4) and asubsequence {ula} of {ul} such that uzk --~ t i n L~(jM). If p = 2, then {u~} is a bounded sequence in the Sobolev space H2(.M). From the compact embedding H2(Ad) C C(Ad), we deduce the existence of a subsequence {ulk} and u C C ( M ) , such that uzk ~ u in L~(Ad). Thus t2~__lC j is precompact. Then by Whyburn's result CJ ~_ lim~__.~ sup C~ is a connected and compact set in X. Moreover, since every S~ is unbounded and fixed Q, the solutions UQ are uniformly bounded in L ~ (AX), for Q < Q, we have that C~ N ( { j } • L ~ (AJ)) r 0, for all j c IN. Now, we prove t h a t the set C y is contained in E. Let us see that for every Q c [Q1, Q4], we have that every (Q, u) E C j verifies that u is a solution of (PQ) (notice that it is true for every Q e (0, Q1] u [Q4, +cr from C~ - C j in these intervals). Let (Q, u) c C j - lim~__.~ sup C3n, that is, there exists a subsequence of (Q~, u~) e C~ such that (Qua, Una) ~ (Q, u) in IR• From estimate (25) and the compact embedding H2(Ad) c L~(AA) (for p = 2 ) a n d V c L ~ ( , M ) (for p > 2), we deduce the existence o f t c L ~ ( A J ) and a subsequence of { (Qua, u~a )} which we call { (Qna, Una ) }, such that

(Q~k,Unk) ~

(Q,u)

in IR • L ~ ( A J ) ,

Since/3~ ~ / 3 in the sense of maximal monotone graphs of IR 2, we have that ~na (Una) ---" Z C /3(U) weakly in L2(3A). Using a Minty's type argument we deduce that u is a solution of the problem (PQ,c). Thus (Q, u) E E and CJ c E. Since for all n and j, C ~ N ( { j } x L~(AA)) # O, there exists {(j, u~)}~e~ such t h a t (j, Un) C C j , that is,

- A p U n + ~;(Un) -- j S ( x ) ~ ( u n )

- C

in Ad.

Using that the operator (Ap + 6)-~ is compact in L ~ ( M ) , there exists a subsequence una --* U in L~(A/I). Thus (j,u) E C j and C j A ({j} x

J.I. Diaz

323

L~(A/I)) r 0. Since j > Q4, Uj is the unique solution of (PQ,c). On the other hand, we know that EN(j, ec) x L~(A//) = EMA(j, cxD)x L~ So, we have obtained a connected unbounded set which starts in (0, G - I ( - C ) ) . The proof ends with the argument used in the proof of Theorem 6 for

Q2 < Q3. m Remark 10. We point out that our results remain true for the more general equation

-div(k(x)lV~l~-2V~) + ~(~) + C e QS(x)~(~) on M, where k(x) is a given bounded function with k(x) _> k0 > 0 a.e.x c A/l, representing the eddy diffusion coefficient. When Ad = 81, it is usually assumed that S(x) = S().) and k(x) = k(A, r with A the latitude and r the longitude. So, in that case, the corresponding solutions are not r Remark 11. By using a shooting method, it is possible to show (see Diaz and Tello [27] that there exist infinitely many equilibrium solutions for some values of Q when we study the one-dimensional problem

-(lu'lp-2u')'+Bu+C c Q~(u) x E (0,1), (P1,Q,C)

u'(0) = u ' ( 1 ) = 0 .

If Q1 < Q < Q2 then (P1,Q,C) has infinitely many solutions. Moreover, there exists K0 C IN such that for every K E IN, K >_ K0 C IN there exists at least a solution which crosses the level U K = - - 1 0 , exactly K times. Remark 12. After my lecture at the Coll~ge de France, Professor J.L. Lions pointed out to me the reference Rahmstorf [63] where a S-shaped diagram bifurcation curve arises in the context of the Atlantic Thermohaline Circulation in reponse to changes in the hydrological cycle.

References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18, 4 (1976), 620-709. [2] A. Ambrosetti, Critical points and non linear variational problems. Supplement au Bulletin de la Societ~ Math~matique de France. M~moire 49 (1992). [3] A. Ambrosseti, M. Calahorrano and F.Dobarro, Global branching for discontinuous problems, Comment. Math. Univ. Carolinae, 31 (1990), 213-222. [4] S.N. Antontsev and J.I. Diaz, New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by energy methods, Soviet Math. Dokl., 38 (1989), 535-539.

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[5] [6] [7] IS] [9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [2o] [21]

Diffusive energy balance models in climatology

D. ARCOYA, AND M. CALAHORRANO,Multivalued non-positone problems, Rend. Mat. Acc. Lincei, 9 (1990), 117-123. D. Arcoya, J.I. Diaz and L. Tello, S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology, Journal of Differential Equations, 150 (1998), 215-225. O. Arino, S. Gautier, and J.P. Penot, A fixed point theorem for sequentially continuous mappings with applications to ordinary differential equations, Funkcialaj Ekvacioj, 27 (1984), 273-279. T. Aubin, Nonlinear analysis on manifolds. Monge-Ampere equations. Springer-Verlag, New York, 1982. M. Badii and J.I.Diaz, Time periodic solutions for a diffusive energy balance model in climatology, J. Mathematical Analysis and Applications, 233 (1999), 713-729. Ph. Benilan, M.G. Crandall and P.Saks, Some L 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions, Appl. Math. Optimization, 17 (1988), 203-224. R. Bermejo, Numerical solution to a two-dimensional diffusive climate model, in Modelado de sistemas en oceanografia, climatologia y ciencias medio-ambientales: Aspectos matems y num@ricos, (A. Valle and C. Par@s eds.), Universidad de Ms (1994), 15-30. R. Bermejo, J.I. Diaz and L. Tello, Article in preparation. K. Bhattacharya, M. Ghil and I.L. Vulis, Internal variability of an energy balance climate model, J. Atmosph. Sci., 39(1982), 1747-1773. H. Brezis, Op~rateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North Holland, Amsterdam, 1973. M.I. Budyko, The effects of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611-619. T. Cazenave and A. Haraux, Introduction aux probl~mes d'~volution semi-lin~aires. Math~matiques et Applications, Ellipses, Paris, 1990. A.S. Dennis, Weather modifications by cloud seeding. Academic Press, 1980. J.I. Diaz , Nonlinear partial differential equations and free boundaries. Pitman, Londres, 1984. J.I. Diaz, Mathematical analysis of some diffusive energy balance climate models, in Mathematics, climate and environment, (J.I. Diaz and J.-L.Lions, eds.), Masson, Paris, (1993), 28-56. J.I. Diaz, On the controllability of some simple climate models, in Environment, economics and their mathematical models, (J.I.Diaz and J.-L. Lions eds.), Masson, (1994), 29-44. J.I. Diaz, On the mathematical treatment of energy balance climate models, in The mathematics of models for climatology and environment, NATO ASI Series, Serie I: Global Environmental Change, 48, (J.I. Diaz ed.), Springer, Berlin, (1996), 217-252.

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[69] L. Tello, Tratamiento matems de algunos modelos no lineales que aparecen en climatologia. Ph. D. Thesis. Univ. Complutense de Madrid, 1996. [70] G.T. Whyburn, Topological analysis, Princeton Univ. Press, Princeton, 1955. [71] I.I. Vrabie, Compactness methods for non linear evolutions, Pitman, London, 1987. [72] X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Aplicable Anal., 42 (1991), 33-59. J. Ildefonso Diaz Departamento de Matems Aplicada Facultad de Matems Universidad Complutense de Madrid 28040 Madrid Spain E-mail: [email protected]

Studies in M a t h e m a t i c s a n d its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 15 UNIQUENESS AND STABILITY IN THE C A U C H Y PROBLEM FOR MAXWELL A N D ELASTICITY SYSTEMS

M. ELLER, V. ISAKOV, G. N AKAMURA AND D. TATARU

1. Preliminaries By x = (XO, X l , . . . , X n ) we denote t h e coordinates in R • R n. For t h e differentiation o p e r a t o r s we set Oj - O / O x j and D j = 7l O / O x j S o m e t i m e s we shall call x0 the time coordinate, a n d use t h e alternate n o t a t i o n t for it. Given a positive scalar function a in R n we define the scalar associate wave o p e r a t o r in R • R n by ['-1a - - aO2t - A .

By H k we denote the classical Sobolev spaces, with the norm d e n o t e d II-llkOften we shall use on these spaces t h e weighted norms,

I]u]]2;~--/ ~

7"2(k-]a])lSaaul2dx

(1.1)

]~t
2. Pseudoconvexity and Carleman estimates T h e aim of this section is to review t h e concept of pseudoconvexity and s t a t e several kinds of C a r l e m a n e s t i m a t e s for scalar equations. T h e Carlem a n estimates have been the main tool in the s t u d y of unique c o n t i n u a t i o n problems; see [5]. While we c a n n o t transfer the unique c o n t i n u a t i o n results directly from equations to systems, it t u r n s out t h a t s o m e t i m e s we can transfer the C a r l e m a n estimates to systems; these in t u r n imply unique continuation results for systems.

Uniqueness and stability in the Cauchy problem ...

330

Thus, let P(x, D) be a partial differential operator with real principal part whose principal symbol we denote by p(x, ~). We will assume that the coefficients of the principal part of P are in C 1 while the other coefficients are bounded and measurable. Let S be a smooth oriented surface in R n+l. Then we can represent it as a level set S - {r = 0} of a smooth function r so that V r ~= 0 on S. The function r is uniquely determined up to multiplication by smooth positive functions. To a smooth function r we associate the operator Pc(x, D, T) = P(x, D + iTVr which is the conjugate of the operator P with respect to the exponential weight e ~r

Pc(x, D, r) = erCp(x, D)e -re The principal symbol of the conjugated operator PC is

p (x,

= p(x,

+i

vr

The Poisson bracket of two symbols P, q is defined as {p,q} = Vep. V ~ q - V ~ p - V e q Let F be a conical subspace of the cotangent bundle T*R ~+1 . Now we are ready to define pseudoconvex functions and surfaces. D e f i n i t i o n 2.1 - a) A smooth function r is pseudo-convex with respect to

P onF if {p,{p,r

V(x,~)CV,~#0,

p(x,~) = 0

(2.1)

b) A smooth oriented surface S = {r = 0} is called pseudo-convex with respect to P on F if {p,{p,r

> 0 V (x,~) e Fs, ~ r O, p(x,~) = {p,r

=0

(2.2)

D e f i n i t i o n 2.2 - a) A smooth function r is strongly pseudo-convex with

respect to P on F if it is pseudoconvex and 1

i--~{I~r162

> 0 V ( x , ~ ) E F, ~ ~ 0, T > 0, p r

0.

(2.3)

b) A smooth oriented surface S is strongly pseudo-convex with respect to P on F if it is pseudoconvex and 1 ~=0, T>0, i-~ {p-c, p,}(x, ~) :> 0 V (x, ~) E Fs, pc(x ' ~) = {pc, r

~) = 0.

(2.4)

M. Eller, V. Isakov, C. Nakamura and D.Tataru

331

The relation between pseudoconvex functions and surfaces is obvious, Remark 2.3. a) An oriented surface S is (strongly) pseudoconvex with respect to P on F iff it is a level set of a function r which is (strongly) pseudoconvex with respect to P on F, so that V r ~ > 0 on F. b) Moreover, if the oriented surface S = {r = 0} is (strongly) pseudoconvex with respect to P on F then e ~r is (strongly) pseudoconvex with respect to P on F for large enough A. c) The (strong) pseudoconvexity condition for both functions and surfaces is stable with respect to small C 2 perturbations. For second order operators the condition (2.4) is void for noncharacteristic surfaces, therefore P r o p o s i t i o n 2.4 - Let P be a second order operator with real principal symbol. If a noncharacteristic oriented surface S is pseudoconvex with respect to P on F then it is strongly pseudoconvex with respect to P on ['. Now we discuss the corresponding Carleman estimates. Since this is all we use later on, in the sequel we only refer to second order partial differential operators P(x, D) with real principal symbol. The classical result (see [4]) is the following one: T h e o r e m 2.5 - Let f~ be a compact subset of R n+l. pseudoconvez with respect to P in f~ then rile'Cull 21;7" < Clle~r

If ~b is strongly

2

(2.5)

whenever u E H 1 is supported in t2 so that the R H S is finite. The substitution v = e~r

reduces the estimate to

2 which is essentially a subelliptic estimate for PC. The strong pseudoconvexity condition therefore expresses the subellipticity of PC in terms of its symbol. If instead D is a bounded domain with smooth boundary and u is a function in Q, then the appropriate estimate should include the Cauchy data of u on the boundary, see Tataru [ll]. T h e o r e m 2.6 - Let f~ be a compact subset of R n. If r is strongly pseudoconvex with respect to P in f~ then

lle' ul121;T,fl --< C(lleTCPull~0)

-1- T(lte~-r

Jr-Ile~'~b~-~t1

whenever u E H 1 is supported in ft so that the R H S is finite.

(2.6)

332

Uniqueness and stability in the Cauchy problem ...

In effect such an estimate might hold even with only one boundary data on the RHS, provided that the corresponding boundary operator satisfies a strong Lopatinskii boundary condition with respect to de, see [11]. If the coefficients of P are analytic with respect to some variables then the unique continuation results can be improved. We state here the Carleman estimate corresponding to the simplest result of this type, namely for the wave equation with time independent coefficients. In this case, we set F = {~0 - 0}, where ~0 is the time Fourier variable and consider functions and operators in R n+l. 2.7 - Let P be a second order hyperbolic operator with time independent coefficients. Furthermore, let r be a quadratic function in which is strongly pseudo-convex with respect to P on F. Then there exist d > O and C > O such that

Theorem

T]le-~D2eTdpul121;T--~ C ( l l e - ~ D ] e r C P ( x , D ) u l l ~ + ]leT(dp--de)?-t[121;7")(2.7) for all u E H 1, supported in ~, provided 7 is large enough and e is su]ficiently small. This result was proved in [10]. In effect it holds even if we only assume that the coefficients of P are analytic in time and that r is analytic. However, the proof is substantially simpler in the special case considered above. The critical point is the conjugation argument, which needs to be carried through also with respect to the Gaussian. With the substitution w - e~+u the estimate (2.7) reduces to

7-][e-~D~w[I 21;r --< C(]]e-~D2op(x,D + iT-Vr

2+[le-derw]] 21;T )

(2.S)

Furthermore

e ~V2ot -- (t + i e - D o ) e - ~ D~ T Since ~zr is linear, this implies that

e ;-~D~op(x,D+iTVr

P(x,D+iTVr162

(2.9)

Hence if we set v = e ~ V2ow then (2.8) reduces to T[[V][ 2I;T --< C[[Pe,r

2 -~-[[e--deTw[]

2I;T)

with P~,r - P(x, V + i ~ - V r

e0t(Vr

If e is small enough then P~,~ is a small perturbation of PC. Then the pseudoconvexity condition for r implies some subellipticity for P~,r in the region [~0[ _< cT. Outside this region, the Gaussian yields exponential decay in z and leads to the last RHS term.

M. Eller, V. Isakov, G. N a k a m u r a and D.Tataru

333

3. Uniqueness and stability for principally scalar systems Here we apply the results in the previous section to the systems of the type

Pjuj + bj(z, t; Vu) + cj(x, t; u) = f j, j = 1, ..., m

(3.1)

Here Pj are second order operators with real principal part and C 1 coeffiON3 cients and other coefficients in Lzo~,bj, cj are linear functions of Vu, u with L~o~-Coefficients , u = (Ul, ..., urn). First we obtain the Carleman estimates for such systems. T h e o r e m 3.1 - Let r be a smooth function which is strongly pseudoconvex with respect to all Pj in a given compact set K . Then

7-11e"%112 < clle~-~fll 2 1,7"

--

,

r > r0

(3.2)

whenever u is supported in K and solves (3.1). Proof. First we apply the Carleman estimates in Theorem 2.5 to uj with respect to Pj. This yields

~tIe~%jll~,~ ~

clle~Pj~jll 2,

and further

~-II~-~ujll~ < cllC~jll 2

2

Summing with respect to j, the first terms on the right are absorbed on the left for sufficiently large ~- and we obtain (3.2). m Next we consider the unique continuation problem for the corresponding homogeneous problem

Pjvj + bj(x, t; Vv) + cj(x, t; v) = O, j = 1, ..., m

(3.3)

T h e o r e m 3.2 - Let S = {r = 0} be an oriented surface which is strongly pseudoconvex with respect to Pj for all j. Then we have unique continuation across S for Hloc solutions u to (3.3). More precisely, given any xo E S and v E Hloc solving (3.3) near xo which is identically zero in {r > 0} near S, this implies that v = 0 near xo.

Proof. W i t h o u t any restriction in generality we assume t h a t x ~ = 0. Since S is strongly pseudo-convex, by R e m a r k 2.3 we can also assume t h a t r is strongly pseudoconvex with respect to Pj. Consider the modified function ~ ( x , t) = r t)+e(r2-1xl2). For small e, r is still strongly pseudoconvex with respect to all Pj near x0. Furthermore,

{r > o} o {r < o} c B(o, ~)

334

Uniqueness and stability in the Cauchy problem ...

Hence, by choosing r sufficiently small we insure that (3.3) holds in {r > 0}. Now let X be a cutoff function which is 1 in ~ > 0 and 0 in ~ < - e r 2. Then the function u = Xv solves (3.1) with f E L 2, supported in supp VX C {r < 0}. Apply the Carleman estimates (3.2) to u. We obtain rlle'r

.,- _<

cile'Wfll 2

Since f is supported in {~ < 0}, letting T go to infinity yields lim II~'~x~ll = 0

T ---~ ( X )

which implies that v = 0 in {r > 0}. Since r > 0 this concludes the proof, m The next step is to establish the corresponding stability estimate. For this it is more interesting to work with boundary value problems. Thus, let ft be a bounded domain in R ~+1 with C 1 boundary Oft. Let S be a part of 0ft where one prescribes the Cauchy data for u

u = go, O~u = gl

onS

(3.4)

Given a smooth function r let

~E = a n { c

< r

O~E -- Of] n {E < r

T h e o r e m 3.3 - Let r be strongly pseudo-convex with respect to all operators Pj on-~ and 0~o C S. Let u C H I ( ~ ) solve the Cauchy problem (3.1),(3.~). Then with some positive constants C, A = A(e) C (0, 1) we have [[Ulll,a~ _< C(e)(F + F ~ M 1-~)

where F = Ilg0l[1,s + Ilglll0,s + []f[[0,ao,

M = II~PIl,ao-

Proof. We want to use the analogue of the Carleman estimate (3.2) but for boundary value problems,

Tlre~uf[21;-r,a

-< c ( [ l e ~ r

0,a -I-

2 .,oa T(llerCU[ll;

N O,Of~)) (3.5)

+ II~ ' ~ 0u II2

for u solving (3.1) in ft. This follows in the same way as (3.2) if we use (2.6) instead of (2.5).

M. Eller, V. Isakov, G. Nakamura and D.Tataru

335

To proceed we need to localize to f~0. Hence, introduce a cut-off function X E C ~ ( R n + I ) , X = 1 on f~/2 and X - 0 outside f~0. T h e n work with Xu. After commuting we get

P j ( x u j ) -- x f j + A1;ju where A1;j is a linear differential operator of first order with measurable bounded coefficients depending on c, supported in ft0. Hence, applying (3.5) to Xu we obtain the inequality 1;"r

<- -

C(c)(lle'

ll 21,r,ao + Iie"~xfjllo2

+

k=0,1

Since X = 1 on ft~/2, this further gives

c(

))lle'%ll 21;~',Ft~/2 k=0,1

For large enough ~- this finally gives

Iluj[ll,,-,a~ ~ C(e)(Fe '-(~-~/2) + M e - ~ / 2 ) where (I) is the s u p r e m u m of r in ft. This inequality holds for all sufficiently large T, say ~- >_ ~-0(e). Increasing C(e) we can take T0(e) = 0. Hence we can minimize the right hand side with respect to T. If M < F , then Theorem 3.3 follows if we simply set T -- 0. Otherwise, we choose T so t h a t

Fer(e~-E/2) = Me-r~/2 and then the right hand side equals

C(e)M1-;~F ~,

E

A = 2---~

and we obtain the conclusion of the theorem with A = e/(2(I)).

ll

If a portion of the b o u n d a r y is strongly pseudoconvex, then the above result can be used locally near S. The corresponding uniqueness result follows easily from Theorem 3.2. C o r o l l a r y 3.4 - Assume that the surface S C Oft is strongly pseudo-convex with respect to P j , j = 1,...,m. If u E HI(f~) solves the system (3.3) with

336

Uniqueness and stability in the C a u c h y p r o b l e m ...

f -- 0 and has zero Cauchy data (3.~) on S then u - 0 in some neighborhood ofS in~.

Another corollary gives an explicit description of an uniqueness domain when S = ~ • ( - T , T) E C 2 in the two following cases: 1) ~/ = 0fY, the space origin is i n ~ ' a n d 2 ) ~'C {-h<xn < 0 , x 2 + . . . + x n2 _ l < r 2 } , S = O~ ~ N {Xn < 0}. Here we let x ~ = x r R n. To formulate the result we need the weight function r t) x 2 + ... + x n2 _ 1 + (Xn --/3) 2 -- 02t2 -- S to define n

>

C o r o l l a r y 3.5 - A s s u m e that the coefficients a(j), j = 1, ..., m of the principal part Pj - [:]a(j) of the system (3.1) satisfy the following conditions 02a(j)(a(j)+Ota(j)/2+a-1/2(j)ltVa(j)l

02a(j) _< 1

) < a(j)+l/2x.Va(j)-l/2/30~a(j)

on ~ , a ( j ) E C 1 ( ~ )

(3.6)

and ft' c B(O; OT), ~ - s = O in case 1) and h(h + 2~) < 0 2 T 2, ~2 + r 2 = s in case 2). A s s u m e that u C H I ( ~ ) solves the system (3.1). Then Ilul[1,a~ <_ C ( F + M I - ~ F ~) where F is Ilfllo,~ + Ilulll,s + IIVull0,s and M u-0,0,u-0 on S then u = O i n t o .

Ilulll,s.

In particular, if

Proof. It is known t h a t under the conditions (3.6) with respect to the principal coefficients the function r - exp(cr/2~) for large a is strictly pseudoconvex in ~ with respect to the wave operators [3a(j) [6]. Moreover the additional conditions on h, 13, 0, r, T imply t h a t (0~t)0 C S. Applying Theorem 3.3 we complete the proof, m We observe t h a t for constant a ( j ) in case 1) the conditions (3.6) simplify to 02a(j) < 1 and ~ ' c B(0; OT). Then Corollary 3.5 gives the sharp description of the uniqueness domain in the lateral Cauchy problem. Generally, when O,~a(j) < 0 on ~ one can satisfy the first condition (3.6) by choosing 13 positive and large. The uniqueness domain ft0 in case 2) is discussed in [7], section 3.4. While applicable to a wide class of systems, including semilinear ones, Theorem 3.2 can be substantially improved when the coefficients do not depend on the time variable.

T h e o r e m 3.6 with respect to with respect to depend on xo. (3.3) across S, S.

- Let S be a C l - s m o o t h surface which is non- characteristic the operators P j , j - 1, ...,m. A s s u m e that Pj are hyperbolic xo. A s s u m e that the coefficients of the s y s t e m (3.3) do not Then we have unique continuation f o r H 1 solutions v to in the sense that if v = 0 on one side of S then v - 0 near

M. Eller, V. Isakov, G. Nakamura and D. Tataru

337

Proof. Denote the two sides of S by S +, respectively S - . Let r be a C 1 function so t h a t S = {r = 0}. Let y E S so that u = 0 in S + near y. Then we want to show t h a t r = 0 in S - near y. W i t h o u t any restriction in generality we can assume that y = 0. To achieve this we use an argument similar to the one in the proof of Theorem 3.2. We start by constructing a quadratic function ~b with the following properties: i) The set ~b > 0 cuts a small neighbourhood of 0 in S - . More precisely {r > 0} FI S - N B(0, 4r) C B(0, 2r) ii) r > 0 iii) r is a small C 1 perturbation of r in B(0, 4r). iv) r is strongly pseudoconvex with respect to all Pj in F = {~0 = 0}. Here r is an arbitrarily small parameter. To achieve this set r

= xVr

25

2AE + ~(xVr

2 - - - ( x 2 - ~2)

Y

Y

where A is a fixed large parameter. The first term insures t h a t X7r ~ X7r in B(0, 4r); the second term yields (i)-(iii); finally, the third term provides the strong pseudoconvexity condition on F while doing no harm to (i)-(iii). For small positive c both (ii) and (iii) are clearly fulfilled, so it remains to verify (i). It suffices to show that r + -4 A75- _~2 > ~ in B(0, 4r) \ B(0, 2r), or that z B ( 0 , 4~) 7~

r

Since

~(x) - ~v~(0) + o(l~l)lxl this follows if r is sufficiently small. It remains to verify the strong pseudoconvexity condition. We have

pc(x, r ~) = p(x, ~) - ~-2p(x, r e ) + 2i~p~V~ therefore 1

~-7{f~,pr

-

p~(x,~)(v2~)p~(x,r

+ ~-2p~(x, vr162

w~)

+o(1r ~ + ~ ) =

__'XSr((P~Vr

+ ~-21P(X'XZr

+ O(r-5 + 1)(15 12 + ~r2 )

Since p(x, Vr # 0 and s is large, the above expression is positive as long as T is away from 0. But when ~- is close to 0, pc is close to p, which

338

U n i q u e n e s s and stability in the C a u c h y p r o b l e m ...

is elliptic on ~0 = 0 because of the hyperbolicity condition. This concludes the proof of (iv). Now we can use the Carleman estimates in (2.7) with respect to the weight function ~p in B(0, 4r). If u solves (3.1) and is supported in B(0, 4r) then we can proceed as in Theorem 3.1 to write (2.7) for each uj and sum them up. The coupling is lower order and, as in T h e o r e m 3.1, can be absorbed into the left hand side after summation. In the end we get

~Jl~-~o~'~ll~l;~ <_ c(ll~-~~'~fll~ + I1~(0-~)~11~;.)

(3.7)

To use this for solutions u to (3.3) we need to t r u n c a t e first. Let X be a cut-off C ~ - function such t h a t it is 1 on B(0, 2r) and 0 outside B(0, 4r). Then the function u = X:v is supported in B(0, 4r) and solves (3.1) with f in L 2, supported in B(0, 4r) \ B(0, 2r). Then (3.7) yields T[[e-~-;'DgerCUll2;r

~ ce ~ ' ,

")' = max{0, ffJ -- de}

(3.8)

where ffJ is the s u p r e m u m of ~p in the support of v. To complete the proof it suffices to make use of the following result of Tataru [10], Proposition 4.1. L e m m a 3.7 - Let u c L 2 be a f u n c t i o n with compact support. Furthermore, let r E C ~ and a s s u m e that

IIe-~Oge'r Thr

~ = 0 i~ { x ~ R~

-~ 0

9r

as 7- ~ oo

> 0}.

Applying this in (3.8) we get v - 0 in {~ > ~/}. Then ff~ _> 7, therefore by the definition of F we get 9 _< 0, i.e. u - v = 0 in {~p > 0}. The proof is complete, m C o r o l l a r y 3.8 - A s s u m e in addition that the principal part is in divergence f o r m and the coefficients bj are C 1. Then the s a m e result holds for L 2 solutions u to (3.3). Proof. Let #(t) be a smooth regularizing kernel supported in [-1, 1], with integral 1. Set t # 5 ( t ) --

(~-1#(~)

and look at the mollified functions u5 -- #5 * u

where the convolution is only with respect to t.

M. Filer, V. Isakov, G. Nakamura and D.Tataru

339

Since (3.1) has time independent coefficients and v solves it, it follows t h a t the functions va solve it as well. We claim t h a t v~ E H I. Clearly Oktu~ c L 2 for all k. Using this in (3.1) we get Pj*uj E L2t (H, --1 oc) where P j contains the part of Pj without any time derivatives. Then Pj* is elliptic, in divergence form, with C 1 coefficients, therefore the elliptic theory yields uj c Lt2 (Hloc) 1 , which proves our claim. On the other hand, supp u n { r > 0} C B(0, 2r) therefore supp uan{~b >_ 0} c B(0, 2r) if 5 is sufficiently small. Then the a r g u m e n t in the proof of the Theorem applies to ua and gives ua = 0 in ~p > 0. Letting 5 -~ 0 we get the same for u, q.e.d. It is convenient to have a global version of the previous Theorem. To formulate the next result we assume t h a t So is a continuous family of C 1 regular non-characteristic mutually disjoint surfaces, 0 E [0, 1]. Set ~0 =

UOE[0,1]S0 C o r o l l a r y 3.8 - Under the conditions of Theorem 3.6 let u be an H 1 solution to the system (3.1) which is zero near $1 so that supp u N fro is compact. Then u = 0 in ~o. The same holds for L 2 solutions if the additional conditions in Corollary 3.8 are fulfilled. Proof.

a) Let s = sup{O E [0, 1]; supp u A So :/= O}

Assume by contradiction t h a t s > O. Then supp uN S0 = 0 for 0 > s, while, by compactness, supp u A S~ :/= O. If x0 E supp u N S~ then, by continuity, we conclude t h a t u is 0 on one side of S~ near x. Then we can use the local argument above to conclude t h a t u = 0 near xo, which contradicts our earlier assumption. I

4. A p p l i c a t i o n s

to the Maxwell

system

The results of the previous can be applied to classical systems of mathematical physics which can be reduced to the systems of the form (3.1) by relatively simple transformations. First we consider the Maxwell system Ot (eE) - curl H - j Ot(pU) + curl E = 0 div (eE) - 4rrp div (#H) - 0

(4.1)

for electric and magnetic fields E -- (El, E2, E3), H = (H1,/-/2, H3) depending on (x, t) E f t c R 4. Here c(x, t), #(x, t), denote correspondingly electric

Uniqueness and stability in the Cauchy problem ...

340

permittivity, magnetic permeability, j(r t) is the density of electrical current and p(x, t) is the electrical charge density. We consider the isotropic case when E, # c C 2 are scalar functions.We are interested in the Cauchy problem for (4.1) when one prescribes the Cauchy data E:E0,

H=H0

onS

(4.2)

where S is a C 2 surface. Uniqueness and stability in the Cauchy problem for the system (4.1) will follow from the results of the previous section and from the following simple lemma. L e m m a 4.1 - For solutions to the Maxwell system (~.1) with j = O, p = 0 we have 0t2(EE) - A ( # - I E )

-}- V(V(~#) - 1 . ~E) - curl ( V # - ' • E) +

#tH) =0. curl (-~-

+

V(V(E#) -1. # H ) - c u r l (V: -1 x H)

(4.3)

and 0 2 t ( # H ) - A(E-1H)

-

curl(etE)-O.

(4.4)

E

Note that this computation requires only C 1 regularity for E, H. If in addition both E and H are at least C 2 then one can rewrite these equations as

(e#O2 - A ) E = QI (E, H) (e#0t2 - A ) H = Q2(E, H) where Q1, Q2 are first order operators with bounded coefficients.

Proof. To reduce the system (4.1) to (4.3), (4.4) we differentiate its first equation with respect to t and use the second equation (4.1) to replace Ot(ttH) by curl E. We obtain Ot2eE + curl (#-1curl E) - curl ( # t H ) - O. # and further Ot2(cE) J- curl (curl ( # - I E ) ) + curl (V# -1 x E) -}- curl ( # t H ) : O. # But curl curl E = - A E + Vdiv E

M. Eller, V. Isakov, G. Nakamura and D. Tataru

341

therefore

02t(eE)- A ( p - I E ) ) +

Vdiv ( # - I E ) -

/It

curl (V# -1 x E) + curl ( - - H ) - O. #

Now using the third equation (4.1) we finally get

Oy(r

(V, -1 • E)+cu l ( #7t H ) = O.

ZX(,-IE))+ V(V(r -1 .r

The equation for H can be obtained in a similar fashion. The equations (4.3), (4.4) form a principally scalar hyperbolic system, so from Theorems 3.2, 3.3, 3.6 and their corollaries we can derive uniqueness and stability results for the C a t c h y problem. The Carleman estimates for the Maxwell's equations have the form --

<

c(lIe~r

- curl H)ll 21 , ~ -

+

[[e"r

+

[[div (eE)][ 21,r hI- Ildiv (#H)[[12,~.)

- curl E)II~,~-

In the presence of the boundary one needs to add another term, namely r[]e~r the conormal derivatives of both E and H can be derived from the equations. By Theorem 3.1, these estimates hold if the coefficients e, p are C 2, provided that r is strongly pseudoconvex with respect to the wave operator t3~,. The main unique continuation result is T h e o r e m 4.2 - Assume that the oriented surface S = {q5 = 0} is strongly pseudo-convex with respect to n~,. If ( E , H ) E Hi(F*) solve the system (4.1) and are zero on S then they are zero near S in {r > 0 } . . Correspondingly, from Lemma 4.1 and Theorem 3.3, we get the stability estimate T h e o r e m 4.3 - Let 0 be strongly pseudo-convex with respect to cq,~ on f~ and (Of~)o C S. Then for a solution (E, H) E H i ( a ) to the Catchy Problem (d.1), (4.2) one has [[E[]I,~ + [[H[[1,~ _< C ( F -t- M I - ~ F )~) (4.5)

with some positive constants C, A < 1 depending on (~. Here F = liE01[1,s + IlH0[[1,s, M = [[E[[1,ao + ][H[[1,~o. Now we give explicit description of uniqueness domains when [~ - ~t~ • ( - T , T) and S = ~/• ( - T , T) in the cases considered in Corollary 3.5.

342

Uniqueness and stability in the Cauchy problem ...

C o r o l l a r y 4.4 - Assume that the coefficients a = e# satisfy the condition (3. 6) and that parameters h,/3, O, T, r also satisfy the conditions of Corollary 3.5. Then (~.5) holds. Finally we give a sharp description of the uniqueness domain in the Cauchy problem when e, #, a do not depend on time; this follows from Theorem 3.6 and Corollary 3.8. T h e o r e m 4.5 - Let S be a C l - s m o o t h surface which is non-characteristic with respect to the operator [:]et," Then we have unique continuation across S for L 2 solutions E, H to (3.3) , in the sense that if v = 0 on one side of S then v = 0 near S.

5.

Applications to

the classical elasticity system

The uniqueness of the continuation for the static elasticity system under C ~ - a s s u m p t i o n s and with arbitrary first order perturbations was proven by Dehman and Robbiano [3]. Recently, Ang,Ikehata,Trong, and Yamamoto [1] reduced a static system with zero order perturbations to a principally diagonal one and obtained sharp uniqueness of the continuation results under reduced smoothness assumptions.The time dependent classical elasticity with arbitrary first order perturbations was considered by Isakov [6] by pseudo-convexity methods. Recently Yamamoto [12] applied principal diagonalization to some inverse source problems for the Maxwell system. A similar reduction is available the classical elasticity system 3

pO2tu - # ( A u + V d i v

u)-V()~div u ) - E

V#.(Vuj+Oju)ej = 0

ina

(5.1)

j=l

for the displacement vector u = (Ul, u2, u3) depending on (x,t) E ~. We will assume that the the density p E C 1(~) and the Lame parameters #, )~ are functions in C2(~).We are interested in the Cauchy problem for this system when one prescribes the Cauchy data u=u0,0~U=Ul

onS

(5.2)

We will extend the system (5.1) for three unknown functions Ul,U2, U3 to a new one for seven unknown functions by introducing two more vector functions v -- div u and w = curl u. Another, more economical reduction requires one to introduce just one auxiliary variable, namely div u. However, this requires some lengthy computations and is relegated to the appendix. For a particular case of this reduction we refer to [2].

M. Eller, V. Isakov, G. Nakamura and D. Tataru

343

To find differential equations for new functions we apply the divergence and the curl operators to the system (5.1). Lemma

5.1

-

Let v = div u, w = curl u. I f u solves (5.1) then p / # O 2 t u - ,/ku 4- A1;l(U, v) = 0

p/(2p + A)O2tv - A v + A2;, (u, v, w) = 0

in ~

(5.3)

p/#O2tw -- ~ w + A3;1 (u, v, w) - 0 where Aj;x are linear differential operators of first order with measurable and bounded coefficients in Ft. In addition, when the p C C2(~-7), #, A E C3(~-7) do not depend on t, then the coefficients of Aj;x do not depend on t and the coefficients of first order derivatives are in C 1(~7). Proof. Dividing the equations (5.1) by p and applying the divergence we obtain 02v - (2# + ,k)/pAv - ( V ( # / p ) + )~V(1/p) + 2/(p)VA + V # / p ) . V v 3

-(v(,/p)

+ v,/p),

div ( 1 / p V A ) v - E

u-

o j ( V # / p ) . ( V u j + Oju) = o

j=l

or using the identity Au

-

Vdiv

O ~ v - (2~ + ; . ) / p A v 1 / # V ( # 2 / p ) 9curl w

- E

u-

curl curl u write it as

2/(~, + 2~)V((~, + 2~)2/p) 9V v +

O j ( V # / p ) . (Vuj + Oju)

-

div ( V A / p ) v = 0

j~3

Similarly applying the curl and using the identity curl ( f u ) - fcurl u + EXf • u leads to the equations Oyw- p/pAw-

( V ( # / p ) ) x (Au + Vv) + v((oj,)/p)

-

V(1/p) • V ( A v ) -

• (oj

+ wj))

= 0

j<:3

Using again the formula Au = V v - curl w we arrive at O~w

-

p / p A w

-

V(,/p)

• (W

-

curl

~;;

+ W)-

V(1/p) x V(Av) - E ( 1 / p O j # O j w + v ( o j # / p ) x (oju + v u j ) ) = 0 j
344

Uniqueness and stability in the Cauchy problem ...

As for the Maxwell system this Lemma and our results on principally scalar systems imply the following uniqueness and stability theorems for the Cauchy problem (5.1),(5.2). T h e o r e m 5.2 - Let r be strongly pseudo-convex with respect to the operators rnp/u, r-lp/(2u+,x) on ~ and (0~)o c S. Then f o r a solution u E H2(~t) to the Cauchy problem (5.1),(5.2) one has

II ll,,a, + Ildiv ulll,a. + licurl Ulll,a. < C(F + MI-'XF "x)

(5.4)

with some positive constants C, A depending on e. Here

F

-II oll ,s + II ,ll,,s,

M

=

II ll ,a.

In particular, when uo = Ul = 0 on S one has u - 0 on ~o.

This result follows from Theorem 3.3. If the part S of the boundary is itself strongly pseudoconvex then we get the following local result" C o r o l l a r y 5.3 - A s s u m e that the surface S is strongly pseudo-convex with respect to the operators Dp/u, [-]p/(2ttTA) in -~. I f u E H2(~) solves the Cauchy problem (5.1),(5.2) with zero no, Ul then u=O near S on ~. Now we give an explicit uniqueness domain when ~t - ~t' • ( - T , T) and S - 7 • ( - T , T) in the cases considered in Corollary 3.5. C o r o l l a r y 5.4 - A s s u m e that ~ , S are the domain and the part of its boundary described in Corollary 3.5. Let the functions a = p / # and a = p / 2 ( # + A) and the parameters h, t9, O, T, r satisfy the conditions (~.3). Let u e H2(~t) solve the Cauchy problem (5.1),(5.2). Then (5.~) holds. In particular if the Cauchy data are zero then u = 0 on ~o. As for the Maxwell system we can improve these results and reduce regularity assumptions about u when the elastic parameters are time independent. T h e o r e m 5.5 - A s s u m e that the coefficients p 6 C2,#, 1 6 C 3 are time independent. Let S be a noncharacteristic surface with respect to both Op/,, E]p/(2,+,x ). Then we have unique continuation across S for H~oc solutions u to (5.1).

Appendix.

A diagonalization

of the classical elasticity system

Let L u - (l +

#)V(V-u) + #Vu + (V. u)Vl- {Vu + (Vu)T}vp

M. Eller, V. Isakov, G. Nakamura and D.Tataru

345

be the spatial part of the elasticity system, where (VU) T denotes the transpose of Vu. By a direct computation we obtain the following lemma: Lemma

Let

A.1 -

= ff

, Z - ~(~ + ~), o - (~ + f f ) v ~ + ~ v ~

and V

-

W

-

V A | V~ + V~ | V~ + (A + ~)V2~ + (V~. V~ + ~A~)I,,

~ { v - ( v o ) T} - v ~ | o.

Then, 7L(7u) = Au + V(~//3V - u + 7 0 - u ) - (V. u ) 7 0 + Wu.

Since V . {TL(Tu)} = A { V . u + 7 / 3 V - u + 7 0 . we

u} - V . {(V.u)7@} + V - { W u } ,

have

VT

In ) 7L(7u)

=

{

AM +

(~ 0)(~o ~) (0---yO o ~)~1+~}~(~: )

--{/kin+l+(

0

V

~/3

VT

~/0 T

+

0

u

where Ik is the k times k identity matrix and

M

89

1 + 73

7 oT /

0

-I

I.

7/3

'

1 +7/3 0 |

W

7 1 +0'/9

89

oT

1 +0'/3

( o _

.,/2

0

o

O00+W

) "

Uniqueness and stability in the Cauchy problem ...

346

Note that for w = ~/u, we have

with Q:

(~/-1

(V,),-1)T)

0

7-1In

"

Hence VT

In ) ~ / L w - {AZn+l'~- (

VT

0

0

~ ) yl

Moreover, from

pg/i)t2 (V 9w) zr- V(fl~/) . i)t2w - vT (~/Lw) = vT (pTOt2w -- 7nw), we obtain

( ~In)

")'{tOOt2W - L w } = S O t 2

()-(~) ~ "w w

In

7Lw

with S__ (t9") ' (V(f~)) T ) 0 (pT)I~ " Summing up, we have proved that

In

(A.1)

~/{pOt2w- Lw} -

S [i:c)t2In+l-S-l{/kln+l-~- (VOT

~) ~1+~/~1 (~w~) 9

A short computation yields

S-1MQ = I42 with

K _ p_ 1 ( )~+2#0

2(V~) T -- /9-1~(V/9) T #/~

Moreover, we have

D "- H - 1 K H -

p-1 ( A -t- 2# 0 \

o)

) "

347

M. Eller, V. Isakov, G. Nakamura and D.Tatnru

with 1 0

H=

2(v~) r - p-x~(Vp)T

--(~+~)In

J

Hence, (A.1) yields the following principally diagonal conjugate to the linear elasticity operator: T h e o r e m A.2 - One has "y{pOt2w - L w } = H { O t 2 I n + l - ( S A I n + l

+ P 1 ) } H -1

P~

=

C[Afn+a, H] + A[AIn+I, B] + A

0

A

=

(SH) -1 = (Aij)l
C

=

H - 1 K = (Cij)I<~,j<2, R = V I B H = (R~j)I<~,j<2,

R'

=

V 2 B H = (Rt~j)l
In

where

o)

V

w

R + R'

)

(Bij)l
The explicit forms of these are the following 9

All - / z i p -1 , 1 89p- 1(,~, _[_/Z)-I (/\ -Jr-5/Z) (~/7/Z)T A12 - ~/Z_/Z 89p-2(/~ _[_/Z)-I (/\ _[_ 2/Z)(Vfl)T A21 - 0, A22 = - ( A + #)-1# 89p--lin B l l -/_t

3

(A -}- 2/Z), B12 = x/Z - 89(v#) T, B21 = 0, B22 = z

/Zl

In

Cll -- p - l ( ~ ~_ 2/Z), C12 --/9-1(2(V/Z) T - p - 1 / Z ( v D ) T ) C21 - 0, C22 -- -]_t()~ -~-/Z)-II n i]~11 -- --~l/Z_ ~ (/Z - - A)V / Z R12

1 _~ (/~ -- 3#)V/Z | V/Z + ~1 p - I/Z- 89(/z - A)V/Z @ Vp

-- ~/Z

+ _ 1 ( ~ + , ) v 2 " _ (~ + , ) ( 1 , _ ~ iv,l~ _ ~,1_~ zx,)i~ R21 __ /Z- 89()~ _~_/Z), R22 __ /Z- 89(/~ _~_/Z)(V/Z)T _ p-l/z 89(,~ _}_/Z)(Vp)T R~I = R,2' - 0, R~, - R I , ,

R~2 = R12

348

Uniqueness and stability in the Cauchy problem ...

References [1] D.D. Ang, M. Ikehata, D.D. Trong, M. Yamamoto, Unique continuation for a stationary isotropic Lame system with variable coefficients Comm. Part. Diff. Equat. 23(1998), 371-385. [2] Chelminski, K. - The Principle of Limiting Absorbtion in Elasticity, Bull. Polish Acad. Sc. Math., 41(1993), 19-30. [3] B. Dehman, L. Robbiano, La propri6t6 du prolongement unique pour un syst~me elliptique, le syst6me de Lam6, J. Math Pures Appl., 72(1993), 745-493. [4] L. H6rmander , Linear partial differential operators, Springer-Verlag, Berlin 1966 [5] L. H6rmander, The analysis of linear partial differential operators I-IV Springer-Verlag, Berlin 1983. [6] V. Isakov, A non-hyperbolic Cauchy problem for EIbD~ and its applications to elasticity theory Comm. Pure Appl. Math., 39(1986), 747-767. [7] V. Isakov, Inverse Problems for PDE, Springer-Verlag, New York 1997. [8] R. Leis, Initial boundary value problems in mathematical physics Wiley, New York 1986. [9] D. Tataru, A priori estimates of Carleman's type in domains with boundary, Journal des Math. Pures et Appl., 73(1994), 355-387. [10] D. Tataru, Unique continuation for solutions to PDE's; between H6rmander's and Holmgren's theorem, Comm. PDE 20(1995), 855884. [11] D.Tataru, Carleman estimates and unique continuation for solutions to boundary value problems J. Math. Pures Appl. 75(1996), 367-408. [12] M. Yamamoto, On an inverse problem of determining source terms in Maxwell's equations with a single measurement in Inverse Problems, Tomography, and Image Processing,241--256, Plenum Press, New York

Matthias Eller Department of Mathematics Tennessee Technological University Cookeville, TN 38505 e-mail :ME 1125 @tnt ech. edu Victor Isakov Department of Mathematics and Statistics, Wichita State University Wichita, KS 67260-0033 e-mail: isakov~twsuvm, uc.twsu.edu

M. Flier, V. Isakov, G. Nakamura and D. Tataru

Gen Nakamura Faculty of Engineering Gunma University 1-5-1 Tenjin-cho, Kiryn-shi, Gunma 376 Japan e-mail: [email protected] Daniel Tataru Dept. of Mathematics Northwestern University Evanston, IL 60208 http://www.math.nwu.edu/tataru

349

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 16

ON THE UNSTABLE SPECTRUM OF THE EULER EQUATION

S. FRIEDLANDER

Introduction In this short paper we survey some recent results concerning the unstable spectrum of the Euler equations linearised about a steady flow of an ideal fluid. We discuss the concept of a "fluid Lyapunov exponent" which determines the essential spectral radius of the linearised Euler operator. A positive lower bound for this exponent is computable for many classes of explicit flows and hence gives an effective method for detecting instability in the essential spectrum. The problem of existence of unstable eigenvalues is more difficult and, at least at present, must be treated on a case by case basis. In the case of certain specific flows containing regions of spatim oscillation it is possible to investigate the discrete unstable spectrum using averaging techniques and to demonstrate the existence of unstable eigenvalues. In the case of plane parallel shear flow with a sinusoidal profile we construct the complete unstable spectrum. We discuss a result which proves under certain rather general conditions that linear instability implies nonlinear instability. We show how this abstract theorem can be used to conclude that some specific examples of oscillating flows with discrete unstable eigenvalues are nonlinearly unstable.

1. The Euler equations for fluid motion The Euler equations for the motion of an invisid incompressible fluid are qt + (q" V ) q = - V P

(1.1)

V . q = 0.

(1.2)

352

On the unstable spectrum of the Euler equation

The evolution of the velocity field q(x, t) is determined by its current state, as well as by the internal pressure forces - V p ( x , t). Let Uo(x) be a steady Euler flow. From (1.1), (1.2) the equilibrium equation for U0 can be written in the form Uo x curlUo = - V H ,

V . Uo = 0

(1.3)

where H is a scalar field known as the reduced pressure. We consider the linearised Euler equations for a small perturbation v(x, t) about a steady flow Uo:

v~ v.~

= =

-(Uo. v)v0

(v. v ) U o - v p -

L~ ] j'

(1.4)

with initial condition

v(~, 0) =

vo(x).

(1.5)

The results t h a t we will present are proved for

Uo(x) c (C~

, V . U0 = 0

Vo(X) c (L:(T")) ~ , V . v 0 = 0

(1.6) (1.7)

where T n = Rn/27r Z ~ is an n-dimensional torus. The applications we will discuss are examples in 2 and 3 dimensions. In this paper we discuss the spectral problem associated with equation (1.4), in particular the nature of the stability spectrum a. The stability spectrum of a viscous flow in a bounded domain is simple: it consists of a sequence of eigenvalues. However the more difficult mathematical problem of the spectrum of inviscid flows still has open questions. Unlike the viscous problem where the relevant Navier-Stokes operator is elliptic, the linearised Euler operator L is degenerate non-elliptic and a continuous spect r u m is usually present. In fluid dynamics the inviscid limit is well known to be a subtle problem that is mathematically and physically challenging [1]. We will give some partial answers concerning the structure of the stability spectrum for the invisid equations (1.4).

2. T h e e s s e n t i a l s p e c t r a l r a d i u s Let e L t be the evolution operator for the linearised Euler equation (1.4). We use the following classification of the elements of the spectrum cr. A point z E a is called a point of the discrete spectrum adisc if it satisfies the following three conditions.

353

S. Friedlander

1. z is an isolated point in a. 2. z has finite multiplicity.

e Lt is closed.

3. The range of z -

If z does not satisfy 1 - 3, it is called an element of the essential spectrum (Tess. Thus (T --- (Tdisc [-J (Tess

and the union is disjoint. We denote by ress the essential spectral radius in L 2, where, following Nussbaum [2] we define ress

(e Lt) -- sup {IzII z e (Tess ( e L t ) } .

The following theorem due to Vishik [3] gives an expression for ress (e Lt) in terms of a geometric quantity t h a t can be considered as a Lyapunov exponent for fluid flow. Theorem

2.1 - One has ress

(e Lt) -- e ~t

where

# -

lim

t--,c~

-1 log t

sup [b (xo, ~o, b0; t)] . xo, ~o, bo bo 9~0 = 0 Ib01 = 1, I~ol = 1

(2.1)

Here the vector b is determined by the following system of ODE's which we call the bicharacteristic amplitude equations.

b

(~

=

Uo(x)

=

-

=

-

~. Ox

(2.2.1)

(2.2.2)

~

b+2

\,,~

b.~

~/1~

(2.2.3)

w~th x(O) = xo, ~(0) = ~o, b(O) = Do.

This theorem is proved by writing the evolution operator e Lt as a product of a pseudodifferential operator and a shift operator along the trajectories of the equilibrium flow U0. This allows the growth of the evolution operator to be studied to precise exponential asymptotics. This approach refines the result of Vishik and Friedlander [4] which proves that e tLt gives a lower bound for ress(eLt). A heuristic derivation of this result is obtained by applying

354

On the unstable spectrum of the Euler equation

a "geometric optics" treatment based on high frequency perturbations to equations (1.4). In the language of geometric optics, equation (2.2.2) is the Eikonal equation and equation (2.2.3) is the transport equation. Equation (2.2.3) is the evolution equation for the amplitude of a high frequency wavelet initially localised at x0, with initial wave number vector ~0. The quantity on the RHS of (2.1) is the Lyapunov exponent corresponding to the maximal exponential growth rate of such an amplitude vector b (t). The result of Theorem 2.1 gives one piece of information concerning the stability spectrum for inviscid flows, namely the maximum growth rate of instability in the essential spectrum. Moreover it implies that any point z in the spectrum a(e Lt) such that [z[ > e t't is necessarily an eigenvalue of finite multiplicity. As we will discuss later, a positive lower bound for the value of the Lyapunov exponent # can be explicitly computed in several examples [5], [6]. Furthermore Theorem 2.1 provides an effective sufficient condition for instability of large classes of inviscid fluid flows. Since expression (2.1) involves the supremum over initial conditions (x0, ~0, b0), it is only necessary to show there exists at least one initial condition for which the solution to the system (2.2) of ODE gives lim t~cx~

1 log ]b] > 0 t

to conclude that # > 0, and hence the unstable essential spectrum is nonempty.

3. E x a m p l e s of i n s t a b i l i t y in t h e e s s e n t i a l s p e c t r u m The idea that exponential stretching of fluid particles could imply instability for the Euler equations is originally due to Arnold [7]. Friedlander and Vishik [6] use the result discussed in Section 2 to prove every flow with exponential stretching somewhere in the flow is linearly unstable. T h e o r e m 3.1 - (3-dimensions). Let the flow Uo have a positive classical Lyapunov exponent at some point xo. Then # > O, and hence the flow Uo is unstable. The proof of this theorem follows from a result obtained from the system (2.2) of ODE, namely: d d-t (bl x b2. ~) - 0 (3.1) where bl and b2 are two linearly independent solutions of (2.2.3) corresponding to a co-tangent vector ~ that satisfies (2.2.2). Since the flow is

S. Friedlander

355

volume preserving the existence of a positive classical Lyapunov exponent (i.e. an exponentially growing tangent vector) implies the existence of an exponentially decaying co-tangent vector ~. From (3.1) we then conclude the existence of at least one exponentially growing amplitude vector b, which implies # > 0. Therefore we note that in 3-dimensions the classical Lyapunov exponent provides a lower bound on #. In 2-dimensions, the system (2.2) of ODE provides an even stronger constraint on the relation between b(t) and ~(t), namely d

d-t (Ib(t)l I~(t)[)= 0.

(3.2)

Hence in 2-dimensions the fluid Lyapunov exponent # and the maximal classical Lyapunov exponent for the dynamical system 2 = Uo(x) are the same.

It follows from Theorem 3.1 that any flow Uo (in 2 or 3 dimensions) with a hyperbolic stagnation point x0 is unstable [i.e. U(xo) = 0 and there exists OU an eigenvalue o f ('0-X-~)~o with positive real part]. There are large classes of fluid flows U0 with such stagnation points. For all such flows re~ > 1. A class of flows with presumably chaotic stream lines was identified by Arnold [8]. An example is the so-called A B C flow Uo = (2, y, ~) where 2 = ~) = ~? =

Asinz+Ccosy Bsinx+Acosz Csiny+Bcosx.

(3.3)

For general values of the constants A, B and C numerical investigations [9], [10] indicate that A B C flows exhibit the phenomenon of Lagrangian chaos which suggests strong exponential stretching. Analytic treatment of A B C flows [11], [12], proves that for certain ranges of A, B and C there is exponential stretching either at hyperbolic points or associated with hyperbolic closed trajectories. The result of Theorem 3.1 then proves that these A B C flows are hydrodynamically unstable. In 3-dimensions (unlike 2-dimensions), it is possible to have flows Uo for which the classical Lyapunov exponents are all zero yet the fluid exponent # is positive. Such an example is constructed in [6]. It is proved that the integrable flow U0 • curl U0 = - V H with V H ~-0 has # positive provided that a certain geometric condition is satisfied by the stream lines. The following (non sharp) condition ensures # > 0: T

fo

~ ~ . V H - ~-g Uo-curl Uo/IVH! 2} dt >_0

(3.4)

356

On the unstable spectrum of the Euler equation

where, for any stream line of the flow as it wraps around the toroidal surface H = H0, T is the period, ~ the curvature, T9 the geodesic torsion and ~ the principal unit normal to the stream line. We have described many fluid flows where it can be shown that # > 0. In a few cases p can be computed explicitly. For example, the 2-dimensional cellular flow U0 = ( - sin x cos y, cos x sin y). In this case # is given by the positive real eigenvalue of the matrix ( - ~ ) at a hyperbolic stagnation point. Thus # = 1 for this simple cellular flow. There are certain classes of 2-dimensional flows for which it follows from (3.2) that # = 0 [13]. In particular # = 0 (i.e. there is no unstable essential spectrum) for (1) 2-dimensional flows with no stagnation point, (2) 2-dimensional plane parallel shear flows.

4. Examples of instability in the discrete spectrum We now turn to the question of existence and distribution of unstable eigenvalues in the discrete spectrum of equation (1.4). As we remarked, the linearised Euler operator is degenerate, non-elliptic and there are no general theorems that may be applied to prove the existence of unstable discrete eigenvalues. However in certain rather special examples it is possible to construct unstable eigenvalues. The spectral problem for the linearised Euler operator is considerably simpler in 2-dimensions rather than 3-dimensions. In particular, in 2dimensions we can define a scalar stream function to replace the divergence free velocity field. We write Uo = k x r e ( x ,

y),

v = k x Vr

y, t ) .

(4.1)

Hence v x Uo - f i v 2 e ( ~ , y ) ,

v x v = fiv 2 r

y, t ) .

(4.2)

Here k is the unit vector perpendicular to the 2-dimensional plane with cartesian co-ordinates (x, y). The equilibrium equation (1.3) will be satisfied when ~ satisfies an elliptic equation of the form V2~ = -F(~).

(4.3)

357

S. Friedlander

Taking the curl of equation (1.4) gives the equation for the evolution of the perturbation vorticity w - V • v: Ot = {Uo,w} + {v, V x Uo}

(4.4)

where { , } denotes the Poisson bracket of two vector fields, i.e.

(4.5)

{A,B} = ( B . V ) A - (A. V ) B .

In general the second Poisson bracket on the RHS of (4.4) is very difficult to analyze. However in 2-dimensions the problem greatly simplifies because k - V ( - ) = 0. We note that for 2-dimensional flows it is proved in [14] that there is a solution of the linearised Euler equation (1.4) of the form e at v(x) with v c L 2, V - v = 0, if and only if there is a non-trivial solution w E H -1 of the spectral problem for the 2-dimensional reduction of equation (4.4) given by w + ( U o - V ) w + (v. V)(V x Uo) = 0. (4.6) We therefore consider the eigenvalue PDE given by subsituting (4.1) - (4.3) into (4.6), namely

v 2r

%~ 0

~ ~ 0 ) (v 2 r

F'

(~)r

(4.7)

We take the boundary conditions to be 2~ periodicity in (x, y). A simple and very classical example that has received much attention in the literature of the past 100 years is plane parallel shear flow (see, for example, [15], [16], [17]). In this case U0 = (U(y), 0) and (4.7) becomes the so-called Rayleigh equation:

(r

+ u(y))

(d2 ) ~

- k2 ~(y) - u"(y)~(y) = o

(4.8)

where we have written

r

y, t) = ~(y) ~k~ ~ .

(4.9)

The celebrated Rayleigh stability criterion [15] says that a necessary condition for instability is the presence of an inflection point in the profile U(y). As we remarked in Section 3, the concept of the "fluid Lyapunov exponent" # given by expression (2.1), can be used to prove that equation (4.8) with periodic boundary conditions has no unstable essential spectrum for any

358

On the unstable spectrum of the Euler equation

profile U(y). It remains to discuss the possibility of discrete unstable eigenvalues (i.e. a such that Re a > 0) associated with equation (4.8) for profiles U(y) that contain at least one inflection point. Meshalkin and Sinai [18], followed by Yudovich [19] investigated the instability of a viscous shear flow U(y) = sin m y using techniques of continued fractions. More recently Friedlander et al [13], [20], [21] showed that these techniques could be used for the invisid equation (4.8) with U(y) - s i n my. Eigenfunctions are constructed in terms of Fourier series that converge to C~-smooth functions for eigenvalues a that satisfy the characteristic equation. We write OO

(~(Y) =

E

an

e iny

.

(4.10)

n=--oo

The recurrence relation equivalent to (4.8) yields the following tridiagonal infinite algebraic system" dn+m = 13r~(a) dn + d n - m , n C Z

where /3n(O') :

2a

(k 2 + n 2)

T"

k2 + n 2 - m 2

(4.11)

(4.12)

and dn - an(k 2 + n 2 - m2).

(4.13)

The system (4.11) is treated using continued fractions to yield the characteristic equation for cr namely (

1 )( 1 ) flJ + [flj+m,~j+2m,-.-] flj-m + [/~j--2m,/~j--am,..-] + 1 = 0 (4.14)

for each integer j - 0, 1 , . . . [m/2]. Here [ , . . . ] denotes an infinite continued fraction, i.e. =

~jTm + /3j+2m + f~j+ 1 a~,, +...

(4.15)

In [20], [21] an analysis is given of all the roots a with Re a > 0, of equation (4.14) for each fixed integer m and 0 <_ j <_ m / 2 . It is shown that for k2 > m 2 - j2 there are no eigenvalues. For k~ < k 2 < m 2 - j2, for a critical wave number kc, there are purely real roots and for 0 < k 2 < kc2 there is a complex conjugate pair of roots. This describes the complete unstable spectrum associated with the flows U - sin my.

359

S. F r i e d l a n d e r

The existence of unstable eigenvalues for equation (4.8) with a general rapidly oscillating profile U ( m y ) , m >> 1, was demonstrated in [21] using homogeneization techniques to compute the spectral asymptotics. It is proved that one root of the asymptotic characteristic equation is given by a / k = + v / ( U 2)

as rn --, c~

(4.16)

where ( 9 } denotes the 27r-average with respect to the fast variable m y . Numerical analysis is used to describe the qualative behaviour of the distribution of roots of the asymptotic characteristic equation in the parameter space of the wave numbers n and k. The following is an example of a stream function 9 satisfying an equation of the form (4.3) which exhibits both the features of exponential stretching at a hyperbolic stagnation point and oscillatory shear flow behaviour. We consider ~= l (cos(x+my)+acos(x-my)) (4.17) m where a is a constant such t h a t 0 < a < 1. This flow has hyperbolic points at x + m y = 2n~r , x -

m y = (2j - 1 )

~r.

Hence by the results of Section 3 there is a non empty unstable essential spectrum associated with this flow. The fluid Lyapunov exponent # can be calculated explicitly in this example to give # -- 2 a 1/2 .

(4.17a)

Again homogenisation techniques can be used to demonstrate the existence of unstable eigenvalues for equation (4.7) with @ given by (4.17) with m >> 1 [22]. We introduce a change of variables ~ = m y and write (4.7) in the form 0

+ x

02 )

[

o]

(sin (x + r/) - a sin (x - r/)) ~0 _ (sin (x + r/) + a sin (x - r/)) N

[0 2

02

~-Sx2+m2~2+(1+m

]

2) r (4.18)

We seek Block eigenfunctions of the form r = e ipn/m G ( x , rl)

(4.19)

On the unstable spectrum of the Euler equation

360

where p is an integer such that p << m, and we seek smooth functions given by a regular expansion in powers of 1/m: -

G(x, ~7)

1

~ ( X , ?7) --" (I)o (X) -~- (~o(X, ?7) -~- - - ((I) 1 (X) -t- ~1 (X, ?~)) -t- 9 9 9

m

with (r""

= fO 27r r~" d r / - 0

and o = or0 +

(4.20)

1 ~

cr 1 -t-

....

The solvability conditions for the hierarchy of equations for Cj gives the ODE that define the averaged part of the eigenfunctions ~j. The equation for (Do becomes d2 a02 ( ~ x 2

+ a2

- p2} ~ 0 + ( 1 2 +ac~ ( d 2 _p2) d'~o

-4asin2x

~ x2

-~x -4ac~

\ax - p 2 / d2~~2 d2 ~

(4.21)

D0-0.

We consider the behaviour of (4.21) with the parameter a. In the limit a = 0 equation (4.21) reduces to the equation governing the spectral asymptotics for oscillatory shear flows. When a = 1 the equation is degenerate at the hyperbolic stagnation points where cos 2x = - 1 . In this case the flow has no shear flow type regions and the degeneracy of (4.21) implies there are no discrete unstable eigenvalues with smooth eigenfunctions. For 0 < a < 1, equation (4.21) has discrete real eigenvalues a0 which are the leading order part of an expansion in powers of ( l / m ) of the eigenvalues of (4.18). For large wave number k, Weyl's formula gives the asymptotic behaviour of a0 as

a2o/k2 ---, (1 - a2)/2

(4.22)

which as a --~ 0 is in agreement with expression (4.16) for the purely sheared flOW.

As we remarked, the fully 3-dimensional vorticity equation (4.4) is very difficult to study. However some "2 1/2" dimensional examples are tractable. For example, the stability problem of a three component velocity of the form U(x, y ) - k • Vqz + k g(q~) (4.23) is governed by an equation for a z-independent perturbation of the form of equation (4.7). A specific example is the "ABC" type flow with 2=cosmy,

~=--

1

m

sinmx, ~=sinmy+m

1

m

cosmx.

(4.24)

S. Friedlnnder

361

For large m, averaging techniques again lead to an equation for the average with respect to my of the leading order eigenfunction ~0(x): ( ) l ( d 2 ) a2o ~d2 _ p2 ~o + -2 ~ _ p2

d2~~ dx 2 -- O.

(4.25)

Hence Cr2o/k2= 1/2 and we conclude there exist discrete unstable eigenvalues. The 3-dimensional general A B C flow given by (3.3) is not integrable. It has been investigated numerically by a number of authors including Henon [9] and Dombre et al [10]. It appears that the Lagrangian trajectories of the flow (3.3) densely fill some open domains of T 3. Stagnation points may occur and when they do there is numerical evidence that they are connected by a web of heteroclinic streamlines. Poinca% sections computed for more general A B C flows include patterns that are qualitively similar to the streamlines of the special flow given by (4.24). This suggests that the unstable spectrum associated with general A B C flows contains both an essential part and discrete eigenvalues.

5. N o n l i n e a r

instability

In the previous sections we have discussed the unstable spectrum of the Euler equation linearised about a given steady flow U0. The following general theorem of Friedlander et al [13] can be applied to prove that some of the examples we considered are also nonlinearly unstable. Consider the evolution equation Wt

--

L w + N ( w ) , w(O) = wo

(5.1)

and fix a pair of Banach spaces X ~-~ Z with a dense embedding. Here L is the generator of a C0-group of operators in s e Lt leaves X invariant for t c R, X c D ( L ) and N is a nonlinear operator N : X -~ Z. We assume there is local existence theorem for (5.1) for w0 C X. Definition of n o n l i n e a r stability: The trivial solution w = 0 of (5.1) is called nonlinearly stable in X if for all ~ > 0 there exists a 5 > 0 such that Ilw(O)llx < 5 implies (i) there exists a unique solution w(t) e L~((0, oc); X) N C([0, oo), Z) and

On the unstable spectrum of the Euler equation

362

(ii) IIw(t)lIx < : for a.e. t e [0, cxD). The trivial solution w = 0 is called nonlinearly unstable if it is not stable.

Note: by this definition the "blowing up" of a solution is a particular case of instability. T h e o r e m 5.1 - Let (5.1) admit a local existence theorem in X . and L satisfy the following conditions. (1)

Let N

IIN(w)liz ~ Collwllx Ilwllz fo~ w e X with [Iwllx < p fo~ ,omr p > o.

(5.2) (2) A spectral "gap" condition, i. e. a(e Lt) = a+ U a _ with a+ ~ r where ~+ c (z e C l ~ M~ < Izl <~A~} ~_ C (z e C I~ ~ < I~1 < ~"~}

(5.3)

with -oc <)~<#<M
andM>O.

(5.4)

Then the trivial solution w = 0 to equation (5.1) is nonlinearly unstable. The proof of this theorem is by contradiction and it uses the gap condition to permit a projection onto the subspace of growing modes that is guaranteed to exist by virtue of the spectral gap. We note that a stronger result, i.e. replacing condition (2) by the condition that the unstable spectrum rT(eLt) is non-empty, may in fact be true but would require a different method of proof. We apply Theorem 5.1 to the Euler equations in the following way. We rewrite ( 1 . 1 ) - (1.2) with the notation

q = u0 + w

(5.5)

where Uo e ( C ~ ( T n ) ) n is a steady solution of (1.1), i.e.

wt V.w

= -

- ( U o . ~7) w _ (w. V)Uo - (w. V) w o.

VP

(5.6) (5.7)

Hence in this notation

Lw N(w)

- ( U o . V ) w - (w. V)Uo - Vp =_ - ( w . V) w - Vq

(5.8) (5.9)

363

S. Friedlander

and instability of the trivial solution w = 0 of the equation wt = L w + N ( w ) corresponds to instability of the steady Euler flow U0. We make the "natural" choices of the spaces X and Z: X Z

=

X s -- {w e ( H S ( T n ) ) n l V . w - - = {w e ( L 2 ( T ~ ) ) ~ I V . w =0}.

0 } , s > ~ -[- 1

(5.10) (5.11)

For this choice of X local existence is a classical result [23] and condition (5.2) follows from the Sobolev embedding theorem. The condition that is difficult to check is the spectral gap condition (5.3) - (5.4). We note that in the case of the Navier-Stokes equations in a finite domain the additional viscous term v~72q ensures that the equations are elliptic. Hence the spectrum is purely discrete and the existence of any unstable eigenvalue ensures the gap condition is satisfied. For the Euler equations, we return to the examples of Sections 3 and 4 where the structure of the unstable spectrum was analysed for certain choices of U0. In Section 4 we discussed how any plane parallel shear flow with a sufficiently oscillating profile has a non-empty discrete unstable spectrum. The results of Section 3 can be used to show that the unstable essential spectrum associated with this flow is empty. Hence the spectral gap condition required by the proof of Theorem 5.1 is satisfied and therefore all shear flows with rapidly oscillating profiles are nonlinearly unstable. We note this result is proved with periodic boundary conditions. The cellular 2-dimensional flow given by (4.17) is also shown to have discrete unstable eigenvalues. In this case the essential unstable spectrum is nonempty. However for a ~= 1 the computable expressions for the essential spectral radius (see (4.17a)) and the spectral asymptotics (see (4.22)) show that there exist unstable eigenvalues outside the essential spectral radius. Hence again we can apply Theorem 5.1 to prove nonlinear instability. References

[1] Constantine, P., A few results and open problems regarding incompressible fluids, Notices of AMS, 42, n ~ 6 (1995), 658-663. [2] Nussbaum, R., The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478. [3] Vishik, M.M., The spectrum of small oscillations of an ideal fluid and Lyapunov exponents, J. Math. Pures et Appl., 75 (1996), 531-557. [4] Vishik, M.M. and Friedlander, S., Dynamo theory methods for hydrodynamic stability, J. Math. Pures et Appl., 72 (1993), 145-180. [5] Friedlander, S. and Vishik, M.M., Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., 66, n ~ 17 (1991), 22042206.

364

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

On the unstable spectrum of the Euler equation

Friedlander, S. and Vishik, M.M., Instability criteria for steady flows of a perfect fluid, Chaos, 2, n ~ 3 (1992), 455-460. Arnold, V.I., Notes on the 3-dimensional flow pattern of a perfect fluid in the presence of small perturbation of the initial velocity field, Appl. Math. Mech., 36, n ~ 2 (1972), 236-242. Arnold, V.I., Sur la topologie des coulements stationnaires des fluides parfaits, C.R. Acad. Sci. Paris, 261 (1965), 17-20. Henon, M., Sur la topologie des lignes de courant dans un cas particulier, C.R. Acad. Sci. Paris, 262 (1966), 312. Dombre, T., Frisch, U., Greene, J.M., Henon, M., Mehr, A. and Soward, A., Chaotic streamlines in A B C flows, J. Fluid Mechanics 167(1986), 353-391. Friedlander, S., Gilbert, A. and Vishik, M.M., Hydrodynamic instability for certain A B C flows, Geophys. Astrophys. Fluid Dynamics, 73(1993), 97-107. Chicone, C., A geometric approach to regular perturbation theory with application to hydrodynamics, Trans. AMS, 347, n ~ 12 (1995), 45594598. Friedlander, S., Strauss, W. and Vishik, M.M., Nonlinear instability in an ideal fluid, Ann. Inst. Henri Poincar, Analyse non Lin~aire 14, n ~ 2 (1997), 187-209. Friedlander, S., Strauss, W. and Vishik, M.M., Robustness of instability for the two dimensional Euler equations, submitted for publication. Rayleigh (Lord), On the stability or instability of certain fluid motions, Proc. Lond. Math. Soc., 9 (1880), 57-70. Tollmein, W., Ein allgenienes kriterium der instabilitat laminarer geschwindigkeitsverteilungen, Nachr. Ges. Wiss. Gottingen Math. Phys., 50(1935), 79-114. Lin, C.C., The Theory of Hydrodynamic Stability, Cambridge University Press (1955). Meshalkin, L. and Sinai, Y., Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid, App. Math. and Mech., 25(1961), 1140-1143. Yudovich, V.I., An example of secondary stationary flow or periodic flow appearing while a laminar flow of a viscous incompressible fluid looses its stability, App. Math. and Mech., 29(1965), 453-467. Friedlander, S. and Howard, L., Instability in parallel flows revisited, Studies in App. Math., to appear Belenkaya, L., Friedlander, S. and Yudovich, V., The unstable spectrum of oscillating shear flows, SIAM J. App. Math., to appear Friedlander, S., Vishik, M.M. and Yudovich, V., The unstable spectrum of a 2-dimensional periodic flow of an ideal fluid, in preparation.

S. Friedlander

365

[23] Temam, R., Local existence of C a solutions of the Euler equations of incompressible perfect fluids, Lecture Notes Math., 565 (1976), 184-194. Susan Freidlander Department of Mathematics (M/C 249) University of Illinois-Chicago Chicago, Illinois 60607 USA E-mail: [email protected]

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 17 DECOMPOSITION EN PROFILS DES SOLUTIONS DE L'EQUATION DES ONDES SEMI LINEAIRE CRITIQUE A L'EXTERIEUR D'UN OBSTACLE STRICTEMENT CONVEXE

I. GALLAGHER ET P. Gt~RARD

R ~ s u m ~ . On s'int~resse ~ l'~quation d'ondes semi lin~aire critique (1)

[-]U n + [Un[4Un -'- 0

darts

IRt • ~,

avec conditions aux limites de Dirichlet, oh ~t est l'ext~rieur d'un domaine strictement convexe de IR 3. En suivant la m~thode introduite par H. Bahouri et P. G~rard dans [1] dans le cas off gt = ]R 3, on d~montre tout d'abord un th~or~me de structure pour des suites de solutions d'~nergie born~e de l'~quation des ondes lin~aire dans ~. La d~monstration fait appel ~ un r~sultat de non concentration pour de telles suites, qui s'obtient en utilisant des mesures semi-classiques. Les estimations de Strichartz d~montr~es par H. Smith et C. Sogge dans [16] permettent d'en d~duire une description des suites de solutions d'~nergie born~e de (1), k des termes de reste pros, petits en ~nergie et en normes de Strichartz. A b s t r a c t . We are interested in the critical semilinear wave equation (1)

[:]Un+iUnl4Un=0

in

IRt x~t,

with Dirichlet boundary conditions, where ~t is the outside of a strictly convex domain of IR 3. Following the method introduced by H. Bahouri and P. G~rard in [1] in the case where ~ = IR 3, we first prove a structure theorem for sequences of solutions to the linear wave equation in ~t. The proof requires a non-concentration result for such sequences, the proof of which involves semi-classical measures. The Strichartz estimates proved "by H. Smith and C. Sogge in [16] enable us to infer a description of sequences of bounded energy solutions to (1), up to small remainder terms, both in energy and in Strichartz norms.

Ddcomposition en protils des solutions de l'dquation des ondes...

368

1. I n t r o d u c t i o n Dans [1], H. Bahouri et P. G~rard ont obtenu un ddveloppement en profils des solutions de l'~quation d'ondes semi lindaire critique suivante: [~]ltn 21-I?.tnl4~tn --- 0

dans

lRt • IR 3,

(u~,

OtUn)lt=O -

((r

I/)n) ,

(1)

~)n)nEIN est une suite born~e de /_:/1 • L2(]R3). Ce d~veloppement s'obtient ~ partir d'une ~tude sur l'~quation libre

oh (~n,

[::]v~ - 0

dans

IRe • IR 3,

(Vn, C~tVn)lt--O

---

(r

Cn),

en utilisant de mani~re cruciale des estimations de type Strichartz (voir [10] par exemple, ou l'estimation (2) ci-dessous). Notre objectif ici est de reprendre cette ~tude dans le cas oh le syst~me (1) est pos~ dans un domaine de l'espace. Le seul cas connu pour lequel on dispose d'estimations de Strichartz est celui de l'ext~rieur d'un obstacle strictement convexe. Ainsi dans [16], H. Smith et C. Sogge ont montr~ que pour tout temps T > 0, et pour tout couple (q, r), v~rifiant 1/q + 3/r = 1/2 et q > 2, il existe une constante CT,~ telle que pour toute fonction f telle que (f, Otf) E C~ T],/~1(~) x L2(gt)), on a

IIfllLa([O,T],Lr(~)) <_ CT,,.

(llVf,~___oll/=(~) -4-[[DflILI([O,T],L=(~))).

(2)

On a not6 /:/01(~t) la fermeture de C~(Ft) pour la norme [IV. IIL2(IR3) dans/:/I(IR3). Dans le cas oh ~ = IR 3, un simple argument d'6chelle permet de montrer que CT, r ne d(~pend pas de T. Dans notre cadre d'6tude, oh ~ est l'ext6rieur d'un obstacle strictement convexe, un article r6cent de H. Smith et C. Sogge (voir [17]) montre que CT, r ne d6pend pas non plus de T. Nous allons donc dans cette 6tude consid6rer un domaine D de ]R 3, r6gulier, ext6rieur d'un obstacle strictement convexe, et 6tudier le probl~me suivant"

{[ oh

-]Un +

]Unl4Un - - 0

dans

IRt • ~,

UnllR~•

= 0

(un, OtU~)lt=O = (p,~, ~ ) ,

(3)

((~:)n,Cn)nE]N est une suite born~e de E(ft) d~f/:/l(ft) • L2(~ )

I1 est de~montr(~ dans [16] l'existence et l'unicit(~ de solutions r~guli~res, globales en temps, 5. (3) pour (~,~,~n) e C ~ ( f t ) • C ~ ( ~ ) . Dans le cas oh (~,~,~n) E E(Ft), on peut montrer, en utilisant les techniques de J. Shatah et M. Struwe (voir [15]) et des estimations de [16], l'existence et l'unicit~ d'une solution un ~ (3), et

Un e C~

HI(Ft))ACI(]R, L2(~))NL~oc(]R, LI~

(4)

I. Gallagher et P. GE/.ard

369

La suite (u~) est de plus born6e dans l'espace d'6nergie. L'un des %sultats de cette 6tude est qu'elle est 6galement born6e dans L~oc(lR , LI~ Cela permet en corollaire d'obtenir une estimation Lipschitz sur le groupe d'6volution non lin6aire associant une donn6e initiale (p~, ~n) dans un born6 de l'espace d'6nergie, ~ la solution Un de (3). Nous renvoyons ~ [4] pour un 6nonc6 pr6cis. Nous ne nous attarderons pas ici sur ce probl~me de r6solution de (3) dans l'espace d'6nergie. Remarquons que la norme L~oc(lR, Ll~ apparaissant dans (4) est naturellement associ6e au syst~me semi lin6aire critique (3): c'est la norme de Strichartz qui permet de consid6rer le terme non lin6aire lu~lau~ comme un terme de force dans L~o~(]R , L2(f~)). Avant d'6noncer les r6sultats que nous allons pr6senter ici, donnons quelques notations: nous appellerons Vn la solution du probl~me lin6aire associ6 k (3): I-Ivn

-

-

-

0

dans ]Rt • ~, VnlIRtxO• -- 0 (Vn, OtVn)lt=O -- (~gn, t~n).

(5)

Rappelons que l'~nergie d'une fonction f est donn6e par E0(f, t)d~t=21/n (lOtf(t, x)i 2 + IVf(t, x)l 2) dx, et ne d~pend pas de t si f v~rifie (5). De m~me, pour une solution de (3) v~rifiant (4), la quantit~ E ( f , t ) d~f 1 / ~ If(t, x)l 6 dx = ~1 /~ (lOtf(t,x) 12 + I V f ( t , x ) 12) dx + -~ est une constante. Le premier r~sultat de ce travail concerne les suites de solutions de (5). Commengons par donner une d6finition.

D~finition 1.1 -Soit (~,r C/~1 xLZ(IR3), etsoit ( h n , x n , t , ) C (IR+)~ x IF[3IN x IR IN, tels qu'il existe ( x ~ , too) C IR 3 x IR, avec

limcx3(hn,Xn,~n) --(O, xoo,~o ). n---+ On appelle onde de concentration lingaire associde ~ (~p,%&,hn,Xn,tn) la solution du probl~me suivant:

(Pn'OtPn)lt--tn--

~n/27)g~ ~( ]ln ) ' ~ 3 - - ~ lg~ (')~/J( hn )

off 7Ju est le p/.ojecteur o/.thogonal

de/-]I(IR 3) st//"/_~/1(~-~).

(6)

370

DEcomposition en profils des solutions de l'dquation des ondes...

Remarques. On omet de specifier, dans la d~finition d'une onde de concentration lin~aire, le domaine dans lequel on r~sout l'~quation d'onde lin~aire (6). On supposera toujours que ce domaine est f~. D'autre part, quitte & extraire une sous-suite et ~ faire une translation en temps, on peut toujours supposer que lim tn - - ~-Cx:~ --Cx~ 011 O. n~oo ~ Dans la suite, nous appellerons donn~e concentrante toute quantit~ du type ( p , r h n , x n , t n ) de la d~finition 1.1. Nous dirons que (hn,xn,tn) et (h~, 5~, t~) sont orthogonales, comme dans [1], si soit

soit

hn = h~

n-.oolimlog \(h~-n~ ]

lim I(xn' tn) -- (Xn, tn)l = +OC. hn

et

n ---, c ~

D'autre part, nous dirons que deux donn~es concentrantes sont ~quivalentes si la diffe@ence des ondes de concentration lin~aires associ~es tend vers z~ro en ~nergie, quand n tend vers l'infini. Nous ferons dans tout ce travail l'hypoth~se de compacit~ & l'infini suivante: lim limsup / ( l ~ 7 ( ~ n ( X ) [ 2 + [~n(x)l 2) dx - O. (7) R - - * c<)

n---* o o

J [x [ ~ R

Enfin nous supposerons pour simplifier que Pn et ! ~ tendent faiblement vers z~ro quand n tend vers l'infini. Nous sommes & present en mesure d'~noncer le premier r~sultat de cette ~tude. Wh~or~me 1 - Soit vn la solution de (5), off (Pn, ~n) est bornde dans E ( ~ ) et vdrifie (7). I1 existe des donnEes concentrantes (p(J), ~(Y), h ~ ) , x ~ ) , t ~ ) ) , pour j C IN \ {0}, orthogonales deux-&-deux, telles que quitte & extraire une sous-suite, la solution Vn se decompose de la mani~re suivante, pour tout g C IN \ {0}: g

(t, x) =

p(d ) (t,

+

j=l

off p(J) est l'onde de concentration associde & (p(J), ~/(J), h~ ), X(nj), t~ )), Off le reste vErifie VT > O,

lim limsup IIW(n~)[ILoo([_T,T];L6(f~)) --O, ~--* O0 n----,oo

I. Gallagher et P. GErard

371

et off les Energies sont orthogonales, au sens off

F~o(Vn) --- ~ Eo(p(nj)) nt- Eo(w(n s j=l

nt- o(1),

n ---~ o o .

Consid~rons ~ present l'~quation non lin~aire (3). Nous allons commencer par ~noncer un th~or~me comparant une onde de concentration lin~aire ?~ la solution de (3) avec m~me donn~e de Cauchy. Dans ce but, donnons quelques d~finitions n~cessaires ~ l'~nonc~ du th~or~me. D~finition 1.2 - Soit (~, r h~,x,~, tn) une donnde concentrante, et soit p~ l'onde de concentration Bndaire associde, au sens de la ddfinition 1.1. Alors l'onde de concentration n o n / i n d a i r e associde g Pn est la soIution de 1'Equation suivante:

{ nqn + [qn]4qn = 0 dans I R t x f~, qnllRtxOa -- 0 (qn, Otqn)tt=o = (P~, Otp~)lt=o .

(S)

Pour toute suite ( h ~ , x n ) convergeant vers (0, x ~ ) , on d~finit le domaine remis ?~ l'dchelle ~'~n d ~ f f~ - --hn

Xn 9

(9)

On verra dans la section suivante que f ~ a d m e t un "domaine limite", quand hn tend vers z~ro, que nous noterons f~oc. D ~ f i n i t i o n 1.3 - Soient M~ et M deux ouverts de ~ 3 . On dit que-]l//n converge vers M quand n tend vers l'infini si les deux propriEtEs suivantes son t vErifi Ees: (i) pour tout compact K de M , il existe N E ]IN tel que pour tout n >_ N, K est inclus dans 2r (ii) pour tout compact K t de CM, il existe N C IN tel que pour tout n >_ N , I ( ~ est inclus dans CM n . On m o n t r e r a dans la section 2.1 le r~sultat suivant: P r o p o s i t i o n 1.4 - Soit le domaine f~n ddfini en (9). Alors quand n tend vers l'infini, ftn prend l'une des limites suivantes, quitte ~ extraire une soussuite: ~, IR 3 ou un demi-espace H, suivant la position de xn p a r rapport Oft. Enfin rappelons que dans [1]-[2], un op~rateur de scattering S ainsi que des op~rateurs d'onde W i sont fitudifis. Dans notre travail, nous aurons besoin d'associer ~ une onde de concentration lin~aire pn deux ondes p~, de la mani~re suivante.

DEcomposition en profils des solutions de l'dquation des ondes...

372

D4finition

1.5

-

Soit pn une onde de concentration Bndaire, associde ~ des

donndes concentrantes (99, ~2, h~, x~, t~). Soit f ~ la limite du domaine f ~ d4t~ni en (9), au sens de la d4~nition 1.3. Si f l ~ est un demi-espace, on note RaM la r4flexion p a r antisym4trie p a r rapport g O f ~ , et PaM

1'opdrateur PaM (f, g) d4f (~)f2Mf, l a M g ) , pour tout (f, g) e /:/I(IR 3) x L2(]R3).

Alors on associe ~ Pn deux ondes de concentration lindaires p~, assocides aux donn6es concentrantes (99+, r hn, x~, t~), avec

(99_, r

d~_f

tn

(99,r

si

lirn ~

PaM W - RaM (99, ~)

si

l i r n ~th ~ - 0

PaM S - 1RaM (99 ~)

si

'

PaM SRaM (99, r

t~

lim ~ n--+ c~

si

-- + o c

= --oc

hT ~

lim ~t~

'

--- -~-(X)

n - - + (X) h n

(99+, I~+) d~f

Pf~Mw+Ra M (99,~)

si

(99 r

si

'

lim - -

n ---+CX3 h / ~

= 0

rn lim - - -- - - e c . n ----+(::~ h n

2 - Soit pn une onde de concentration lindaire, associde ~ des donndes concentrantes (99, r et soit qn l'onde de concentration non lindaire associ6e g~Pn. A1ors qn est bornde dans L~oc(IR, LI~ et vdrifie les propridtds suivantes, avec les notations de la ddfinition 1.5: pour tout temps T > 0, la limite quand A tend vers l'infini de

Th~or~me

/ lim sup ( n--,c~

Eo(p~ - qn, t) nt- IlPX - qnllLS([--T, tn--Xh,,],Ll~

sup

k.-T<_t<_t,--Xh.,,.

et de lim sup ( n-+oc

sup

\t..+Ah,,

Eo(p + - qn, t) + lip+ - q~[IL~([t,,+~h..,T],LlO(~)))


est nulle. Remarques.

Le rdsultat que nous ddmontrerons sera en fait une version plus precise du thdor~me 2, car nous obtiendrons aussi une description de l'onde de concentration non lin~aire qn autour du t e m p s de concentration tn, c'est-5~-dire pour t C [tn -/~hn, tn + )~hn] avec les n o t a t i o n s du th~or~me 2. Ce r6sultat, associ6 au th6or~me 1, p e r m e t d'obtenir une d6composition en profils pour la solution u~ du probl~me de d @ a r t (3) (voir [4]): on utilise

I. GaIlagher et P. G&ard

373

pour cela l'orthogonalit~ des p~) ainsi que des donn~es concentrantes. La d~composition s'obtient facilement pour des normes de Strichartz petites, puis le th@or~me 2 permet, par un argument de d~formation en temps semblable au cas trait~ dans [1], de d~duire la d~composition g

ve e

\ {0},

Un (t, x) = E qU)(t, x) -t- w (~) (t, x) + r (e) (t, x), j=l

o2 r (e) tend vers z@ro en norme ~nergie et en normes de Strichartz. Les q(J) sont des ondes de concentration non lin~aires, uniform~ment born@es en ~nergie et dans L~oe(IR; Ll~ Nous n'entrerons pas dans les d~tails ici, mais nous concentrerons sur la d~monstration des th~or~mes 1 et 2. Nous renvoyons g [4] pour plus de pr~cisions. D'autre part, comme nous l'avons remarqu~ plus haut, on s'est restreint au cas de l'ext~rieur d'un domaine strictement convexe car c'est le seul cas connu pour lequel on dispose d'une estimation de type Strichartz pour les solutions de l'@quation des ondes, cruciale dans notre d~marche. N@anmoins, supposer q u ' u n e telle estimation soit vraie dans d'autres cadres, d'autres probl~mes apparaissent: n o t a m m e n t l'orthogonalit~ en temps des ondes de concentration lin~aires repose sur un r~sultat de non-concentration de la norme L 6 de la solution de l'@quation lin~aire (voir la section 2.3 ci-dessous), pour lequel la forme de f~ choisie ici joue un rSle important. Le plan de l'&ude est le suivant. Dans la section 2 ci-dessous, nous 6tudions t o u t d ' a b o r d le domaine remis g l'6chelle ~t~ d6fini en (9), puis nous ~nongons un r~sultat de propagation de la (h~)-oscillation stricte, et enfin nous donnons un r4sultat fondamental de non concentration de la norme L6(~t) pour des ondes de concentration lin4aires. Quelques ~14ments de d4monstrations sont pr&ent4s. Dans la section 3, nous d4montrons le th~or~me 1, tandis que le th~or~me 2 est d~montr~ en section 4. Nous noterons toutes les constantes universelles par la m@me lettre C. D'autre part, sauf mention du contraire, toutes les fonctions d4finies sur un domaine de IR a seront prolong~es par z~ro en des fonctions d4finies dans l'espace IR 3 entier.

2. l ~ t u d e p r d l i m i n a i r e L'objet de cette section est d'~noncer quelques r~sultats pr~liminaires auxquels nous ferons appel par la suite. Nous en donnons les idles de d~monstration, et renvoyons g [4] pour les d~tails.

374

Ddcomposition en profils des solutions de l'dquation des ondes...

2.1. Passage h la limite dans des domaines remis h l'~chelle Nous avons d~fini en (9) le domaine remis ~ l'~chelle ~ . Nous allons (~tudier ici l'~volution de ce domaine quand n tend vers l'infini, et notamment d~montrer la proposition 1.4; rappelons que la notion de convergence du domaine g/~ a ~t~ donn~e en d~finition 1.3. La proposition suivante d~crit les diff~rentes configurations possibles pour ~or la limite de ~ , et p%cise la proposition 1.4. Avant d'~noncer le %sultat, donnons quelques notations. Rappelons que l'on a not~ xcr c IR3 la limite de la suite (x~). On choisit une fonction 4) telle que

a = {x e

I

> 0}.

Si x ~ C O~t, on note alors w = V(I)(x~) et a =

lim

hn

, limite qui

existe dans IR quitte ~ extraire une sous-suite. Enfin on d~finit les ensembles

H~,~ d~g {y e I R 3 1 y . w > - a } H + ~ , ~ d_6fIR3

et

si a e IR,

(10)

g_cc,w d~_f 0.

P r o p o s i t i o n 2.6 - Le domaine f~n ddfini en (9) a d m e t l'une des limites suiyantes quand n tend yers l'infini: (i) si xoo E ~, alors ~oo - IR 3 et pour tous (p, r C i2/1 x L2(IR3), pour toute suite (tn) E IR IN, l'onde de concentration lindaire pn associde ~t (99, r hn, Xn, tn) ydrifie lim E o ( p ~ ) =

IIV~ll~=

(ii) si xc~ C c~, alors le domaine limite vdrifie ~oo = fi et toutes les donndes concentrantes sont dquiyalentes ~ zdro. (iii) si xoo c Oft, a10rs le domaine 1imite ydrifie ~oo = H ~'~ et toute donnde concentrante (p, r h~,x~, t~) est dquiyalente ~ la donnde concentrante (T)H~,~, 1H~,~r hn,xn, tn). Ddmonstration. Nous n'allons pas ~crire tous les calculs ici: les deux premiers cas se d~montrent de faqon tr~s simple, nous allons donc seulement donner les arguments p e r m e t t a n t d'obtenir le cas (iii). Par d~finition, y c ]R 3 est un ~l~ment de ~ . si et seulement si (I)(x~ + h~y) > 0. Commenqons par supposer que c~ = +c~, et soit K un compact de IR 3. Alors la formule de Taylor nous permet d'~crire, uniform~ment en y, Vy E K,

~(x~ + hny) _- (~(x~._.___~)+ V ( ~ ( x ~ ) . y + o(1), hn h~

n -~ cr

375

I. Gallagher et P. Gdrard

Pour n assez grand, on a donc O(x~ +h~y) > 0, donc y 6 ftn, et donc f t ~ = IR 3. Le m~me type de raisonnement p e r m e t de montrer que si a = - o o , alors flo~ = O. Reste donc ~ 4tudier le cas off a c IR. Si K est un compact de H ~'~, alors par d4finition de H ~'~ en (10), on a VyeK,

V~(x~).y=-a+m(y),

avec

re(y) >0,

donc pour n assez grand, uniform4ment en y 6 K , on a ~(Xn + h~y) > O. Inversement, si K est un compact de c H ~'~, alors pour tout y E K , pour n assez grand uniform4ment en y, on a ~(xn + hny) < 0, et le r4sultat s'en d4duit imm4diatement" f t ~ -- H ~'~. Pour conclure la d4monstration de la proposition, consid4rons une donn4e concentrante (p, r hn, xn, tn), d'onde de concentration lin4aire associ4e Pn, et soit p,~'~ l'onde de concentration lin4aire associ4e ~ ( ~ ' ~ , ! ~ ' ~ , h~,

Xn,tn), avec (9~a'w, ~)a,~o) d4=f(~H~,~qp, 1H~,~ r Par conservation de l'4nergie, on peut 4crire

2

Xn,,)) _ 7)a(99( X ~hnX n )) I EO(Pn - p~'W) : hn3 /]R IV (~)f~(q~pc~'w(x - xn 3 hn .... V(lfl(x)r

+h~3/~3

dx

x-xn X--Xn )!2 An ) - lft(x)~)(- hn ......) dx

_

dy

I1 suffit donc ~ pr4sent de d4montrer la convergence de Pan f v e r s / ) a ~ f dans/:/1(IR3), Mnsi que celle de l a n g vers l a ~ g darts L2(IR3), pour t o u t couple (f, g) c/2/1 x L2(IR3). 2.7 - Soit fin un domaine convergeant vers f l ~ , dans le sens de Ia ddfinition 1.3, off f~oo est un demi-espace, @ ou IR 3. Alors on ales propridtds suiyantes:

Lemme

(ii)

Vf E/2/1 (IR3),

(iii)

Vg e L2(IR3),

Ddmonstration du lemrne.

lirn 79anf = T)a~ f lim l a . . g = l a ~ g

n---,cx~

dans dans

IJt1(IR 3) ; L2(IR3).

Les cas off f t ~ = IR 3 ou 0 sont 4vidents. Dans le cas off fto~ est un demi-espace, il est facile de d4montrer le point (i)" on

376

Ddcomposition en profils des solutions de l'dquation des ondes...

suppose par exemple que f t ~ est d~fini par f t ~ = {x E IR 3 ix3 > 0}, et l'on consid~re une fonction p C C ~ ( f ~ ) . Alors pour toute fonction u c t:tl (IR3) support~e dans ~oo, la fonction u ~ d~f = u . p ~, oh pe d~f = 1/E3p(./E), tend vers u dans/:/1(IR3), et est support~e dans f t ~ , donc (i) est d~montr~. Le point (ii) se d~montre de la mani~re suivante: on constate facilement d~f

que pour tout domaine M, et pour toute fonction f , la fonction v = T ' M f est d~finie comme la solution de Av--Af

dans

M

vlo M ~- O. d~f

Alors il suffit de d~finir v~ = P ~ n f , et soir v une de ses limites au sens des distributions; on a pour toute fonction r C C ~ ( f 2 ~ )

IR3AV n-~ dx =/~ts Aye dx, puisque pour n assez grand, on a r ~ C ~ (gt~). P a r passage ~ la limite faible dans la condition aux limites de Dirichlet, on en d~duit que v = 7 ) a ~ f . Finalement la convergence forte est due au fait que

IVv.I 2 dx

=

/IR ~ v n A ~ n dx

=

-

-

Vvn" Vfdx, 3

f

f

et donc comme ] V v . V f dx - - ] [Vv[ 2 dx, le r~sultat suit. JIR 3 J]R3 Le point (iii) s'obtient de mani~re similaire, nous laissons les d~tails au lecteur. [] Ce lemme termine la d~monstration de la proposition 2.6. [] Le r~sultat suivant sera utilis~ fr~quemment dans la suite; sa d~monstration est du m(~me type que les calculs ci-dessus conduisant au point (ii) du lemme, nous ne le d~montrerons donc pas ici. 2.8 - Soit Mn un ouvert de ]R 3, de limite M. Soit (fo, fn~) une suite bornde de E ( M n ) , convergeant faiblement (resp. fortement) vers un couple (fO, f l ) dans/-:/I(IR 3) x L2(IR3), avec (fO, f l ) C /_:/1 X L 2 ( M ) . Alors la solution de Proposition

{ Dfn--O

dans

IRt x Mn ,

f nlIR,, • OMn = 0

O fn)t =o = (fo, fx)

I. Gallagher et P. Gdrard

377

converge faiblement (resp. fortement) vers f duns Llo~(IR; /[/l(]n3)), e t 0tfn converge faiblement (resp. fortement) vers Otf dans L~oc(IR; L2(]R3)), off dans I R t • ff~txOM = 0 (f, Otf)lt=O -- (fo, . D'autre part, si M~ est l'extdrieur d'un domaine strictement convexe, alors la convergence a lieu aussi dans L~o~(~;

fl~

2.2. Propagation de la (h~)-oscillation Commenqons par rappeler la d6finition d'une fonction strictement (hn)oscillante. Dans la suite nous noterons A l'op6rateur auto-adjoint non born6 suivant: 7)(A) = { (u,/t) c/:/1 • L2(gt) !/t E H01(gt), An c L2(fl)}, Ad~f ( 0 = A

1) 0 "

D ~ f i n i t i o n 2.9 - Soit (hn) une suite de rdels strictement positifs, tendant vers O, et soit (fn,gn) une suite bornde dans /2/1 X L2(]R3). La suite (f~, gn)est dite (h~)-oscillante si lira lim ,,ll[[llAl>hn---z-(fn'gn)[[ftX• -,

R---~cx) h---~0

La suite (fn, g~) est dite strictement (h~)-oscillante si elle vdrifie d'autre part s--~0 h - - * 0

-- ~'s

'

L 2 ( I R a)

Remarque. Nous dirons qu'une suite (v~) bernie dans C~ T],/:/l(a)) telle que Otv~ est born6e dans C~ ~st (strictement) (h~)oscillante si (Vvn, OtVn) l'est au sens de la d~finition 2.9. Nous ne rappellerons pas ici la notion de composante strictement (h~)oscillante d'une suite bernie de L2(IR3), et renvoyons ~ [1], Lemme 3.2 (iii), pour une d~finition. Enon~ons maintenant le %sultat suivant, de propagation de la (h~)-oscillation stricte. Nous laissons sa d~monstration au lecteur (voir [4]). P r o p o s i t i o n 2.10 - Soft (Pn,~n) une suite bornde de E(ft), telle que la suite ( ~ 7 ~ n , ~ n ) est (strictement) (h~)-oscillante. Soit v~ la solution de (5) associde; aIors (Vvn, OtVn)(t) est (strictement) (hn)-Oscillante, uniform~ment en t pour tout t E IR.

Ddcomposition en profils des solutions de l'dquation des ondes...

378

D'autre part, soient V n et vn les solutions de l'dquation des ondes lindaire, 1 assocides a des donndes (VOn,Vn) et (v~'~ ~1) respectirement, off (v~On,~ln) est o v~). Alors pour toute la composante strictement (hn)-oscillante de (%, suite (tn), ( V ~ (t~), Ot~n (tn)) est la composante strictement (h~)-oscillante de (VVn(tn), OtVn(tn)). 2.3. Un r6sultat de non concentration Nous allons dans cette section donner quelques idles de la d~monstration du r~sultat suivant.

P r o p o s i t i o n 2.11 - Soit pn une onde de concentration lJndaire associde une donnde concentrante (~, ~, hn,xn, 0). Alors pour tout intervalle de temps I c IR borng, pour route suite (sn) telle que lim E~/hn = +oc, on a n---*c~

lim [[PnlIL~(I\[-~,~]),L~(a) = O.

n-,c~

Ddmonstration. Nous allons en donner le principe g~n~ral, mais n'entrerons pas dans les d~tails des calculs (on renvoie g [4] pour les arguments precis). Commenqons par remarquer que la proposition 2.11 est d~montr~e si l'on montre que (i) pour tout temps T 7~ 0, et pour toute suite T~ ~ T, on a lim [Ipn(Tn)llL6(~) = 0;

n - - - - ~ (:X)

(ii)

pour toute suite r lim

--~ 0, avec r flP (

~ +ec, on a

)ltLO( ) - 0 .

n---~ oo

Le cas (ii) peut ~tre consid~r~ comme une version d~g~n~r~e du cas (i), et se d~montre de faqon analogue, mais plus simple (par changement d'~chelle); nous n'y reviendrons donc pas. La m~thode de d~monstration de (i) s'appuie sur le principe de concentration-compacit~ de P.-L. Lions (voir [11]-[12])" introduisons la densit~ d'~nergie

en(t,x) d~f (latpn(t,x)12 + ]V~p~(t,x)12) dx, qui v&ifie

Oren

--

div~ (0tp~ V ~p~ ).

Alors si eo~ est un point d'adh~rence de e~, alors eo~ est continue en temps, valeurs mesure, et par le principe de concentration-compacit~, il suffit de d~montrer que

eoo(t)({xo}) -- O,

Vxo e ~,

Vt # O.

(11)

I. Gallagher et P. G~rard

379

Le calcul de e ~ se fait par utilisation des mesures semi-classiques de Wigner (voir [7], [8], [13])" soit # la mesure semi-classique associ~e ~ (Vt,xpn). I1 est bien connu que

L'r X ]P,.~

#(t,x, dT, d~) <_ e~(t)dt,

que le support de # est tel que (~.2 _ ]~]2)# = O, et que # v~rifie l'~quation de t r a n s p o r t (~a~ - ~. v ~ ) ~ = o. Notre d~marche va ~tre d'exhiber une mesure # . telle que # , _< #, et

IR,,,-x IR ~

(12)

tt. (t, x, dT, d~) = e. (t)dt,

oh e. est continue en temps, pour t -~ 0, et v~rifie

~.(t)(~)-

I}V~oo~{{~

2(~)

§ IIl~oor

L ~2(IR3)-

Par application de la conservation de l'~nergie et de la vitesse finie de propagation, on peut montrer que

vt r 0,

~.(t) - ~ (t),

et si l'on connai]t # . explicitement, on peut en dgduire (11). Le calcul de p . se fait de manihre diff~rente suivant la position de x ~ par r a p p o r t au bord 0~. Rappelons que d'apr~s la proposition 2.6, on peut supposer que x ~ E t2. Nous allons expliquer bri~vement la m~thode d'obtention de p . , suivant que x ~ E ft ou x ~ E 0t2. Dans le cas oh x ~ E t], on prolonge p~ par zgro en dehors de t2, en une suite notre u~. La mesure # . ~voqu~e ci-dessus est la mesure semiclassique associ~e ~ Vt,xu~, dont on calcule l'expression en utilisant le riot du billard Gt au-dessus de ~" on peut montrer que

f (G_t(x,T,~))p.(t, dx, d~-,d~) = / f(x,T,~)d#o(X,T,~) avec

2 d~ i

(13)

380

Ddcomposition en profils des solutions de l'dquation des ondes...

On montre que - ,0 ({(x,

e fl •

{0} •

oh px est la projection sur x. Alors par (13), on en d~duit que =

oh f+ e LI(IR3), et oh E+(t) d~f {~ 6 IR 3 IpxClt(x~,-t-[~l,~ ) = xo}. I1 est alors facile de constater, par des arguments g~om6triques sur ft, que E+ (t) est soit vide, soit une demi-droite, ce qui implique (11). Dans le cas oh x ~ E Oft, on montre que

#, = 5 (x - x ~ - t ~ )

,

(14)

oh I0 est une mesure, absolument continue par rapport ~ la mesure de Lebesgue sur le c6ne {'r 2 -

I~12, ~--'TN ( x ~ ) < 0 } , oh N(x)d~signe la nor-

male unitaire sortante en x c 0ft. En outre, la mesure A0 a pour masse totale (15) TM(Ao) - IIVT)a~ ~ 1 1 ~ ( ~ ) + Illa~ ~11~ ( ~ ) 9 Montrons comment on exhibe cette mesure I0: on prolonge pn par %flexion, dans un voisinage de x ~ E Oft. La suite v~ ainsi obtenue v~rifie une ~quation du type pO2t Vn -- div(A 9V V n ) --" O, oh p et A sont continues, %guli~res par morceaux, avec Oft pour interface. En outre, on a p(x) = 1 et A(x) = Id dans ft. On consid~re alors la mesure s e m i - c l a s s i q u e , associ6e ~ (v/-fiOtvn, v/-AVxvn), dont on montre qu'elle est continue en temps, et v~rifie ~, ({x E 0ft, ~.

N(x) -r 0, 7- -r 0}) = 0.

D ' a u t r e part, cette mesure v~rifie une 6quation de transport le long du 1 champ Hamiltonien associ6 au symbole ~ (T 2 - 1A~. ~). La difficult~ est due P au fait que les coefficients de cette ~quation de transport sont discontinus (Vp et VA). N6anmoins, leur composante normale est continue, ce qui suffit pour d6finir localement un riot Gt pour cette 6quation, tel que si vlt= o _> v,, alors v(t) >_ Gt(v,). On montre alors que vlt=0 s'6crit 6 ( x - x ~ ) A ( r , ~ ) , oh A est tel que Ao(T,~C) d_~f lf.N<0A(r,~c ) v6rifie (15). Enfin on remarque -r

I. Gallagher et P. Gdrard

que si } - N

381

< 0, alors G t ( x ~ , T, ~) = ( x ~ + t ~-, % ~) reste au-dessus de ft, T

ce qui permet d'en d4duire (12), et donc le %sultat cherch4, avec # . d4finie en (14). Nous renvoyons ~ [4] pour des d4tails, m

3. L e p r o b l ~ m e l i n 4 a i r e : d 4 m o n s t r a t i o n

du th@or~me 1

Cette section est d4volue ~ la d4monstration du th4or~me 1 4nonc4 en introduction. La d4marche, inspir4e de [1], est la suivante: dans une premiere partie, les 4chelles h~ ) de concentration sont extraites; dans la seconde partie, nous d4composons une fonction strictement (hn)-oscillante, au sens de la d4finition 2.9, en une somme d'ondes de concentration lin4aires orthogonales, avec donn4e en un temps t (k) concentr4e en un point x (k) . Avec les r4sultats de la section 2, la m4thode est alors tr~s semblable au cas de tout l'espace trait4 dans [1]; nous n'entrerons pas dans les d4tails ici, et renvoyons s [4] pour les arguments pr4cis. 3.1. E x t r a c t i o n des 4chelles D'apr~s le th4or~me de structure d4montr4 par P. G4rard dans [6], on peut d4composer les donn4es initiales de la manibre suivante"

V(/9 n ( X ) --- E V~(nJ)(X) j=l

=

+ r

-~- V(I)(n~)(X),

(x),

j=l

oh les ~0) et les ~(J) sont strictement (h(nJ))-oscillants, les h(nj) sont orthogonaux deux-~-deux, oh IlVqani[22(1R~) = E

I t V ~ ) 11~2(~R3)+ I I V ~ ) [[~2(IR3) + o(1)

n ---~ (N:)~

j=l

n ---~ (:X)~ j=l

et oh lim limsup]]V~(ne)llBO

~---~(:x:) n--+oo

2,c~

=

lim limsup II~(~)[IBO

~--,(:x:) n--+(:x~

2,oo

= 0.

382

Ddcomposition en profils des solutions de l'dquation des ondes...

Rappelons que pour toute fonction f , on d~finit

llfll/~o

,or

d,~=fs u p kEZ

(/2

IfA(~)] 2 d~ k
oh f est la transform~e de Fourier de f. Mais dans [9], P. G6rard, Y. Meyer et F. Oru ont d6mont% que pour toute fonction f , on a 1/3

IlfllL~(~t~) <- CIIVfllL~(r~)llVfllBo

2/3

9

Alors par le %sultat de propagation d~mont% en section 2.2, et par conservation de l'~nergie, on peut finalement montrer la proposition suivante. P r o p o s i t i o n 3.12 - Sous les hypotheses du thdorbme 1, la fonction vn peut, quitte ~ extraire une sous-suite, se ddcomposer de la manibre suivante: e

w e IN \ {0},

vn (t, ~) = ~ V(nj) (t, x) + ~ ) ( t , x), j=l

off V(nj) est une solution de l'dquation des ondes lindaire dans f~, strictement (h(nJ))-oscillante, et off les h~ ) sont orthogonaux deux-a-deux. On a en outre

VT > O,

lim limsup [IW(ng)IIL~([_T,T],L6(Ft))--O, ~--+ c ~

n - - , cx:)

et E~

= E E~ j=l

VT > 0,

Vj # k,

-~- E~

+ o(1),

n---~ (:x:).

Enfin l 'on a lim ]Iv(J)V(nk)I]L~([_T,TI,L3(a))

: O.

n---+~

Ddmonstration. Seul le dernier r6sultat de cette proposition reste ~ ~tre d6mont%. I1 repose sur le lemme suivant. L e m m e 3.13 - S o i e n t h (j) et h (k) deux dchelles orthogonMes, et (f(J))

et (f(k)) deux suites de fonctions borndes dans /:/1(IRa), respectivement strictement (h~)) - et (h(~k))-oscillantes au sens off lim l i m s u p ......[[ll~l>R/h~J'~-J)[[L2(aa) = 0, R---*c~ n---,c~

I. Gallagher et P. Gdrard et

383

~-~olimlimsuplln_~r ll~l~/h(~)~)llL~(~) ~ =

et de m~me en remplafant j par k. Alors lim Ill2 )f(k) IIL~(a) = 0.

n ---+(2~

Ce lemme conduit ~videmment directement s la proposition 3.12, puisque les suites Vv(J)(t (j), .)et Vv(k)(t (k), .), pour des suites quelconques (t~)) et (t (k)) de r6els, v~rifient les hypotheses du lemme 3.13 et la proposition 3.12 est d6montr6e, m

Ddmonstration du lemme 3.13. Supposons par exemple que lim h(nk)/h(~j) = n --+cx)

O. Commen~ons par supposer que les fonctions f(j) et f(k) ont un spectre tel que respectivement h(nJ)i~i C C(oj) et h(~k)]~I ~ C(ok), o~ C(Oj) et C(k) sont des couronnes fixes de IR. On a alors

Ilf~(J)IIL~oR~) _ et

Ch~)liVf(~J)llg~(~)

ilf~(k) IIL~(~) --< Ch(~k)IIVL(k) IIL~(~)

et donc

IIL(k)I1L~(~) _< Ch(~k).

IIL(5)llL~(~) ~ Ch(nj) et

Alors par l'in~galit~ de Hausdorff-Young, on a

Donc finalement 1/2

1

h~ ) ( h ( ~ )

<

Le lemme est d6montr6 dans le cas d'un spectre localis6. Sinon, les hypotheses de stricte oscillation faites sur f(J) et f(k) permettent d'approximer ces fonctions dans/2/1 (I~3) par de telles fonctions localis6es, et d'obtenir le r~sultat,

m

DEcomposition en profils des solutions de l'Equation des ondes...

384

3.2. E x t r a c t i o n des t e m p s et des c o e u r s de concentration Chacune des fonctions v(~j) obtenue dans la proposition 3.12 ci-dessus va 8tre d~compos~e en une somme d'ondes de concentration lin~aires orthogonales. Pour simplifier les notations, nous allons fixer une ~chelle de concentration h, et consid~rer une famille de fonctions (Vh), strictement (h)-oscillante. P r o p o s i t i o n 3.14 - Soit (Vh) une famille de fonctions strictement (h)oscillante. Quitte g extra/re une sous-suite, ii existe des ondes de concentration linEaires p(hk) associEes a des donndes (9~(k) , ~2(k) , h, X(hk) , t(k)), orthogonales deux-g-deux, telles que

ve e ~ \ {0},

vh (t, x) = Z p~)(t, x) + ~ ) ( t , ~), k=l

avec orthogonalitd des Energies: Eo(vh) = ~

Eo(p(hk)) + E0(~(he)) + o(1),

h-,0,

k:l

et avec VT > O,

lim lim II~(he) llL~([_T,T];L0(fl)) = O.

~--~ oc h - - , O

DEmonstration. La m~thode de d~monstration suit les calculs de [1], qui s'inspirent de la d~marche de [14]. On commence par d~finir, pour tout couple (x(hJ),t(hJ)), l'op~rateur D(hj) suivant"

m(j)f(y) d__6f(hl/2f(t(hJ)X(hJ)nuhy),h3/2Otf(t(J)x(hJ)~_hy)). On d~finit alors une fonction dite d'exhaustion 6, pour toute famille de fonctions (fh), de la mani~re suivante:

a(fh) d ~ f sup { flV~fl~(~)+ If,Jl~(~),Dhfh " (~,*),}, (xh),(th)

oh la limite ci-dessus s'entend s extraction de sous-suite pr~s. L e m m e 3.15 - Pour toute famille de fonctions (fh) bornde en dnergie, on a

VT > O, h]ir~IIAIIz,~(t-TTJ;',~(n))--
I. Gallagher et P.

GErard

385

Ddmonstration.

I1 suffit d'utiliser la d6composition de [6] pour fh(th, "), suite born6e dans /:/1(IR3), ainsi que l'injection de/;/1(IR3) dans L6(IR3). Nous laissons les d~!tails au lecteur, m La construction des fonctions p(hk) se fait par r6currence: d'apr6s le lemme 3.15, on peut supposer que 6(Vh) :> 0. On choisit alors la donn6e concentrante (99(1), ~p(1)x(1),t(1) ) tel que 1 -~(~(Vh)

IIV99(1) I]~2(IRa) -~-I]~Z;(1)l122(IR3) > et, quitte A extraire une sous-suite,

D(hl)Vh __~ (99(1)r Alors Ph(1) est l'onde de concentration lin~aire associ6e ~ la donn~e concentrante (99(1), ~2(1), h,x(1), t(1)). Le lemme suivant se d~montre par un simple calcul, utilisant la proposition 2.6 et le lernme 2.7. L e m m e 3.16 -

Soit w (h1) d=e f y

EO(Vh)

h - - ~ h,(1)"

Alors

-- E0(v(h 1)) -}- E0(W(h 1)) nu o(1),

h --, 0.

On poursuit alors le d6veloppement par rdcurrence: on suppose que

K-1 Vh(t,x) = ~ p(k)(t,x) -t- W(K-1)(t,x), k=l avec orthogonalit~ en ~nergie, off chacun des

p(hk)

est une onde de con-

centration lin~aire avec orthogonalit6 des temps t(hk) et des coeurs x (k) de concentration. Alors on exhibe par la m~me technique que ci-dessus une nouvelle onde de concentration lin~aire p(hK), de la mani~re suivante: on peut, d'apr~s le lemme 3.15, supposer que 5(w (K-l)) > 0, et l'on d~finit la donn~e concentrante (~o(K), ~ ( g ) ,

x (K), t (g))

I[V99(K) II~2(IR3)-Jr-[Ir

de telle sorte que 1

3) :> ~(~(W (K-l))

(16)

et, quitte 5~ extraire une sous-suite,

D(h/{) Wh(K-l) __x (99(K) ~)(/6)). Alors p(/{) est l'onde de concentration lin~aire associ~e ~ la donn~e concentrante (99(K), g2(K),h,x (K),

t (K)),

et l'on a, comme dans le lemme 3.16

386

Ddcomposition en profils des solutions de l'dquation des ondes...

ci-dessus, K

Eo(vh) = E E~

+ E~

+ o(1),

h --~ 0,

k--1

K -- Vh-- E k=l

vec

L'orthogonalit~ des temps t (k) de concentration est due ~ la proposition 2.11. Celle des coeurs est laiss~e au lecteur: elle se montre par un argument classique de changement d'~chelle en espace, en utilisant la proposition 2.8. Reste alors ~ d~montrer que VT > 0,

lim lim ,,IIw[K) ,, ,~ IIL~([-T, TI,L6(a)) - - O .

K-.cx~ h-+0

Ce r6sultat resulte du fait que la s6rie de terme g6n6ral IIVp(k)ll 2L2(IR3) -~2 I1~(k) [IL2(IR 2 3) est convergente, et done IIV~ (k) IIL2(IR3) -]- I]~/)(k) 1122(IR3) tend vers z6ro quand k tend vers l'infini. en (16), on a

Mais par d6finition de (~(K) ~(K))

(~(W(hK - l ) ) ~ 2 (]Iv~(K) II~2(IR3) -~-]'~(K)]]~2(IR3)) , donc lim

h--~0

5(w (K-l)) tend vers zdro quand K tend vers l'infini. Le lemme 3.15

donne alors le r~sultat. La proposition 3.14 est d~montr~e,

4. L e p r o b l ~ m e

non lindaire: dc!monstration

m

du th~or~me

2

Nous allons proc~der en plusieurs ~tapes, qui nous p e r m e t t r o n t d'obtenir un r~sultat plus precis que celui ~nonc~ dans le th~or~me, avec notamment une ~tude autour du temps de concentration tn; comme dans la section pr~c~dente, nous allons pour simplifier les notations consid~rer une famille (Ph) d'ondes de concentration lin~aires. D'autre part, nous ne nous int~resserons qu'au seul cas off lim th h-~0 -h- = + ~ ' et laissons les deux autres cas ( - c ~ ou 0) au lecteur. Les lemmes 4.17, 4.18 et 4.19 suivants conduisent directement au th~or~me.

I. Gallagher et P. G&ard

387

4.1. Avant le t e m p s de c o n c e n t r a t i o n L e m m e 4.17 - Sous les hypothbses du thdorbme 2, on a

/ lim lim {

sup

A--,c~ h--,O \--T<_t<_th--Ah

Eo(Ph -- qh, t) +

[Iph - qhllLS([--T, th--ahl,L~~

= O.

Ddmonstration. Le r~sultat est simplement dfi ~ la proposition 2.11 et du th~or~me de lin~arisation de [5]. m 4.2. A u t o u r d u t e m p s de c o n c e n t r a t i o n Avant d'~noncer le r6sultat que nous nous proposons de d~montrer, nous avons besoin d'introduire une onde non lin6aire, notre qh~. Tout d'abord, d~finissons l'onde remise ~ l'~chelle suivante:

Ph(s,y) d~f hl/2ph(t h -t- hs, xh Jr hy). La fonction Ph v~rifie une ~quation d'ondes lin~aire dans le domaine fth d~f

~t--Xh

9On lui associe la solution de l'~quation lin~aire dans le domaine h limite ~oo: DP=0

dans IR, x f t ~ , Pl~sxOa~ = 0 (P, O~P)ls=0 = ( P a ~ ~a, l a ~ r

On d~finit alors Q~, la solution de l'~quation non lin~aire suivante:

{ n Q x+[Q~14Q x = O dans I R s x f t ~ , QxIIR~ x O~o~ (QX, OsQ ~')!~=_~ = (P, O~P)ls=-x.

-

-

Enfin q2 est d6finie par q~ (t, X) d~f

1 Q~(X-Xh hi~ 2 h

'

t--th) h

"

Nous allons montrer le r~sultat suivant. L e m m e 4.18 - Sous les hypothbses du thdorbme 2, on a lira lim Ilqh - q-~llh,X = O,

,k--, c~ h--*0

off l'on a notd "

Itfllh,

d~f

-

sup tE[th--Ah,thq-Ah]

E o ( f , t) + llfi]LS([th_~h,th+iXh],LlO(]R3)).

Ddcomposition en profils des solutions de l'dquation des ondes...

388

Ddmonstration. Elle va se fake en deux 6tapes. Commenqons par d~finir la solution du syst~me suivant: DQ~h + IQ~I4Q~ = 0

dans

(Q~, O~Q~),~=_~

~ -0 IRs x a h , QhllRsxOf~h -= (Ph, o ~ P h ) ~ _ _ ~ .

Nous allons commencer par d~montrer que lim ~Qh~ - Qa h--,0 ~"

II~ = 0,

w > 0,

(17)

d6f

sup E o ( f , s ) + [If[[L~([_,X,,X],L~O(IR3)). La convergence forte ~[-~,~] en norme ~nergie s'obtient par un argument identique ~ celui permettant de d~montrer la proposition 2.8 ci-dessus (voir [4]). Pour ce qui est de la norme Strichartz, le %sultat s'obtient par les calculs suivants: l'in~galit~ de H61der fournit avec [If[[,x --

IIQ~ Q~ 115 55([__~,~],510) < IIQ~ -- Q)'llL~([--~,),I,L~

--

--

-

QXII4

54([--~,~],L12)

"

Mais Qh~ et Q~ v6rifient l'estimation de Strichartz, et comme l'6nergie de Q X - Q ) ' tend vers z~ro quand h tend vers z~ro, pour tout A > O, on peut h ~crire

L5 ([-~,~],L 1o) __ < C~(h) avec lim r

( IIQ~II2~([-x,xl,/,o) + IIQhllx

(

205 [-A,A] L

,i IO)

)

= 0 pour tout A > O. On conclut alors par bootstrap

h--+O

superlin6aire (voir par exemple [I], lemme 2.2) ~ partir de l'estimation

IIQ~ - Q~ll~([-~,~l,/,o) -< ce~(h)llQ~'ll~~ +cr

-

Q~IIu~ Ls([-A,AI,L~~

"

Pour terminer la d6monstration du lemme, on introduit la fonction Rhx d~f Qh _Q,Xh' avec

Qh(s, y) d~f hl/2qh(t h + hs, xh nt- hy). Alors R~ v~rifie l'dquation suivante:

{en2

hi ~ h + Q2) -IQ21 ~Qh~ -- 0 dans + In2 + Q~'~(n~

IRs x Dh,

t ~ h,x l I R s x Of~h ---- 0

(~, o~)1~=-~

= ( Q ~ , O~Q~)I~=-~ - (P~, o ~ P ~ ) I , = _ ~ .

D'apr~s le lemme 4.17, on a lim lim Eo(R~h,-A) - 0,

X--+o~ h---, 0

I. Gallagher et P. Gbrard

389

donc en notant de mani~re g~n~rique s(A, h) une fonction v6rifiant lim lim r

A--*c~ h---~0

h) - 0,

l'estimation de Strichartz (2) fournit, pour tout To > -A,

IIR~IILS([_A,Tol,LlO(IR3)) < Cr

h)

5

j=l < C

5-j ilR~hllJLs([_A,To],LIO(IR3))IIQ~]]L~([_A,To],L~O(IR3))

~(A,h)-t-__

)

. (lS)

I1 est facile de voir, par un changement d'~chelle, que la constante C ne d~pend ni de h, ni de A. Mais d'apr~s (17), on peut remplacer dans (18) la fonction Qh~ par Q~. En outre, la fonction Q~ converge, quand A tend vers l'infini, vers Q - dans Ls(]R, LI~ avec v1Q-

-t- I Q - 1 4 Q

-

dans IR~ x f t ~ , Q~R~xOa~ -- 0 lim E o ( Q - - P, s) = O.

= 0 8----> ~

(20

Cette convergence a lieu pour tout temps, car le domaine limite ~ est soit l'espace IR3 entier, soit un demi-espace, pour lesquels on sait que les solutions de l'~quation des ondes non lin~aire critique sont dans L5 (IR, Ll~ (voir [3], le cas du demi-espace d~coulant du c a s ] R 3 par r~flexion antisym~trique). On a donc en fait

liR llL0([- ,ToJ,L o( )) --< Cs( , h) 5

5--j

j--1 et l'on va conclure par d~formation en temps (voir aussi [2]): si le temps To est assez petit, uniform6ment en h et en A, clots l'estimation ci-dessus implique directement, par bootstrap superlin6aire, que lim lim ,,,o,,IIR~IIL~([-A,Tol,LI~

A-+oo h---,0

-- O.

D6finissons ~ pr6sent = sup { T E IR ] A--.cx~ lim h~O lim ]]R~[ILS([_A,T],LlO(IR3))--0}. rmax d6f

390

Ddcomposition en profils des solutions de l'dquation des ondes...

On a Tm~x >_ To; soit T1 < Tm~x, alors on peut 6crire

]lR~hllL~([_~,Tm~l,L~O(n~)) ~ C~(A, h) + e l i R h~ il 5 5 ([T1 ,Tmax ] , i 10 (in3))II Q - I I t 5 ([T1 ,Tmax ] , i 10 (iRa)).

I1 suffit alors de choisir 7'1 assez proche de Tmax, uniform~ment en h et en )~, pour conclure que lim l i m A--.c~ h - . 0

]]R~IILS([_A,Tma~],LlO(IR3)) : O,

et par l'in~galit~ d'~nergie, lim lim

sup

Eo(R~h, s) = O.

A- .c<) h--,0 - A < s < T m a x

Le m6me type de calcul permet de montrer que sir] > 0 est assez petit, alors lim lim IIR~llLS([Tmax Tmax+n] LlO(IR")) = 0, A---,c~ h--,0 ' ' ce qui contredit la maximalit~ de Tmax; enfin on montre de la m(~me mani~re que lim lim ,,,~,,IIR~AIILS([-A,+oc[,L~~ - - 0, A--~co h---~0

et avec l'in6galit6 d'e~nergie, le lemme est d~montr6. D

4.3. Apr~s le temps de concentration L e m m e 4.19 - Sous les hypothSses du thdorSme 2, on a

/ lim lim |

sup A---*oo h---+O \ t h T A h < t < T

Eo(p + - qh, t) + lip + -- qhllL~([th+Ah,T],ilo(a))) = O.

D~monstration. Elle est tr~s simple, au vu des deux lemmes p%c~!dents: il suffit d'appliquer les %sultats de scattering obtenus dans [1]-[2], qui s'~tendent s notre cadre par %flexion antisym~trique si f ~ est un demiespace, ll

R~f~rences [1] H. Bahouri et P. G~rard, High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Mathematics, 121 (1999), 131-175.

I. Gallagher et P. G@rard

[2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [121 [131 [14] [15] [16] [17]

391

H. Bahouri et P. G@rard, Concentration effects in critical nonlinear wave equations, Geometrical Optics and Related Topics (F. Colombini and N. Lerner eds.), Progress in Nonlinear Differential Equations and Applications, 32 (1997), Birkhs Boston, 17-30. H. Bahouri et J. Shatah, Global estimate for the critical semilinear wave equation, Annales de l'Institut Henri Poincar~, Analyse non lin@aire, 15 (1998), 6, 783-789. I. Gallagher et P. G~rard, Profile decomposition for the wave equation outside a convex obstacle, Journal de Math~matiques Pures et Appliqu~es, 80, 1, pages 1-49, 2001. P. G~rard, Oscillations and concentration effects in semilinear dispersire wave equations, Journal of Functional Analysis, 141 (1996), 60-98. P. G~rard, Description du d~faut de compacit@ de l'injection de Sobolev, ESAIM ContrSle Optimal et Calcul des Variations, 3 (1998), 213-233, (version @lectronique: http://www.emath.fr/cocv/). P. G~rard et E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Mathematical Journal, 71 (1993), 559-607,. P. G~rard, P. Markowitch, N. Mauser et F. Poupaud: Homogenization limits and Wigner transforms, Communications on Pure and Applied Mathematics, 50 (1997), 323-379,. P. G@rard, Y. Meyer et F. Oru, In@galit~s de Sobolev pr~cis~es, S~minaire l~quations aux D~riv~es Partielles, l~cole Polytechnique, d~cembre 1996. J. Ginibre et G. Velo, Generalized Strichartz inequalities for the wave equation, Journal of Functional Analysis, 133 (1995), 1,50-68. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Annales de l'Institut Henri Poincar@, 1 (1984), 109-145. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, II, Revista Matematica Iberoamericana, I (1985), 145-201. P.-L. Lions et T. Paul, Sur les mesures de Wigner, Revista Matematica Iberoamericana, 9 (1993),553-618. G. M6tivier et S. Schochet, Trilinear resonant interactions of semilinear hyperbolic waves, Duke Mathematical Journal, 95 (1998), 2, 241-304. J. Shatah et M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, International Mathematics Research Notices, 7 (1994), 303-309. H. Smith et C. Sogge, On the critical semilinear wave equation outside convex obstacles, Journal of the American Mathematical Society, 8 (1995), 4, 879-916. H. Smith et C. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, preprint.

392

Ddcomposition en profils des solutions de l'dquation des ondes...

Isabelle Gallagher D~partement de Math~matiques Universit~ de Paris-Sud 91405 Orsay Cedex and Centre de Math~matiques t~cole Polytechnique 91128 Palaiseau Cedex, France E-mail: Isabelle. Gallagher@math. polytechnique, fr Patrick G6rard D~partement de Math~matiques Universit~ de Paris-Sud 91405 Orsay Cedex, France E-mail: Patrick. [email protected]. fr

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 18

U P W I N D D I S C R E T I Z A T I O N S OF A S T E A D Y GRADE-TWO FLUID MODEL IN TWO DIMENSIONS

V. GIRAULT AND L. R. SCOTT

1. I n t r o d u c t i o n A fluid of grade two is a particular non-Newtonian Rivlin-Ericksen fluid (cf. [28]) whose equation of motion is 0 Ot

u)

u + curl(u

+

u) • u (1.1)

- (a~ + a 2 ) A ( u . V u) + 2(a~ + a 2 ) u . V ( A u ) + V i5 = f . This system of equations is completed by the condition of incompressibility: div u = 0,

(1.2)

and suitable initial and b o u n d a r y conditions. Here f is an an external force (usually gravity), u is the velocity,/5 is the pressure, ~ is the viscosity and a l and a2 are material stress moduli, the three parameters being constant and divided by the density. It is considered an appropriate model for the motion of a water solution of polymers. Dunn and Fosdick prove in [13] that, to be consistent with t h e r m o d y namics, the viscosity and normal stress moduli must satisfy // > 0

,

O~1

> 0

and a l + a2 = 0.

The reader can refer to [14] for a thorough discussion on the sign of a l . W i t h these assumptions, setting a = a l , (1.1) simplifies and leads to the equation of motion 0 0-~(u - a A u) - tJA u + c u r l ( u - a A u) x u + V p = f , where the modified pressure p is related to i5 by 1 1 p -- i5 -4- ~ u . u -- c~(u- ~ u -4- ~ t r A2),

394

Upwind discretizations of a steady grade-two fluid

where A1 = V u + (V u) t is the symmetric gradient tensor. Interestingly, in [21, 22], Holm, Marsden and Ratiu derive these equations with v = 0 and c~1 + a2 = 0, as a model of turbulence. They are called averaged-Euler equations, and c~1 is an averaged length scale. It can also be interpreted as a measure of dispersion and in this respect, these equations describe a dispersive fluid model (cf. [18, 21]). The equations of a grade-two fluid model have been studied by many .authors (Videman gives in [31] a very extensive list of references), but the best construction of solutions for the problem, with homogeneous Dirichlet boundary conditions and mildly smooth data, is given by Ouazar in [26] and by Cioranescu and Ouazar in [7, 8]. These authors prove existence of solutions, with H 3 regularity in space, by looking for a velocity u such that z = curl(u - a A u), has L 2 regularity in space, introducing z as an auxiliary variable and discretizing the equations of motion (in variational form) by Galerkin's method in the basis of the eigenfunctions of the operator c u r l c u r l ( u - c~ A u). This excellent choice of basis allows one to recover estimates from the transport equation O~Z a - ~ + vz + c~(u. Vz - z . Vu) = v c u r l u + a c u r l f.

(1.4)

Whenever c u r l f belongs to L2(~) 3, this construction is optimal because, in contrast to fixed-point arguments, it uses all the information conveyed by (1.1)-(1.4). Thus, it allows one to derive global existence of solutions with minimal restrictions on the size of the data, cf. [3] and [9]. A fixed-point argument cannot use all four equations because they are redundant. It is particularly important to preserve (1.1) since it implies that the energy is bounded without restrictions on the data. This point will be crucial for the numerical analysis of schemes discretizing (1.1). The transport equation (1.4) substantially simplifies in two dimensions since the second nonlinear term z . V u vanishes. In this case, z = (0, 0, z) with z = curl(u - ~ A u), where curl is the operator curlv =

OVl

OV2

Ox2

Oxl

Hence z is necessarily orthogonal to u = (Ul, u2, 0). This vanishing term has a very important consequence: all the analysis can be performed without having to derive an a priori estimate for u in W 1,~ (f~)2. The same property will hold in the discrete case, provided the discrete scheme is suitably chosen.

V. Girault and L. R. Scott

395

In this article, we propose finite-element schemes for solving numerically the equations of a steady two dimensional grade-two fluid model, with a non-homogeneous tangential boundary condition. Defining z as above, the equation of motion becomes -uAu+zxu+Vp=f

inft,

(1.5)

the incompressibility condition is unchanged: divu=0

inf',

(1.6)

the boundary condition is u=g

on Oft w i t h g . n = 0 ,

(1.7)

where n denotes the unit exterior normal to Oft, and the transport equation becomes uz + a u . Vz = ucurl u + acurl f.

(1.8)

Girault and Scott in [16] prove that (1.5)-(1.8) always has a solution u in HI(f~) 2 and p in L2(ft), on a Lipschitz-continuous domain, without restriction on the size of the data, provided curl f belongs to L2 (ft), thus extending to rough data a result of Ouazar [26]. This unconditional existence result relies entirely on the fact that u does not need to be bounded in W 1'~ (f~)2. Similarly, our finite-element schemes are chosen so the numerical analysis can be performed without having to derive a uniform W 1,~ estimate for the discrete velocity. As expected, the difficulties arise from the transport equation (1.8). As is observed in [17] and [10], a straightforward argument shows that either the discrete velocity must have exactly zero divergence, or its non-zero divergence must be compensated by an extra stabilizing term in the transport equation or by a compatibility condition between the spaces of discrete pressure and discrete auxiliary variable z. Roughly speaking, let Xh, Mh and Zh be discrete spaces for the velocity, pressure and variable z and, as usual, let us discretize (1.6) by

Vqh E Mh, (qh, div U h ) = O.

(1.9)

Clearly, if we want to derive an unconditional a priori estimate from the discrete analogue of (1.8), we must be able to eliminate the nonlinear term. But even in the simplest case, Green's formula gives

/a (Uh" V Zh)Zh dx = --~1/ (diVUh)(Zh)2dx.

396

Upwind discretizations of a steady grade-two fluid

Hence, we can eliminate this right-hand side either by adding to the lefthand side of (1.8) a stabilizing, consistent term, so that it becomes uz + a u . Vz + 1 (div u)z = vcurl u + acurl f,

(1.10)

or by asking that (Zh) 2 e Mh,

(1.11)

and applying (1.9). Keeping this in mind, we propose to discretize (1.10) or (1.8) by an upwind scheme based on the discontinuous Galerkin method of degree one introduced by Lesaint and Raviart in [23]. This means that in each element of the triangulation, Zh is a polynomial of degree one, without continuity requirement on interelement boundaries. On one hand, if the form (1.10) is used for discretizing the transport equation, then we can approximate the velocity and pressure by the standard 1 P 2 - / P 1 Hood-Taylor scheme, where f~k denotes the space of polynomials of degree k in two variables (cf. for example [15]). On the other hand, if we discretize the transport equation in the form (1.8), then (1.11) implies that Ph must be a polynomial of degree two, discontinuous across elements. In addition, the fact that the pressure and velocity spaces must satisfy a uniform discrete inf-sup condition implies that each component of Uh can be a polynomial of degree three plus two bubble functions of degree four, with continuity requirement on interelement boundaries (cf. [15]). Thus, denoting by C(Uh; zh, Oh) the discrete nonlinear part of (1.10) in variational form (el. (3.5)), our scheme is: Find Uh in Xh + gh, Ph in Mh and Zh in Zh such that Y v h C X h , ~ ' ( V u h , V V h ) + ( Z h XUh,Vh)--(ph, d i v V h ) = ( f , Vh), Yqh E Mh , ( qh, div Uh) = 0, VOh ~ Zh , v (Zh, Oh) + C(Uh; Zh, Oh) -- ~ (curl Uh, Oh) + C~(curl f, 0h).

(1.12) (1.13)

(1.14)

Here ga is a suitable approximation of g and the functions of Xa vanish on 0Ft. Without restriction on the size of the data, we establish that this scheme always has a discrete solution in a Lipschitz polygonal domain and that this solution converges strongly to a solution of the exact problem. Furthermore, if the domain is convex and the data small, this solution can be computed by a converging successive approximation algorithm, with arbitrary starting guess. In addition, we prove an error inequality that leads to

V. Girault and L. R. Scott

397

error estimates when the solution is sufficiently smooth. For both velocitypressure discretizations, the error is of the order of h 3/2, a result that remains valid as a tends to zero. With the lP2 - lPl Hood-Taylor scheme, this is the best that can be achieved, considering that the discretization of the transport equation loses inevitably a factor h 1/2. For the iP3 - iP2 scheme with discontinuous pressure, whose transport equation is simpler, this result is disappointing considering that the interpolation error for the 'velocity and pressure is of order h 3. These two results complete those of Girault and Scott in [17].

Remark 1.1 Another possibility is that Zh be constant in each element of the triangulation. Then (1.11) implies that Ph must also be a piecewise ,constant and we can associate with it the incomplete lP2 finite-element ,of Bernardi and Raugel [15] for the velocity, or even the non-conforminig element of Crouzeix and Raviart [12]. Otherwise, if we use the stabilizing term of (1.10), we can discretize the velocity and pressure with the "minielement" of Arnold, Brezzi and Fortin [15]. The analysis below extends to these examples and it can be shown that their error is the order of h 1/2. Remark 1.2 The results presented here are much more valuable than what Baia and Sequeira derive in [2]. Their analysis is of very limited use because, in order to guarantee the convergence of their algorithm (or even any algorithm), they must start with a first guess that has an error of order h 3/2. And since they prove no a priori estimate, they cannot construct this first guess, which in fact amounts to solving their problem directly. The remainder of this paper is divided into three sections. Section 1 briefly recalls the analysis of the exact problem (1.5)-(1.8) and compares it with the formulation proposed by [2]. The finite-element schemes are described in Section 2 and their error is estimated in Section 3. We end this introduction by recalling some notation and basic functional results. For any non-negative integer rn and number r >_ 1, recall the classical Sobolev space (cf. Adams [1] or Ne6as [25]) W'~'r(gt) = {v e Lr(s

; Okv e L~(f~) Vlkl _< m } ,

equipped with the seminorm

IVlw'~'~(~) -- [ E fg21Okvlrdx] 1/r Ikl-m and norm (for which it is a Banach space)

O~k~rn

Upwind discretizations of a steady grade-two fluid

398

with the usual extension when r - c~. The reader can refer to [20] and [24] for extensions of this definition to non-integral values of m. When r = 2, this space is the Hilbert space Hm(f~). In particular, the scalar product of L2(~) is denoted by (., .). The definitions of these spaces are extended straightforwardly to vectors, with the same notation, but with the following modification for the norms in the non-Hilbert case. Let u = (Ul,U2); then we set

IlullL~(a) = where

II. II

Ilu(x) I1" d x

,

denotes the Euclidean vector norm.

For vanishing boundary values, we define

H1 ( ~ ) - { v e i l 1(~); vlo~ =0}. We shall often use Sobolev's imbeddings: for any real number p > 1, there exists a constant Sp such that

Vv c Hg(~), IIvlIL~(~) <---S~IvlH,(~).

(1.15)

When p -- 2, this reduces to Poincard's inequality and $2 is Poincard's constant. For tangential boundary values, we define H : ~ ( ~ ) - {v e H1(~)2" v - n = 0 on af~}

(1.16)

A straightforward application of Peetre-Tartar's Theorem (cf. [27] and [30] or [15]) shows that the analogue of Sobolev's imbeddings holds in H~(fl) for any real number p > 1"

Vv e

H~(~), Ilvll~,(~) _< ~IvlH~(~).

(1.17)

In particular, for p = 2, the mapping v H Ivlgl(a) is a norm on H~(f~), equivalent to the H 1 norm and ~52 is the analogue of Poincard's constant. We shall also use the standard spaces for Navier-Stokes equations V = {v E Hl(fl)2 ; d i v v = 0 in fl}, W-

{v e H ~ , ( ~ ) ; d i v v -

0 in ~ } ,

Lo2(~) - {q e L2(f~) ; f q dx = 0}, Jn and also the space H(curl, ~ ) =

{v e L2(f~)2 ; curlv e L2(f~)}.

V. Girault and L. R. Scott

399

Finally, recall the standard lifting Wg in W of g: it is the solution of the non-homogeneous Stokes problem: -Awg+Vpg=0

and d i v w g = 0

inf~

,

wg=gon0f~.

(1.18)

It satisfies the bound, with a constant T that depends only on the domain (cf. for instance [15]) IWglHl(fl) < TligllH~/~(oa).

(1.19)

2. T h e e x a c t p r o b l e m Let fl be a Lipschitz polygon in two dimensions with boundary 0 ~ (cf. [20]). We consider the steady grade-two fluid model: Find a vector-valued velocity u, a scalar pressure p, and an auxiliary scalar function z, solution of (1.5)(1.8), namely -~Au+zxu+Vp=f inf,, divu-0 u=g

inf,,

on0f~withg.n=0,

vz + a u . Vz = ucurlu + acurl f , where z x u = (-zu2, zul). For simplicity, we call it Problem P. Here u > 0, and for the sake of generality, we allow a to take negative values. It can be readily checked that z = (0, 0, z) and u are related by z = c u r l ( u - a A u), so that (1.5) is indeed the steady-state version of (1.1), with a l + a2 = 0. The following theorem is established in [16] and [17]. T h e o r e m 2.3 - Let ~ be a Lipschitz polygon. For all v > O, all real numbers a, all f C H(curl, F/) and all tangential vectors g c H1/2(0~) 2, Problem P has at least one solution (u, p, z). All solutions of Problem P satisfy the following estimates:

lulH (a) _< S2 IlfllL=(a) + TIIgllHx/ (oa)(X + s4V 4 IlzllL (a)), V

(2.1)

1

(2.2) + S 41ulH (a)llzllL (a)), ]lZllL~(a) < V~IUlH,(a) + 1_~_1ilcurl fllL~(a),

(2.3)

400

Upwind discretizations of a steady grade-two fluid

I1,~u. v zllL~(.) _< ~' v/2 lulH.(a) + I'~111curlfllL~(.),

(2.4)

where fl > 0 is the isomorphism constant of the divergence operator (cf. [15] or [5]), Sp and Sp are defined in (1.15) and (1.17) respectively and T is the constant of (1.19). 1 Furthermore, for any t > -~, we have

w > o , lulH,(~) -<~ S2 Ilfllz~(~)+ IlzllL~(~) ~2

C

~

E

ii1+ t Jig,,M,/~. (oa) +-IIzEIL~<~), /2

[-~llcurlfllL~(~) v~~S211flIL~(~) +2

(2.5)

(2.6)

-~- (2V~) l + t C ~-Ilg[I TM

H1/2(Ogt) ,

where C depends only on t and Ft. The above a priori estimates substantially simplify when g = 0; we obtain

Ilzllz=(~) ~ ~S=v Ilfllz=(~) + I_~ Ilcurl fllz~-(~),

(2.7)

lUlH'(~) ~ __S2 IlfllL=(~), /2

(2.s)

I(S~IIflIL~

§

IIZlIL~(~)

(2.9)

It is worthwhile comparing Problem P with the formulation proposed by [2] (el. also [31]) for the problem with homogeneous boundary conditions. This formulation is based on the fact that f can be written as a divergence: f -

div F.

Then, introducing an auxiliary pressure 7r and tensor o, the equations are -Au+V~=diva

,

divu=0,

u ---- 0 on 0 ~ ,

va+au. Va-aa.(Vu)

t = F-azr(Vu) t - u | + ~(V u)~(V u + (V u)~).

(2.1o) (2.11)

(2.12)

V. Girault and L. R. Scott

401

The equivalence between (2.10)-(2.12) and the original grade-two fluid model iis established in [31]. This formulation is not as interesting as ours, even in three dimensions, because it loses the unconditional energy estimate (2.8). Thus, in contrast to Problem P, its analysis requires in particular that u be bounded in W l , ~ ( f t ) 2. This has serious consequences: on one hand, existence of solutions cannot be derived without heavy restrictions on the size of the data, and on the other hand, the numerical discretization and programming are unnecessarily difficult, and rigorous proofs for its numerical analysis are non-existent.

3. A d i s c o n t i n u o u s

upwind scheme

Let h > 0 be a discretization parameter and let 7-h be a regular family of triangulations of ft, consisting of triangles K with maximum mesh size h (cf. [6], [5])" there exists a constant a0, independent of h, such that h~VK c :Yh, - ~ _ a0,

(3.1)

flK

where hK is the diameter and PK is the diameter of the ball inscribed in K. We first recall how upwinding can be achieved by the discontinuous Galerkin approximation introduced in [23]. Consider the discontinuous finite-element space

Zh = {Oh E L2(~) ; VK E "Yh, OhlK E J~D1}.

(3.2)

As interpolant, we shall mostly use an approximation operator (cf. [11], [29], [4]) Rh C s Zh N C~ for any number p __ 1, such that, for m - 0 , 1 and 0 _< 1 _< 1 VZ C W I + I ' P ( ~ )

, [Rh(Z) - Z]w,~,p(f~) ~___C h l + l - m l z I w z + l , p ( f t )

.

(3.3)

Let Uh be a discrete velocity in H~(ft), and for each triangle K, let

OK_ = {x c OK; C~Uh. n < 0).

(3.4)

Note that, when running over all triangles K of Th, OK_ only involves interior segments of Th, because Uh" n -- 0 on 0~. Then we approximate

Upwind discretizations of a steady grade-two fluid

402

the nonlinear terms a [ ( u . V z,O)+ l ( ( d i v u ) z , 0)] by a / ~ (div Uh ) ZhOh dx

+ Z

(f~ ~(u~. v z~)0~ ax

(3.5)

K ETh

-~- ~fOK_ IOLUh"nl(Zhnt -- z~xt)oihntd8)' where the superscript int (resp. ext) refers to the trace on the segment OK of the function taken inside (resp. outside) K. Note also that when summing over all triangles, OK_ is not counted twice because Uh. n changes sign across adjacent elements. Rather, in the above sum, the boundary integrations are taken over complete interior segments.

3.1. The Hood-Taylor scheme Let us first recall the standard Hood-Taylor discretization of the velocity and pressure. The discrete pressure space is

Mh -- {qh e Hl(~t)M L02(a) ; VK e Th, qhlg e ~~

(3.6)

and we interpolate the functions of L02(~t) by a regularization operator analogous to Rh, rh E s Mh), such that for 0 _< 1 _< 2, Vq c Hl(~)) n L0e(gt), Ilrh(q) - qll/~(~)

~- C hl[qIu~(~).

(3.7)

The discrete velocity space is X~,T

-

{v e c~

; VK e ~ , vlK e ~ , v. nlo~ - o } ,

X h : X h , T C1 g 1 (gt) 2 ,

f

W h --

{V E X h , T ; Vq C Mh , .In q div v dx

= 0},

(3.8)

(3.9)

y. - w~ n H](n) ~ . If all triangles K of Th have at most one edge on the boundary 0~2, the pair of spaces (Xh, Mh) satisfies a uniform discrete inf-sup condition (el. [15]). But as in [17] and [19], we can obtain better results from a local infsup condition that yields an approximation operator satisfying sharp local estimates. Indeed, we can prove that there exists an operator Ph C

V. Girault and L. R. Scott

403

s Ns such that the support of Ph(v) is contained in a neighborhood of the support of v, and

Vv e H~(a), IIv- Ph (v) llL, (a) <_ C h2/PlvlH~(a),

(3.1o)

Vv e g ~ ( a ) n W~,~(a) ~ , [Iv - Ph(v)llWr~,p(a ) < C hS-m]]v]]ws,p(a),

(3.11)

for all numbers p >_ 2 and all real numbers s with 1 < s < 3, m - 0, 1, and VK e Th, Vv e H~(gt), Vqh e M h , / g qh d i v ( P h ( v ) - v ) d x = 0.

(3.12)

Let gh -- Ph(r) where r is any lifting of g in W. This lifting is only a theoretical convenience because on one hand, gh can be constructed directly by interpolating g appropriately on Oft and on the other hand, gh does not depend on the particular lifting chosen; in addition, gh satisfies

IlghllH1/:(oa) = IIPh(wg)llH1/:(Oa) < IlPh(wg)llH,(a) _< Co IWg[H,(a) < Co T IlgllH1/~(oa) ,

(3.13)

where T is the constant of (1.19) and Co and all subsequent constants Ci are independent of h. Then, as written in the introduction, our discrete scheme is" find Uh in X h + gh, Ph in Mh and Zh in Zh solution of (1.12)-(1.14):

V'Vh E X h , l](V Uh, V Vh) Jr- (Z h X Uh, Vh) -- (Ph, d i v v h ) = (f, Vh) , Vqh C Mh , ( qh , div Uh) -- O, VOh C Z h , v (Zh, Oh) + C(Uh; Zh, Oh) -- V (curl Uh, Oh) + a (curl f, Oh), where c is defined by (3.5). As in [17], the trace preserving properties of Ph and its sharp local estimates allow one to construct a Leray-Hopf's lifting of gh satisfying: L e m m a 3.4 - For any g c H l / 2 ( 0 ~ ) 2 such that g . n = 0 and for any real number ~ > O, there exists a lifting Uh,g of g such that

lUh,glHl(a) _~ c~-l/2-1/tl]gllH1/:(Oa)

, 2 ~ t < c~,

(3.14)

and if hb < s, where hb denotes the m a x i m u m mesh length of triangles in a tubular neighborhood of the boundary, then for all v C Ho~(~) 2 and for t < s < c~, (recall that I]" I] denotes the Euclidean vector norm),

IIIlu..~ll Ilvllll,_..(~) < cell2-11~llgllH~,':~(oa)lVlH~(a),

(3.15)

where the constants C depend on t or s, but are independent of h, ~ and g.

404

Upwind discretizations of a steady grade-two fluid

In order to prove existence of solutions of (1.12)-(1.14), let us recall the :following identity established by Lesaint and Raviart in [23]" L e m m a 3.5 - For all vh in Xh, for all Zh and Oh in Zh, we have

~(v~; z~, 0~) = ~

(- f~ ~(v~. v O~)z~ ~x

K ~Th

Io~vh. nlz~Xt(0~xt-

+Z

0hnt)ds)

(3.16)

(div Uh)OhZh dx

(3.17)

K_

a f (div Uh)0h Zh dx. 2 J~ Note that when Oh is in Ht(~t), (3.16) reduces to

a/a C(Vh; Zh, Oh) -- -- / ~ a(Vh" V Oh)Zh dx - -~ Note also that, when

0 h -- Z h C Z h ,

c(vh; zh, zh) - ~1 g ~ ~

then

f o ~ _ [avh " nl(z~Xt -- zihnt)2ds"

(3.18)

Therefore, choosing Oh = Zh in (1.14) and applying (3.18), we obtain

I~uh. nl(z~ x t - z~nt)2ds

Ilzhll~(~) + 5 ~ KeT-h

K_

(3.19)

= ~ (curl Uh, Zh) + ~ (curl f, Zh). Equation (3.19) allows us to prove the following existence theorem. T h e o r e m 3.6 - There exists a constant C1 ~ O, independent of h, such that for all v > 0 and a C Kl, for all f in H(curl, ~) and all g in H1/2(0~) 2 satisfying g . n - O, if hb < Clv2+tllg[[-2-t H1/2(0~)

,

for some t > 0

(3.20)

then the discrete problem (1.12)-(1.1~) has at least one solution (Uh,Ph, Zh) in (Xh + gh) • Mh • Zh and every solution satisfies the following a priori estimates:

lUhIH~<~> ~ ~5'2Ilf]lL~(~) + K1 (h)Co Tllgl]H~/~(o~), v

(3.21)

V. Girault and L. R. Scott

IlPhlIL~(~) ~ ~-* (S211flIL~(~) + v'CoTllg[[Hl/~(0~)

405

(3.22)

+ S4S41U~IH~(~)Ilzhll~(~)),

PlzhllL~(~) ~ V~[UhlHl(~) ~- I-~!llcurlfpJL~(~), 1

~K~~fO,C [auh'n[

(z~X t

--

z~nt )

2ds

(3.23)

(3.24)

(V/-2/]lUh[HI(Ft) d-[a[[lcurlf[[L2(a))[[Zh[lL2(gt), where Co is the constant of (3.13), ~* is the constant of the discrete inf-sup condition, and

Kl(h)-

1+

s~4 [[Zh[[L2(~). V

In addition, we have for any real number s > -~

~

+ 2v/2S2 [[fllL2(a) /2

(3.25)

+ 2 [am[]]curl fllL2(~),

/2

where the constant C2 depends on s and t, but not on h or v.

Extracting subsequences (that we still denote by the index h), the uniform a priori estimate (3.25) combined with (3.21), (3.22), (3.24) show that, on one hand, (uh,Ph, Zh) converge weakly to functions (u,p, z) in H~(gt) x L2(~t) x L2(~t), and on the other hand, the quantity ~'~Ke~-h fOK_ Ia u h " nl(z~ xt -- zihnt)2ds converges to a non-negative number, say S. Passing to the limit in (1.12), we see that (u,p,z) satisfies (1.5). Next the strong convergence of Uh is easily established, as in [17]. Owing to this strong convergence, using Rh(O) with smooth 9 for test function in (1.14), and applying (3.17), we can readily prove that (u,p, z) is a solution of Problem P. Using again the strong convergence of Uh and the weak convergence of Zh, and comparing with (1.8), we see that the right-hand side of (3.19) converges to v [[zl122(a). Here, we need the following consequence of Green's formula established in [16]:

/(

u- V z ) z d x = O,

Upwind discretizations of a steady grade-two fluid

406

for z a solution of (1.8). Therefore h--,olimu Ilzhll2
S

=

[Izll

9

Then the lower semi-continuity of the norm for the weak topology implies that lim IlzhllL (a)= IlzllL
This yields on one hand the strong convergence of Zh and on the other hand the fact that S = 0. The following theorem sums up these convergence results. T h e o r e m 3.7 - Suppose that hb satisfies (3.20), and let (Uh,Ph, Zh) be a solution of (1.12)-(1.1~). Then we can extract a subsequence of h, still denoted by h, such that lim

Uh

---

h---,O

U strongly in W ,

lim Ph = P strongly in L2(f~)

h---,O

lim Zh = z strongly in L2(ft), h---,O

lim E h--,0

K ETh

fo

laUh. nl(z~ xt - z~nt)2ds - 0 in Bzt, K_

where (u, p, z) is a solution of Problem P.

3.2. T h e if)3 - if)2 scheme with discontinuous p r e s s u r e We describe briefly this scheme, since its analysis follows closely that of the Hood-Taylor method. In view of (1.11), we discretize pressures in the space Mh = {qh e L02(f~) ; VK e Th, qh[g e / P 2 } ,

(3.26)

and we interpolate the functions of Lo2(ft) by a local projection on each triangle K , rh C s Mh), such that for 0 _< 1 _< 3, Yq e H I ( a ) n L02(Ft), []rh(q) -- q]]L~(a) <---C hZlqlH~(a).

(3.27)

Then, in order to satisfy a uniform discrete inf-sup condition, we discretize each component of velocities in each K, in the polynomial space 79 = ~ 3 + b K ~ l ,

(3.28)

V. Girault and L. R. Scott

407

where bK -- AIA2A3 is a bubble function and ~ 1 - {ClXl + c 2 x 2 ; Y c l , c 2

e ~}.

Then, we discretize velocities in the space

Xh,T = {v e C~ ; VK e Th, VlK ~ p2, v - nlo~ = o}, Xh = Xh,T A H01(ft) 2 ,

(3.29)

with Wh and Vh defined as in (3.9). The fact that the functions of Wh satisfy Vh" n = 0 implies VK C Th, VVh C Wh, Vq 6 B:~2 , /K(divvh)qdx : 0.

(3.30)

Hence V K C 'Th, VV h C

W h , MZh, Oh C Zh , /K(diVVh)ZhOh dx = 0.

(3.31)

It is shown in [15] that the pair of spaces (Xh, Mh) satisfies a uniform discrete inf-sup condition, but as in [19], combining the construction of [15] and [17], we can prove that there exists an operator Ph with the same local properties and satisfying (3.12), (3.10) and (3.11) with 1 _ s _ 4. By virtue of (3.31), the stabilizing term in c is no longer necessary and (3.5) is replaced by

~(u~; z~, 0~) : ~

(/~ ~(u~. v z~)O~~x

K 6"rh P

+/

(3.32)

I~u~. ~l(~nt _ ~e~tj%'~nta~)"

JO K_ With (3.32), the scheme (1.12)-(1.14) is simpler, but the finite-element functions for the velocity and pressure are more complex. All the results of this section remain valid for this scheme; the proof of Theorem 3.7 being easier owing to (3.31).

4. E r r o r e s t i m a t e s

Estimating the error of (1.12)-(1.14) is not straightforward because of the strong nonlinearity of the equations. First, we derive the following bound:

luh-ul~,(~) _< 2 IPh(u) - ulH~(~) + S~llPh(u)llL4(~)llzh -- zllL~(~) /] + S4[IZIIL2(~t)IIPh(u ) -- U l l L a ( a ) -F vf2llrh(p) -pllL2(a) . v

v

(4.1)

Upwind discretizations of a steady grade-two fluid

408

B u t this is not sufficient. As mentioned in the introduction, we do not need a W 1'~ estimate for U h - u but we do need W 1,r estimates with some 2 < r < oc. Such estimates stem from the following decoupling error formula, proven in [17], t h a t extends (4.1). It is valid for both schemes, with possibly different constants. T h e o r e m 4.8 - Assume that f~ is convex, g belongs to H1/2+8(Of~) 2 for some s e (0, 1/2), and in addition to (3.1) and (3.20), Th satisfies:

(4.2)

V K c Th, PK >_ "Yh~ , with a constant ~/ > 0 independent of h. independent of h, such that

-}- ~

1(

Then there exist constants Ci,

C3lPh(U) -- utile(a) + x/2 llrh(p) -PIlL'(a)

/Jmin

(4.3)

/2

+ liz~ - zliL~(a)(c~ w h e r e Pmin i s

)

the minimum of

+ C~hl/~),

PK for

all K in Th.

Remark ~.9 The bound (4.3) is one in a family of estimates established by [17] in Wl'r(gt) 2 for all r in (2, o0). The assumptions on the boundary d a t a g, the domain and the triangulation depend upon the value of r. The choice r = 2 + 1/4 is convenient here because on one hand, it is sufficient for deriving the error inequalities below and on the other hand, the condition (4.2) allows for a wide range of refinements of the triangulation. Remark ~.10 Since, W1'2+1/4(~) in continuously imbedded into L~176 an easy variant of Theorem 4.8 gives with possibly different constants but under the same assumptions:

Ilu~ - uffL~(a) < IfPh(u) - utiLe(a) + , / 91

(c~ip~(u) - uf.,(a)

/3min

+ v~ll~(p) /y

- pfr~(~)) + ffz~ - z l l ~ ( a ) (c~ +

(4.4)

c~h'/~).

Now, in order to prove an error inequality for Z h -- Z, we assume t h a t z belongs to HI(Ft). It is shown in [17] t h a t this assumption holds on a convex domain, when g and f are sufficiently smooth and small. It is restrictive, but so far, relaxing this assumption appears difficult. The following arguments are written when c is in the form (3.5); they simplify when the stabilizing

V. Girault and L. R. Scott

409

term is missing. Taking the difference between (1.8) and (1.14), inserting any element Ch of Zh, choosing 8h = Zh--~h, and applying (3.16) and (3.18), we obtain

1 fo

IOLUh" n l ( ( Z h

-- ~h)ext

--

(zh -- ~h)int)2d8

K ETh + E (--OZ/K U h ' ~ ( Z h , - - ~ h ) ( ~ h - - z ) d x K ~q-u + ~ K _ [(~Uh " n'((Zh -- ~h)ext -- (Zh -- ~h)int)(~h -- Z)extds) (4.5)

62/a div(uh -- U)(~h

+-~ a ]~

--

Z)(Zh -- ~h)dX

div(uh -- U)Z(Zh -- ~h)dx

+ a ~ (Uh -- u). V Z(Zh -- ~h)dx = ~,(~ - Ch, zh - Ch) + ~ , ( c u r l ( u h

- u), z~ -

r

Clearly, the most troublesome terms in the left-hand side of (4.5) are the third and fourth terms. In each element, the third term can be split into: KUh

V(Zh -- ~h)(~h -- z ) d x = f K ( U h -- u ) . V(Zh -- ~h)(~h -- z ) d x + / K u . V(Zh -- ~h)(~h -- z ) d x .

Now, instead of choosing ~h -- R h ( z ) , that clearly cannot give anything better in the second term than an error of the order of h, let us take advantage of the discontinuity of the space Zh and choose ~h -- ~h(Z), the L 2 projection of z on iPl in each triangle K: ~h(Z) C iPl, such that

Vq E ~ 1 , /K(~Oh(Z) -- z)q d x -- O. This operator has locally the same accuracy as Rh. Moreover, we have for any constant vector c:

]~

u. V(z~-

Qh(z))(~ (z)- z)dx = f ( u - c ) . V(zh - ~ (z))(~h (z)- z)dx,

410

Upwind discretizations of a s t e a d y grade-two fluid

because V(Zh -- ~h(Z)) is also a constant vector. Thus a local inverse inequality and (3.1) yield: f ]a ./ ( u - c). V(Zh -- Qh(Z))(~h(Z) - z ) d x I Cl

-- ZIIL2(K)]IZ h -- ~h(Z) I]L2(K)

]OL]I[ u - - C I I L ~ ( K ) I I ~ O h ( Z )

PK < C2~ol~llUlw,,~(g)ll~(z)-

zllL~.(g)llzh

--

(4.6)

~(z)llL~(g),

where ci denote various constants independent of h. Similarly, applying the same local inverse inequality and the approximation property of Qh, we obtain f ]~ ./ (Uh -- u ) . V ( z h -- Qh(Z))(~h(Z) - z ) d x I

(4.7)

The fourth term can be split into" f

/

]~uh. nl((z~ - Q~(z)) ext- (z~ - e~(z))int)(e~(z)- z)eXtd~

KeTh JDK_ <

-2

1/o

--

[OLUh 9 n ] ( ( Z h

-- Qh(Z))

ext --

(Zh -- Oh(z))int)2d8

K_

+21 f0K_

IC~Uh"n] ((0h(Z)-

z)eXt)2ds"

The first term cancels with the first boundary term in the left-hand side of (4.5) and by virtue of a scaling argument, the second term is bounded by 9

_

ext

d8

(4.8)

K_ -< - ho-20c4[o~l IlUhllLoo(O.,K)]Qh(Z)- 21 2 H

where WK denotes the union of triangles adjacent to K. involving the divergence is bounded as follows"

~

d v(u -

1 (WK ) '

The first term

Q (z))dx I (4.9)

< c5 ~-! Ildiv(uh -

U)llL=+,/~(a)lZlHl(a)llZh

--

Q~(z)IIL=(a),

V. Girault and L. R. Scott

411

with a similar bound for the second term. Finally, the last term is bounded by:

J~/(u~

- u). v z(z~ - Q~(z))~xJ

(4.10)

< I'~1 IzlH~(~)lluh - ullL~(~)llzh - ~h(z)llL=(~). Hence collecting (4.6)-(4.10), substituting them into (4.5), and applying Jansen's inequality, we obtain:

I~l ~

~

+ Jul~ + cTh -]al -/2

[]UhIIL~(~t)

Iz --

LOh(Z)]2H1 (~t)

(4.11)

+ ~ - 7la12 1 1 z l l ~ 1 ( ~ ) J[div(uh -- u)JJ 2~+~4(~) + 2(11~ - Q~(z)IIL(~) + 21u~ - ul~i(~)). By substituting (4.1), (4.3) and (4.4) into (4.11), we derive the following error inequality. T h e o r e m 4.11 - Let ~ be a convex polygon and suppose Th satisfies (3.1), (~.2) and (3.20). Assume that the data f and g are suJ~ficiently smooth and small so that u E Wl'~(gt) 2 and z C g 1 (~t), and a and/2 are such that

I~1 (C4 -~- Cbhi/4)lizllH~(~)(~/-~ +

-7-

2~/~s)+ 2 S 4 IlPh(u)llL~(~) < 1 7

-~'

(4.12)

with the constants C~ of (~.3) and (~.~) and c~ of (~.11) (which implies that Problem P has a unique solution), then there exists a constant C, depending on the data but independent of h, such that each solution of (1.12)-(1.1~) satisfies the error bound ]JZh --

ZIIL2(Ct) ~

C(llgh(u ) 1

--

II[JL~(Ft) - ~ - J g h ( u ) -

UIW1,2A-1/4(~-~)

+ yiTb-(IPh(u)- UlH~(~) + Ilrh(p)- PlIL~(~)) /')min

(4.13)

+ [IPh(u)- ulIL~(~) + II~h(Z)- zlii~(~)

+ h ~ / ~ l ~ ( z ) - zlH.(~)) 9 Clearly, the last term in (4.13) is dominating; it accounts for the loss of h 1/2 in the final error term. Thus, under the assumptions of Theorem 4.11, if z

Upwind discretizations of a steady grade-two fluid

412

belongs to H 2 (~), u to H 3 (~)2 and p to H 2 (~), we have

lUh -- UlHI(~ ) +

Ilph --pllL2(~-~)

-~-]]Zh

--

ZlIL2(~~) ~

C h 3/2 .

(4.14)

We finish this section by addressing the computation of (Uh,Ph, Zh). Consider the following successive-approximations algorithm. Let z~ be an arbitrary element of Zh (e.g. z~ = 0); then for k >_ 0 we compute the sequence u k c Xh § gh, pk C Mh and z k+l C Zh by solving first the generalized Stokes problem: MVh E X h , tJ(Vuk, V V h ) + (z k x uk, v h ) - (pkh,divvh ) = (f, Vh),

Vqh C Mh , (qh, div Ukh) -- O, and next the linear transport equation: V~h e Z h ,

b' (zhk+l,~h)-[c(uk;zk+l,{Oh)

--

v ( c u r l u ~ , 0 h ) + a (curlf, 0h).

As in [171, the uniform a priori estimate (3.25), combined with (3.21), (3.22), (3.24) allow us to prove that there exists a subsequence of k tending to infinity such that (Ukh,Pkh,Zhk) converges uniformly without restriction on the data or the domain. Furthermore, under the sufficient conditions of Theorem 4.11, the limit is a solution (Uh,Ph, Zh) of (1.12)-(1.14).

References

[1] Adams, R. A., Sobolev Spaces, Academic Press, New York, NY, 1975. [2] Baia, M. and Sequeira, A., A finite element approximation for the steady solution of a second-grade fluid model, J. Comp. and Appl. Math., 111 (1999), 281-295. [3] Bernard, J. M., Stationary problem of second-grade fluids in three dimensions: existence, uniqueness and regularity, Math. Meth. Appl. Sci., 22 (1999), pp. 655-687. [4] Bernardi, C. and Girault, V., A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal., 35, 5 (1998), pp. 1893-1916. [5] Brenner, S. and Scott, L. R., The Mathematical Theory of Finite Element Methods, TAM 15, Springer-Verlag, Berlin, 1994. [6] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [7] Cioranescu, D. and Ouazar, E. H., Existence et unicit6 pour les fluides de second grade, Note CRAS 298, S6rie I (1984), 285-287.

V. Girault and L. R. Scott

Is] [9]

[10]

[11] [12] [13] [14] [15] [16]

[17] [is] [19] [20] [21] [22]

413

Cioranescu, D. and Ouazar, E. H., Existence and uniqueness for fluids of second grade, in Nonlinear Partial Differential Equations, Coll~ge de France Seminar, Pitman 109, Boston, MA (1984), 178-197. Cioranescu, D. and Girault, V., Weak and classical solutions of a family of second grade fluids, Int. J. Non-Linear Mech. 32 (1987), 317-335. Cioranescu, D., Girault, V., Glowinski, R. and Scott, L. R., Some theoretical and numerical aspects of grade-two fluid models, in Partial Differential Equations- Theory and Numerical Solution, (W. Jaeger, J. Necas, O. John, K. Najzar, J. Stara, eds.), Research Notes in Mathematics 406, Chapman & Hall/CRC, New York, NY, 99-110, 1999. Clement, P., Approximation by finite element functions using local regularization, RAIRO, Anal. Num. R-2 (1975), 77-84. M. Crouzeix and P. A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes problem, RAIRO Anal. Num~r., 8, (1973), 33-76. Dunn, J. E. and Fosdick, R. L., Thermodynamics, stability, and boundedness of fluids of complexity two and fluids of second grade, Arch. Rational Mech. Anal. 56, 3 (1974), 191-252. Dunn, J. E. and Rajagopal, K. R., Fluids of differential type: Critical review and thermodynamic analysis, Int. J. Engng Sci. 33, 5 (1995), 689-729. Girault, V. and Raviart, P. A., Finite Element Methods for NavierStokes Equations. Theory and Algorithms, SCM 5, Springer-Verlag, Berlin, 1986. Girault, V. and Scott, L. R., Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition, J. Math. Pures Appl. 78, 10 (1999), 981-1011. Girault, V. and Scott, L. R., Finite-element discretizations of a twodimensional grade-two fluid model, M2AN 35, 6, (2001), 1007-1053. Girault, V. and Scott, L. R., Stability of dispersive model equations for fluid flow, to appear in Collected Lectures on the Preservation of Stability Under Discretization, eds. D. Estep and S. Tavener, 2002. Girault, V. and Scott, L. R., A quasi-local interpolation operator preserving the discrete divergence, in preparation. Grisvard, P., Elliptic Problems in Nonsmooth Domains, Pitman Monographs and Studies in Mathematics, 24, Pitman, Boston, MA, 1985. Holm, D. D., Marsden, J. E. and Ratiu, T. S., Euler-Poincar~ models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett. 349 (1998), 4173-4177. Holm, D. D., Marsden, J. E., Ratiu, T. S., The Euler-Poincar~ equations and semidirect products with applications to continuum theories, Adv. in Math. 137 (1998), 1-81.

414

Upwind discretizations of a steady grade-two fluid

[23]

[24] [25] [26] [27] [2s] [29] [30] [31]

Lesaint, P. and Raviart, P. A., On a finite element method for solving the neutron transport equation, in Mathematical Aspects of finite Elements in Partial Differential Equations, 89-122, Academic Press, New York, NY, 1974. Lions, J. L. and Magenes, E., Probl~mes aux Limites non Homog~nes et Applications, I, Dunod, Paris, 1968. NeSas, J., Les M~thodes directes en th~orie des ~quations elliptiques, Masson, Paris, 1967. Ouazar, E. H., Sur les Fluides de Second Grade, Th~se de 3~me Cycle de l'Universit~ Pierre et Marie Curie, Paris VI, 1981. Peetre, J., Espaces d'interpolation et th~or~me de Soboleff, Ann. Inst. Fourier 16 (1966), 279-317. Rivlin, R. S. and Ericksen, J. L., Stress-deformation relations for isotropic materials, Arch. Rational Mech. Anal. 4 (1955), 323-425. Scott, L. R. and Zhang, S., Finite element interpolation of non-smooth functions satisfying boundary conditions Math. Comp., 54 (1990), 483-493. Tartar, L., Topics in Nonlinear Analysis, Publications Math~matiques d'Orsay, Universit~ Paris-Sud, Orsay, 1978. Videmann, J. H., Mathematical analysis of visco-elastic non-newtonian fluids, Thesis, University of Lisbon, 1997.

Vivette Girault Laboratoire Jacques-Louis Lions Universit~ Pierre et Marie Curie 175 rue du Chevaleret 75013 Paris France E-mail: girault @ann.j ussieu, fr L. R. Scott Department of Mathematics University of Chicago Chicago, Illinois USA E-mail: [email protected]

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 19 STABILITY OF THIN LAYER A P P R O X I M A T I O N OF E L E C T R O M A G N E T I C WAVES S C A T T E R I N G BY LINEAR AND NON LINEAR COATINGS

H. HADDAR AND P. JOLY

1. Introduction The scattering of electromagnetic waves by thin coatings, using effective boundary conditions have been widely studied in the case of harmonic Maxwell equations (see [1], [4], and references therein). We give an extension to the time dependent problem, where obtaining the stability in time is fundamental. We apply a general technique based on an asymptotic expansion of the solution with respect to the thickness of the coating ([17], [1]). The main motivation for using these effective boundary conditions comes from numerical considerations. Roughly speaking, as the obtained approximate model is posed on the exterior domain (i.e. not including the thin layer), we eliminate the geometrical constraints for meshing. This reduces (especially for layers that are very thin compared with the wavelength of the incident wave) the size of the discrete model, and thus the time of computations. We notice also that this method may constitute a valuable alternative to mesh refinement around the obstacle. This article is divided into three parts. In the first one, the coating is made of linear material and the boundary is planar. This simplified case enables us to present, in a simple way, the formal construction of the effective boundary conditions. Once stability conditions are derived, one can perform an asymptotic analysis to establish error estimates. In the second part, we treat the case of curved and regular boundaries. We explain how to take into account the geometric characteristics of the boundaries. The expression and the stability of the third order condition constitute the main result of this part. In the last section, our work is generalized to a class of non linear materials of ferromagnetic type. Only the formal derivation and the stability analysis are discussed. The error estimates will be the subject of a forthcoming paper.

416

Stabifity of thin layer approximation of electromagnetic waves...

2. T h i n l a y e r approximation: the linear case 2.1. D e s c r i p t i o n of the model We consider a linear material characterized by its electric permittivity r and its magnetic permeability #f that occupies a layer f~ C 1~ 3 of a thickness ~. This layer separates two media: the exterior domain f~v of characteristics (~o,/to) and the interior domain f~, namely a perfectly conducting scatterer. We denote by F the boundary of f~. and F ~ the boundary of fig (see fig. 1). We assume that the material is homogeneous and that its characteristics do not depend on r/, (the non-homogeneous case where the characteristics depend on the thickness coordinate and vary slowly with respect to the tangential coordinate can be treated with minor modifications).

Figure 1: Problem presentation We denote by (E0, H0) the electromagnetic field at t -- 0. We assume that it is compactly supported in f~. Let (E~, H~) (resp. (El~, H~)) be the electromagnetic field in ft~ (resp. in f~f~). They satisfy: { r

- rot H~ - 0,

#o cOtH~ + rot E~ - 0

(E~, H~)(t - 0) - (Eo, Ho) { E~xn

E~xn,

H~xn-Hf

on f~v x l:t+,(1

)

on fly. ~xn

on F x R+

(2)

where n is a unit vector normal to F, { Cf OtEf~ - rot H~ - 0, (Ef~, nf~)(t - 0) - (0, 0)

#f OtHf~ + rot E~ - 0

on gtf~ x 1~+ on ~ ,

'

(3)

H. Haddar and P. Joly

417

and finally, the "Dirichlet" boundary condition on F ~ (where n is a unit vector normal to F n), Ef~ x n - 0 ,

on F ~ x l R +.

(4)

The problem constituted by (1), (2), (3) and (4) is the exact transmission problem. As we consider the situations where r/ << A (the incident wavelength), numerical methods applied to this problem will be penalized by the computation of (Ef~, H~) (that requires fine meshing). The idea to overcome this difficulty, is to replace the equations (2), (3) and (4) by an effective boundary condition posed on the exterior boundary F such that: the electromagnetic field, satisfying Maxwell's equations on f~ coupled with this effective boundary condition, constitutes a good approximation of (E~, Hv~). More precisely, our construction leads to the hierarchy of effective boundary conditions indexed by an integer k

Ev~ x n - B # ( n x

(:Hv~ xn))

on Fxl::t +,

(5)

where (E~, H~) is the approximate electromagnetic field on f~v, satisfying #o OtI--I~+ rot E~ - 0

{ ~o OtEv~ - rot HW - 0,

(F,v~, H~)(t - 0) - (E0, H0)

on Fly x IR+,

(6)

on f~v,

and where B~ is a tangential operator such that (at least in a formal way)

II

-

In such a case, the effective boundary condition (5) is said to be of order k. The identity (7) is referred to the c o n s i s t e n c y p r o p e r t y of the construction. A fundamental notion linked to this property is the s t a b i l i t y with respect to the small parameter 7. For problem (1), (2), (3), (4) one gets a uniform bound, with respect to t > 0 and ~, of the electromagnetic energy in f~v, namely s H~(t)) _ E~(Eo, no), (8) where, s H) ~o [[EI[2 L2(f~. ) -~- 2/~ [[H[ 2 , 2 )" Relation (S) is obtained from the energy identity -

$~(E~(t), H~(t)) + s where E~ (E ,H ) - ~2

H~(t)) - $~(Eo, Ho),

lIEll~:(~;)+ ~ IlHII2L=(a~)-

(9)

W e deduce that estimation

(7) requires at least a stability identity, for the solution of (6), of the form Ev(F,~(t), H~(t)) _< Cv(E0, n 0 ) + o(~k- 89

(10)

In fact, identity (10) will constitute the basic requirement of the construction of B~.

418

Stability of thin layer approximation of electromagnetic waves...

2.2. Formal construction: the case of a plane boundary In this section we assume that the unitary normal vector field n on F, outwardly directed to Dr, is constant (see figure 2).

Figure 2" Problem presentation: the case of a plane boundary

2.2.1. Scaling, asymptotic expansion and principle of the effective conditions We apply first the well known technique of scaling with respect to the thickness coordinate of the thin layer, which removes the dependence of the geometry on the small parameter ~. Let us proceed in an intrinsic way, so that the link with the case of curved boundary shall be more clear later. We use the parametric coordinates system of 12fn (fig. 2). xe~tf~:

; (Xr,S) e F •

such that: X - X r + S n .

(11)

In the case of straight boundary, this map is well defined for every ~. The scaling consists on the change of variables" (Xr, s) e F • [0, r/] ~

(xr, v) e F • [0, 1] such that: s - 7 ? v.

(12)

S o m e n o t a t i o n s . We denote by Vr the derivative operator with respect to Xr. For instance, if (7-1,7-2) E 1~3 x R 3 is an orthonormal basis of F, and (~1, ~2) is the system of coordinates associated to this basis, we haveVr - (0~1 .)7-1 + (0~2 ")7-2. It is easy to see then that the expression of the derivative operator V in l~ 3 becomes in the coordinates system (xr, v),

10vn Let v be a tangential vector field defined on F. We define the operator divr by: divr v - XTr "v - (c9~1v)" 7-1 + (0~2 v)" 7 "2.

419

H. Haddar and P. Joly

We recall that divr is the adjoint o f - V r We introduce also: rotr v - divr (v x n),

for the L 2 scalar product on F.

(13)

ro-*tr u - (Vr u) x n,

where v (resp. u) is a vectorial (resp. scalar) field defined on F. rotr and ro-*tr are adjoint operators for the L 2 scalar product on F. Furthermore, divr ro-*tr = 0 and rotr Vr - 0. We shall distinguish the normal and the tangential part of a vector field v E l ~ 3. v - Hs, v + Hi v, where IIj, v - n x (v x n) and II~ v - (v. n) n.

(14)

Throughout this paper, the variable t will always indicate the time variable but may sometimes be omitted from the list of variables. On the other hand Xr E p 3 will always denote the tangential coordinate belonging to F. S c a l e d e q u a t i o n s . Let E be a vector field defined on ~t~. We define 1~ on the scaled domain Fx]0, 1[ by l~(xr, y) - E(x) where (Xr, ~) and x are linked by (11) and (12). Working in the orthonormal basis ( r 1, T2,n), one checks that

rot E - rotr (I~. n) + (rotr I~)n -

0~(E

•

~).

(15)

This identity enables us to rewrite equations (2), (3) and (4) in the scaled domain. For simplicity, we shall denote (in abusive way) by (El~, H~) the electromagnetic field in Fx]0, 1[, (E~, H~)(x, t) ~ (]~, I:I~)(Xr, ~', t) - (El~, Hf~)(Xr, v, t). So, equations (2), (3) and (4) become respectively,

II,,E3(x~, t) = n,,ET(x~, 0, t), nHH~(xr , t) for (Xr,t) E F •

-

II,,HT(x~, 0, t)

(16)

+.

O.(H~ • ~ ) -

c, OtE~ - rotr (H~. n) - (rotr H ~ ) n + #, OtH~ + rotr (E~. n) + (rotr E ~ ) n -

1

0

0v(E~ • n ) - 0

on F x ]0, l [ x R +

(17)

(E~, H~)(t - 0) - (0, 0), on F• ]0, 1[, E~(xr, 1) x n = 0 for (xr,t) e F x lit + ,

(18)

420

Stability of thin layer approximation of electromagnetic w a v e s . . .

and the exact transmission problem is constituted now by equations (1), (16), (17)and (18). A s y m p t o t i c expansion. We seek a solution of the form: { (E~,H~)-(E ~

~

~/(E 1,H 1) + ~2(E 2,H 2) + . . .

(E~,Hf~ ) - ( E ~

~

~/(E 1,H 1) + r/2(E 2,H 2) + . . -

(19)

and proceed to formal identification after the substitution of (19) into (1), (16), (17) and (18). This process allows a recursive determination of the different terms. From (1) and (16) we get for all k _ 0, 60 OtE k - rot H k - 0,

(E k, H~)(t - 0) -

#o OtH k + rot E k - 0,

(Eo,Ho) (0,0)

II,Ek(xr)-

n,,E~(x~,

if k - 0 ,

(20)

if k > 0 ,

0) for Xr e F,

that can be interpreted as Maxwell's equations with non-homogeneous boundary conditions on F. They permit to determine (E k, H k) for each k _ 0, if we are able to compute the boundary value II..Ek(xr, 0). This will be possible using (17) and (18). For k - 0, we get { 0~(E ~

E~

0~(H ~ • n) - 0 ,

'

1)•

H~

for Xr E F,

0) x n -

H~

(21)

x n,

while for k > 0, we have 0u(E k • n) - #fcgtH k-1 + rotr (E k-l" n) + (rotr E k-l) n, Ek(xr,1) x n - - 0 ,

(22)

for Xr C F.

and { 0~ (Hfk x n) - -ef OrE k-1 + rotr (H k - l - n) + (rotr H k-l) n, Hk(xr,0) •

xn,

(23)

for Xr e F.

Equations (21), (22) and (23) can be solved like the following. For k - 0, equation (21) shows that,

{ E~ H~

• ~ - 0, u) x n - H~

For higher order terms, we apply

(x;,~) e r• x n,

1[,

(Xr, u) e Fx]0, 1[.

(24)

H. Haddar and P. Joly

I

1/i

421

L e m m a 0.1 - For all k > O, and for all (Xr, v) 6 Fx]0, 1[, 0tEk(xr, v) • n --

Cf

(~f#fOtt + rotr rotr ) n,.H~-l(x~, ~) d~,

(25)

(E~ (x~, ~) • n ) ( t - 0) - 0.

0t H k (Xr, v) x n - 0t H k (xr) x n

_1

#3

(e~#fOtt + rotr rotr ) II,iEk-1 (xr, ~) d~,

(26)

(H~ (Xr, ~) • n)(t = 0) - 0.

Proof. Let us prove (25). Applying II H to the first equation of (22) shows that: 0~(E k x n) - #f 0 t ( H , n k-l) + rotr (E k - l " n). (27) We take now the scalar product of the first equation of (23) by n. n-ro~tr - 0 we obtain: efOt (E k - l " n) -- rotr H k-1.

As

(2s)

Combining (27) and (28) yields Or(OrE k x n) -

1 ) k-1 #~Ott + -- rotr rotr Hijn f ~f

We get identity (25) by integrating the previous equation between ~ and 1 and by using the derivative in time of the boundary condition given in (22). Identity (26) is obtained in a similar way. m Remark 2.1. From equation (28) one can easily deduce that the normal components (E k. n) and (H k- n), k >_ 0, are determined by ef0t (E k. n) - rotr HL.Hk,

#f Ot (H k- n) - - r o t r II,,E~,

(E~..)

(H k- n) (t - 0) - 0.

(t - 0) - 0.

(29)

These equations will be useful later in order to prove the error estimates. Construction principle of the effective boundary conditions. In order to obtain an approximation of order k of the scattered field (EW, HW) a good candidate would be the truncated expansion --k uk

k-1

( E , , H v) - ~ i--0

r]i (E~, H~)

422

Stability of thin layer approximation of electromagnetic w a v e s . . .

(cf. theorem 0.2) . According to (20) , (Ev, --k --k H~) satisfies Maxwell's equations on fl~ coupled with the boundary condition on F --k

k-1

Hll E . (Xr) - ~-'~i=0

~i

i

xr e F.

IIttE,(xr, 0),

(30)

On the other hand, we notice that, having Ht.E ~ and HttH ~ by (24), equations (25) and (26) are nothing but first order differential equations with respect to v that permit to compute step by step IIttE k and H,tH k for each order k as functions of the boundary values of IItt H k on F. So one can express the right hand side of (30) as a function of the boundary values of HHH~1 0 < i < k - 1. Then the question is: can we express it as a function --k

of (H v x n)? The response is no, in general. Meanwhile, we may establish formally the existence of a linear tangential operator B~ such that -Ek v x n - B

k, ( i i -H- k~ ) + O ( r l k)

on

F.

To obtain our effective boundary condition of order k, we chose to omit 0(~? k) in the previous identity. That is why the approximate solution, de"~

""

--k

mk

noted by (E~, H~), will be different from (E,, H , ) . We shall discuss the validity of this choice via the stability study. We are going to give the details of this construction for k - 213, 4. Remark 2.2. As we assumed that the thin layer characteristics are independent of the the thickness 77, the limit problem, when ~ goes to 01 is obtained by omitting the thin layer and applying the Dirichlet boundary condition (4) on F. This coincides with the effective boundary condition of order 1" combine the first equation of (24) with the system (20) written for k-0. 2.2.2. E f f e c t i v e b o u n d a r y

condition of order 2 and 3

According to condition (30) we can derive the effective boundary conditions of order 2 by computing E ~ (Xr, 0) x n and E 1 (Xr, 0) x n. E ~(Xr, 0) x n - 0 by (24). From (25), written for k = 1, and from the second equation of (24), we deduce that 1 0tEl (xr, v ) x n - - - - ( 1 - v ) ( c f # ~ O t t + r o t r r o t r ) I I j t H ~

(31)

Cf

We have then: --2

T]

OtEv(xr ) x n - - - -

--2

(r #fOrt + rotr rotr ) IIttHv(xr ) + O(r/2)

~f

(where O(r/2) - ~Cf (~f #f(gtt + rotr rotr

1

) II,,Hl(x~)).

H. Haddar and P.

Joly

423

If we denote by (Ev~,H~) the approximate scattered field, the effective boundary condition of order 2 is: atEv~

x

n

-

-

~

6f

(~f #, art -I- rotr rotr ) rill Hvn on F x R +.

(32)

In this particular case of a straight boundary, condition (32) is in fact of order 3 (but will be false with curved boundary or non linear materials (see section or section 0.0.1)). Indeed, in order to compute (E2(xr,0) x n) using equation (25), we need to compute (H,H~(xr,y)). Since II,E ~ - 0, equation (26) shows that (after time integration)

II,,H~(x~,v)- II..Hl(x~),

(x~,v) e Fx]0, 1[.

(33)

So we have from (25) written for k = 2, 1

cOtE2(xr, L,) x n -

(1 - ~,) (~f #~Ott + rgt~ rot~ ) II,,H~(x~).

(34)

~f

We deduce from (24), (31) and (34) that --3

?7

OtEv(xr) x n - - - -

Cf

--3

(r #fOrt + rotr rotr ) H,Hv(xr ) + O(~3),

which gives, when omitting O(~3), the same expression as (32). The fundamental point is that the coupling between this condition and Maxwell's equation leads to a stable problem. T h e o r e m 0.1 - Sufficiently regular solutions (EW,HW) of {(6), (32)} satisfy the a priori energy estimate:

s { dt

(t) fi , (t)) + ,

(t) } - 0,

(35)

N

where, if we set, ~o" (Xr, t) - H, H~ (Xr, t), (Xr, t) E F x R +, #f

2

1

E~'(t) = ~ II~'(t)llL. + ~-~

fo

Li2

rOtr ~v(.i-) dT L2

Identity (10) is then well satisfied. Proof. In the classical energy identity for Maxwell's equation,

d---tg~ (t) -

(E' x n ) . ~v dxr

(36)

424

Stabifity of thin layer approximation of electromagnetic waves...

we explicit the right hand side using (32). We integrate in time this condition, take the scalar product with ~o~ and integrate over F. We use the duality between the operators rotr and rotr to deduce that: (En x n). ~on dxr - -r/~-~Ern(t).

(37)

Identity (35)is yielded by (36) and (37).

m

2.2.3. Effective b o u n d a r y c o n d i t i o n of o r d e r 4 We need to compute (E3(xr, 0) • n) using equation (25), written for k - 3, which requires (H..H2(xr, u)). Equation (26), implies for k - 2 0tH2(xr, u) x n - 0 t H 2 ( x r ) x n - -

1/o?

ef ][/f 0tt + rotr rotr)H..Efl (Xr, ~) d~.

Pf

(38) From (31) we have, 0tH,Ef~(xr, u ) _ 1 (1 - u)(ef#f0u - Vr divr ) (H~

• n).

et

(39)

We apply the operator (of #f0u + rotr rotr ) to (39) (using rotr Vr - 0) and integrate in time. Hence (ef #f 0tt + rotr rotr ) II,,Ef1(xr, u)

(40)

= (1 - u)#f 0t (el #f 0u - Vr divr + rotr rotr ) ( n ~ (xr) • n). Combining this identity with (38) yields Hf2(Xr,U) •

H~(xr) •

( e . # . O t t - ~ ' r ) (H~

•

(41)

where Ar -- Vr divr - r o t r rotr, is the Laplace-Beltrami operator. From (41) and (25) we finally deduce Ot Ef3 (Xr, u) • n -

(

n,,H~(x~)-

1

(1 - u) (el #f 0u + rotr rotr ) sf ~1 (1 + u - v ) 8f ~fd~tt - /~r II,,H~

)

)9

(42)

Combining (42) with (24), (31) and (34), we obtain the formal identity: --4

OtEv(xr ) x n =

r] (ef#fcgt t + rotr rotr ) ( 1 -

ef

m4

~ ( e l # f O r t - /~r ) ) I I , , H v ( x r ) + O(r] 4) 3

(43)

H. Haddar and P. Joly

425

which gives the following effective boundary condition of order 4 on F x R +

OtEnv x n - -~-- (e~ #fOrt + rotr rotr ) (1 - ~ (el # f O r t - A r ) ) II,,~I~v ~.f

3

"

(44) A n i n s t a b i l i t y r e s u l t . Contrary to the case of the second order condition, The condition (44) coupled with Maxwell's equations does not lead to a stable problem. In particular, one cannot obtain a priori estimate like in

(10).

Let us study for that the case of the 2D problem. We denote by (x, y, z) the space coordinates system, and assume that all the fields do not depend on the z variable. For sake of simplicity, we make e%= ef - 1 and #0 - #f 1 and assume that f~v - y < 0. If we consider (E~, H~) satisfying Maxwell's system (6) coupled with the boundary conditions (44), the we observe that the z component of Ev~ (resp. of H~) is a solution of the initial boundary value problem 7)~ (resp. 7~nn).

O t t u - (Oxx + Oyy)u - O, V~

u + 7 7 ( 1 - ~ - 3( O t t - O x x ) ) O n u - O , ('U,, Or?2) -- (UO, ~t 1),

t>O

y--O,

t > O, y < O,

OnU + rl(Ott - 0 ~ ) ( 1 - ~-(Ott - O~))u - 0 3 o

u) -

(45)

t -- O, y < O.

O t t u - (O~ + Oyy)u - O, V~n

t > O, y < O,

t -

t > O, y - 0[46)

o, y < o.

These problems correspond to the classical decomposition of the electromagnetic waves into two independent polarizations: T.E. (system 7)nn) and T.M. (system 7)~). T h e o r e m 0.2 - For fixed rl, the problems 7)~ and T)n~ are well posed in the sense of Kreiss [5]. However both of them are strongly unstable in the following sense: there exists (u~(y), u~(y)) a sequence of initial data and un(y, t), the corresponding solution, such that:

where C E R + and is independent of 7, limn~0 JJunJJL2(]_oo,O[x]O,T[) = +oo, VT > 0.

Proof. We seek a solution of P~ or P~ of the form: y, t) -

),

426

Stability of thin layer approximation of electromagnetic waves...

where, k E It, s = a + i w , (a,w) E R 2, and fi E L 2 ( - c ~ , 0 ) . equation implies that our particular solution is of the form ~(y)

Ae

,

The wave

AER,

with Re((s 2 + k2) 89 > 0. Let us set a - ~(s 2 + k 2) 89 One checks that the effective boundary condition of either Pd or ?n leads to the characteristic equation: l + a ( 1 - ~ 1 ~2 ) - 0 . (47) This equation admits one real solution n0 > 0. The two other ones have negative real part, so they are not admissible. Hence acceptable waves t~ 2 satisfy: s 2 + k 2 - ~ , which is equivalent to (w-0anda2+k2-a2)

( ~ - or~

a-0andk

2 - w 2-a~

.

(48)

Let ~ be fixed. Relation (48) shows that plane wave solutions of P~ or Pn~ are such that Re(s) < ~o. 1/ This means that the boundary value problems ?~ and P~ are well posed in the sense of greiss (see [5]). -

-

However, one notices that when 7/ goes to 0, the higher bound of the

Re(s) goes to +oc. This means the existence of plane wave solutions that blow up for t > 0, when rl --+ 0. For instance, let us consider the worst case, corresponding to k - 0 (which is nothing but the 1D approximation) and a - a~ = ~o. v We fix m E IN, and set f ' ( y ) = e~'~ y ' y < 0. The sequence of initial data

:"(~) /"(U)

-

IIs'(y)ll...

, u ~ ( y ) - II:'ll,m

satisfy {IJu0iiHm + IiuoiiH m } < 1 + ~ < c , when 77 --+ 0. They correspond to the plane wave solutions: -

(~--~o)m+l

IIf~JIL2(O'T)

Therefore

=

.f,

whereas

~ 12-~o exp( no~7T

IIu~?IIL2(]_cx),O[x]O,TD"+ "1"-00 when rI --+ 0.

permits to

(t)

Ilu~l[L~(]_~,0[)IIf'llL~(o,T).

We have: IJu'TlIi2(]_~,o[• When rl-+ 0 : I I u ~ l l L 2

s"i i s(y+t) .ll.m

I

H. Haddar and P. Joly

427

S t a b l e c o n d i t i o n of o r d e r 4 We are going to built, from the unstable fourth order condition (44), another condition that gives the same order of approximation, but leads to stable problems. This is reminiscent of the techniques used for absorbing boundary conditions [3]. Let us consider first the simple case of the 1D approximation: i.e. we omit the tangential derivatives in the effective boundary conditions. We take the problem (45) as example. The boundary condition becomes when we apply Fourier transform with respect to t: 1 2 032 )On~t - + r/(1 + jr/

(49)

0

where w is the dual variable of t. Comparing this condition with the stable condition of order 2: fi + rl0n~ = 0, we observe that the instability comes from the substitution of (1 + ~1 ?]2 032 ) for 1. A natural idea is to replace (1 + 1 ?72032 j~ by another function g(r/w) such that" g(~/03) - 1 + y1 T]2032 + O((~w) 3 ). As (49) is obtained in a formal way by neglecting the terms of order greater than 4 (with respect to 7), we keep unchanged the order of approximation. We chose g(r/03) - 1/(1 - ~1 T]2032 ). Hence, instead of using (49), we suggest to use the condition (1 _ 1 2~n 2 ^ w )u + n O ~ - O,

that corresponds to (1 + ln20~t)u + nO~u - o.

This new condition leads to stable initial boundary value problems. This result will be shown later in the general case via energy estimates, but we can already see that the previous construction of unstable plane wave solutions fails: the characteristic equation (47) becomes I + ~ 1 ~2 + ~ - - 0 ,

~ -- v/s,

and has no roots with positive real part (which can easily checked). For the 3D case, we would like to apply the same ideas to (44). Unfortunately, working directly on the expression of (44) do not lead to the appropriate condition with regards to energy estimates. That is why we begin by writing differently the differential operator in the right hand side of (44). As, rotr Vr = 0, we have (el #f 0tt + rgtr rotr ) (1 - ~ (~f #f 0u - / ~ r ) ) -

e~#~Ott ( 1 - ~ (efptfOtt-Nr + r o t r r o t r ) ) +ro-*trrotr ( 1 -

~ro-*trrotr)

(50)

428

Stabifity of thin layer approximation of electromagnetic waves...

Let P be one of the two differential operators (El pf O f t - / ~ r + rotr rotr ) and (rotr rotr ). Like previously, we apply to (50) the formal approximation

(1-~P)- (I+~P)-1 -~- O(~4), and obtain the condition (51), where we need to introduce two new variables r and r on F, and where ~ ( x r , t ) - I I , H ' ( x r , t ) , (xr,t) E F x R +.

OtE~v x n - - rl (~ #fOtt~2 ~ 4- rotr rotr Cv) ~f

onF•

( 1 + ~3r o t r roUt) Cn - ~ n (1+

~2 - (/ ef~#f r(~t 3

+ r~

r~

+.

(51)

Cn - ~ n,

To the third equation of (51) we associate the initial conditions r

= o) = o,

= o) = 0

(52)

Such doing, we have constructed a boundary condition, that when coupled with Maxwell's equations, leads to a stable problem: T h e o r e m 0.3 - Sufficiently regular solutions (E~, Hv~) of { (6), (51), (52)} satisfy the a priori energy estimate:

d d--t { s (E~v(t), I--I~(t)) + rl S~r (t) } - O, where, S~r (t) - ~2 IIr r~32

and

(~2 gl(r

111 f0 (rotr Cge) (T) d7 iiL22 + + ~1 g2(r )~

2+ ~

(53)

81 (Ca (t)) -- r #f IlOtr ~ (t)ll L2 2 + Ildivr Cn (t)IlL2 2 + 2 IIrotr Cn (t) II2L2~ t 2 g2(r -- ~o (rotr rotr Ca) (T)dT . L2

Proof. We first write the L2(F) scalar product of the first equation of (51) by ~ fr ( E ' x n ) . ~ dxr = - ~ (#f fr ~n . OtCndxr + ~1 fr ~" . fot-rOtr rOtr r

dT dxr) .

(54)

H. Haddar and P. Joly

429

When we take L2(F) scalar product of the second equation of (51) by /0 t (rotr rotr r

dT

and

by part, we get

fr ~P~ " ( fo r~

r~ r ~ dT) dxr

=

Cn dT

2 dt

fO rotr

ir

L 2 + ~3

ir

fot rotr -~ rotr

Cn

ff')

(55)

dT- L2

and when we take L2(F) scalar product of the third equation of (51) by Ore ~ and integrate by part, we get

/ ~''Otr

1d dxr - ~ dt

( II

r

,

2 (t)llL2 + -~s162

)

.

(56)

The energy identity of theorem 0.3 is obtained by using (56) and (55) in (54) and substituting (54)in (36). m 2.3. E r r o r estimates Let k 6 {1, 2, 3, 4}. We recall that the effective condition of order k on the boundary F is the Dirichlet boundary condition when k - 1, the condition (32) when k - 2,3 and the condition (51)-(52) when k - 4. We shall determine, using the stability results (theorems 0.1 and 0.3), the order of approximation between the exact solution (E~, H~) and the approximate one (E~v,H~). Our main result is summarized by theorem 0.4 below. N

In fact, rather than working directly on the difference (E~ - Ev~, H~ - H~), we shall insert the asymptotic expansion (19). So we consider (E k, H k, E k, nk)k>0 the sequence (of sufficiently regular fields) satisfying (20), (24), (25), (26) and (29). Remark 2.3. At least, when (Eo, Ho) is regular (let say in ~P(~tv)) the existence of the expansion (E k, H k, E k, Hk)k>0 can be easily shown by a recurrence on k, as explained in the construction of the equations (20), (24), (25), (26) and (29). Moreover, one can check that (E k, H k) are polynomial functions with respect to v, of degree lesser than k. T h e o r e m 0.4 - Let k 6 {0, 1,2,3}. If (Ev~,H~) is a sufficiently regular solution of (6) coupled with the effective condition of order (k + 1) and (E~, Hv~) is a sufficiently regular solution of the exact transmission problem, then, there exists for all 0 < T < +co a constant Ck(T) independent of ~, but depending on {(E~176 ~ ~ 9 9 9 (E k , H k,E k , n k)} and T, such that: sup O
Stabifity of thin layer approximation of electromagnetic w a v e s . . .

430

Proof. The proof of this theorem uses the results of the subsections below: L e m m a 0.2 gives estimates on the difference between the exact solution and the t r u n c a t e d series, meanwhile lemmas 0.3, 0.4 and 0.5 give similar estimates but related to the approximate solutions. Admitting (for a while) these results, the present theorem is deduced by 9 k = 0 : apply l e m m a 0.2 to k = 0. 9 k = 1, 2, 3: combine the result o f l e m m a 0.2 with the result o f l e m m a 2.k using triangular inequality, m

Remark 2.4. T h e o r e m 0.4 gives very little information about the constant Ck(T). One can show for instance t h a t if Eo and Ho are in H3k+2(~tv) and satisfies div (Eo) = div (Ho) = 0, then we can chose

Ck(T)

=

C(T){ItEolIH~+= +

[IH011H~+~},

where C(T) is a polynomial function of T. This result (even if it may not be optimal) requires some sharp estimates that are very technical and do not add a substantial interest to the result. 2.3.1. C o n s i s t e n c y

of the asymptotic

expansion

(19)

L e m m a 0.2 - Let 0 < T < +oc. There exists a constant Ck(T) independent ofrl such that, V 0 < t < T, 2

-Zi=o

_~_~ IlUv~(t)_

E . ( t ) L:

k

T]i

i

~_Ck(T)?~2k+l.

2

The constant Ck(T) depends only on T and ( n k, Hk), and can be expressed by (61).

Proof. Let us set: k

k

e~ - E~ - ) - ~ i = o

~]i

i

Ev,

e k - E7 - ~ i =ko ~7i E~, i

k o r/i Hv, i hk -- H~v - ~ i = hk

-

H~f

-

k

~-,i=o

~i

i

Hr.

k hv, k e k and hfk depends on 77 (which is not indicated Of course each of ev, explicitly to simplify the notations). Using (1) and (20), one observes t h a t Co Otek - r o t h k - 0, (e k, h k)(t - 0) - (0, 0).

#o Ot h k + rot e vk _ 0 , on ftv ,

(57)

H. Haddar and P. Joly

431

while using (17), (21), (22) and (23) leads to ?](cf0tefk -- rotr (h k" n) - (rotr hk)n) + 0~(h k x n) - @+lFk,

~(#f0th k + ro-*tr (e k" n) + (rotr ek)n) --0v(e k x n) -- @+1Gk~58) (e~, h~)(t

-

0)

-

(0, 0),

where we have set F k - e, 0rE k - rotr (H k- n) - (rotr H k ) n , G k - #, 0tH k + rotr (E k" n) + (rotr E k ) n . The boundary conditions in (16), (18), (20), (21), (22) and (23) show that f o r x r 6 I" ek(xr) X n - - e k(xr, 0) x n, hk(xr) X n = h k(xr, 0) x n, ek(xr, 1) X n -

0,

(59)

x~ e r .

We remark that using (29), one has F k - ef 0 t H , E k - rotr ( n k" n)

and

G k - #f 0 t H , H k + rotr (E k" n).

Let us choose CI(T) such that sup

sup

O
(~[[Fk(u,t)l[

L2(F) + ~

I[Gk (1],t)]IL2(F)) -- C1 ( r ) ,

and consider the classical energy identity of the system {(57), (58), (59)} k k g(t) _ so2 Ilev(t)ll2L 2 + ~ Ilhv(t)ll2L 2 + rl {e_x 2 Ilek(t)ll 2L 2 + ~ Ilhk(t)llL2 2 }

Applying Cauchy-Schwarz inequality to the right hand side of the previous identity yields

~(t) < Ilk+ 89C1 (T)

/o

X/g(T) dv.

(60)

C1 (T) t) 2 which proves By Gronwall lemma we conclude that S (t) < (~1 @+ 89 the lemma with (61) Ck(T) - (~1 C1 (T) T) e m

Stability of thin layer approximation of electromagnetic waves...

432

2.3.2. C o n s i s t e n c y

of the second and third order conditions

0.3 - Let (Ev~, nv~ ) be a suj~ciently regular solution of (6) coupled with the effective condition of order 2. Then, there exists for all 0 < T < +cx~, a constant C(T) independent of ?] such that, V 0 < t < T,

Lemma

2 IlEa(t) ~~

'7~ Ev(t) ~ ]1~L2 ~-

-- E i =10

9

IIH~(t)

1 o r]i H ,i( t ) -~7~i=

II

< C(T)?}3 L2 --

The constant C(T) depends only on T and H~ and can be chosen like in (65). Proof. We introduce the fields (~v, h i ) (that depend on ?])

-e 1v _ ~ ,

-

(EO+~E 1) , ~1

--

fi,

_

(HO+~H 1) ,

and set" ~ i ( x r , t ) -- II,,h~(x~,t), ~ ( x ~ , t ) - I I . , H l ( x ~ , t ) (x~,t) e r • [0, ~ [ . Combining (32) with (24) and (31), shows that ( ~ , h ~ ) satisfies

r Ot~lv

/to Oth I + rot -1 e v - 0, on ~tv,

rot h I - 0,

(~1, h~)(t - 0) - (0, 0), on ~ ,

Ot'~ • n - - ~

Cf

(62)

(el/tfOtt + rotr rotr ) (~1 _ ?]~al), on F.

Following the same steps as for the proof of identity (35) of theorem 0.1, leads to

E(t) ~o2 Ii~l(t)II ~~ + ~

--1 h~(t) ~-

-~

~: 2

(t)

L~_[_1~

t for~

1 ( f : rOtr ~O1 dO) 9rotr ~1 /tf 0tr 1 " ~1 _[_ ~ff

__ ?]2

~1

d~

2

dxr dT.

Integrating by p a r t in time the second term of the right hand side yields

.~_ ?]2

].Lf0tr 1 9~1 _ 1 . rotr ~O1 gf

rotr ~1 dO

dxr dw. (63)

Let us chose now

cl(T) sup t~T C2(T)-

sup ( ~

t
IlI:rotr l d ll F ) []rOtr ~ o I ( t ) I ] L 2 ( F ) q - ~

[[Cgt~oI(t)]IL2(F))

H. Haddar and P. Joly

433

Applying Cauchy-Schwarz inequality to (63) shows that s

<_ r/~ C1 (T) x / ~

V@(T) dT.

+ r/~ C2(T)

Consider now, for t _ 0, the function, X(t) - ] v / ~ obtain from (64), that

X(t) < r/3C3 (T) + r/~C2(T)

(64)

1 ~ CI(T)[ 2, we

X / ~ ( 7 ) ) dT,

where C3(T) - ~C~(T) 1 2 + T C~ (T)C2(T). Applying Gronwall's lemma to the previous inequality yields

X(t) < { ~ (vYC3(T) + 2C2(T)t)} 2, which proves the lemma with

1 (T C2(T) + C1 (T))} 2 C(T) - { x/C3 (T) + -~

(65) i

L e m m a 0.4 - Let (Evn, nvn ) be a sufficiently regular solution of (6) coupled with the effective condition of order 3. Then, there exists for all 0 < T < +cx~, a constant C(T) independent of rI such that, V 0 <_ t <_ T, 2

- z, =o

2

+

li , -<

The constant C(T) depends only on T and H 2. Proof. Like in the previous proof, we introduce -ee~ = Evn - (E ~ + r/E 1 + r/e El), and set:

h~ - :H~ - (H ~ + r/H 1 + r/2 H~).

t~2 (xr, t) - 1-It.h~ (xr, t),

~o2 (xr, t) - HII H~ (Xr, t)

(Xr, t) e

r x [0, , v

Using the relation (34) we see that (~2, h~) satisfies the following equations, that are similar to (62). s

0t ''2 e~ -- rot h v2 - O,

(~2,h2)(t-0)-

Po Othv2 + rot --2 e~ _ 0, on fly,

(0,0), on fl~,

Ot'~2v x n - - ~ (6~ pfOtt + rotr rotr ) (~2 _ r/2~02), on F, ~f The proof follows exactly the same steps as the proof of the previous lemma. One has only to substitute T]3 (resp. ~2, ~O2) for r/2 (resp. ~1, ~1). i

Stability of thin layer approximation of electromagnetic waves...

434

2.3.3. C o n s i s t e n c y of t h e f o u r t h o r d e r c o n d i t i o n , v

~..

L e m m a 0.5 - Let (Ev~, nv~ ) be a sufficiently regular solution of (6) coupled with the effective condition o/order ~. Then, there exists for all 0 < T < 4-00, a constant C(T) independent of ~1 such that, V 0 < t < T, 2

<_ C(T) ~17. The constant C(T) depends only on T and (Hlv, H 2 , H 3) and can be chosen like in (72). Proof. Like previously, we denote

---3ev_ ~v~ _ (EOv 4- 77E 1 4- ?72 E2v 4- ?73 E3), ~3 _ ~v~ _ (H 0 + 77H~ + r/2 H 2 + 7/3 H 3). Let us denote also by ~k the trace of HI.H k on F, for k - 0, . . . , 3 . order to write the equations satisfied by (~3, h 3) we need to introduce , r

r

(l)0 __ ~0 and to set,

_qal , r

(~1 __ ~1

In

=qa2 1 (~ ef#f0tt _ /~r 4 - r o t r r o t r ) qo~,

(1)2 __ (p2

l rotr rotr qo~

r- 3 _ r 1 6 2 1 6 2 1 6 2

r-3 _ r _ ( r 1 6 2

~2r ).

One notices that r (resp. r i - 0, 1, 2, are independent of 7/and are nothing but the first three terms of the asymptotic expansion of r (resp.

N3

---3

r introduced by the condition (51). However, r and r We verify, using (42) and the first equation of (51), that Ot.~3

X n

-

- - -T] (Cf~ft~tt~)3 - +rotrrotrr

~f

3)

depend on ~.

1~q- .

onFx

(66)

Let ~3 be the trace of II..h 3 on F. We deduce, from the second equation of

~-3

(51) and the definition of r

( 1 4 - 3 rotr rotr)

--

4-

G~

onFx

, -~3

and from the third equation of (51) and the definition of r

(67)

H. Haddar and P. Joly

435

In (67)"

G~ - lrotr rotr (~O1 -4- ?7~2) _ ~3,

and in (68)"

L~-g

1( e f # f 0 t t - A r-

)

+r&rrotr

(~o~+rl~o 2) -~o 3.

In conclusion, (g~, h~)satisfies

COOre 3 - rot h~ "3 = 0, #o Oth~ "3,+ rot -3 e~ _ 0, on f~v x R + "3 "3 (%, h.)(t = 0) = (0, 0), on fl~

(69)

-3 e v x n and ~3 verify (66), (67) and (68) on F x R +. The derivation of an energy estimate for (69) follows the same steps as theorem 0.3. If we set

E(t) _~o

- 2

-~ hv(t) L2 + 7

L2+~ +36~

~2

(t)

L2

2~f

ftr~

ef#f Ilcgtr 8(t)]lL 2 + Ildivr r (t)l[L 2 + 2 llrotr + -6ef ~ for&r rotr

then we get E(t)

-

?74

L (t)][L 2

dr L~-

~0t ~r {roUt G~(v)'fo rotr r-..3 dtc+L~(7")'Otr ---3(T) } dxr d~

Integrating by parts, with respect to T, we obtain ~(t)

i - - ?74 L L ~ ( t ) "

~3 (t) dxr +

(70)

,IF

r]4 fo fr {rotr G~ (T). f0 rOtr

dtr OtL~(T)-

(T) } dxr

dT.

If we set

1( )-sup sup

~_1 t~_T

C 2 ( T ) - sup sup (x/~f ~_1 t~_T

I]rotrG~(t)]lL2(r)+ ~

]]OtL~(t)[]L2(r)),

then, applying Cauchy-Schwarz inequality to the right member of (70) shows that, Vt > 0,

~(t) _( ?~}C1 ( T ) ~

-~- ?~}C2(T)

~0t X/E(T) dT.

(71)

Stability of thin layer approximation of electromagnetic waves...

436

We notice that this identity is similar to (64). The same technique based on Gronwall's lemma enables us to prove the present lemma with

C(T) - { x/C3 (T) + I(TC2(T)+CI(T))}e;

(72)

C3(T) - zC1 1 (T) 2 + T C1 (T)Ce (T).

m 2.4. Effective construction: regular boundary The principle of deriving effective boundary conditions in the case of curved boundary is the same as in the case of straight boundary. However, it is technically much more complicated because one needs to take into account the geometrical properties of the boundary. e3

J

F~7 -- e 2

el

Figure 3" Local description of the thin layer We assume that F is a regular surface (at least of class C2), and we denote also by n the unitary normal vector field to F outwardly directed to f~v.

2.4.1. Some geometrical properties of regular surfaces let us first introduce some tangential differential operators on on F, that we shall use to express the differential operator rot in the parametric coordinates. After giving some recalls of differential geometry (the definitions below are close to that used in [14], see also [15]), we shall prove some specific results in connection with our formal construction of the effective boundary conditions (we have tried to make this section self-contained). For r/small enough (see Remark 2.5. below), the parametric representation (11) of f~f~ remains valid. The only difference is that n depends now o n Xr"

x e f~f~ ~

(Xr, s) e F x [0, 7/], such that" x -

Xr + s n(xr),

(73)

437

H. Haddar and P. July

where Xr is defined by ]Xr - x[ - min lY - x]. For each Xr E F we denote yEF

by Txr the tangent plane to F on xr, which is the plane orthogonal to n(Xr). The projection operators H• and II.i are still defined by (14) but they depend Xr. Let ~ = (~1, ~2) ~ Xr E F be a parameterization of a neighborhood of Xr; for ~ E O C IR2. We introduce the covariant base (r~)~=1,2 of Txr: ~'~ = 0 ~ Xr. The contravariant (or dual) base (T~)~=l,2 is defined by: ~-~ 9~-~ - ~ for ~ - 1, 2, where 5~ is the Kronecker symbol, and T ~- n = 0. We shall denote by (ei)i=1,2,3 (--(ei)i=1,2,3, by convention) the canonical orthonormal basis of R 3. We use in the sequel the convention of summation for repeated subscripts, where Greek subscripts vary between 1 and 2 and Roman subscripts vary between 1 and 3. The curvature tensor C: We introduce fi the vector field defined on Ftf~ by: fi(x) = n(xr) where x E ~f~ and Xr are linked by (73). We define the curvature tensors Cs, s E [0, r/I, and C by

c~ (x~) - (~ ~)(x;, ~),

C = Co.

(74)

For every Xr C F, the operator C(xr) is symmetric, and has 0 as an eigenvalue associated to n and two other eigenvalues (cl,c2), called principal curvatures, that are associated to tangential eigenvectors (7"1, ~'2). This uperator satisfies also the following useful property, where H - ~1 tr C, is the mean curvature, c . (v • n) + (C. v) x n = 2 H

v •

n,

V v ~ R 3,

(75)

that can be checked out, by using the orthonormal basis (7"1, ~'2, n) of R 3, where ec~ is a unitary eigenvector of C associated to ci (i = 1, 2). Moreover, using the parameterization (~1,~2) and differentiating n with respect to (~1, ~2) we get: O~n =era. Remark 2.5. Assuming F regular enough implies in particular the existence of a lower bound, c > 0 of Xr ~ min(Icl (Xr)l, Ic2(Xr)l), on F. The mapping (73) is then an isomorphism for r / < c -1. The covariant derivative operator Vr : Let v be a scalar function defined on F. We define ~ on ~ by: ~(x) = V(Xr), x and Xr related by (73). The operator Vr is defined by

(vr ,)(x~) = (w)(x~, 0),

Xr E F

(76)

Stability of thin layer approximation of electromagnetic waves...

438

It is a tangential operator that can be expressed in the contravariant basis of Txr, by V~ v - (0~o v)~. (77) To prove (77), we use (73) and differentiate ~ with respect to (~1, ~2, s). We get

(0~o ~)(x~, ~) - (W)(x~, s). ( ~ + ~ c ~ ) , (0~)(x~, ~) - (W)(x~,

~). ~.

Making s - 0 and using (asO) - 0 (see the definition of ~) show that (Vr v).z'~ - Or and (Vr v ) . n - 0. We deduce (77) since (T 1, T2,II) is the dual base of ( r l , r2, n).

Consider now a tangential vector field v defined on F and a field of symmetric matrices 74 defined on F such that T~n - 0. We define (divr v), ((T~Vr). v) and (divr T~) by the following expressions, where, denotes the scalar product in R 3 ((79) and (80) are original definitions), divr v - (Vr ( v . ei)) .ei - (Or v ) . r " ,

(78)

(nVr).v

(79)

- ( ( n V r ) ( v . ei)) .ei - ( 0 f . v ) - ( 7 ~ T"),

divr 7~ - (Vr 9( n e ' ) ) 9ei - (0~. 7~)T".

(80)

Remark that the definition of divr v coincides with the definition of Vr 9v (n

-

n,).

As in the case of straight boundary, the operators rotr and ro-*tr are defined, from divr and V r , by relation (13), that we recall here rotr v -

divr (v x n),

ro~tr u -

(Vr u) x n,

where v (resp. u) is a vectorial (resp. scalar) field defined on F. We still have (we omit the proof)" rotr Vr - divr rotr - 0. Also divr (resp. rotr ) is the adjoint o f - V r (resp. rotr ) for the L2(F) scalar product. We prove now the basic result used in the formal derivation of the effective boundary conditions (it differs from the classical representation of the curl operator (see for example [14]) by making explicit the dependence on s). L e m m a 0.6 - Let v be a vector field defined on f~7. Let (Xr, s) E Fx]0, r/[ be the parametric representation o/ ~ . Then we have rotv-Tr sv-

Os(vxn),

~h~r~, %'v - [(n, v~ ). (v • n)] n + [n~ V~ (v. n)] • n and Tie defined by:

7~s (l-I, + s C) - Hii, 7~s n -- 0.

( n , C v) • n,

439

H. H a d d a r and P. J o l y

Proof. For a scalar function u defined on f~fn, one gets (using (73)),

0~o u - ( V u ) . (n,, + ~ c ) r . ,

G u - (Vu). n. As TEe(H, + s C ) - H,, T E s n - (H, + s C ) n - 0, TEs and (H, + s C ) are symmetric, we deduce that (TCs~'1,7~sr2,n) is the dual basis of ((H, + s C)vl, (H, + s C)v2, n). Hence, (Sl)

V u - (O{~ u) T G r " + ( O s u ) n .

From rot v - e i x V ( v . ei), we obtain using (81), rot v -

(nsV a) x ( O ~ v ) - (OsV) x n.

Let us set B~ v - (TCs~-~) x ( 0 r (B~ ~) x ~ - - ((o~o

~). ~)

n~

(82)

and prove that Brs - Trs. ~

=

-(o~o

=

-(o~o

(~. n) - c ~ . (~. ~) n ~ ~

~)n~

~

+ (c~.

~) n ~ " .

As the tangential and symmetric tensors TCs and d commute, the previous equality yields, n., (B~ ~) - ~ x ((B~ ~) x ~) - I n , v ~ ( v . ~)] x ~ - ( n ~ c v).

(83)

On the other hand, (Bs v ) . n - ((0f~v) x n).7~sT a - (Of~ (v x n)).TEsTa + v-(TEST a x Cra). By applying the lemma 0.7 below, to .4 - d and B - 7~s, we deduce that the last term in the previous equality is 0. Consequently, this equality becomes, using (79), (B~ v ) . n - (ns Vr )" (v x n).

(84)

Identities (83) and (84) show that B s - "Frs. Hence, lemma 0.6 is proved by (82).

l

L e m m a 0.7 - Let A and 13 be two s y m m e t r i c and real matrixes in s such that A n -

13 n -

0 and having the s a m e e i g e n w c t o r s .

3, R 3)

We have:

~4~'~ x 13r ~ - O.

Let (r162 be the eigenvectors of ,4 and B and (a~,a2) (resp. (b~, b2)) the corresponding eigenvalues of ,4 (resp. /3). Let us set: ~-~ = Ttr r -[- Ttr r and r ~ - T~'1 r + T~'2 r Then Proof.

Ar~ x Br ~

+ a2 T~,2r

X (bl T ~ ' 1 r

-

(al T~,I r

=

(al b2 T~,IT ~'2 -- a2 bl T~,2T ~'1) r Xr

~'2r

Stabifity of thin layer approximation of electromagnetic waves...

440

By definition of the dual base ( r ~, r 2 ) , we have, setting 5 = 1/(T~,~ 72,2 -7"1,2 T2,1 ), T 1,1 _. (~7.2,2 '

T 1,2 __ __(~T2,1 '

T2,1 __ _(~7.1,2,

T 2 ' 2 --" (~7"1,1.

T h e n , one easily checks t h a t Tn,1 T n'2 - - O a n d Tn,2 T n ' l = O. We prove finally the following geometrical identity t h a t enables us later to write in s y m m e t r i c way the effective b o u n d a r y condition of order three. Lemma

0.8 -

We have divr ( 2 H H., - d) = - 2 G n,

where G = det C, is the Gaussian curvature. Proof. Following the definition (80), we have, divr ( 2 H II..-C) = ( 0 ~ ( 2 H H i , -

C))r ~. So divr (2HH.. - d ) . n -

[(0e~(2HH,. - d ) ) v s ]

9n - [ ( 0 e ~ ( 2 H I I H - d ) ) n ] . l

= [(C - 2 H H,.)0~ n ]. v s - [ (C - 2 H H..) d vs ]- v s - t r ( d 2) - t r ( d ) 2 = -~ To conclude, we are going to show t h a t divr ( 2 H H,, - C ) . v ~ E {1, 2} (which means: I I , d i v r ( 2 H II, - d) - 0).

- 0, where

divr ( 2 H II,, - C ) - r Z - [ ( 0 ~ ( 2 H II,, - d ) ) ~ . s ]. ~.Z _ [ ( 0 ~ ( 2 H II,, - C))~-Z] = 0 0 ( e l l ) - [(0e~ C ) r ~ ] . 7-s We have (using O 0 0 ~ n [(o~o c ) ~ ] .

~-

0~.Oon )

-

[o~o (c ~ ) ] .

~

-

[coco r,]. ~-~

=

[ o o (c ~ ) ] .

~

-

[coco r,]. ~-,

=

o~, (c

whence divr ( 2 H I I , - d ) . r Z Let us set: a n _ ( c ~ ) .

~. ~)

- (c ~ o ~

-0OTs

+ c ~-O~o

~,),

- - (d v s 0 O r s + d 7-s0~ ~'Z) - - B . 7"n. Since C 7-s - anS 7.n and C r s _ a s 7-n,

B - a s Tn 0 0 r s + a ns r ~ 0 ~ vZ - - a s~ o e ~ + (=

~

a~" ~ o e o

v~

+ 0 ~ TZ) a~ 7"n

B u t we have 0 0 7"s - 0 ~ ( 0 ~ x r ) divr ( 2 H H,. - C) .7-~ - 0.

-

0 ~ 7"Z. Consequently, B -

(Sh) 0 and m

H. H a d d a r and P. J o l y

441

2.4.2. Scaling and asymptotic expansion Scaling. As in the case of the straight boundary, the scaling corresponds to the change of variables (12). According to lemma 0.6, the differential operator rot becomes in the new coordinates systems (Xr, u) E F x]0, 1[: 7~rv + s X 0~. Notice that the operator 7~r~ is not singular with respect to rl. Equation (17) becomes 1 0~(Hfn x n ) er Ot Enf - 7~r ~ Hnf + -~

0,

#~OtHn~ + 7~r~'En~ - 1 0v(E~ x n ) -

0.

(86)

Our problem is constituted now by equations (1), (16), (86) and (18).

Asymptotic expansion. We use the ansatz (19). Moreover, we need an analogous expansion of the operator 7~rv. We use, for instance, a Taylor expansion of 7~nv with respect to the parameter (~u g)" ']~-,v -- nil -t- ~i~176(--,/] e) i, so we have T~rv -- ~ 0 ( - r w ) i ~v

Tr/, where,

(87)

- [(e ~ v , ) . (v x ~)] ~ + [e ~ v ~ ( v . ~)] x ~ - (e ~+~ ~) x n.

The formal identification lets unchanged equations (20) and (21). However, equations (22) and (23) change to the following ones, where k > 0, 69u(E k+l x n ) -

, f 0 t n k + ~ i =k 0 ( - l ] ) i .-[-.ii~k-i ,r--'f ,

Ek+l(Xr,1) x n - - 0 ,

{

(88)

for Xr e F.

0v(H k+l x n) - - e , cgtEk + E i = 0 ( - v)' '-Fr'H~k-i, H k + l ( x r , 0 ) x n - - n k + l ( x r ) X n,

(89)

for Xr e F.

The zero order terms are determined by (24). To determine the higher order terms, we use the lemma 0.9. We introduce the notations, where i is an integer, rotr('~ u - (Ci Vr u) x n,

rotr('~v - (C i Vr )" (V X n).

442

Stability of thin layer approximation of electromagnetic w a v e s . . .

L e m m a 0.9 - For all k > O, we have for all (Xr, u) E Fx]0, 1[, 0t

E k+l

(xr, u) x n -

-fl

#, Ouli,Hk(xr, ~) d~

Y~i=o Y ~ j = o ( -~)

-

H j (Xr, ~) d~.

rgtr(k-') r~

Y~i:o(-~) k-i (C k-i+1 0tE~(xr, ~)) x n d~.

(90) 0tHk+l

(

Xr u) x n ,

-

v 1

fo

- fo

--

k

(9-t H v k+ 1

Z,=o

(Xr) x n -

f0 ef0ttH"Ek(xr,~)d~ v

i

)-~j=o (_ ~ ) k - j rgtr(~-,)rOtr(,-j)E j (xr ~) d~ (C k-i+l 0tH~(xr, ~)) x n d~

k

(91) Proof. We are going to prove (90) only. One deduces (91) by analogous arguments. By applying H.. to the first equation of (88) we obtain

0v(E k+l x n) -

a,0t(rI,,H~)

ki=o( _u)k_i (C

k + Y~i=o(--u)k-irot(r k-') (E~. n)

i+1 0tEf(xr,~)) i

x

n.

(92)

On the other hand, applying II. to the first equation of (89) shows that for i>0, ~f0,(E~ 9n ) - ~ j = ~ o ( - U ) ~-J rotr('-~) n fj . (93) Equation (90) is obtained from f~ 0t(92) d~, by using the expression of Ot (E~. n) in (93), and by using the boundary condition at u - 1, given in (8s).

m

0.0.1

2.4.3. E f f e c t i v e C o n s t r u c t i o n

Of course the principle is the same as the case of the straight boundary. C o n d i t i o n o f o r d e r 2. The terms H,E ~ and H , H ~ are given by (24). Making k = 0 in (90) and using (24) shows that

OtE~(xr,u) x n -

- ( 1 - u)

( #fOrt + ~ r o t r rotr ) II, Hv(xr 0 ),

(94)

which is exactly the same expression as (31). Hence, the condition of order 2 in the case of curved boundary is given by (32). Indeed, the stability theorem 0.1 still applies. However, the condition of order 3 has no longer the same expression.

H. Haddar and P. Joly

443

C o n d i t i o n of o r d e r 3. Make k - 0 in (91). Using (24) we get,

II.,H~(x~)- vcn~

IIHH~(xr,v ) -

(x~,v) e r•

1[.

(95)

Consider (90) when k = 1, OtE2(xr,v) x n

-

- f l #fOttlittHl(xr,~ ) d~ -4- f~ (COtE~(xr,~)) x n d~ _ fl

1 rotr rotr H I (xr ~) d~

1 (r~t(1) rotr + rotr rotr(x))H ~ (Xr, ~) d~, + f l ~ ~ff

(96) I I , H ~ is given by (24), IIttH~ is given by (95) and we have COtE~ - COtIIttE 1, where OtHttEI~ is given by (39). We obtain after simplifications and using

(75), Ot E 2 (Xr, V) X n -- # f O t t ((1 - v)II,,H~(x~) - (1 - v2)(c - H)II..H~ (xr)) 1 rotr ~,

rotr ((1 - v) II,,H~(x~)1 ~1(1

+7,1 ~1(1__

(97) 0 H~(xr))

V2 ) ~gt~ ~otp) H~(x~). 0

The ultimate simplification is due to lemma 0.8 and is given by the following lemma: L e m m a 0.10 - Let v be a tangential vector field defined on F. We have the identity: rotr(1) v + rotr (C v) - 2H rotr v Proof. By definition, rotr(1) v rotr(~) v + rotr (C v)

(C Vr )" (v x n). Hence, using (75),

=

( C V r ) " (v x n) + Vr 9( ( 2 H n l l -

C)(v x n ) )

=

2H divr (v x n) + (divr (2H Hi, - e))" (v x n),

and according to lemma 0.8, (divr (2H IIll - C))" (v x n) - 0.

m

Applying this lemma to relation (97) yields 0tEfe(xr, v) x n = -#~ 0tt ((1 - v)II,~H~(x~)-(1 - v2)(8 - H)II..H~ 1 ((1 - v ) r o t r rotr H,.Hlv(xr) - (1 - v 2) rotr g r o t r Hi.H~

~f

(98)

444

Stabifity of thin layer approximation of electromagnetic w a v e s . . .

Combining (98) with (94) and (24) leads finally to the following third order condition

OtE~ • n - -~-- (~r #f [1 - rl(C - H)]Ott + rotr (1 - r/H)rotr ) II,.H:~, Cf

(99) on F • 1~+. Notice that this condition differs from the second order one by corrector terms involving the geometry characteristics C and H, only. In the case of straight boundary, C = 0 and H = 0, the two conditions are of course identical. We give now the stability result for this condition. N

N

Theorem 0.5 - Sufficiently regular solutions (E~,Hv~ of { (6), (99)} satisfy the a priori energy estimate: d d--t {$~(F,~(t), H~(t)) + ~$r~(t)} - 0, N

where, if we set, qo~ (xr, t) - II.. H~ (xr, t), Xr e F, g~ (t) - -~

-~7

-

.OtcP'7

+~

(1-~H)

Iforotr qon dT I dxr

We have stability as defined by (10) when, r/ i~f (max (1Cl + c21, Ic1

and where

Cl

-

-

C2[)) < 2,

and c2 are the eigenvalues of C.

Proof. The proof of the a priori estimate is rather straightforward. It follows the same steps as the proof of theorem 0.1 and is based on the symmetry of [1 - r/(C - H)]. The stability (defined by (10)) is obtained when the eigenvalues of 1 r](C-H) and the real ( 1 - 0 H ) are positive; i.e. when ~ m a x (ICl + c21, Ic1 - 521 2 a.e. on F. In this case we have $r~ (t) _> 0. II

3. Thin layer approximation: the non linear case We generalize the construction of thin layer approximations to the case of non linear materials of ferromagnetic type. We consider here directly the case of a curved boundary.

H. Haddar and P. Joly

445

3.1. Description of the model We keep the same notation as in section. However the material of the thin layer ~t~ is no longer linear and obeys to the following equations"

{

r OtE~ - rot H~ - 0,

#f Ot(H~ + MT) + rot E~ - 0, on f~,

(ET, H~)(t - 0) - (0, 0),

(100) where M~ is called the magnetization field and is linked to H~ through the ferromagnetic law s as follows:

0tM~ - s

H,no,; x),

M~ (x, 0) - M~(x),

x E a~,

(101)

H~o, - H~ - VM~I,n(M~; x), where, for a.e. x 6 ~ , (m,h) 6 l~ 3 X l~ 3 ~ s h; x) E l~ 3 is a C c~ function and m , ~ On(m; x) 6 R is C cr and positive function, s satisfying:

(i) (ii) (iii)

h:

; s (m, h; x) is linear, m E l ~ 3 X E ~'~f~

s

re, h E R 3 x E f ~

(102)

s (m, h; x) . h >_ 0, m E R 3, xEf~fv.

The diffraction problem is constituted by the equations (1), (2), (100), (101) and (4). E x a m p l e . The Landau-Lifshitz law of ferromagnetic materials, without exchange field (see [13] or [16]), is a particular case of this general framework. It corresponds to: s

a:(x) I m x ( h x m ) , h; x) - 7 h x m + iMP(x)

(I)n(m; x) -- g1 i n ~ ( x ) _

ml +1g K 2 (x)

Im - (pn (x). m) pn (x)[ 2 ,

where, 7 is the absolute value of the gyromagnetic factor, that is a universal constant, K~ and a n are positive scalar functions, M~ is the initial magnetization, H~ is a given static magnetic field, and finally, pn is a unit vector called easy axis of the magnetization. One easily checks that this law satisfies the required properties. We have for example: s

x). h-

~',(x) I Ih x ml iMp(x)

> _ o.

446

Stability of thin layer approximation of electromagnetic w a v e s . . .

Apriori estimates. The stability of the coupling between Maxwell equations and the ferromagnetic law relies on the following fundamental a priori estimates. The first one is a consequence of the nonlinear law itself. According to the property (102) (ii), if we take the scalar product of (101) by M~, we see that OtlMT(x,t)[ 2-0

~" IMT(x,t) l - I M ~ ( x ) l

Vt_>0anda.e.x

E ~7. (103) The second one is the equivalent to the classic energy estimate for Maxwell's system:

~

s (E~(t),I-I~(t)) + g~ (E~(t) H~(t)) + p~

(M~(t)) dx

< 0. (104)

Proof. Using the continuity relations (2), we have: d~gV (E~(t), I-I~(t)) + g n (E~(t),I-I~(t)) - -#~

O t M ~ . I-I~ dx.

(105)

7 On the other hand, property (102) (iii) yields, for a.e. x ~_ f ~ ,

0~MT. H7 - 0~MT. (n~o, + VM~'(MT)) _> 0~'(M,'). Consequently - #f

7 Ot MT

. H 7 dx <_ -

#f

d ~ -~

7

O,7(M~(t))dx

Identity (104) follows from (105) and (106).

(106) m

3.2. Construction of the effective boundary conditions Scaling. As in the linear case, the scaling is done through the transformation: x E Ft~ '. ; (Xr,V) E F • [0,1], such that x - X r + y y n . We still keep calling (E~, H~, M~) the unknowns defined on the scaled domain F• 1[. The system (100) is transformed to: 1 0v(H~ • n ) c f Ot E ~ - 7"~rVH ~ + -~

0

#f Ot(H~ + M ~ ) + 7~r~E~ - ~1 0~(E~ • n) - 0 ,

(107)

where the operator 7~r" is given by the lemma (0.6). To write the ferromagnetic law on the scaled domain, we assume that the characteristics of

447

H. Haddar and P. Joly

the thin layer do not depend on the thickness ~. That is, there exists, s ~, and M0 functions defined on Fx]0, 1[ and independent of ~, such that: for x E gtf~; x - Xr + 7/~ n, s

x)-s

Xr,~),

O~(m; x ) = O ( m , h ;

Xr,~),

m, h e R 3, m e R a,

M~(x) - M0(xr, v) From now on, we will not indicate explicitly the dependence of s and 9 on the variables (Xr,~). The system (101) becomes

{

OtM~ - s

H~ot) ,

M ~ ( t - 0 ) - M0, on Fx]0, 1[,

(108)

H~ot - H~ - VMO(Mf~).

P r i n c i p l e of t h e c o n s t r u c t i o n . We use the ansatz (19) and the analogous one for Mf~" M~-M ~ l+r/2M 2+ --. (109) as well as the Taylor expansion (87) of the operator 7~rV. To adapt the construction of the linear case to the present one, the idea is to consider first the field M~ as given source term for (107). This permits to derive effective conditions of order k of the form Ev~

xn-B~(n,,fiv ~)+7~k(M~)

on

F x R +,

(110)

where B~ is the same operator as in the linear case, 7~k is a linear operator and M~ is a certain approximation of M~ defined on Fx]0, l [ x R +. Let us recall that B$ = - e~l (el #f Oft + rotr rotr )

(111) B3~ = -e-~r/ (el #f [1 - rl(C - H)]au + rotr (1 - rIH)rotr ) To determine the operator T~k, we use the same procedure as the construction of B~, explained in the linear case. So the first step requires the identified equations associated to the asymptotic expansions. The second step of the formal construction is to characterize Mf~ by using the ferromagnetic law. Indeed, the main difficulty consists on proving the stability in time. F o r m a l identification. Comparing with the linear case, in the identification process, only equation (88) changes to: k q-iFk_i Ov(E k+l x n) = #fOt(H k + M k) + Ei=o(-L,) i , r ' - f

Ek+l(Xr,1 ) X n -

0,

for Xr E F.

(112)

448

Stability of thin layer approximation of electromagnetic waves...

In a similar way, and keeping the notation of the linear case, II,E ~ and H , H ~ are still given by (24) and L e m m a 0.11 For all k >_ O, we have for (Xr, v) E Fx]O, 1[, cOtEk+ 1

Xr, v) x n - - f l Pf Oft (H,H~ +

(

k Y~-j=o(-~) i k - j r~ - f : ~1 Y~-i=o

+fl

n,M, k) (x~, {) d{ r~

Hj (xr, ~) d~

i=o(-{) k-i (C k-i+1 0tE~(xr, {)) x n de

(113) c0tn k+l

(x~.~,)

• n - a,H~+l(x~) v 1

k

• n-

f [ ~.a.II..E.~(x~.,q

d,'

i

- fo 7, E~=o Y~-j=o (_~)k -J rotr(~-), rotr(~-j) E~ (xr, ~c) d~

- f 0 ~ i =k o ( - ~ ) k-i (C k-i+l 0tH~(xr, ~)) x n d~ -- fo Y~i=0 k ( __{)k-i rotr(k-') 0, ( M ,i. ll)(Xr , {) d{ .

(114) 3.3. C o n d i t i o n of o r d e r 2

We need to find the analogous to (94). Making k - 0 in (113) and using (24) yields Ot

E 1 (Xr, u) x n

= - (1 - v) -

(

1 -, ) o # f O t t + ~ r o t r rotr H , H v ( x r )

L 1 #,OttII, g ~

(115)

d~

Let M~ be an approximation of order one, that will be specified later, of the magnetization Mfv. We can write in a formal way: M ~ - M ~ + 0(7/). If we combine this with (115) and (24), we can write an effective boundary condition of order 2, like: .... c0tEv~(Xr) x n -

...

r1

.--...

B~(H,Hv~)(Xr)- r/#f J0 cgttIIliM~c(Xr,V ) dr.

(116)

Now, the question is how to determine M~, using the ferromagnetic law s Let us give the general principle of the answer (i.e. how to determine M~,

H. Haddar and P. Joly

449

an approximation of order k of M~)" suppose that H~ is an approximation of order k of H~" H~ - I-I~ + rlkH~, where H~ is uniformly bounded with respect to rl. Thus

OtM~ - s

H~ - VM(I)(M?))+ r/ks

H~).

As M~ has a uniform bound with respect to ~ (by (103)), we conclude (at least formally) that s H~) is also bounded uniformly with respect to Vl. So s VM(~(M~)) constitute an approximation of order k of cOrMS. A natural way to define M~ is then impose . v

cgtg ~ - s

H7 - ~TM(I)(M~)), M7(t -- 0) -- M0.

(117)

Doing so, we see that the definition of M~ amounts the one of HT. This can be done using the asymptotic expansion k-1

H7 - E

~li H~ + O(rl k)

(118)

i--0

To determine H k, we split it into two parts" IIliH k that can be computed by (114) and H~H k that can be computed by applying H~ to (112), that yields 1

Ot (H k- n) -- -Or (M k" n) _ __ E i k= 0 ( - u ) i rotr(') E k - i

(119)

#f

We are going to see that applying expression (118) is sufficient to derive a stable effective boundary condition of order 2, while the condition of order 3 requires slight modifications. We come back now to the second order condition, where, according to the previous considerations, we need an approximation of order 1 of H~. Combining (24) and (119) applied to k = 0, we see that

HT(xr, u)

-

n,H~

+ n~ (M0 - M ~

=

H.. n ~ (xr) + II• (M0 - M~)(xr, v) + O(~/)

u) + O(~/)

(120)

450

Stability of thin layer approximation of electromagnetic w a v e s . . . N

N

So, setting HT(xr,v) - n,~H~,(x~) + H• (116) and (117) leads to following condition

- M~)(Xr,V) and applying

0tEv~(Xr) x n - B2V(H..Hv~)(xr)- ~/#f ]0 IIitM~(xr, v ) dr, where Mf~ satisfies for a.e. (Xr, v) E F x]0, 1[, OtM~ = s

H7o~), Mf~ = O) = M0,

fiTo,(X~, v) = n..H~,(x~)+ n~(Mo - MT)(x~, v ) - V~,(~(MT)(x~, v). (121) According to the following theorem, this condition is stable. Theorem 0.6-

Regular solutions ( E ~ , H ~ , M ~ ) o f {(6), (121)} satisfy,

for all t >_ O,

Ig ~ l - I M o l

a.e. F•

and

_d {E~(~,(t) fi~,(t))+ ~ E~,(t)+ ~ E,,(t)} < 0 dt ' - " where, if we set, cp'7(Xr, t) - IIHfi~(Xr, t), (Xr, t) e ~f

2

$~r (t) - -2 ]]r

"~" ~

l l]ft

F x

It +,

2

]]JO (rOtr r

, L2

C ( t ) - 7#* II H~ (Mo - ~ 7 ) ( t ) I 1 ~ + , , fr •

ll O(MT(t)) dxr dr.

Proof. As (121) has the same structure than (101), property (103) still applies on the scaled domain. From Maxwell's equations and (121), we get (see the proof of theorem 35) 1

d{s dt

'

0tH, MT(xr v)'~n(xr)d~'dxr.

ffI~(t))+~s

(122) For a.e. (xr, v) C F• OtH,.~I~. r

1[,

_ Ot~'I~. r

_ Ot~.inf. (fi~o~ + VM(I)(M:7) - H.(M0 - ~fn))

So, using property (102)(ii), Otn.,M'~. ~"

1 >_Or{ O(M~) + ~[ 111(Mo - M~)] 2 )

Integrating over Fx]0, 1[, shows -.,

JfF

~01 O~IIj.M~(x~, ~~ ~,). ~,'(x~) d~, ax~ ___ - ~ d C(t)

(123)

H. Haddar and P. Joly

451

The energy estimate of theorem (0.6) is deduced by identifying the right hand side of (122) with the left hand side of (123). m 3.4. C o n d i t i o n of o r d e r 3 Let us begin by the "Maxwell" part (like for the condition of order two). We point out the differences with the linear case. Equation (95) changes to II,Hf~ (xr, v) - I I , H ~ ( x r ) - vC HO(xr) -

fo

Vr ((Mr~ - Mo)" n ) ( x r , ~) d~, (124)

and relation (98) becomes

OtE~ (x~, ~) • ~ -

-#fOrt

((1 - v)II,Hlv(Xr) - (1 - v2)(C - H ) H . , H ~

- • g f ( (1 - v ) r o t r rotr IIi, H l ( x r ) - (1 - v 2) rotr H r o t r H,,H~

-#,Ott

f l ( n . . M f l ( x r , ~) + f l (2H _ C)ii.,MO(xr, T)

+#fOrt fl f:

Vr ( M ~ n ) ( x r ,

dT) d(

T)dT d~. (125)

If we regroup (125), (115) and (24), written for v - 0, and if we set Mf~ = M ~ + ~M~ + 0(72), an approximation of order 2 of the magnetization M~ (to be specified later), we can write the condition of order three in the following way. 0tE~(xr) • n -

B~(H,H~)(Xr)

-~

fo1[1 + r~u ( 2 / - / -

~

Oft

+ ~ .~ o~

~01 (1 -

r

-

(x~, u) du

(126)

~

~,) v~ ( M ~ . n)(x~, v) a~,

We used the fact that fo f (~) (f2 g(v) dr) d~ - fo (fo~ .f (v) dr) g(~) d~. To obtain an approximation of order 2 of Mf~ we need to compute 111Hf1 . Equation (119), written for k - 1 and combined with (115), leads to (we use rotr Vr - 0) Hfl(xr, v) 9n - - ( M 1. n ) ( x r , v) + (1 - v ) d i v r

~

(n,,H~

-

(127) ldivr (II, (Mo - M~

{) d{.

Stability of thin layer approximation of electromagnetic waves...

452

We obtain the following approximation of Hf~ by regrouping (120), (124) and (127):

Hr~(xr, ~') = [1 - ~ uC] II,,H~n(Xr) + rl

Vr ((M0 - M ~ ) . n)(x~, ~) d~,

+ H i (M0 - Mf~)(xr, u) + ~ (1 - u) { divr (IIliH~)(xr) } n

(128) Contrary to the case of the second order condition, when we use (128) (of course without O(~2)) to determine Mf~ via (117) and we couple this with (126) in order to built a third order condition, we are unable to prove stability via energy estimate like in theorem 0.6 (this doesn't mean the instability of the obtained condition). However, we are able derive another expression of the third order condition, that differs from the initial one by terms of order ~3, but that enables us to get energy estimates. The next modifications are suggested by difficulties encountered when trying to prove energy estimates. In (126), we replace the operator [1 +fl u ( 2 H - C ) ] by (1 + ~ u 2 H ) ( 1 - ~ uC). As the two expressions differs by a O(rl 2) term of order 7/2, the new condition differs from the initial one by O(~/3) terms, which is consistent with the order of (126). Remark that T ] r rl d e f

-

n (1 + r/u 2H)

(129)

is the development of order 3 of the volume element df~fn. Meanwhile unstead of using (128) we set, using the usual notation qo~ (xr, t) - II HHn (xr, t), Hfn(xr,u)

-

[1 - ~ u C ] ~ o n ( x r ) +

VrCX(xr,~)d~ (130)

+ (r where r r

,

u ) + ~ ( 1~-o nu()Xd ir v) )r r n

n,

is a scalar field defined on F x]0, 1[ by" - (Mo - Mf ) 9n

rrl

1 divr (IIjl(Mo - M~))(Xr f) df ~

9

(131)

Comparing the right hand side of (128) to the right hand side (130), we see that Hf~ _ ~fn + O(y2),

H. Haddar and P. Joly

453

which satisfies the requirement of the construction of the third order condition. In conclusion, if we set r

u) - rn(1 - r/uC)II,M~(x~,

u) -

rl (1 - u) Vr ( M ~ . n ) ( x r , u), (132)

the new third order condition can be rewritten as

N

- Bg(q0n)(Xr)

x n

where r

is related to M~n by (132), and M~ satisfies a.e. Fx]0, 1[,

OtMnf - s

n, n,not),

-

r/#,

/ol

0tEvn(Xr)

OttCn(Xr,U) du,

Mtn(t - 0) - M0,

H,not - H~ - ~TM~(Mfr/).

The field H~ is given by (130). (133) The main justification of this condition is the following a priori estimate.

1 theorem 0.6 remains valid when we replace T h e o r e m 0.7 - For 71 < if-re' the effective boundary condition (121), by condition (133), and the expressions o[ Cnr and E~ by Ern(t) - m2 fr[1 - ~(C - H)]OtqO n . Otqondxr

1

+5-~et f r ( 1 - ~ g ) _

f

.fat

(rotr ~O~)(T)dT

2

(134)

dxr

1

x]0,1[ Proof. Like in the case of the second order condition, we have the classical identity

d .trig2() + ~grn(t)} - - # f dt

; / o 10tCn(Xr

We are going to compute the right hand side of of Cn in (132), one has O r e '7 . ~o '7 -

r '7 0 t M ~

.

[1 -

~ u C ] ~ o '7

u)-q0n(Xr) du dxr.

(135). Using

- ~(1 - u ) V ~

(OtM~

.

(135)

the expression

n). ~.

(136)

[ 1 - 7 Y C]q0n can be expressed using (130), where H~ - H~o, + VMO(M~) as shown by (133). Sue5 doing, we get r n OtMnt 9[1 - r/uC]~o ~ - OtM~. {r n (H,no,

-r

n + VMq,(M~)) - r / ( f o Vr r

d~) - 7(1 - u)(divr ~o") n}.

(137)

454

Stability of thin layer approximation of electromagnetic waves...

1 r~? For ~ < WH-T' > 0. Thus, property (102)(iii) yields rV 0tM~. H~ot _> 0. Then using (137) in (136), yields, after integrating over F • [0, 1] and multiplying by -#f,

-#f

/~/o I Ore "1. ~'~ du dxr _< -#f ~-~ ~/~/o ~(I)(M~(t)) r v dxr du

+#f

0t(M7" n) r" du dxr

r

(/o~Vr r d~) 9Otl-I,~I~ du dxr +rl #f /01(1 - u) {/~ Vr (0M~. n). ~v + (0tM7 9n) divr (~a') dxr ) du +7/#f fr fo1

(138) The last term in (138) is zero (integrate by parts). Let A be the value of the the second line of (138). Using the expression (131) of r one has A = #f (Ix + 7//2) where,

/~fo 1r -

n) r ~ du dxr

~/o 11~ 0t

i~

Hz (Mo - MT) r ' du dxr

+~? J(r J~010 t ( ( M o - MT). n ) ( j f 1 divr (II,, ( M o - M~))d~)du dxr and,

12-- ~r ~O1 (~o" Vr r

d~) " OtII,, ~IT du dxr

=

~r~olCvn'Ot(~x

-

/~/ol ((Mo - MT). ~ n) Ot (/1 divr (II,(Mo - ~I7) ) d~) du dxr - 7/

divr(IIjl(Mo-~I~))d~)

/~/ol,r-~ 5,o/~ 1 divr (Hjl(Mo t

dudxr

~I~)) d~

i~ du dxr.

This gives

'/~/o 11~ Ir

A - -#f ~-~

2 r '7 du dxr

which, combined with (138), shows that

-#f

/~/o 1Ore ~ . ~a'~ du dxr

< -~g'f(t)

(139)

H. Haddar and P. Joly

455

We obtain the desired energy estimate by combining (139) and (135).

References [1] A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 58 (1996), 1664-1693. [2] F. Collino, Conditions absorbantes d'ordre ~lev~ pour des modules de propagation d'ondes dans des domaines rectangulaires, I.N.R.I.A, no. 1794 (1992). [3] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651. [4] B. Engquist and J. C. Nedelec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers,Ecole PolytechniqueCMAP (France),278 (1993). [5] H. Kreiss, Initial Boundary Value Problems for Hyperbolic Systems, Comm. on Pure and Appl. Math.,13 (1970),277-298. [6] B. Gustafson, H. Kreiss, and A. Sundstrom, Stability theory of difference approximations for mixed initial boundary value problems, Math. Comp.,26 (1972), 649-686. [7] A. Haraux, Nonlinear evolution equations-global behavior of solutions, Springer-Verlag, 1981. [8] J.L. Joly, G. Metivier, and J. Rauch, Global solutions to Maxwell equations in a ferromagnetic medium, S~minaire EDP, Ecole polytechnique (France) ,1996-1997, no. 11. [9] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations, IEEE Trans. Antennas and Propagat., AP-14 (1966), 302-307. [10] A. Visintin, On Landau-Lifchitz equations for ferromagnetism,Japan J. Appl. Math.,2 (1985), 69-84. [11] L. N. Trefethen, Group velocity interpretation of the stability theory of Gustafsson, Kreiss and SundstrSm, J. Comp. Phys., 49 (1983), 199217. [12] P. Joly and O. Vacus, Mathematical and numerical studies of ld non linear ferromagnetic materials, In Numerical Methods in Engineering 96, ECCOMAS, 1996. [13] P. Joly and O. Vacus, Mathematical and numerical studies of non-linear ferromagnetic materials,M2AN, (1997). [14] I. Terasse, R~solution math~matique des ~quations de Maxwell instationnaires par une m~thode de potentiels retard~s, Ecole polytechnique (France), 1993. [15] Y. Choquet-Bruhat, G~om~trie diff~rentielle et syst~me ext~rieur, DUNOD, Paris, 1968.

456

Stability of thin layer approximation of electromagnetic waves...

[16] O. Vacus, Mod@lisation de la propagation d'ondes en milieu ferromagn@tique, Ecole Centrale de Paris, 1997. [17] H. Haddar and P. Joly, An Asymptotic Approach of the Scattering of Electromagnetic Waves by Thin Ferromagnetic Coatings,Mathematical and Numerical Aspects of Wave Propagation,Siam, (1998) ,June. [18] H. Haddar and P. Joly,Conditions @quivalentes pour des couches minces ferromagn@tiques, @tude du probl~me monodimensionnel,I.N.R.I.A.,3431, (1998),May, Th~me 4. [19] H. Haddar and P. Joly, Effective Boundary Conditions For Thin Ferromagnetic Layers; the 1D Model. , (99) ,Submitted to Siam J. Appl. Math.. [20] H. Haddar and P. Joly, Stability of thin layer approximation of electromagnetic waves scattering by linear and non linear coatings, (2000) ,preprint. [21] H. Haddar and P. Joly, Electromagnetic waves in laminar ferromagnetic medium. The Homogenized Problem., Mathematical and Numerical Aspects of Wave Propagation, Siam, (2000) ,July. Houssem Haddar and Patrick Joly INRIA, Domaine de Voluceau-Rocquencourt BP 105 78153 Le Chesnay C@dex France E-mail: houssem.haddar@inria, fr, [email protected]

Studies in Mathematics a n d its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter

20

R E M A R Q U E S SUR LA LIMITE c~ ~ 0 P O U R LES FLUIDES DE G R A D E 2

D. IFTIMIE

R 6 s u m 6 . On consid~re la limite a ~ 0 dans l'~quation des fluides de grade 2. On montre la convergence faible des solutions vers une solution faible de l'6quation de Navier-Stokes, en supposant que les donn~es initiales convergent faiblement dans L 2. A b s t r a c t . We consider the limit a --+ 0 for the equation of the second grade fluids. We prove that weak convergence of the solutions to a weak solution of the Navier-Stokes equation holds by assuming t h a t the initial d a t a weakly converges in L 2.

1. Introduction I1 existe dans la nature des fluides qui n'ob~issent pas aux classiques ~quations de Navier-Stokes. Des modules plus compliqu~s ont dfi ~tre d~velopp~s pour les ~tudier. Ainsi, Rivlin et Ericksen [10] introduisent les fluides de type diff~rentiel. Un cas particulier de ces fluides est constitu~ par les fluides de grade 2. L'analyse de Dunn et Fosdick [6] montre que l'~quation d'un tel fluide est donn~e par Ot(u - a A u ) - v A u + ~-~.(u - a A u ) j V u j J +u. V(u-

a/ku) - -Vp

+ f,

(1)

div u - 0, oh a > 0 est une constante mat~rielle, v > 0 est la viscosit~ du fluide, u le champ de vitesses et p la pression. Pour a - 0, on obtient les ~quations classiques de Navier-Stokes Otu- vAu + u. Vu - -Vp

+ f,

divu-0,

(2)

458

Remarques sur la limite a ~ 0 pour les fluides de grade 2

de sorte que l'~quation du fluide de grade 2 est une g~n~ralisation simple des ~quations de Navier-Stokes. Les premiers r~sultats math~matiques pour les fluides de grade 2 ont ~t~ obtenus par Cioranescu et Ouazar [4]. Ces auteurs montrent l'existence et l'unicit~ des solutions, globale en dimension 2 et locale en dimension 3, pour des donn~es initiales appartenant ~ H 3. L'existence et l'unicit~ globale des solutions tridimensionnelles ont ~t~ obtenues par Cioranescu et Girault [3] pour des donn~es initiales petites dans H 3. La m~thode de d~monstration repose sur des estimations d'~nergie. Un autre point de vue est adopt~ par Galdi, Grobbelaar van Dalsen, Sauer [7] et Galdi, Sequeira [8]. Ces travaux utilisent une m~thode de point fixe pour obtenir des r~sultats similaires. Tous ces r~sultats sont ~nonc~s dans des domaines born~s mais l'extension ~" ne semble pas poser de difficultY. Une question qui se pose naturellement est de savoir si les solutions des ~quations du fluide de grade 2 convergent vers une solution des ~quations de Navier-Stokes lorsque a --+ 0. La r~ponse n'est pas ~vidente et ne d~coule pas des travaux precedents car toutes les estimations pr~c~dentes "explosent" lorsque a -+ 0. Le but de ce travail est de montrer que la convergence vers une solution des ~quations de Navier-Stokes a bien lieu, et cela sous des hypotheses tr~s g~n~rales. La seule hypoth~se "artificielle" sera la borne Ca -1/2 pour la norme H 1 de la donn~e initiale. Avant d'~noncer nos r~sultats, rappelons un r~sultat classique d'existence des solutions faibles pour l'~quation de Navier-Stokes qui est du ~ Leray [9], voir aussi [5], [12]. On appelle solution faible des ~quations de Navier-Stokes sur [0, T) un champ de vecteurs de divergence nulle

u e C~([O,T);L2)nL~oc([O,T);H 1) qui v~rifie l'~quation (2) au sens des distributions. Le th~or~me classique de Leray affirme l'existence d'une telle solution, unique en dimension 2, d~s lors que u0 e n 2, divu0 = 0 et f e n~oc([O,T);g-1); de plus, on peut supposer que cette solution v~rifie l'in~galit~ d'~nergie suivante: [lu(t)ll~ + 2 ,

IIVu(T)II~ dT <_ Ilu(0)ll~ + 2

(f(T), U(T))dT,

(3)

pour tout t < T. Une relation similaire a lieu pour le fluide de grade 2. On multiplie (1) par u et l'on int~gre. Cela implique, apr~s quelques integrations par parties, l'estimation H 1 suivante: [lu(t)[l~,2 + aIIVu(t)ll~,2 + 2u

~00t liVu(T)II~2 d r

_< fluollL +

llVuollL + 2

(f (T), U(7) ) dT. Jo

(4)

459

D. Iftimie

On voit tout de suite que ces estimations donnent des informations '% priori" pour des normes H 1 en espace seulement, i.e. seules les d~riv~es d'ordre 1 en espace peuvent ~tre "contr61~es". Par consequent, pour pouvoir passer ~ la limite dans (1) avec l'information (4) seulement, il faut mettre l'~quation sons une forme oh les termes non-lindaires soient des produits de d~riv~es de u d'ordre inf~rieur oh ~gal ~ 1 ou des d~riv~es de tels produits. Cette forme sera la suivante:

j,k

j,k

-

o

(O

ujWj) - - V p

+ f.

(5)

j,k

Remarquons enfin que, pour montrer que les termes suppl~mentaires par r a p p o r t ~ l'~quation de Navier-Stokes convergent vers 0 au sens des distributions, l'hypoth~se v > 0 est importante. On montre le th~or~me suivant" T h 6 o r h m e 1.1 - Considdrons l'dquation d'un fluide de grade 2 posde dans I~n, n > 2. Soient v > 0 et f e L~oc([O, T);/_/-1) fixds et u ak (0) une suite de donndes initiales ~ divergence nulle correspondant ~ une suite ak --+ 0 tels que a) il existe ~0 e L 2 tel que u a~ (0) - - ~0 ]aiblement dans L2; 1/2

b) la suite a k u

c~u

(0) est bornde dans H1;

c) il existe une solution (an sens des distributions) u a~ e C~ ([0, T); g 1) de (5) avec ~ = ak, ayant comme donnde initiale u a~ (0) et vdrifiant l'indgalitd d'dnergie (4). Alors, il existe une solution ]aible ~ de l'dquation de Navier-Stokes sur [0, T) avec donnde initiale ~(0) - uo et une sons-suite ua~(~) telles que pour tout 0 < T on air u ~(~) ~ ~ dans Lcc(O,O;L2) ]aible* et dans L2(O,O;H 1) ]aible.

a e m a r q u e 1.2. Le fait que u ~ E C~ ([0, T); H 1) et que u ~k v~rifie l'in~galit~ d'~nergie est a u t o m a t i q u e m e n t v~rifi~ pour des solutions obtenues par r~gularisation et passage ~ la limite. Ainsi, la seule hypoth~se "restrictive" est la borne sur IlVu~k(0)lln 2. Cette borne ne dit pas que les donn~es initiales restent born~es dans H I , elle dit seulement que ces donn~es initiales n'explosent pas trop vite en norme H I lorsque o~k ---~ 0.

460

Remarques sur la limite a -~ 0 pour les fluides de grade 2

Remarque 1.3. En dimension 2, comme on a unicit6 des solutions faibles de l'6quation de Navier-Stokes, il s'ensuit que la conclusion reste vraie pour toute la suite ak au lieu d'une sous-suite a~(k). Remarque 1.4.

Pour que la limite ~ satisfasse l'in6galit6 d'6nergie (3), il suf-

fit d'ajouter les hypotheses u ~k (0) -~ u0 fortement dans L 2 et ak/2U ak (0) 0 fortement dans H I. Remarque 1.5. La preuve qu'on va donner n'utilise pas de mani~re essentielle le fait qu'on se place dans l'espace entier au lieu d'un domaine borne. En effet, les estimations '% priori" (4) et l'6quation 6quivalente (5) qui sont les ing%dients essentiels de la d6monstration, restent vrais dans des domaines born6s. Ensuite, les techniques du passage ~ la limite peuvent ~tre remplac6es par des techniques similaires adapt6es aux domaines born6es sans trop de modifications.

2. Pr61iminaires On note par H s l'espace de Sobolev suivant:

H 8 - { g ' I ~ n -+ C;

I]g[l~/,- jf~n(1 + 1~12)sl~(~c)lu d ~ < +c~},

off ~ d6signe la transform6e de Fourier de g e t I" I la norme euclidienne. La version homog~ne de ces espaces est

/:/8 = {g. i~n --+ C; ~ e L~oc(IRn )

et ]lgllH, - JfR,~ I~128Ig({)]2

d~ < + ~ }.

L'espace homog~ne/:/s est un espace de Banach pour s < n/2. Pour des fonctions ~ valeurs vectorielles h 9I~n -+ ~m, on dira que h E H 8 si et seulement si chaque composante hi de h appartient ~ H 8 et on notera m

IrhJl . -

IIh, IP .. i--1

On utilisera la m~me notation pour les espaces de Sobolev homog~nes. On d6signe par (., .) le produit scalaire L 2, le produit de dualit6 entre H s et H -8 ou encore le produit de dualit6 entre H s e t H -s. Pour des fonctions valeurs vectorielles g, h" I~n --+ I~m , on notera m

(g, h) -

h,) i--1

461

D. I f t i m i e

Le projecteur de Leray I? d6signe la projection orthogonale L 2 sur les champs de vecteurs ~ divergence nulle. Le th~or~me de produit suivant est classique (voir par exemple [1])" T h ~ o r ~ m e 2.1 - S o i e n t s et t des rdels tels que s + t > O, s < n / 2 , t < n / 2 . I1 existe u n e c o n s t a n t e C > 0 telle que s i u E H s et v E H t alors UV E H s + t - n / 2 et

II~vll..+,-:/~ < Cllull.. Ilvll-,. s7 I.I < ~ / 2 ~t ~ > 0, il existe une c o n s t a n t e C ' > 0 telle que s i u v E H - s alors u v E H -nl2-e et

E H set

II~vll.--/~-. < C'llull-. Ilvll.-.. Enfin, on a l e lemme suivant tr~s simple: L e m m e 2.2 - S o i t H u n espace de Hilbert et A C H u n sons e n s e m b l e dense. Si Un est u n e suite borage de H telle que (Un, a) -~ (v, a) p o u r t o u t d l d m e n t a de A , alors Un converge f a i b l e m e n t vers v.

Rappelons l'6quation du fluide de grade 2: Otv - v A u + u . V v + E v j V u j J

= -Vp

+ f,

v - u - aAu.

(6)

Dans un premier temps, on va donner une autre forme ~ cette 6quation oh la %gularit6 H 1 en espace pour u suffira pour donner un sens ~ l'~quation. Pour cela, remarquons d'abord que u . Vv est un vecteur dont la /&me composante vaut

j

j

j,k

= ~ , u~Oj~, - ~ ~ j

(7)

0j(~j0~,)

j,k

= ~ ~oj~, - ,~ ~ ojo~(~jo~,) + ~ Z o ~ ( o ~ o ~ , ) . j

j,k

j,k

De m&me, l a / & m e composante de ~-~j V j V U j e s t 6gale

j

j -

j,k

!Olul ~ - . ~ Ok(Okuja~j) + ~ ~ OkujO~Ok~j 2 '

i,k

J,k

1

= ~O~(lul ~ + ~lV~l ~) - ~ ~ j,k

o~(o~jO~uj).

(s)

Remarques sur la limite a --+ 0 pour les fluides de grade 2

462

Les relations (7) et (8) utilis~es dans (6) donnent une forme ~quivalente

(6). j,k

j,k

-

Z

o (O ujW, j) - - v p + S,

(9)

j,k

1 off on a incorpor~ clans la pression le terme ~V(lul 2 + O/}~Tul2) qui apparMt dans ~j vjVuj. Remarquons qu'il suffit de supposer que u E L~oc(0, T; H I) pour d~finir l'~quation (9) au sens des distributions. En effet, Otv et /ku sont toujours d~finis si u est une distribution. Ensuite, un terme du type Du D~u, off Du et D~u d~signent une composante de u ou une d~riv~e d'ordre 1 d'une telle composante, est dans L~oc(O, T; L I) et, par consequent, d~finit une distribution. Ses d~riv~es aussi et on a ainsi ~puis~ tous les termes de (9). Nous travaillerons d~sormais sur l'~quation (9). On a mentionn~ dans l'introduction que toute solution obtenue par un proc~d~ de r~gularisation v~rifie l'in~galit~ d'~nergie (4). En effet, si l'on multiplie formellement (6) par u et qu'on int~gre, on trouve, apr~s quelques integrations par parties, la relation (4) (voir aussi [4]). aigoureusement, (4) sera v~rifi~e par la solution approch~e (ce sera m~me une ~galit~). Le terme de droite passe k la limite sans probl~me. Pour le terme de gauche, apr~s avoir fait les extractions habituelles, il suffit d'utiliser le fait que si Xm -+ x faiblement dans un espace de Hilbert, alors Iix[I < lim inf Iixml[. En ce qui concerne la continuit~ faible s valeurs dans H 1, on a classiquement que u E L~176 H 1) (par extraction d'une suite convergente dans L ~ 1) faible*). A partir de l'~quation, il est facile de voir que Otu E LI(0,~; H - k ) , pour k assez grand; par consequent u ~ C(0,~; H - k ) . Comme H k est dense dans H 1, le lemme 2.2 implique u e Cw ([0, 0]; g 1) pour tout 0 < T.

3. Preuve du th~or~me 1.1 La preuve s'inspire fortement de la d~monstration de l'existence des solutions faibles de l'(!quation de Navier-Stokes, et plus pr~cis~ment de la partie concernant le passage ~ la limite, tel qu'on peut la trouver dans [2], voir aussi [11]. Ici, la difficult~ consiste en l'obtention d'estimations ind~pendantes de a et de montrer que les termes suppl~mentaires par rapport ~ l'~quation de Navier-Stokes convergent vers 0 au sens des distributions. Dans la suite, l'hypoth~se v > 0 joue un r61e important. Ainsi, C d~signera une constante ind~pendante de v et a, qui peut changer d'une

D. Iftimie

463

indgalitd h l'autre. Pour alldger la rddaction, on notera a - ak et u -- u s = u ~k . Toutes les limites qui suivent ont lieu pour a - ak -+ O. La premiere dtape consiste en l'obtention d'estimations inddpendantes de a. 3.1. E s t i m a t i o n s u n i f o r m e s e n c~

Soit 0 < T fixd. On va utiliser l'indgalitd d'dnergie (4). Remarquons que

fot (f,

u) <

fot Ilfll~-~llullw <__~u for IlVu[l~ + ~1 for Ilfl [n-~ 9

En utilisant cette relation dans (4) on trouve

Ilu(t)ll2L2 + allVu(t)ll2L2 + u f0 t

IIVu(~-)l122d T <w IlUollL2 2

lfo'

+ ~llVuol}~= + -//

Ilfll~-x.

(10)

En tenant compte des hypotheses a), b) et du fait que f e n2oc ([0, T);/f/-1) on obtient de l'indgalitd ci-dessus que, pour tout 0 < T,

n~176 dans L2(O,O;L2);

u est bornd dans

(lla)

Vu est bornd

(llb)

al/2Vu

est bornd dans L~176 0; L2).

(llc)

Avant de pouvoir passer h la limite, il nous faut une certaine convergence forte. Une convergence forte peut s'obtenir si l'on dispose de l'dquicontinuitd en temps qui, h son tour, peut s'obtenir h partir d'estimations sur Otu. Revenons h l'dquation (9). On va estimer Otv dans un certain espace H -k, k assez grand, t~tudions chaque terme de (9), h l'aide des informations (11) et du thdor~me de produit 2.1:

9 uAu

est bornd dans L~176 0; H -2) car u l'est dans L~176 0; L2);

9 I1~" V~IIH-,-~/~ _< CllulIL~IlVUlIL~ donc

9 de m6me,

u- Vu

est bornd dans L2(0,0; H - l - a / 2 ) ;

UjOkU est

bornd dans L2(0, 0; H -1-d/2) donc

aEOjOk(UjOkU) ~ 0 j,k

dans

L2(0,0;H-3-d/2);

Remarques sur la limite c~ --~ 0 pour les fluides de grade 2

464

9

IlOkuj0~ullH-~-~/~ <_ CIIWlI~ done c~1/20kuj OkU est born6 dans L2(0, 0; H-l-d~2), d'oh

a E Oj(OkUjOkU) ~ 0 dans

L2(0,0; H-2-d/2);

j,k

9 de m~me,

a EOk(OkUjVUJ) --+ 0 dans

L2(0,0;H-2-d/2).

j,k

En ce qui concerne le terme de pression, le plus simple est d'appliquer la projection de Leray I? h (9) pour obtenir

j,k

j,k

+ ~ Z o~(o~jwj) + s). j,k

Comme I? est une projection orthogonale dans tout espace de Sobolev H s, il d~coule de la discussion ci-dessus que

Otv est born~ dans L2(0,0; H-a-d~2).

(12)

Or, on a la relation suivante:

Ila~ulIH" ~ Ila~VlIH',

(13)

pour tout s E ll~. En effet, si l'on note ~) - Ot(1-a)s/2v et ~i = Ot(1-a)s/2u, on

&

,,o~11~. -I,~,IL - I , ~ - ~A~,,~ = I1~11~.+ ~ I I A ~ I I ~ - 2~<~, a~> = I1~11~=+ ~lla~ll~= + 2~(W, W> = I1~11~=+ ~ l l a ~ l l ~ + 2~IIWlI~ > I1~11~=-II0~ull~. On d~duit de (12) et (13) que

Otu est born~ dans L2(0,0; H-3-d/2).

(14)

On dispose maintenant de tous les ~l~ments n~cessaires pour passer k la limite.

D. Iftimie

465

3.2. P a s s a g e h la l i m i t e On a d~j~ vu que

j,k

j,k

j,k

dans n2(0, o ; g - 3 - d / 2 ) ,

(15)

donc la convergence a lieu aussi au sens des distributions. Avec les informations (lla) et (llb) on peut extraire une sous-suite, encore notre u, telle que u -~ ~ dans L ~ ( 0 , O; L 2) faible*

(16)

u --~ ~ dans L 2 (0, O; H 1) faible,

(17)

et

pour tout 0 < T. De plus, grace ~ (14), on peut aussi supposer que

Otu ~ Ot~ dans n2(0, 0; g -3-d/2) faible.

(18)

Comme u --4 ~ au sens des distributions, on aura Ot/ku --40t/k~ au sens des distributions donc aOt/~u --4 0 au sens des distributions. Les seuls termes qui restent dans (9) sont exactement les mSmes que ceux de l'~quation de Navier-Stokes. Les mSmes arguments s'appliquent donc ici. I1 n'est pas n~cessaire de les reproduire en d~tail; on en donnera les grandes lignes seulement. Soit 0 fix~ et t, t t E [0, 0] arbitraires. On part de t ~ -

-

d'ofi P

lib(t)

-

<

t ~

<

it -

En se rappelant (14), il s'ensuit que les u - u s sont ~quicontinus (par rapport ~ a) dans C([0, 8]; H-3-d/2). Par le th~or~me d'Ascoli, pour tout compact K on peut extraire une sous-suite, encore notre u, telle que U]g converge fortement dans C([0, 0]; H-4-d/2(K)). En prenant une suite croissante de r~els 0m -4 T, de compacts Km - B(0, m), des sous-suites successives et en extrayant une suite diagonale, on voit qu'on peut supposer que

466

Remarques sur la fimite a -4 0 pour les fluides de grade 2

ulv converge fortement dans C([0,9]; H-4-d/2(U)) pour tout 9 < T et U ouvert relativement compact. Une in~galit~ d'interpolation simple ainsi que la relation (lla) montrent maintenant que ulv --9 u[u fortement dans C([0, 9]; Hs(U)) pour tout s < 0, 0 < T, et U ouvert relativement compact. C ~ ((0, T) x ll~n). On a, apr~s une integration par parties,

(19) Soit ~p E

Comme le domaine d'int~gration ci-dessus est un compact de (0, T) • I~n , on peut utiliser les relations (17) et (19) pour d~duire que T

T

ce qui revient h dire que u . V u - 4 ~. V~ au sens des distributions. Enfin, on a clairement que Otu --90tu et Au ~ A ~ au sens des distributions. Comme la limite d'une suite de gradients est un gradient, on obtient, apr~s passage h la limite dans (9),

O t ~ - u A ~ + ~. V ~ -

- V p + f,

donc ~ est une solution de l'~quation de Navier-Stokes qui v~rifie

C LIo~176

Otu E L~oc([O,T);H-3-d/2),

oh on a utilis~ (16), (17) et (18). La continuit~ faible r~sulte de la continuit~ forte ~ e C ([0, T); H -3-d/2) (consequence de Ot~ e L~oc ([0, T); H -3-d/2)), de l'appartenance de ~/~ L~oc ([0, T); L 2) et du lemme 2.2: e Cw ([0, T); Le). La donn~e initiale de ~ v~rifie, grace/~ la relation (19),

~(0)lv -~ ~(0)lv, au sens des distributions pour tout U ouvert relativement compact. Comme on a aussi que

u(0) -~ ~0,

D. Iftimie

467

au sens des distributions, il s'ensuit que uolu - u(0)lu pour tout U ouvert relativement compact. Par cons@quent, Uo - ~(0). Pour terminer la preuve, il reste ~ montrer la Remarque 1.4. Supposons que u(0) --4 u0 fortement dans L 2 et al/2u(0) tend vers 0 dans H 1. Sous ces hypotheses, en utilisant aussi (17), la partie droite de (4) tend vers

II oll = + 2

(f (T), ~(T)) d~-.

Quant ~ la partie de gauche, remarquons d'abord que par (lla), (19) et par le lemme 2.2 on a que u(t) ~ ~(t) faiblement dans L 2 pour tout t C [0, T). Rappelons maintenant que si Xm -4 x faiblement dans un espace de Hilbert, alors Ilxll < liminf IlXm[I. I1 ne reste plus qu'~ utiliser (17) pour minorer la limite sup~rieure du terme de gauche de (4) par

II (t)ll + 2y fot ]]V~(T)iI~2 dT, ce qui conclut la preuve de l'in6galit@ d'@nergie (3) pour ~. Le th6or~me 1.1 est compl~tement d@montr@, m Remarque 3.1. D'autres r@gularit@s en espace peuvent @tre consid~r@es pour le terme de force f. Si l'on supposait par exemple que f e L2oc([O, T); L2), il suffirait de majorer f t (f, u/ <_ fot ilull~ 2 + fot Ilfll~2, de modifier ensuite l'in@galit@ (10) et d'appliquer le lemme de Gronwall pour obtenir les m@mes estimations a priori (11).

R6f6rences [1] J.-Y. Chemin, Fluides parfaits incompressibles. Ast@risque, 230, 1995. [2] J.-Y. Chemin, M@thodes math@matiques en m@canique des fluides, I. 1997. Cours de DEA et Preprint Laboratoire d'Analyse Num@rique A97004. [3] D. Cioranescu et V. Girault, Weak and classical solutions of a family of second grade fluids. Internat. J. Non-Linear Mech., 32, 2 (1997), 317-335. [4] D. Cioranescu et E. H. Ouazar, Existence and uniqueness for fluids of second grade, in Nonlinear partial differential equations and their applications. Coll~ge de France seminar, Vol. VI (Paris, 1982/1983), 178-197. Boston, MA, Pitman, 1984. [5] P. Constantin et C. Foia~, Navier-Stokes equations. Chicago, University of Chicago Press, 1988.

468

R e m a r q u e s sur la limite ~ ~ 0 p o u r les fluides de grade 2

[6] J.E. Dunn et R.L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade. Arch. Rational Mech. Anal., 56 (1974), 191-252. [7] G. P. Galdi, M. Grobbelaar-van Dalsen, et N. Sauer, Existence and uniqueness of classical solutions of the equations of motion for secondgrade fluids. Arch. Rational Mech. Anal., 124, 3 (1993), 221-237. IS] G. P. Galdi et A. Sequeira, Further existence results for classical solutions of the equations of a second-grade fluid. Arch. Rational Mech. Anal., 128, 4 (1994), 297-312. [9] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math., 63 (1934), 193-248. [10] R.S. Rivlin et J.L. Ericksen, Stress-deformation relations for isotropic materials. J. Rational Mech. Anal., 4 (1955), 323-425. [11] M. E. Taylor, Partial differential equations. III, New York, SpringerVerlag, 1997. [12] R. Temam, Navier-Stokes equations, Amsterdam, North-Holland, 1984. Dragos Iftimie IRMAR Universit~ de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex France E-mail: [email protected]

l.fr

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 21 REMARKS ON THE KOMPANEETS EQUATION, A SIMPLIFIED MODEL OF THE FOKKER-PLANCK EQUATION

O. KAVIAN

1. Introduction In this talk we present a few results regarding the Kompaneets equation described below. These results are part of a joint work with C.D. Levermore, which will appear in more details elsewhere. The equation we study concerns a physical model in which radiation interacts with the electrons and ions of a fully ionized plasma, primarily through scattering with electrons and emission-absorption. In each of these processes energy is exchanged between the photon and particle fields. In the low density-high temperature regime, scattering is the dominant mechanism for the energy exchange. This process was quantified by A.S. Kompaneets [6] for the weakly relativistic limit. In this model, for a spatially homogeneous plasma held at a constant temperature 0, the radiation distribution f at time t is only a function of x : - hv/O, which is the photon energy normalized by the temperature (h is Planck's constant and ~ the photon frequency). This radiation distribution f is defined so that the total photon number density is given by:

N[f] "=

f(t,x)x2dx.

(1.1)

For a scattering process this quantity must be conserved. The Kompaneets equation governing f is:

O t f = x-20~[c~(x)(Oxf + f + f2)]

for 0 < x < c~

(1.2)

where a(x) > 0 on (0, c~). Classically a(x) = x 4 and that is the case we will treat here. Physically f must satisfy the boundary conditions : lim a(x)(Oxf + f + f2)(x) = lim a(x)(Oxf + f + f2)(x) = 0, X---~O

X--+OO

(1.3)

470

Remarks on the Kompaneets equation

which express the conservation of photon number density. These nonlinear boundary conditions can be satisfied by seeking a solution in an appropriate functional space. We point out here the fact that equation (1.2), written formally in an expanded form, reads:

o~f = x 2 0 ~ f + (~20~f + 4xo~f) + 4~f + ~20~(f2) + 4~2f 2.

(1.4)

As far as the order of the principal part of the differential operator and the degree and the type of nonlinearity are considered, equation (1.4) reminds of the Burger's equation

0~ = O~u + 0~(~). However it is clear that the singular term X2Oxxf, as x ---, 0 +, makes equation (1.4) more delicate to handle. But we attract the reader's attention to the fact that this is not the only difficulty with equation (1.4). Indeed, the terms x20~f and x f add some degree of difficulty to the appropriate study of that equation, especially if one wishes to prove a local or global existence theorem in a Banach space of the type LP(d#) for some p E [1, +c~] and some positive measure d#. In an attempt to convince the reader, we may give the following example of a linear equation, which contains terms of the above mentioned nature (however slightly different because we want to show an explicite example). For x E ]R consider the linear equation 1

{ o~(t, x) = o ~ - xO~ + 4~2~

(1.5)

~(o, ~) = ~o(x). Now the principal part of the differential operator is Ox~u, which is not singular, but if one tries to prove a global existence result for a given u0 C LP(]R) for 1 < p < +c~, one sees that the usual techniques do not apply. For example when uo(x) := e x p ( - x 2 / 4 ) , then one may show that the unique solution of (1.5) in LP(]R) is given by

u(t,x) "= (2t + 1)-1/2et/2 exp/'[ ( 2 t - 1)x2}\ \ 4(2t + 1) / " Clearly this solution blows up in LP(IR), and this for

any p, at time Tmax -

1/2. This is to emphasize that, in some cases, it is not a good idea to consider such terms as xOxu or x2u/4 as perturbations of Oxxu, the principal part of the differential operator. It is also clear that if one considers the solution

O. Kavian

471

of (1.5) in the weighted Lebesgue space LP(d#), where d# "= e x p ( - x 2 / 2 ) , with the linear operator u ~-~ Ox~u-xOxu, such difficulties are easily avoided. Regarding the Kompaneets equation (1.2)-(1.3), we are going to construct a solution in the Hilbert space

{ /0

L2 "= u ;

lu(x)12xe~dx < c~

}

(1.6)

endowed with its natural norm and scalar product. To do so we consider the unbounded linear operator ( B , D ( B ) ) defined by

Bu "= x G u + ( x - 1)u = e-Xx20~(x-le~u),

(1.7)

D(B) "= {u E L2 ; Bu E L2}.

We will show that B is closed and that D(B) = D(B*). Then, as one can be convinced after some simple calculations using the fact that

B* u = - x O x u - 3 u ,

and

B* Bu = - x 2 0 ~ x u - ( x 2+3x)O~u-(4x-3)u,

we write equation (1.2) in the following form:

I Off + B * B f = - B * ( f + x f 2) f(t,.)eD(B*B)

on (0, T) x (0, c~) for t e (0, T)

(1.8)

f(0, . ) = knit(')In this setting the boundary conditions (1.3) are derived from the fact that we seek solutions satisfying f(t) E D(B*B), when fInit is an appropriate given initial data. Our main result is the following. T h e o r e m 1.1. - For any given fInit E D(B*B)), there exists T > 0 such

that qu tio (1.8)

u.iq,,

olutio. f e C([O, TI,D(B*B)). D

oti.g

by Tmax "-- T m a x ( f I n i t ) the supremum of all such T's, then either Tm~x = +ce or Tmax < (x~ and lim [[xu(t)lTo~- +oo. t--~Tmax

Also a comparison principle holds for solutions of (1.8), i.e. if two intial data fInit, fInit E D(B*B) satisfy fInit ~ fInit then the corresponding solutions satisfy f (t) <_ f (t)

for

t < min(Tmax(fInir

Tmax(fInit)).

YVe prove also that unlike Burger's equation, the Kompaneets solutions which blow up in finite time:

equation has

472 Propositionl.2. and

Remarks on the Kompaneets equation - Assume that fInit E D ( B * B ) ) , is such that fInit k 0

/

oo flnit(x) x 4 e - x d x > 2,

then the corresponding solution blows up in a finite time Tmax < oo. One m a y prove global existence of solutions for a large class of initial data. First recall t h a t the equilibria for (1.2) are obtained by setting

Gf + f + f2=O, yielding the one p a r a m e t e r family of positive solutions f = ~ , 1

~(x)

9 e x+~'

-

1

for # >_ 0,

which are the classical Bose-Einstein equilibria distributions. Now it is clear t h a t by the comparison principle stated in theorem 1.1, if 0 <_ fInit ~ ~/z for some # > 0 then Tmax = +oo. However we point out the fact that our functional space setting is not optimal, as the Bose-Einstein equilibrium ~0 does not belong to L2. Before going into the details of the proof of these results, we would like to mention t h a t the main ingredient in the proof of T h e o r e m 1.1 is the following Nash type lemma, which, we believe, is interesting in its own right. L e m m a 1.3. - Let H be a Hilbert space, ( A , D ( A ) ) a densely defined, dosed linear operator acting on H and a Banach space X such that D ( A ) N X is dense in H. Assume that A*A generates the continuous semigroup S(t) := e x p ( - t A * A ) which satisfies

V~o e x,

IIS(t)~ollx _< co II~ollx.

(1.9)

Then the following properties are equivalent: (i) There exist Cl > 0 and fl > 0 such that for all u c D(A) r3 X one has

[llt[[~/+/3 _~ C1 [IAu]]~ IIu[l~x.

(1.10)

(ii) There exist c2 > 0 and fl > 0 such that for all uo C X one has Vt > o,

]lS(t)uollH <_ c2 t -1/~ Iluollx.

(1.11)

O. Kawian

473

This result is essentially well known, although we give here a slightly different and general expression for it. Indeed since the works of J. Nash [7], M. Fukushima [4], N. Varopoulos [10] and E.B Fabes s W. Strook [3], it is known that there is an equivalence between the regularizing effect (or more precisely the ultracontractivity) of a heat semigroup on one hand, and Nash or Sobolev inequalities on the other hand. More precisely (see for instance E.B. Davies [2]), assume that L = L* is a positive selfadjoint operator acting on L2(d#) and such that the semigroup S(t) :-- e - t L is positivity preserving and maps continuously Ll(d#) into itself, then denoting I]" lip the norm in LP(d#), the following properties are equivalent: 1). there exist two constants Cl > 0 a n d / 3 > 0 such that for all p E D(L) n Ll(dp) one has

11~!!~(2/3+4)/~__'(Cl !1 114/z ( L ~ I ~ )

"

2). there exist two constants c2 > 0 and fl > 0 such that for all ~ c

L l(d#) and all t > 0 one has

Ils(t)~lloo _< c2 t-~/2ll~lll. Moreover if in either of above properties one has /3 > 2 , then they are equivalent to the following Sobolev embedding: 3). there exist two constants c3 > 0 and t~ > 2 such that for all ~ E D(L) one has ]]PI[2~/(Z-2) _< c3 (Lp]p). (The n u m b e r / 3 may be thought of as a geometrical dimension for the underlying measure space). The remainder of this note is organized as follows. In Section 2 we establish some of the properties of the operator B defined in (1.7) and we prove lemma 1.3. In Section 3 we prove local existence in time and uniqueness in an appropriate space using semigroups and fixed point theorem. In Section 4 we study the qualitative properties of the solutions and we discuss global existence for some solutions, as well as demonstrating finite time blow-up for some initial data.

Remarks on the Kompaneets equation

474 2. P r e l i m i n a r y

results

In order to solve the Kompaneets equation (1.2), we introduce the Hilbert space L2 :=

{ fo lu(x)12xe~dx } u ;

< oo

(2.1)

with its natural inner product ( u [ v ) : = f o u(x)v(x) xe~dx and associated norm. On this space we consider the differential operator (B, D(B)) acting on L2 and defined by

B u "- e-~x20~ ( ~ u )

= xO~u + (x - 1 ) u

(2.2)

with domain

D(B) :=

{

u e L2, 9

/o

xhe -

~

I(

O~ e~u(x) X

)l

2d x

< oe

}

.

(2.3)

In order to determine the adjoint of ( B , D ( B ) ) we introduce an operator (B1,D(B1)) (which actually is the formal adjoint of B) by setting

B l u := -(xOxu + 3u) = -x-2Oz(xau), D ( B I ) := {u e L2 ; B l u r L2}.

(2.4)

As usual for a closed operator (B, D(B)) we define its graph norm by

II~llD(m := (11~112+ IlBull 2) 1 / 2

for

u C D(B).

L e m m a 2.1. - (B, D(B)) and ( B 1 , D ( B , ) ) a r e densely clel~ned closed operators in L2 and C~(O, oo) is dense in D(B) and in D(B1) endowed with

their respective graph norms. Moreover for any u c D(B) U D(B1) one has the boundary limits lira xu(x) = 0 = lira e x/2 xa/2u(x). x--+0

(2.5)

x--* Cx:~

In particular there exists a constant C > 0 such that for u E D(B) (resp. u e D(B1)) one has II(x + xa/2e~/2)u[Io~ < CIlullD(m (resp.

II(z

+

xa/2e~/~)ull~ < CIi~IID(B,)).

Proof. The density of C ~ ( 0 , oe) in D ( B ) , or D(B1), equipped with the graph norm is straightforward. Let u be in D(B); for any 0 < xa < x2 < oe we have the basic estimate : X2

eX'uXx'l=Xx

(eX,X,x )dxl = J

3xox xl

1

(2.6)

O. Kavian

475

Using the rough estimate obtained from

( f ~ 2 --~dx) e~ 1/2 <eX2 /2 ( fz ~ d x l / 2 1

--

1 "-~)

-

ex2/2

--

2X21'

(2.7)

we begin with the small x limit. From the above rough estimate and (2.6) we get XllU(Xl)I

Letting

--- x 2

X l --~

oxl l UXl( X 1)1

I

~ X2 eX2u(X2)

9

1

x2/2

0 in the right hand side, followed by lim

x---+O

x2

12xeZdx)

X2 X2 ~

0,

yields the result

xu(x) = O.

The large x limit is obtained in two steps. First, (2.7) and (2.6) give

eZ2/2]u(x2)l

--e

_x2/2e~2u(x2) x2

X2

<_

e l (Xl) + Zl

Letting x2 --+ c~, followed by

Xl -'~

lira

( ~1

IB (x)l 1

(:X:),we obtain the intermediate result

e~/2lu(x)l

x---,c~

X

= 0.

Returning to the basic estimate (2.6), set xl := x2/2 and multiply by 5/2 -~2/2 x2 e to get

x3/2e~" /2 lu(x2 ) [ ~_zx 2 _.}_(xSe_~ 2

2 eX.zdx )1/2 . ( /~21Bul2xeXdx)

1/2

Letting x2 --* oc, the first term on the right hand side vanishes by the above intermediate result, while the second term vanishes because Bu C L2 and limsup

x5 e -x2

eX

--~dx < oc.

X2--~OO

This establishes the result for u E D(B); if u c the same and based on the basic estimate (2.6).

D(B1) the proof is much m

476

R e m a r k s on the K o m p a n e e t s equation

Lemma

2.2. - T h e adjoint B* o f B in L2 is given by -1 B * u = --~O~(x3u) = - ( x O ~ u + 3u)

(2.8)

and its d o m a i n is D ( B * ) "--

{

u e n2 "

and ( B * , D ( B * ) ) = ( B I , D ( B 1 ) ) . b o u n d a r y conditions (2.5).

/o

7glO~(x3u)12dx < oo

}

.

(2.9)

In particular any u e D ( B * ) satisfies the

Proof. From lemma 2.1, we have t h a t for any u c D ( B ) and v e D(B1): ( v l B u ) -- ( B l v l u ) = -

-

(eZ-----~u] -4X oo [x2eZv(x)u(X)]o O. -

' X

Oz(xv 3) dx

-

Therefore we may conclude t h a t v E D ( B * ) and u C D(B~), thus the inclusions ( B I , D ( B I ) ) c ( B * , D ( B * ) a n d ( B , D ( B ) ) c ( B I , D ( B ~ ) ) in the sense of operators. Now let v E D(B*); then for any ~ e C ~ ( 0 , oc) we have (vlB~) = ( B * v l ~ ) by the definition of B*. In particular this implies v C Hllc(0, c~) (i.e. O~v is locally square integrable for the Lebesgue measure on (0, oo)). Repeating the calculation above, one sees t h a t (vlBqa) = (Blvlp); from which we infer B l v C L2, i.e. v E D(B1). 1 Lemma

2.3. - We have D ( B ) - D ( B * ) and for any u e D ( B ) one has

IIB*~ll 2 - I I B ~ I I 2 + I!xl/2~112, Also for u E D ( B B * )

IlB*ull 2 _< 21lBu[I 2 + 5ll~ll 2

(2.10)

we have B B * u = B * B u + xu.

Proof. For u c C ~ ( 0 , oo) we have B B * u B * B u + xu, which yields the first identity for u c C ~ (0, oo). On the other hand note t h a t ( B* + B )u - - 4 u + xu,

and (B* - I ) ( B - I ) u + x u = B * B u - (B* + B ) u + u + x u - B * B u + 5u

which implies that

IIx~/2~ll 2 _~ I I ( B - I)ull 2 + [[xl/2tt[[ 2

--[IBu[I

2 -[- 51IUll 2"

This, combined with the first identity, gives the claimed inequality. The operators B* and B being closed, by a density argument one sees t h a t D(B)D ( B * ) and that relations (2.10) are valid for all u E D ( B ) . m We have also a Poinca% inequality for B*, namely we have:

O. Kavian

477

L e m m a 2.4. - For any u C D ( B * ) one has

3[lull 2 _< I[B*ull 2. Proof. For u c C ~ ( 0 , oc), the result follows from the identity B B * u = 3u + B~B2u, where B2u := xe-XOx(e-Xu) and B~u = - x - l O x ( x 2 u ) (recall that the adjoints are taken with respect to scalar product in L2). I From the above we can conclude that the selfadjoint operators B * B and B B * generate contractive analytic semigroups on L2; they are denoted by e - t B * B and e - t B B * respectively. Some important properties of these analytic semigroups are gathered in the following lemma. L e m m a 2.5. - (i) For any u E D ( B * ) = D ( B ) one has e-tBB*u C D ( B * ) and B* e-tBB*u = e -tB* B B* u. (ii) For any u C L2 and t > 0 one has D ( B ) , with the estimates IIB*e-tBB*ull < ct-1/21tui],

and

e-tBB*u

E D ( B * ) and e-tB*Bu E

lIBe-tB*Bu[I < ct-1/2]iu]l .

(iii) If u C L2, and u > 0 then e-tBB*u > 0 and e-tB*Bu > O. We omit the proof, which is standard (see for instance M. Reed & B. Simon [9]). Using the operators B and B*, equation (1.2) (with c~(x) "= x 4) can be written in the form Off = - B * ( B f + f + x f 2) f(0, x) = fInit(x) ~__0,

(2.11)

and in its mild formulation f (t) = e -tB* B finit

-

-

~0 t B*e -t-s)BB* ( f + x f 2 ) ( s ) d s ,

(2.12)

where we are using lemma 2.5 (i) to help write the integral term, but more is needed to control the nonlinear term. Actually we will need to prove a regularizing property of the semigroup e -tBB*, namely t h a t it is ultracontractive in an appropriate sense. As it is known, a useful property of the classical heat semigroup in ]R g is t h a t any initial data in L 1 (JR N) is mapped into Lc~(IR N) for any time t > 0. Using this property one can develop existence and uniqueness theories for semilinear perturbations of the

478

Remarks on the K o m p a n e e t s equation

heat equation. This so-called ultracontractivity property may be proved by knowing the heat kernel and using Young's inequality for convolutions. However (see E.B. Davies [2] for a general discussion) for general semigroups one may use the equivalence of properties 1), 2) and 3) given at the end of section w 1. Here we establish a Nash inequality related to B* in order to prove the ultracontactivity of the semigroup e -tBB* . We begin with the proof of the general lemma 1.3. P r o o f of Lemma 1.3. Assume that property (i) of lemma 1.3 is satisfied. Let uo E X A D ( A ) be given and set u(t) : - e-tA*Auo be the solution of Otu - - A * A s ,

u(O) - so.

Then if ~(t) := Ilu(t)ll~, one has ~'(t) = - 2 1 1 A u ( t ) l l fact that Ilu(t)llx <_ coll~ollx, we have:

2.

Using (1.10) and the

-~'(t) > 2~yllIu(t)IIx~ll~(t)II~ +n > 2ci-lCo~II~ollx z ~(t) (2+~)/2. This can be written in the form 2~-1(~(t)-Z/2)' > 2c~-lcoZlluoll X- z from which we infer that

~(t)-z/2 > ~ci-~coZll~ollx~t + ~(o) -~/2 >_/3~i-lCoZll~ollx~t. Clearly this yields (1.11), that is Ilu(t)llH (C1/~)I/J3Co

<_ c2t-1/Zlluollx

with c2 "=

.

Conversely, assume that property (ii) of lemma 1.3 is satisfied. Then for uo E X N D ( A ) we have, with the above notation for u(t)

c~t-2/~ll~oll2x

> (u(t)lu(t))H --I1~o11~

+

f0 t ~ll~(~)ll~d~ d

> 11~o11~-2/o t IIAS(~)IISd~ > I!~o11~

_

-

2tllAuoll 2H

where we have used the fact that u(t) satisfies Otu = - A * A u and therefore IIAu(t)llH < IIAuollH (it is enough to take the scalar product with Otu). Finally we obtain

I1~o11~ < 2tllAuoll~ + c~t-2/~lluoll X2 "

479

O. Kavian

Upon minimizing the right hand side for t > 0, i.e. setting

( c,~lluoll~-) ~/(~+2) t.--

DIIAuolI2H

one obtains inequality (1.10) of the lemma, m In order to get the regularizing property of the semigroup e - t u B * , we apply lemma 1.3 to A := B* by proving inequality (1.10) when x

{

.-

~ ; II,~llx " -

/o

lu(x)leX/2dx < c~

}

.

(2.13)

We begin by seeking a pointwise estimate for u E D ( B * ) . L e m m a 2.6. - For any u C D ( B * ) = D ( B ) we .have

e-X~2 l u ( ~ ) i _< v " i ....... x

IIB*uJl 1/2 I1~il 1/2

Proof. For u E C ~ (0, c~) we have y3u(y)Ogy(y3u(y))dy = 2

(X3U(X)) 2 -- --2

yhu(y)B*u(y)dy

y 4 e - y , lu(y)B*u(y)]yeYdy.

~ 2 X

Now for 4 <__x < y one has y 4 e - y ~_ x4e - x . So for x >_ 4 we get lu(x)[ 2 < 2x-2e-~llullilB*uil , which is the desired result for x >_ 4. beginning with the identity (X3U(X)) 2 - - 2

'0 x

When x < 4 we argue similarly

y3u(y)Oy(y3u(y))dy.

m Next we prove t h a t the semigroup e -tBB* acts on X. L e m m a 2.7. - The semigroup e -tBB* induces a contractive, p o s i t i y i t y _preserving semigroup on X ; more precisely we have

lle-*BB*uoIIx

< e-Tt/4lluollx.

Remarks on the Kompaneets equation

480

Proof. It is sufficient to prove the lemma for uo E X such that Uo _> 0. Now for uo E C ~ ( 0 , cx~), and uo > 0 set u(t) := e-tBS*uo, that is Otu = - B B * u and u(0) = uo. One has u(t) > 0 and u(t) E D(BB*) while d

dt

f0

u(t)eX/2dx

BB* u eX/2dx oo

e x

= fO x2e-X/2Ox("~Ox(x3u))dx" But integrating by parts twice yields _d_ / u(t,x)eX/2dx = fo ~176 u(t,x)x30z ( 1~-~eX/2(2- gx ) dx

dt

= - fo ~176 u(t,x) IX-~ + (x -4 4)2]eX/2d x < - 7 fo ~ u(t x)eX/2dx, 4 and the lemma follows for uo _> 0. 1 Finally in the following lemma we state the desired regularizing property of the semigroup e - t B B * . L e m m a 2.8. - For any u E X N D(B*) one has

II~[I6 < 4[[B*~l1211ull 4 X

~

and for ali t > 0 and uo E X one has Ile-'BB'Uoll _< t-1/4 Iluollx.

(2.14)

Proof. Using the uniform estimate given in lemma 2.6 we have I1~112 =

/0

[u(x)12xeXdx =

/0

x}u(x)l lu(x)leXdx

< v~fIB*ulll/2llul]~/2

/o

[u(x)leX/2dx,

~nd hence 1[~113/2 _ v~llB*~lll/211~lIx. Next, using lemm~ 2.7, ~s ~ corollary of lemma 1.3 we obtain (2.14). 1 Combining this result and lemma 2.5 (ii) we have the following regularization result: C o r o l l a r y 2.9 . - For u0 E X and all t > 0 one has

IIB*e-tBB*uoII <_ ct-3/411UoIIx.

O. Kavian

481

3. E x i s t e n c e of local s o l u t i o n s In order to solve the mild formulation of the Kompaneets equation given by (2.12) we use the classical approach of the fixed point method, when the time t belongs to a sufficiently small interval [0, T]. Given T > 0, we denote by C([O,T],L2) the space of continuous functions from [0, T] into L2, equipped with the norm

II~IIc([0,T1) :-- sup II~(t)ll. 0
For u E L2 we define the mapping: (~(u))(t) :=

jl

t B , e - ( t - s ) B B (u(s) + xu2(s))ds.

L e m m a 3.1. - For any T C([0, T], L2) into itself and

(3.1)

> O, fro is a locally Lipschitz map from

I1~(~) - ~(~)!I _< e(T, I!~ + vllc~)- 11~- vllc~ with

e(T, I1~ + ~11) <

c( T1/2 + T1/4[lu +

~[Ic(to,T])).

Proof. Using lemma 2.5 (ii) we bound the linear term of the integrand by:

llB*e-(t-~)BB*u(s)ll <_ c ( t - s)-l/211u(s)l t and the nonlinear term by using corollary 2.9

IIB*e-(t-~)BB*xu2(s)I! <_ c(t -- s)-3/411xu2(s)llx <_ c ( t -

$)-3/4 llu(s)ll 2.

From this one infers that t ~ (~(u))(t) is HSlder continuous of exponent 1/4. Now for u, v E C([0, T], L2), using the above estimates we may write:

It~(u)(t) - ,I,(v)(t) !1 <_cT~/2 I!~- ~tlc(to,Tl) nu c T 1 / 4 1 [ u nu v[lc([o,T])llu - vllc([o,T]).

(Here we have estimated roughly all the constants, which are not relevant in this study). II We can now state the existence and uniqueness result for the mild formulation (1.8), the proof of which is classical.

482

Remarks on the Kompaneets equation

T h e o r e m 3.2. - For any fInit E L2 there exists T > 0 such that a unique solution f of the mild formulation (2.12) exists and f E C([0, T], L2). Concerning existence of strong solutions when fInit E D(B*), we proceed as follows. First we set 9 ~(u) "= B* (u + xu2),

(1.13)

then we recall that (2.12) can be written as t

f(t) = e-tB'B finit

-

-

o•o

e-(t-s)B'Bff~(f(s))ds.

(3.2)

Using standard Hilbert space techniques, which we omit (cf. M. Reed and B. Simon [9]), we have the following characterization of the domain of (BB*)I/2: L e m m a . Define A "= (BB*) :/2. Then D ( B ) = D(A), the inverse A -1 exists and l I B * A - l u l l - Ilull. With a slight modification of the proof given in A. Pazy [8] (theorem 3.1, section 6.3) one may state the following T h e o r e m 3.3. - Let ~ and A be defined as above. Then for any fInit C D ( B ) , there exists T > 0 such that equation (2.12) has a unique solution f e C([O,T],L2)n CI((O,T),L2). Moreover f e C ( [ O , T ] , D ( B ) ) a n d in particular f is a strong solution to

Otf -- - B * ( B f + f -4- x f 2) f ( o , ~) - fi.it

(~).

on (0, T) x (0, oo)

on (0, oo)

Moreover if Tmax is the maximal time of existence, then either Tmax = +oo or else one has Tmax < +cxD and limtTT,,ax IlB f(t)ll = +oo. Proof. We begin by checking that q : D(A) ~ L2 is locally Lipschitz. Using the pointwise estimate of lemma 2.6 and the fact that B*(xu 2) = 2xuB*u + u, we have easily I1~(~) - ~(v)ll ~ c (1 + IfB*ull + liB*vii)[IB*u - B'vii and also

II,I'(U-l~) - r

~ C (1 + II~ll + Ilvll)11~ - vii

(3.3)

O. Kavian

483

for some constant C > 0 (we use also the fact that [lull _< IIB*ull by lemma 2.4). Now for u c C([0, T], L2) we define: t

(F(u))(t) := A e - t B B * f I n i t -

Ae-(t-s)B*St~(A-lu(s))ds.

It is straightforward to show that

tlF(u)(t) - F(v)(t)] I <_ C . (1 + Ilu]lc([o,T]) + IIvllc([o,T])) x ilu - vllc(to,Tl)

(t - s ) - l / 2 d s

and therefore

[IF(u)- F(v)llc~ <_ c (1 + II~IIc([o,T])+ IIVlIc(Eo,TI)) T 1/2 ilu- vtlc([o,T]). From this one classically infers that, for T small enough, there exists a unique u e C([0, T], L2) such that t

u(t) -- Ae-tB*B fInit -

Ae-(t-s)A*Aq2(A-lu(s))ds.

(3.4)

Now one may check t h a t as t H u(t) is Hhlder continuous on [0,T] --. L2, and 9 is locally Lipschitz continuous, t ~-. ~ ( A - l u ( t ) ) is also Hhlder continuous. But knowing that e -tB*B is an analytic semigroup the function t

f ( t ) :--- e - t B * B fInit --

e-(t-s)B*Bt~(A-lu(s))ds

(3.5)

is a strong solution to the initial value problem

Otf + B*B f = t~(A-lu(t)) f (O) = k.i~. (This means that f c C([O,T],L2)N CI((O,T),L2)). One sees also that because of the uniqueness of the solution to (3.4), f defined by (3.5) satisfies f(t) = A - i t ( t ) and therefore t

f(t)

= e-tB*B

fini t --

and the proof of the theorem is done.

e-(t-*)B*Bt~(f(s))ds m

Remarks on the Kompaneets equation

484

4. Qualitative properties, global existence or blow up In order to prove a comparison result, we observe first t h a t B being a first order differential operator, if y) c D(B), then y)+ := max(y), 0) E D(B) and B(Y) +) = By)l[~>0]. Recall also t h a t if y) c D ( B ) , then by lemma 2.1 we have I[xy)[[cr < Cl[y)[lD(S), for some constant C > 0.

Proposition

4.1. - Assume that fl,Init _~ f2,Init are two initial data in D(B) and denote by fl and f2 the corresponding solutions in Cl((o, T), L2) N C ( [ O , T ] , D ( B ) ) for s o m e T > O. T h e n one has fx ~ f2 on

[0, T] x (0, cx3).

In particular if fInit ~ O, then f >_ O. Proof. The argument is a classical one: set g := fl - f2 and q(t,x) := x ( f i -F f2); recall t h a t due to the above remarks q E LCc((O,T) x (0, oc)). As g satisfies the equation

Otg + B*Bg = - B * (g + qg), upon multplying this by g+ in the sense of the scalar product of L2 we get

d~

Ig+(t'x)[2xeXdx + -

/o /o

_< e

/0

IB(g+)(t'xl2xeXdx-

g+B(g+)(t,x)xe~dx -

IB(g +)(t,

/o

q(t,x)g+B(g+)(t,x)xe~dx

zl2ze~dz

+ (c(~)+ I]q(t,-)ll~)

Ig+(t,x)[2xe~dx

where we have used Young's inequality a/3 _< ~a 2 + Ce/3 2. Choosing ~ = 1/2 for instance, Gronwall lemma and the fact t h a t g+(O,x) - 0 imlpy g+ (t, x) -- 0, t h a t is fl _< f2II The same observation leads to the following: Lemma

4.2. - ff f is a solution such that Tmax < cx~, then l i m s u p I[zf(t, ")[1~ = +oc. t TTm~x

Proof. Indeed, assuming that Tm~x < oc and Ilxfll~ ~ c for some constant C > 0 and all t < Tma• multiplying equation (2.11) by f, using the approach developed above we get d~

[f(t,z)

,2xeXdx + -~1/o

]B f(t,x)[2xeXdx <_ C

/o

[f(t,z)[ezeXdx,

O. Kavian

485

which first implies, via the Gronwall lemma, an upper bound for IIf(t)]l <_ M := M(Tm~x) < ce for t < Tm~x. Now using the fact that B * ( x f 2) = - 6 x f 2 + 2 x 2 f ~ - 2 x f B f , we obtain, via the estimates of lemma 2.1, for some constant C > 0 depending on M, [l~(f(s))ll <_ C(1 + IIBf(s)ll), and

IIB f(t)IF <_ IIBe-tB'B finitil + C

jfot (t -

s)-

/21lB f ( s ) l ] d s ,

where we have also used the regularizing property of the semigroup e - t B * B given by lemma 2.5 (ii). It is clear that the generalized Gronwall lemma yields a bound on IIBf(t)ll for t < Tmax. This contradicts the assumption Tmax < C<). m We point out that the comparison result together with lemma 4.2 gives also a criteria for global existence. Indeed if for instance for some # > 0 we have flnit ~ ~tt(X) :-- (e x+tt -- 1) -1, then the solution is global, because this implies that f ( t , x ) <_ ~ , ( x ) and therefore for all t > 0 we have Ilxf(t, ")11 <-

llx .ll

<

Regarding the blow up results, we give here a simple criteria for blow up of solutions, and we refer to [5] for other results. It is clear that the criteria below applies not only to the solutions constructed above, but also to any solution for which the quantities which are manipulated in the following lemma, make sense. 4.3. - Denote p(x) := e -~. Assume that f is a nonnegative solution of equation (2.11), such that t H f o f(t, x ) ~ ( x ) x 2 d x is of classe C 1 on an interval [0, T] and such that

Lemma

~0

+ / ( t , x ) ) f ( t , x ) p ( x ) x 4 d x < ce

~

for ~ll t E [0, T]. Then if

j~~ oo

a0 :--a0(fInit)"'-

fInit(x)~(x)x2dx > 2

there exists T, (ao) < oo such that T ( T, (ao). Proof. Let h(t) "- f o f (t, x)~(x)x2dx. One checks t h a t h'(t) -- = _

x4(Ozf + f -t- f2)Ox~(x)dx

~00~176

/o

+

/o

( f + f2)x4

(z)d

Remarks on the Kompaneets equation

486

where we have performed another integration by parts on the first term involving O~f. As O~(x4~(x)) = (4x 3 - x4)~(x), we get finally

h'(t) =

/0

f(t,x)(2x 2 - 4x)x2~(x)dx +

Upon using the fact that 2x 2 inequality

/0

f(t,x)2x4~(x)dx.

4x > - 2 in the first term and Jensen's

J~o f(t'x)2x4~(x)dx >-

(/0

f(t'x)x2~(x)dx

in the second term, we end up with the differential inequality ht(t) >_ - 2 h ( t ) + h(t) 2, or equivalently

(e2th(t))' >_e -2t (e2th(t)) 2 " This implies that e -2t

1

0 < h--~ < h(0)

1 2 ~

e -2t 2 '

and therefore T < T.(ao)"= -log((h(0) - 2)/h(0))/2. One may check also that the test function ~(x) " - e -x may be replaced by a function of the type ~(x) "- e - ~ for some )~ > 0. In this case one may see that if the initial data satisfies:

j~

oo fInit (x)e- ~x x 2dx >

then the corresponding solution blows up in a finite time.

B

References [1]. R.E. Caflisch and C.D. Levermore, Equilibrium for radiation in a homgeneous plasma Phys. Fluids, 29 (1986), 748-752 [2]. E.B. Davies, Heat Kernels and Spectral Theory Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, UK, 1989 [3]. E.B. Fabes and D.W. Strook, A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash, Arch. Rat. Mech. Analysis, 96 (1986), 327-338

487

O. Kavian

[4]. [5]. [6]. [7]. Is]. [9]. [10].

M. Fukushima, On an Lp estimate of resolvents of Markov processes~ Research Inst. Math. Science, Kyoto Univ., 13 (1977), 277-284 O. Kavian and C.D. Levermore, On the Kompaneets equation, a simplified model of the Fokker-Planck equationIn preparation A.S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4 (1957), 730-737 (translated from J. Exptl. Theoret. Phys. (USSR), 31 (1956), 876-885) J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954 A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Math. Science 44, Springer, New-York, 1983 M. Reed and B. Simon, Methods of Modern Mathematical Physics, (volume IV, Analysis of Operators)Academic Press, New York 1978 N.Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Func. Analysis, 63 (1985), 240-260 Otared Kavian Laboratoire de Math~matiques Universit~ de Versailles 45, avenue des Etats Unis 78035 Versailles cedex France E-mail:

Appliqu~es (UMR

7641)

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 22

S I N G U L A R P E R T U R B A T I O N S W I T H O U T LIMIT IN THE E N E R G Y SPACE. CONVERGENCE AND COMPUTATION OF THE ASSOCIATED LAYERS

D.

LEGUILLON,E.

SANCHEZ-PALENCIA and C. DE SOUZA

1. Introduction We consider a class of elliptic singular p e r t u r b a t i o n problems depending on a small parameter ~. The energy space V for z > 0 is strictly contained in the energy space Va for the limit problem (E = 0). Obviously, the dual spaces are such t h a t V~ is strictly contained in V'. Classical singular p e r t u r b a t i o n theory is concerned with the case when the loading f is contained in Va~, SO t h a t the problem for ~ > 0, as well as the limit problem ~ - 0 make sense in the variational formulation. We consider here the case when f E V ' but f r t, where the variational formulation makes sense for ~ > 0 but does not for ~ - 0. The energy of the solution tends to infinity as r tends to 0. Two examples in dimension one, borrowed from [9] are considered in sections 3 and 5, using the formal asymptotic m e t h o d of " matched asymptotic expansions" [14]. The solutions exhibit an i m p o r t a n t layer phenomenon, and the energy of the solution concentrates in such a layer as ~ - 0. Sections 4 and 6 are devoted to a rigorous justification of the above formal results. To this end, the exact problem for u s in the variable x is written in terms of a new variable y - x/G, the " inner variable" for the description of the layer. The p e r t u r b a t i o n problem changes drastically, becoming a sequence of problems in domains depending on r which tend to an u n b o u n d e d limit domain. Moreover, the different terms of the expression of the energy are changed in different ways, so t h a t a new concept of energy appears in the variable y, and this energy remains bounded (after an appropriate scaling of the unknown u e, in the example of sections 5 and 6). Then the convergence is proved using an elementary estimate of the new energy. The problems examined here may be considered as a first a t t e m p t to u n d e r s t a n d asymptotic properties of thin elastic shells, where the structure

490

Singular perturbations without limit in the energy space...

of the limit space V~ depends on the shape and the fixation conditions of the shell. In some cases, the space V~ is very large, going even out of the space of distribution and accordingly V~ is very small, so that loadings f do not belong to V~ (see [6,11,12], and [7] for a model problem). The corresponding difficulties of the numerical computation are mentioned in [4,13]. The special structure of the solutions u ~ with small ~ needs a mesh refinement in the region of the layers, as explained in [9] for the one-dimensional examples of sections 3 and 5. A first attempt to elaborate strategies of numerical computation for the problems of shells is done in section 8 for a problem analogous to that of section 3 in dimension 2. The loading f is a ~I distribution localized along a curve g. One may think to use refined meshes in the vicinity of C, in particular anisotropic meshes involving flatened triangles in the direction normal to the layer as used in other problems involving layers [1,2,3]. Instead of this, we used an iterative adaptive mesh procedure. Starting from a conventional F.E. mesh, new refined meshes are sucessively derived using numerical estimates of the computed solution on the previous mesh [5,8]. Such a procedure has been successfully used for computing fluid mechanics problems involving shock waves, boundary layers and wakes. Traditionally, automatic mesh generator produce "isotropic" meshes where triangles are as close as possible to equilateral ones. In the present case it will lead to meshes containing a drastic large number of elements. Thus, we select a mesh generator (BL2D [8]) able to constructanisotropic meshes, i.e. triangles having a large aspect ratio.

2. Singular perturbations Let V be a real Hilbert space, a(u, v) and b(u, v) two continuous and symmetric bilinear forms oil V. In addition, the form b is coercive, so that it may be taken as scalar product in V. Let the form a satisfy

a(v, v) > O, a(v, v) = 0 ~

(2.1) v = O.

In other words, a 1/2 is a norm on V. Let Va be the completion of V with this norm. Obviously, V, Va and their duals V', Va~ satisfy Y C V~, V~ C V' (2.2) with dense and continuous embedings. We consider the following family of problems with parameter s c (0, 1] : Problem P~ : Let f c V/, find u ~ c V satisfying

a(u E, v) + s 2 b(u ~, v) = < f, v >,

Vv e V.

(2.3)

D. Leguillon, E. Sanchez-Palencia and C. de Souza

491

Obviously, u E exists and is unique. Its energy is defined by 1 [a(u~ , u ~) + s 2b(u e, ue)] . E(u E) = -~

(2.4)

In usual examples, this energy is an integral on a certain domain, and the energy in a part of this domain makes sense. Problem P0 : let f E V~. Find u EVa satisfying

a(u, v ) = < f, v >,

Vv c V,.

(2.5)

with the energy 1

E(u) = -~a(u, u).

(2.6)

T h e o r e m 1.1 - Let f E V~ be fixed. The solutions u ~ and u of P~ and P0 are then well defined and satisfy u e --~ u

in Va strongly

E(~ ~)

, E(~)

(2.7)

(2.8)

We shall not give here the proof of this classical result which may be found, for instance in [13] sect. VI.1.4. In particular we note that (2.8) is a corollary of (2.7), taking v = u s or u in (2.3) and (2.5) respectively. Moreover, ,are have (see [6] or[7]) T h e o r e m 1.2 - Let f C V' and u s be the solution of P~, then a) E(u ~) is bounded iff f c V~', b) If f r V'~, then E(u ~) -----,oc. Remark 1.3. When f ~ Va~, the limit problem does not make sense as a variational problem, but it may happen in elliptic problems that it does in the Lions and Magenes sense [10]. 3. F i r s t e x a m p l e i n d i m e n s i o n

one

Let us consider V =/-/2(0, 1) and the forms 1

a(u, v) =

jr0 ~0

u'(x)v'(x)dx,

(3.1)

~"(x)v"(:~)dx.

(3.2)

1

b(~, v) =

492

Singular perturbations without limit in the energy space...

The completion of V is Va = H0~(0, 1). We shall identify H = L2(0, 1) to its dual, so that V' = H - 2 ( 0 , 1), V" = H - l ( 0 , 1). (3.3) Let us take f = 5~/2, where 51/2 denotes the Dirac distribution at x = 1/2. We note that

e v',

r v'.

(3.4)

and we are in the situation of Theorem 1.2.b. The problem P~ is: d2

d4 ) +

--

u

~X 4

= 51/2 '

du ~ du ~ ue(0) = uE(1) = ~-x (0) -- -~x (1) = 0

(3.5)

(3.6)

The limit problem P0 does not make sense as a variational problem because of (3.4). Nevertheless, as 6~/2 is smooth near the boundary, it belongs to the space ~ - 2 of [10] and P0 is a "Lions-Magenes problem" (see also Remark 1.3) : d2 dx 2 u = 6~/2 (3.7) u(0) = u(1) = 0

(3.8)

whose solution is U(X)

/ __ ~

X

t x--1

for 0 < x < 1/2 for 1/2 < x < 1

(3.9)

which presents a " Heaviside step" at x = 1/2. The formalasymptotic expansion of u ~ is easily obtained by the method of the matched asymptotic expansions [14]. It appears t h a t u ~ exhibits an internal layer in the vicinity of x = 1//2 which is described by an inner asymptotic expansion in terms of the "inner variable" (3.10) y ~ ~x - ~ 1/2 the leading term of this asymptotic expansion is ley

vOtyjr, =

1

1

..~e-Y m _

2 which is the " smoothed" step.

for y < 0 (3.11)

1 for y > 0

D. Leguillon, E. Sanchez-Palencia and C. de Souza

493

There are also " small boundary layers" in the vicinity of x = 0 and x = 1 described in terms of x/e and (1 - x)/e respectively, accounting for the lost boundary conditions on the derivative (see (3.6) and (3.8)). Out of these three layers, the convergence of u e to u is uniform. It is then seen that the energies of u e in the layer (3.11), in the small layers and out of the layers are of orders O(e-1), O(e) and O(1) respectively. We then see that the total energy tends to infinity, according to Theorem 1.2.b ; moreover it " concentrates asymptotically" in the layer (3.11). 4. C o n v e r g e n c e

to the layer

The layer (an internal layer) (3.11) is a well defined function of the variable y. We show that, writting down the problemP~ in the variable y, it is possible to prove that the solutions converge to (3.11); this convergence holds in the topology of some energy space of functions of y, which does not coincide (at least concerning limits as e --. 0) with the energy spaces for the variable x. Let us write explicitly the variational formulation of P~ in the case of Section 3" Find u ~ e H02(0, 1) such that, Yv e H02(0, 1)

fo l(u~'v' + e2uE'v")dx _ --~-~x dv (1/2)

(4.1)

Then we define

v (y) =

+ 1/2).

(4.2)

The problem for U ~ is obtained from (4.1) using (4.2) and an analogous formula for the test function. This gives, after multiplying by e Findu ~ c V~ such that, VW c V~

l/2s [(OyUE)(OyW) + (02yUE)(O2yW)] dy = -OyW(O)

(4.3)

i/2e

where VE - H02(-1/(2e), 1/(2e)). Here, it is useful to consider the functions of this space continued with value zero for l Y i> 1/(2e). In order to define an appropriate " limit space" we note that, as e tends to zero, the " boundary conditions" are sent to infinity, which is not very easy to handle. We also note that the right hand side of (4.3) only contains the trace of the first derivative, so that it is not modified by adding a constant to the test functions. Consequently, we are passing to the limit" up to additive constants". To this end, for each V~ we also consider the space Ve of the functions of H02(-1/2e, 1/2e) defined up to

494

Singular perturbations without limit in the energy space...

an additive constant. The spaces are ordered by embeding as ~ decreases. Then we define the " limit space" 12 as the completion for the norm

II w 112-

F

~

[(0~w) 2 +

(0uW) 2] dy,

(4.4)

(X)

of the space (4.5) S

The " limit problem" writes- Find U E 12 such t h a t VW E 12

/_§

[(OyU)(OyW) + (02U)(O2W)] dy = - 0 y W ( 0 ) .

(4.6)

We note t h a t it is a classical variational problem in the Hilbert space 12 as the right hand side is obviously a continuous functional on it. Then the solution U is well defined (we shall see later t h a t it is the layer (3.11) up to an additive constant). Then we have 4.1 - Let U s c Vr be the solution of (4.3) defined up to an additive constant, and U the solution of (4.6). Then :

Theorem

rs ----, U

in 12 strongly

(4.7)

Proof. Take W = U s in (4_3). Then, considering for each U s the corresponding equivalence class U s defined up to an additive constant, we have

Ff u~ IIv~ c

(4.8)

and after extracting a subsequence 9 U c-

, U*

in 12 weakly

(4.9)

which implies t h a t the first and second derivatives converge in L2(IN) weakly. Let us fix W belonging to a certain V~ (and then to the Vs with smaller ~) in (4.3). We may write ~ s and 17d instead of U s and W. Then, passing to the limit (4.9) we obtain (4.6) with U* instead of U.As the considered W are in a space dense in 12, we see t h a t U* = U, i.e. the subsequence (and then the whole sequence) tends to U. It only remains to prove that the

D. Leguillon, E. Sanchez-Palencia and C. de Souza

495

convergence in (4.9) is strong. Let us denote by B the bilinear form in the left hand side of either (4.3)or (4.6). We have: II 8 ~ - u I1~= B ( 8 ~ - U, 8 ~ - u )

= B ( U ~, Ue) -

2B(U, (7E) + B(U, U)

(4.10)

= - o ~ : ~(o) + 2 o ~ : ~(o) - o ~ u ( o )

where we used (4.3) with W = U ~ and (4.6) with W - U ~ and W = U. But the right hand side of (4.10) tends to zero by virtue of (4.9) as it involves a continuous functional on 1/. m It is not hard to check t h a t (3.11) (up to an aditive constant) is the solution of (4.6). Indeed, the equation associated with (4.6) is

( - 0 ~ + 0 4) u = ~'(y)

(4.11)

and we note that (3.11) is a solution of (4.11) for y ~: 0. Moreover, at y = 0,, has a discontinuity of the second derivative, which implies a 5' term for the fourth derivative, so t h a t v~ solves (4.11). Finally, we must check that v ~ in (3.11) (up to an additive constant) isan element of V. Indeed, the completion process passing from (4.5) to V allows functions tending to two different constants at +c~ and - o o , whereas (4.5) only contains functions vanishing (i.e. equal to a single constant) for large I Y I- Let U tend to the different constants at +oo and - o o . It may be approximated by functions UL tending to the same constant using a matching on a large interval of length L --+ oo ; the first and second derivatives of UL are of order O(L -1) and O ( L - 2 ) , respectively. Then

v~

[I U - UL i1~= O [L(L -2 + L-4)] --+ 0.

(4.12)

As a result, the boundary layer (3.11) is the limit of the functions U~ in the topology of 1~, i.e. in the energy of the " inner problem" for the variable y (which is not the energy for the variable x). 5. S e c o n d e x a m p l e

in dimension

one

We are now considering a second example with (3.1)-(3.3), when the right hand side is given by

f(x) = x - p - 2 + (1 - x) -p-2.

(5.1)

with some p E (0,1/2). We note that in this case f is a second derivative of a function of L 2, so t h a t it belongs to V' = H -2. Nevertheless, it is singular

496

Singular p e r t u r b a t i o n s w i t h o u t limit in the energy s p a c e . . .

at the boundary of the domain, and then it does not belong to the =-2 space and the Lions-Magenens theory does not apply. Moreover, we shall see that the limit of u ~ does not exist in a usualsense. We must perform a re-scaling to prove the convergence to the corresponding boundary layer. The equation is always (3.5) with the right hand side (5.1). The (formal) asymptotic expansion of the solution u e takes the form : ~tCout - -

s

r = s - P v ~ (y) + uin

(5.2)

"~ ...

...

,

Y=

X/E

(5.3)

where "out" and "in" denote the outer and inner matched expansions. Of course there is an analogous inner expansion in the vicinity of x -- 1. Note that the first one is such that its leading term is constant with respect to x; ~y is a constant coming from the study of the boundary layer(5.3). As for the inner expansion, v ~ is the unique solution of to d2

d4 )

+

v~

=

y e (0,

(5.a)

dv o

v~

=

(o) = o

v ~ is bounded on (0, c~).

(5.5)

(5.6)

It appears that (5.4)-(5.6) has a unique solution which tends to a certain constant 3' as y ~ +r This is the constant which appears in the outer expansion (5.2). It is apparent that the boundary layer is somehow "autonomous", as (5.4)-(5.6) is a well-posed problem with a given right hand side. The outer expansion (5.2) is in some sort a "sequel" of the boundary layers: in fact its leading term is nothing but the horizontal asymptotic of the function v ~(y). It is noticeable that an accurate finite element computation of u z needs a very small mesh step h in the layers, whereas a coarse mesh out of the layers works very well. Moreover, if the mesh step in the layers is not sufficiently fine, the computation of the layer is obviously poor, but, in addition, the region out of the layers (which depends on them, as we just pointed out) is also inaccurately computed. Concerning the energy, a simple computation of (5.3) shows that the energy in the layers is of order O(~--2p--1), whereas out of the layers,as the leading term in (5.2) has a vanishing energy, it is of order o(E-2P). We observe again that the total energy tends to infinity and it concentrates in the layers.

D. Leguillon, E. Sanchez-Palencia and C. de Souza

497

6. C o n v e r g e n c e in t h e s e c o n d e x a m p l e First we give a variational formulation of the problem (5.4)-(5.6). Let V be the completion of the space of function of H02(0, oc) which vanish for sufficiently large y, with the norm [(0yw) 2 + (0~w) 2] dy.

II w I [ ~ =

(6.1)

It is easily checked as at the end of Section 4, that V contains functions tending to a constant different from zero at infinity. Then, the variational formulation of (5.4)-(5.6) is: Findv ~

VwCV (6.2)

(v ~ ~)v =

y-~-2~(y) ay

We must check the following result: L e m m a 6.1 - The right hand side of (6.2) is a continuous functional on V.

Proof.

As the function y-p-2 is locMy of class H -2, its behaviour at infinity must only be checked. We may consider

fro+~ p(y)w(y) dy where p is a smooth function, equal t o may choose p such that

y-p-2 for

Jo

+ ~ ~(y)dy

Let us construct

~(y) =

~

(6.3) sufficiently large y. We

0.

(6.4)

~(~)d~.

(6.5)

Y

which is smooth and satisfies ~(0) = 0

(6.6)

y-p-2 9 (y) =

p+l

for large y.

(6.7)

498

Singular p e r t u r b a t i o n s w i t h o u t limit in the energy space...

Let us take w C Ho2 (0, c~) vanishing for large y. We have ~(y)w(y)dy

=

~' ( y ) w ( y ) d y

=

~ ( y ) w ' (y)dy

(6.8)

< II 9 IIL~II ~ ' I1~< c

Ii ~ Ilv

which proves the l e m m a as w is any function in a dense set of )2. I In order to prove the convergence to the layer at x = 0, we write the problem PE ( ( 3 . 1 ) - ( 3 . 3 ) ) with the right hand side (5.1) after the c h a n g e : u ~(x) = e - P U E(y),

y = x/e

(6.9)

namely: Find U e e/-/02(0, l / z ) s u c h t h a t for all W e/-/02(1, l / z )

fo 1/~ [(o~u~)(o~w)+ (o~u~)(o~w)] dy

[,,. + (1

.]

(6.10)

The function U E m a y be continued with value zero for y > l / e , so t h a t it is element of ~. T h e proof of the convergence is then analogous to t h a t of Section 4, and even simpler, as we do not deal with equivalence classes. T h e o r e m 6.1 - L e t U E and v ~ be the solutions to (6.10) and (6.2) respectively. Then, U ~ ~ v ~ in V weakly (6.1 1) R e m a r k 6.2. In Theorem 6.1 the convergence is only weak, and the reasoning (4.10) for the strong convergence does not work because of the presence of the term f near y = 1/z. Nevertheless, by the linearity of the problem, we m a y decompose it into two problems for the two terms of f in (5.1). For each one (the solution of which is somehow analogous to that of Section 5), we m a y prove strong convergence in P to the corresponding layer. Oppositely, when we consider simultaneously the two terms in (5.1), strong convergence does not hold, as the limit is the layer in the neighbourhood of x = 1, whereas U ~ bears the energy of both layers. 7. A n e x a m p l e

in dimension

two

In this section we consider a problem analogous to t h a t of Section 3 but in Ft - (0, 1) x (0, 1) of the (Xl,X2) plane. The equation is (~2A2 - A)u ~ -- f.

(7.1)

D. Leguillon, E. Sanchez-Palencia and C. de Souza

499

The chosen boundary conditions are not the Dirichlet ones, but special ones allowing on one hand anti-symmetry continuation for x E (1,2) and on the other hand, such t h a t the solutions for f independent of y is it self independent of y. The sake of such a choice is obviously to compare twodimensional solutions with the previously obtained one-dimensional ones. This conditions are u=0 for x = 0, (7.2)

s(Ou/On) = 0 u=0

(7.3)

forx=l

s2Au=0

Ou/Ou=O

for x = 0

(7.4)

forx=l fory=0,

E2(OAu/On) = 0

(7.5) y=l

(7.6)

for y = 0, y = 1

(7.7)

Clearly conditions with the factor ~ only are concerned with s > 0. The spaces V and V~ a r e : V=

{v e g 2 ( ~ ) , u satisfies (7.2), (7.3), (7.4), (7.6)} Y~ = {v 6 g l ( f t ) , u satisfies (7.2), (7.4)}

(7.8) (7.9)

Let us define the bilinear form a e (u, v ) = s 2 / a A u . A v dx + J~ V u . V v dx.

(7.10)

The space Va is classical. As for V, we have L e m m a 7.1 - There exist two constants cl and c2 such that for any u, v in V: l aE(u,v) I<<_Cl (~2 !1 it I]V II V llV + !1 it ]IV. il V IlV.) (7.11)

a~(v,v) >_ c2 (a2 ]1 v I[~ + I! v lifo) Prod. then

(7.12)

Inequality (7.11) is obvious. To prove (7.12), we note that if v C V, Av = ~ e L2(ft)

and v satisfies (7.2), (7.4), (7.6). Each one of Shapiro-Lopatinskii condition for (7.13) (i.e. it dition for the elliptic problem (7.13)") so t h a t regularity theory for elliptic problems hold true

11

(11

(7.13) these conditions fulfils the is a " good boundary conthe classical inequalities of and we have

IlL + ti

9

500

Singular perturbations without limit in the

energy

space...

Moreover, for v E V the Poincar6 inequality gives : ]] V ]]~2 ~ C4 I[ VV []~2

(7.15)

and (7.12) follows from (7.14) and (7.15). It should be noticed that regularity theory holds true in the above problem, even if the boundary of is not smooth. Indeed, the corners have amplitude 7r/2, and a reasoning of continuation by symmetry or antisymmetry shows that the problem is not singular, and behaves as on a straight boundary. I As for the " loading" f, we shall take a so t h a t

6'(C) distribution

on a curve C,

t ~

< f, v > - -

Jc (Ov/On)ds,

(7.16)

where s and n denote the arc and the normal to C. Obviously, by the trace theorem, 6'(C) e V', 6'(C) ~ Vd (7.17)

8. Numerical approximation The two-idimensional problem (7.1)-(7.7) with the right handside member (7.16), (7.17) is still a "Lions-Magenes" problem (see Section 3 and remark 1.3). As in the one dimensional case [9], it can be reasonably expected that an almost classical finite element procedure will solve the problem, provided some precautions. First of all, the element must be able to take into account the special right handside member (7.16). Obviously, Lagrange elements are not wellfitted to this formulation. Zienkiewicz and Taylor [15] propose an Hermite triangular element with corner nodes (9DOF) which "performs excellently" in problems of plate and shell bending. It does not ensure the C 1 continuity, normal slope is discontinuous through the triangles edges, however it includes derivatives as degrees of freedom at nodes and allows us to compute expression (7.16). As emphasized in the one-dimensional case [9], the second precaution is to select an appropriate mesh, the problem being of course more complicated in two dimensions. It will be the aim of the following in the present section. Computations are carried out on the square [0, 1] • [0, 1]. As a test, the line C in (7.16) is the right edge X 1 = 1 of the square. Thus, it can be easily seen that the solution is independent of x2 and coincides with the one-dimensional solution derived from (3.5), (3.6) (see Section 7). It must be pointed out that conditions (7.4) and (7.5) play the role of antis y m m e t r y conditions, thus the solution extended by anti-symmetry to the

D. Leguillon, E. Sanchez-Palencia and C. de Souza

501

domain [0, 2] • [0, 1] is exactly that exhibited in Section 3 after having being stretched. The computation of (7.16) does not afford special difficulties since C is made of the successive triangle edges forming the right side of the square. Moreover, there is no lack of normal slope continuity through C in the present anti-symmetrical case. Thus, OrlOn in (7.16) is numerically defined without ambiguity. A first computation has been carried out on a coarse mesh with 155 triangles (figure 4-(0)), for ~ = 0.01. As anticipated, results are far from being satisfactory (as observed for instance from the isovalues o f Figure 5(0), corresponding to the second numerical example). Obviously, it is the coarseness which is responsible for the inaccuracy. A strongly and uniformly refined mesh with 25600 triangles, as illustrated in Figure 1, would be able to provide us with better results, but the computation cost would increase drastically.

Figure 1: A uniformly refined isotropic mesh (25600 triangles) The right approach is clearly to refine the mesh only in the vicinity of the layer which location is well known in the present case. It can be done (Figure 2) using a structured mesh with 684 triangles.

Figure 2: A refined structured anisotropic mesh (684 triangles)

502

Singular perturbations without limit in the energy space...

A similar result can be achieved by using an automatic mesh generator (BL2D [8]) with a prescribed map of triangle sizes. It is illustrated in Figure 3 with a mesh made of 3031 triangles. There is a strong refinement along the right edge which covers the singular layer. A smaller refinement has also been required along the left edge to take into account the weaker boundary layer in this area (as already mentioned in Section 3). It is to be pointed out t h a t triangles in the right part of Figure 3 are by far smaller than those of Figure 1, although the total number of triangles is itself smaller.

Figure 3: An automatically (BL2D) refined isotropic mesh (3031 triaagies) Independently of the procedure used to generate the meshes, there is a fundamental difference between Figures 2 and 3. For most of mesh generators, as the one used to produce Figure 3, more a triangle is equilateral and more it is optimal, this a consequence of an interpolation error estimate. Thus triangles have roughly the same diameter in both x l and x2 directions. Using a different procedure, Figure 2 has been obtained by subdividing the edges in an appropriate way. A refined mesh is prescribed on the right side by small divisions of the horizontal axis, since it is known that it is the location of the layer. On the other hand, the knowledge of the test solution (independent of x2) brings us to define coarse divisions on the vertical axis. It is clear, in this very particular case, that the high aspect ratio of the elements located within the layer is not at the origin of additional errors (as a m a t t e r of fact, a single vertical element would suffice to get a reliable solution). Such a mesh is called anisotropic, there is a privileged direction. On the contrary, a mesh favouring equilateral triangles is baptised isotropic.

D. Leguillon, E. Sanchez-Palencia and C. de Souza

503

This remark and the observation of meshes obtained by [5, Chap.12], concerning problems of fluid mechanics with shock waves, give raise to a new question: could it be possible to generate automatically anisotropic meshes taking into account such layers, in the general case, i.e. even if their location is unknown ? The procedure proposed in [5, Chap.12] and [8], is an iterative one. A first calculation is performed on a coarse isotropic mesh. A metric is extracted at each node from the computed solution. This metric consists in a 2 x 2 symmetric positive definite matrix. Its orthogonM eigenvectors form a local basis and the inverse of the eigenvalues define shrink coefficients. In such a metric, if the eigenvalues differ significantly, shrinked equilateral triangles become thin in the direction of the eigenvector associated with the larger eigenvalue. Triangles seem to arrange in a parrallel direction to the other eigenvector. Based on these metrics, an anisotropic new mesh is generated and then an (expected) improved solution is computed and so on. It must be pointed out that such a mesh generation is a complicated procedure, in particular "interpolated" metrics must be defined between nodes (see the guide [8] for a complete explanation). The main difficulty to use the above iterative procedure is then to define at each node a metric extracted from a computed solution. As suggested in [5, Chap.12], the hessian matrix of the solution (i.e., the matrix of the second order derivatives) will be used. Thus, in a first step the second order derivatives of the solution 02u/Ox 2, 02U/OX2 and 02U/OXlOX2 a r e computed at each node (an average between neighbouring elements). Then, eigenvectors vl and v2 and eigenvalues A1 and A2 are derived. Since the hessian is not necessarily a positive matrix, in a third step, a new matrix is built having the eigenvectors Vl and v2 and the eigenvalues IAll and IA21. This matrix calculated at each node provides the procedure with appropriate metrics. From a mechanical point of view, it means that the meaningful parameter to refine a mesh is the stress gradient (i.e., the second order derivatives of the displacements). The above procedure has been checked in the present case of singular layers. The line C in (7.16) is chosen within the domain of computation, it is defined by x2 = - 3 x l + 2. The computation of (7.16) is carried out on the successive segments at the intersection of C and the triangles using a three-point quadrature formula. Of course, we consider at the beginning of the process that the location of the layer around C is not known. Thus, no special mesh refinement can be initially prescribed. Figure 4 shows a "film" of the 4 iterations required to generate a satisfying mesh for s -- 0.01 (the initial isotropic step called iteration 0 is included in the account). Meshes contain a reasonably low number of elements, respectively 155, 197, 314 and 685 triangles.

504

Singular perturbations without limit in the energy space...

0

1

2

3

\

I Figure 4: The four succesive meshes generated by the iterative process (BL2D), the first one is isotropic the next ones become more and more anisotropic ((0) 155, (1) 197, (2)314 and (3) 685 triangles)

D. Leguillon, E. Sanchez-Palencia and C. de Souza

505

Figure 5 gives an insight into the numerical solution improvement which can be expected from such an iterative remeshing procedure.

Figure 5: The isovalues of the computed solution corresponding to the first (coarse isotropic mesh 4 (0)) and last ("converged" anisot~opic mesh 4 (3)) iterations The last Figure 6 exhibits the final l l t h iteration in the situation of a thiner layer arising from the case s - 0.001 (528 triangles). Despite the high aspect ratio of the triangles, never any warning, pointing out the matrix illconditioning, has been displayed during computations (performed in double precision).

1/

Figure 6: The 11 th anisotropic iteration (BL2D) in case of a very thin layer (528 triangles) Two difficulties remain in this procedure, first there is no criterion to stop automatically the iterations, such a criterion would probably imply the concept of optimal mesh which seems to be out of reach (at the time

506

Singular perturbations without limit in the energy space...

being). Iterations have been stopped at a guess. Secondly, a parameter is missing to define the triangles. In addition to the metrics, the procedure requires an estimation of a reference element size to perform the shrinks. An arbitrary parameter selected by the user has been added in oder to get the above results. This parameter differs in the two cases s = 0.01 and = 0.001. A c k n o w l e d g e m e n t s . The authors are indebted to P. L. George and P. Laug from I.N.R.I.A. (Rocquencourt, France) for fruitful discussions and explanations on the automatic mesh refinement procedures. P. Laug has initiated and designed BL2D and P. L. George is the author of successful books on mesh generation procedures.

References [1] Apel T. and Lube G., Anisotropic mesh refinement in stabilizedGalerkin methods, Numer. Math., 74 (1996), 261-282. [2] Apel T. and Lube G., Anisotropic mesh refinement for a singularly perturbed reaction-diffusion model problem, Appl. Numer. Math., 26 (1998), 415-433. [3] Apel T. and Nicaise S., Elliptic problems in domains with edges: anisotropic regularity and anisotropic finite element meshes, in Partial differential equations and functional analysis, in memory of P. Grisvard, J. C~a, D. Chesnais, G. Geymonat and J.L. Lions eds., Birkhauser, Boston, 18-34 (1996). [4] Chapelle D. and Bathe K. J., Fundamental considerations for thefinite element analysis of shell structures, Comp. and Structures, 66 (1998), 19-36. [5] George P. L. and Bourouchaki H., Triangulation de Delaunay et maillage. Application aux ~l~ments finis, Hermes, Paris 1997. [6] Gerard P. and Sanchez-Palencia E., Sensitivity phenomena for certain thin elastic shells with edges, Math. Meth. Appl. Sci. (to appear). [7] Karamian P., Sanchez-Hubert J. and Sanchez-Palencia E., A model problem for boundary layers on thin elastic shells, Math. Modelling and Num. Anal. (to appear). [8] Laug P. and Bourouchaki H., The BL2D mesh generator: beginner'sguide, user's and programmer's manual, Rapport technique n~ I.N.R.I.A., Rocquencourt, France (1996) [9] Leguillon D., Sanchez-Hubert J. and Sanchez-Palencia E., Model problem of singular perturbation without limit in the space of finite energy and its computation, C. R. Acad. Sci. Paris, s6rie IIb, 327 (1999), 485-492. [10] Lions J.L. and Magenes E., Probl~mes aux limites nonhomog~nes et applications, vol. 1, Dunod, Paris 1968.

D. Leguillon, E. Sanchez-Palencia and C. de Souza

[11]

[12] [13] [14] [15]

507

Lions J.L. and Sanchez-Palencia E., Probl~mes sensitifs et coques ~lastiques minces, in Partial differential equations and functional analysis, in memory of P. Grisvard, J. C~a, D. Chesnais, G. Geymonat and J.L. Lions eds., Birkhauser, Boston, 207-220 (1996). Sanchez-Hubert J. and Sanchez-Palencia E., Coques ~lastiques minces. Propri~t~s asymptotiques, Masson, Paris 1997. Sanchez-Hubert J. and Sanchez-Palencia E., Pathological phenomena in computation of thin elastic shells, Trans. Canad. Soc. Mech. Engng., 2(4B) (1999), 435-446. Van Dyke, Perturbation methods in fluid mechanics, Academic Press, New-York 1964. Zienkiewicz O. C. and Taylor R. L., The finite element method. Fourth edition, Vol.2, Mac Graw Hill, London 1991. Dominique Leguillon, Evariste Sanchez-Palencia and Carlos de Souza Laboratoire de Mod~lisation en M~canique UMR 7607 CNRS-Universit~ Pierre et Marie Curie 4 place Jussieu 75252 Paris Cedex 05

France E-mail: [email protected], [email protected], [email protected]

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 23 O P T I M A L DESIGN OF G R A D I E N T FIELDS W I T H A P P L I C A T I O N S TO ELECTROSTATICS

ROBERT LIPTON AND ANI P. VELO

1. Introduction Consider a two dimensional design domain ~, containing two isotropic dielectric materials. The dielectric permittivity is specified by s(x) and is piece-wise constant taking the values a and /~ where /~ > a > 0. For a prescribed charge density f the associated electric potential 9 satisfies the Poisson equation given by - d i v (s(x)V~) = f,

(1)

and qo = 0 on the boundary of f~. In order to include the broadest class of charge densities we suppose that f lies in W-I'2(f~) and that ~ is a W~ '2 (~) solution of the Poisson equation. The associated electric field in the domain is - V ~ . We introduce a "target" electric field l~. For a given charge density, our objective is to design a two phase dielectric that supports an electric field - V ~ that is as close as possible to ]~. Here l~ = -V~3, where ~3 is a potential in Wl'2(gt). Placing a constraint on the amount of the better dielectric/~, the design problem is to minimize the difference

/I

V~-

V~312dx,

(2)

over all configurations of the two dielectrics. In general, material layout problems of this type fail to have an optimal design given by a configuration of the two materials. Instead one must study the behavior of minimizing sequences of configurations. The purpose of the analysis given here is to provide the methodology for the recovery of optimal configurations when they exist and to identify minimizing sequences of configurations for (2) otherwise. We introduce a tractable method for the numerical computation of minimizing sequences of configurations. These minimizing sequences are associated with materials with graded dielectric properties that may exhibit a fine scale structure composed of layers of the

510

Optimal design of gradient fields with applications to electrostatics

two dielectrics. Moreover, for a dense class of target fields we are able to characterize all fine scale structure that can appear in minimizing sequences of configurations, see Theorem 1 of this Section. We illustrate our approach through numerical examples provided in Section 6. The examples illustrate how the electric field can be controlled using functionally graded materials.

2. Background and main theoretical results The nonexistence of an optimal configuration for the design problem coincides with the appearance of minimizing sequences containing regions of finite measure where the dielectric permittivity becomes highly oscillatory. As one follows these minimizing sequences the dielectric permittivity oscillates between the values a and/3 on progressively finer scales. To describe this mathematically we denote the subset of the design domain ~ containing the/3 dielectric by w. The characteristic function of this set is written as X where X = 1 for x in w and X = 0 otherwise. The piece-wise constant dielectric permittivity is given by r

- r

~ fix + a(1 - X).

(3)

Oscillation of a sequence of designs {wV}~=l is described by the weak L ~ ( ~ ) star convergence of the associated sequence of characteristic functions {X~}~=1 to a density t? in L ~ ( ~ ) where 0 _< t? _< 1. The issue of nonexistence of optimal configurations for problems of material layout has been the object of much interest. The classic example is illustrated in the problem of minimizing the energy dissipation associated with configurations of two materials. In the context of two phase dielectric materials the energy dissipation for a configuration is given by ~ ~(x)V~. V~dx. The problem of nonexistence was resolved in an elegant fashion by extending the design space to include all effective dielectric permittivities that could be obtained through oscillation, see [10], [11] and [6]. The crucial connection between minimizing sequences of configurations and optimal designs in the extended design space is established through a continuity property of the energy dissipation given in [3]. This continuity property is an example of the theory of compensated compactness developed in [7] and [12]. Here continuity is given in the context of weakly convergent sequences in L2(~) 2. Indeed, consider sequences {@V(x)~7~oV}v~176and {~7~gV}vC~=l, such that - d i v (r ~) = f. If these sequences weakly converge to the limits

R. Lipton and A.P. Velo

511

c~176 ~176 and Vqp~ where c ~176 is an effective tensor in the extended space of designs then one has the continuity expressed by lim f c~(x)V~ ~. V~ ~ dx - f c ~ ( x ) V ~ ~ . V~ ~176 dx. J~ Ja

/,l--+oo

For the design problem treated here we can attempt to resolve the nonexistence problem by extending the design space to include effective properties. However unlike the energy dissipation and other continuous functionals treated earlier, the objective functional given by (2) is not continuous with respect to weak convergence. Thus additional theoretical work is required to provide the connection between an extended space of designs and minimizing sequences of configurations. In this presentation we outline a methodology for the identification of minimizing sequences of configurations. The method is based on a careful extension of the design space and by replacing (2) with a suitable "relaxed" functional that is associated with the extended design space. It is evident that any attempt to identify minimizing sequences of configurations must account for the possibility of oscillations in the sequence of gradients associated with minimizing sequences of designs. As above an oscillatory sequence of gradients is characterized by the weak convergence of the sequence in n2(~) 2. To fix ideas, let { V ~ } ~ = I denote a weakly converging sequence of gradients associated with a minimizing sequence of designs. The weak limit of the sequence is denoted by VqS, and one has lim f [ V ~ - Vq3]2 dx Ja

v--+oo

12-'+OO

Jn

Ja

The oscillatory behavior of minimizing sequences is naturally linked to the dependence of the limit lim f [ V ~ - ~ z ~ ] 2 dx, J~

(5)

v--~ co

on the weak limits V qS, ~, together with other moments of measures associated with weak limits of geometric quantities, (e.g., the H measure introduced by L. Tartar [14]). Our methodology for identifying minimizing sequences is based upon on writing (5) as an explicit function of the relevant weak limits. Although at this time we are unable to produce a formula for every type of oscillation we note that an explicit closed form expression is available when the oscillations consist of layers of the two materials. The formula follows directly from the corrector theory of homogenization given

512

Optimal design oE gradient fields with applications to electrostatics

in [1] and [9]. To be precise we introduce the characteristic function X(x, t). Here X(x, t) is piece-wise constant in the first variable and unit periodic in the scalar t variable. As is done in homogenization theory [1], we define a locally layered material by X(x) - X ( x , x . n). Here n - n(x) represents the normal to the layers given by n - (cos (-y(x) ) , sin(-y(x))). An oscillatory sequence is given by X v(x) - X(x, v x - n ) , v - 1, 2 . . . , co. For this case we have the closed form expression given by lim~_~oo f~ IVg~ - V~h]2 dx - fn R('/(x))H(O(x))RT(~/(x))V~(x)

9V~5(x) dx.

(6)

Here R(~) is the orthogonal matrix associated with a rotation of "y radians and the matrix H(0) is a function of the density 0 given by 1 _ ~)2t~(1

H(O) -

_ O)h~

0

0) 0

'

(7)

where he - (l:aO _~_~)--1 is the harmonic mean of the two dielectric permittivities. Here, the sequence of gradients { V ~ } ~ = I is related to the sequence of configurations through the equilibrium condition - d i v ( s ( X ~ ) V ~ ) = f.

(s)

The "homogenized " equilibrium equation satisfied by the weak limit ~ is given by - d i v (e(0(x),-y(x))V~5) = f, (9) where =

(10)

and the diagonal tensor A(O) is given by

A(0) -

h0 0

0 mo

) '

with me = a (1 - t9) +/~ ~. Here the tensor c(t~,-),) is the G-limit associated with the sequence of dielectric tensors {e(X ~) }~=1, see [9]. (Since the dielectric tensors are symmetric, the G-convergence [18] and the H-convergence [9] of any sequence of dielectric tensors is the same.) The methodology presented here uses the explicit formula given by (6). Our approach is to replace X and e(X) with the new design variables ~,

R. Lipton and A.P. Velo

513

3', and r 3') given by (10). In addition we introduce the new objective functional given by

RF(O, ~/, ~(0, 7), V~3) - ./o [V~ - Vq3[2 dx + f R(7(x))H(O(x))RT(7(x))Vqa 9Vqa dx, Jn

(11)

where the state variable is the W~ '2 solution of - d i v (r

7(x))Vqa) - f.

(12)

In order to state our results we formulate the original design problem in a precise way. We introduce the constant O, such that 0 < 0 < 1. The space of admissible configurations and associated dielectric permittivities is denoted by ado, and

ado - {X" /n X dx < 0 meas(f~)}.

(13)

The objective functional is denoted by F(X, c(X), Vq3) and is given by

F(X, ~(X), V~3) - fn IVqa - Vq3I2 dx,

(14)

where the state variable ~ is a solution of (1). The original design problem is formulated as P - inf F(X, ~(X), V~3). (15) xEado

The admissible space of designs for the new design problem is given by Do - { (0, 7, ~(0, 7)) I t? e L~(f~; [0, 1]); 7 e LC~(f~; [0, 27r]) 9

f O dx < 0 measlY; ~(0(x), 7(x)) - R(7(x)) A(0(x)) RT(~(x)) }, (16) and the new design problem is formulated as

RP -

inf

( O,"r,e( O,"r) ) e D e

RF(O, 7, r

"7), Vq3).

We point out that the extended space of designs Do contains the original space of designs ado. Indeed, choosing 0 - X we have e(0,7 ) - ~(X), H(O) - 0 and

F(X, ~(X), Vq3) - RF(O, 7, c(0, 3'), Vq3).

(18)

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Optimal design of gradient fields with applications to electrostatics

The first result that we describe is given in Theorem 8. It states that for every q5 in Wl'2(ft) and for every f in W-l'2(ft) that

P-

RP.

(19)

In deriving (19) we identify a class of minimizing sequences for P that is tractable for numerical computation. Our approach is based upon a discrete approximation of the design space Do. We consider any partition T~ of f~ consisting of a finite number of pair-wise disjoint subdomains fti C ft, i - 1 , . . . , N(t~) such that" N(~)

f~ -- U fti and i'-i

max i=l

(diam(f~i)) < ~.

.... N(t~)

(20)

--

We fix the partition T~ and the discrete approximation D~ is given by the piece-wise constant functions O~(x), ~/~(x) taking constant values in each subdomain. We denote the restriction of O~(x) and -y~(x) to ft~ by 0~ and 7~ respectively. Here 0 _ 0~ <_ 1, 0 _ -/~ < 27r, and EN__(1) (0~ meas ( f ~ i ) ) - O meas (f~).

(21)

The piece-wise constant dielectric permittivity tensor is given by

e(9/~(x), O~(x)) - R('y~(x))A(O~(x))RT (7~(x)),

(22)

and the associated state variable qa~ solves the Poisson equation - d i v (s(~,~(x), 0~(x))V99 ~) - f.

(23)

It is clear that D~ is contained in the larger set Do and the design problem posed on this smaller set of designs is written

RP ~ =

inf

RF(O ~, "y~, e(O~, ~/~), VqS).

(24)

It is shown that a minimizing vector of design variables ( ~ , ~ ) exists for this problem, see Theorem 4. Most importantly, it follows from (6) that there exists a recovery sequence of configurations X" in ado for which lim F(X", e(X~), VqS) - R P ~,

/2---+(X)

(25)

see Theorem 5. For any given partition T~ we consider its refinements, i.e., the nested family of partitions {T~}~>0 that includes T~. We show that the optimal design vectors associated with the refinements represent a minimizing sequence of designs for the problem RP, see Theorem 7. Moreover,

R. Lipton and A.P. Velo

515

using (25) we are able to recover the explicit form of minimizing sequences for the original problem P, see Theorem 5 and equation (57). We point out that the choice of the initial partition T~ is arbitrary so this method generates minimizing sequences of designs for any initial choice of partition. It is instructive to write RF(O,-y,~(tg, V), V~) in a form where ~(0, 3') appears explicitly. Manipulation gives

(moz - c(o, 7)) (too1 R(7)H(tg)RT(v) - (1 - 0)fl(/3 - a) + /~

7))

"

(26)

It is clear from (11) and (26) that if ~(0, ")')VqD = moV99 then

f

R(7)H(O)RT(v)VqD 9Vqodx - O.

It now follows from (6) that the gradients VqD" associated with a recovery sequence of configurations {~(X~) }~=0 G-converging to ~(0, V) converge strongly to V qp. Conversely if the gradients V q0~ associated with a recovery sequence of configurations converge strongly to V99 then the term

f

R(~/)H(O)RT(7)V99 9V99dx

vanishes and ~(0,7)Vv9 = meVqo follows from (26). In this context we mention that the earlier work of [5] focuses on the energy dissipation to show that the condition ~evqo - c*VqD is necessary and sufficient for the strong convergence of gradients associated with sequences {~e}v~__0 G-converging to ~e and weak L ~ s t a r converging to ~*. For a dense set of target fields we show that our method accounts for all oscillations appearing in minimizing sequences of configurations. We consider target fields of the form -V~,

~ e W~'2(~)

(27)

and the relaxed version of the original problem is given by T h e o r e m 1. - There exists a dense G5 subset K of W 1'2 (f~) such that for

inK, (1) There exists a minimizer of R P in Do, (2) P = RP, (3) Any cluster point of any minimizing sequence in ado of P is a minimizer of R P and any minimizer of R P in Do is a limit of a minimizing sequence for P.

516

Optimal design of gradient fields with applications to electrostatics

(4) Let ~ be the potential associated with the minimizer (0, 9, 6(~,'~)) o] R P , then RF(~, ~/, ~(~, 9), V~) -

IV~ - V~I 2 dx and ~(~, ~)V~ - m~V~.

Here the convergence of sequences of designs are with respect to the Gconvergence [18]. The class of targets appearing in Theorem I is motivated by the following theorem of L. Tartar [2], [13], which is an improvement of a result of M. Edelstein [4]. T h e o r e m 2. - Let S be a non-empty strongly closed subset of a Hilbert space H. Then there exists a dense G~ subset K o3t H such that for any x E K, the minimizing sequences {Cn}nCr E S of the ]unction c -+ IIx - c]I are Cauchy sequences. In particular the subset of points of H with a unique projection on S contains a dense G~ subset, as it contains K. With Theorem 2 in mind we can take advantage of the geometry of the set of effective tensors for two dimensional problems and establish Theorem 1. This topic is taken up in Section 5 where Theorem 1 is proved. The recent work of P. Pedregal [15], [16] approaches similar design problems from a different perspective. In that work the equilibrium equation for the potential, together with the resource constraint is incorporated into the cost functional and a new type of envelope for the augmented functional is introduced. The envelope is shown to be weakly lower semicontinuous in wl'2(f~) x L~(f~) see [15], [16] and can be thought of as a constrained quasiconvexification of the original augmented functional. The constrained quasiconvex envelope can be expressed in terms of a class .A of gradient Young measures, see [15]. To proceed further, the envelope needs to be given in terms of explicit formulas. This requires knowledge of the set ,4. However at this stage the characterization of ,4 is not known. In principle the methods of this paper can be used to deduce the part of .A containing Young measures associated with simple one scale laminates, see [15]. Theorem 1 shows that the knowledge of gradient Young measures associated with layered microstructures (a.k.a. laminates) is sufficient for the computation of the constrained quasi convex hull when the target fields are in the class K C W 1,2. We point out that the discrete problem given by (24) is of interest on its own right. From a practical perspective there is a prohibitive manufacturing cost incurred when attempting to make a graded material with possibly different anisotropic dielectric properties at every point. Instead there is

R. Lipton and A.P. Velo

517

a smallest scale n over which the dielectric properties change. The scale is set by the manufacturing cost. Practically speaking one partitions the design domain into subdomains of diameter n and inside these subdomains one optimizes the dielectric properties. This approach to the design of graded materials is naturally incorporated in the formulation of the discrete problem given here and is discussed in the context of the numerical examples given in Section 6. 3. T h e d i s c r e t e p r o b l e m In this Section we analyze the design problem on the discretized space of designs. The existence of an optimal design is established in this space. Next we apply the corrector theory to exhibit a recovery sequence of configurations of the two dielectrics. We consider any partition Tn of ~ consisting of a finite number of pairwise disjoint subdomains ~i C ~, i - 1 , . . . , N(~) such thatN(n)

-

U ~i and i--1

max

i-'l ..Y(n)

(diam(~i)) < ~.

'"

We fix the partition Tn and the discrete approximation D~ is described by equations (20-23) given in the introduction. The design problem over the discrete space is given by (24). Existence of the optimal design is established using the direct method of the calculus of variations. We start by introducing the type of convergence relevant to the discrete problem. A design (0 n, ~,n ~(0n, ~n)) in D~ can be identified with the vector (~,~'~) for i - 1 , . . . , N(~) in R 2g(n). Thus D~ is identified with a compact subset of R 2N(n) and convergence of designs in D~) is given by sequential convergence in R 2g(n). Existence of an optimal design will follow once we show that the functional RF(O n, 7n ~(0n,.yn), V~) is continuous with respect to sequential convergence in R 2g(n). T h e o r e m 3. - Given a sequence of designs {(On'n, 7n,n)}C~n=l and a design (0n,~n) such that lim (0n,~, 7~,~) _ ( ~ ~,~) (28) n-+(x)

as elements of R 2g(n), then lim RF(O n'n, 7~'~, c(0 n'~, ~/~'~), V~) - RF(O n, ar~/n, ~(~n, ~,n), V~). (29) n--~oo

Proof. The state variable associated with the limit design (0n, ~n) is denoted by ~ and is the W~ '2 (~) solution of - d i v (~(~n, ~n)V~n) _ f.

(30)

518

Optimal design of gradient fields with applications to electrostatics

The convergence of the sequence of designs given by (28) implies that the associated conductivities {e(O~'n, 3,~'n) }~cr i converge to e(0~, ~ ) almost everywhere. From the theory of G-convergence [18] we also know that the sequence G-converges to r ~, ~ ) . The definition of G-convergence implies that the state variables ~p~ associated with the sequence {r 1 converge weakly in W~'2(f~) to ~ . In order to establish the continuity given by (29) we first show that the sequence 9~r~converges strongly to 97~ in w0i'2(f~). To do this we recall the formula (10) for e(0, 7) to easily see that 0 < a <

<

We apply this estimate to obtain,

/ alVqo ~ - V~J2dx (32)

+ ~ e(o~'n' ,),~,n)V~'~. V~'Cdx.

(33)

Passing to the limit as n --+ oo in (32,33) , we apply the well known properties of G-convergence together with the almost everywhere convergence of {e(O~,n, ,~'~'n~~ and the Lebesgue convergence theorem to find that lim II V~,~ - VV3~ 11~2= O,

(34)

n---4 o o

and strong convergence of ~r~ to ~ in W~ '2 follows. From (7) we easily obtain the following estimate for the sequence

{R(7~,n(x))H(O~'n(x))R T (7~'n (X)) } ncr given by

f12 1

1

]2

R(7~'n(x))H(O~'n(x))RT (7~'n(x))rl "rl < --~(-~ - -~) ]rl ,

(35)

for every vector ~/in R 2. Moreover, from the convergence given by (28) the sequence converges almost everywhere to

R(~/~ (x) )H(O ~ (x) )R T ( ~ (x) ).

R. Lipton and A.P. Velo

519

Finally since V ~ --+ V ~ ~ strongly in L 2 we apply (35) together with the Lebesgue convergence theorem to conclude that: lim RF(9 ~'~, 7 ~'n, r

= limn_~[ / a I V ~ - V @ 1 2 d x + / a

~'n, ~/~,n), V~) -

R(7~'n)H(O~'n)RT(7~'n) V ~ . V ~

dx]

= fo lye - v,l dx + [o (36)

= RF(O ~, 9~, s(9~, 9~), V@). which proves the theorem. Collecting our results we have shown that

T h e o r e m 4. - There exists an optimal design (9~, ~ , ~(9~, 6/~)) in D~ for

the discrete problem, i.e., Rp ~ = =

RF(O ~, 9~, 6(0 ~, ~,~), V~) min RF(O ~, 7 ~, ~(0~7~), V~). (e-,~-,e(0~'))eD~

(37)

Now we show how to construct a sequence of configurations described by with the sequence of characteristic functions {Xn}~=l for which lim F(X n, r

V~) - RP ~.

(38)

n - - - } (~b

In view of Theorem 4 it is sufficient to consider any design given by (9~, 7~) in D~ and show how to construct a sequence {x~'n}~=l for which lim

F(X ~'n, a(x~'n), V~ ) - RF(9 ~, ~/~,e(9 ~, ~/~), V~).

(39)

n--+ c~

We start by observing that for 9 - 0 or 9 - 1 that ~(9,7) - a I or ~ I respectively, where I is the 2 x 2 identity. Thus for a design specified by (9~,7~) we proceed to construct the sequence {X~'n}n~__l in the following way. In the subdomains fli for which 0~ - 0 we set X ~'n - 0, n - 1, 2 . . . c~ and in the subdomains Fti for which 9~ - 1 we set X~''~ - 1, n - 1, 2 . . . co. Next we consider the subdomains ~i where 0 < 0~ < 1. In these subdomains we have 0 < ~k < 2~ and we set X~'n - # ( n x . n ( ~ ) ) , where tt(t) is a periodic function on the real line of period unity taking the values 1 for 0 _< t < 9~

520

Optimal design of gradient fields with applications to electrostatics

and 0 for 0~ < t < 1 and n(7~) = (cos Tk,sinTk). construction in the following equation, Xn'n -

0, 1,

#(nx. n(7~)),

We summarize our

in f~i for which 0~ - 0 , in f~i for which 0~ - 1, in fli for which 0 < 0~ < 1.

(40)

The associated dielectric permittivity e(X n,n) corresponds to pure a dielectric in the subdomains f~i where 0 n = 0, pure ~ dielectric in the subdomains f~i where 0 n = 1, and layers of a and fl dielectric with layer normal in the direction (cos 7 k, sin 7 k) in the subdomains where 0 < 0~ < 1. The associated state variables qon,n are the W~ '2 (f~) solutions to the equilibrium equation - d i v (e(Xn'~)Vg~ n'n) = f. (41) The sequence e(X n,n) G-converges to e(0 n, 7n) [9], hence the sequence 9~n,n converges weakly in w l ' 2 ( f l ) to the state variable 9~n associated with the design (0 n, 7n). With this construction in mind we state the following Theorem that guarantees the existence of recovery sequences of configurations. 5. - Given a design (0 n, 7 n) in D~ the sequence of configurations {X n'n}~=l given by (4 O) is a recovery sequence, i.e.,

Theorem

lim F(X n'~, e(X~'n), VqS) = RF(O n, 7 n, e( theta n, 7n), V~).

n--if(x)

(42)

Proof. We have lim F(X n'n, e(xn'n))

--

~ IV~O~ -- V~3I2 dx

n---), (x)

+ n-,~limf~ IV~ ~ ' n - VqD~I2 dx

(43)

Applying the corrector theory of homogenization given in [9] we can write

~ iV~~'n-

lim n-+(x)

=

V(pn[ 2 dx

~ I(P~'~- / ) V g ~ n +

zn'nl2 dx.

(44)

where pn,n is the corrector matrix associated with X "~,n and on each subdomain f~i it is given by:

pn,n _ R ( < )

,~

0

)+Zx~'n ]

[ a(1--X~'

0

1

RT(Tr), and pn,n ~ I in L 2

521

R. Lipton and A.P. Velo

as n -+ co. Since pn,n E L~(fl) 2x2 it follows from the corrector theorem of F. Murat and L. Tartar [9] that z n,n -+ 0 strongly in L 2. As a consequence lim /~ I(P ~'~ - I)XT~ ~ + z~'nl2 dx n--+oo

_1)2 0 / RT(7~)V~ n" V~ndx I

li~m N ~ ) I / ~ i----1

0

- ~ R(vn)H(On)RT(Tn)V~n" V~ndx,

(46)

and the Theorem follows.

4. Minimizing sequences of configurations In this Section we identify minimizing sequences of design vectors for the problem RP. These sequences are associated with the refinements of a given partition Tn. Next we employ Theorem 5 to deduce that R P = P and identify a special class of minimizing sequences of configurations for P. We recall that a nested family of partitions {Tn]n<~ of ~ is a family that satisfies" 2

(47)

For any given partition T~ the sequence of refinements of this partition is denoted by {Tn)n<~ and is a nested family of partitions as described by (47). We show that the space of discrete designs D~ associated with the refinements of T~ is dense in Do. T h e o r e m 6. - The system of designs {D~ } n-+0 is dense in De. Indeed, for every (0, V, e(0, "7)) C Do, there exists a sequence (0 n, Vn, e(O~, Vn)) E D~ for which

On --+ O, .yn _+ V a.e. in ~2 and e(0 n, Vn) G-converges to e(O, V) as ~ -+ O,

(4s) furthermore: lim RF(O n, V n, e(On, 7n), V@) = RF(O, V, e(O, V), V@) .

n-+0

(49)

522

Optimal design of gradient fields with appfications to electrostatics

Proof For a given design (0, 7, ~(0, 7)) E Do we choose any partition Tn of 12 and consider its refinements {Ta}n<_n. For any refinement Tn , we construct (0 n, 7n, ~(0 n, 7~)) E D~ as described below: Oi,r - meas1 f~i f n ~ O(x) dx

-

1 /a meas 12i

7(x)

dx

6(0 ~, 7 n) - R(7~)A(O~)RT(7~)

on ~i.

We consider the intersection of Lebesgue points for the functions O(x) and 7(x). On this set we have:

On(x) --+ O(x), 7n(x) --+ 7(x)

as n --+ O.

This delivers the convergence

6(O~(x), 7n(x)) --+ e(O(x), 7(x)) - R ( 7 ) A ( O ) R T ( 7 ) a.e. in ~ as ~ --+ 0

and R ( T n ) H ( O n ) R T ( T n ) --+ R ( 7 ) H ( O ) R T ( 7 ) a.e. in 12 as ~ -+ O.

From the properties of G-convergence [18] we deduce as in Theorem 3 that ~(0n(x), 7n(x)) G-converges to ~(0(x), 7(x)) and this establishes (48). This implies that the sequence of state variables ~n, satisfying ~n E W~'2(~) and - d i v (~(0n, 7n)V~n) _ f, (50) converges weakly in Wo '2 (~) to the Wo '2 (12) solution ~ of - d i v (~(0, 7)V~) - f.

(51)

Following the same arguments given in the proof of Theorem 3, we see that the sequence {~n}n>0 converges strongly in W~'2(gt) to ~. Moreover, the same estimate as given in (34) holds for the sequence { R ( T n ) H ( O n ) R T (7 n) }n>o

523

R. Lipton and A.P. Welo

and we can proceed along the same lines as in the proof of Theorem 3 to show that lim RF(~ ~, .7~ , e(0 ~, 7~), V@) - RF(O, 7, ~(0, 7), V@).

(52)

to--+0

We now identify minimizing sequences of designs for the R P problem. We consider any nested family of partitions denoted by {T~}~>0. For each value of n we consider the optimal design for the discrete problem R P ~ denoted by (0~, ~ , e(0 ~, ~ ) ) . T h e o r e m 7. - T h e sequence {(0~,~,~(0~, ~))}~>o, is a minimizing se-

quence for the R P problem and satisfies the monotonicity condition: for n < n', R P ~ = RF(-~ ~, ~ , ~(~, zy,~), V~)

< _ R p ~' _ RF(~ 'r

,

,

),

and lim RF(-~ '~, ~'r 6(~ ~, z/,~), V@) - RP.

~---~0

Proof. The monotonicity follows immediately from the fact that n < n' implies that D O C D~). We note that the monotonicity property implies the existence of the limit lim RF(O '~, -~, ~(~'~, ~'~), V@).

t~--+0

Since D~ C Do we have:

R P < RF(~ ~, ~/'r ~(0~, ~/'~), V~),

(53)

for every n > 0. On the other hand, for a nested family of partitions {T~}~>0 and for any given (0, 7, r 7)) in Do, it follows from Theorem 6 that there exists a sequence {(0 ~, 7 ~, r ~, 7~))}~>0 for which:

RF(~ ~, ~ , ~(~, zy,~), V~) _ RE(O h, 7 ~, ~(0 ~, 7~), V~),

(54)

and lim ~--+0+

RF (-0~,-~, ~(0~, ~ ) , V~)

<_

lim

RF(O ~, 9/~, ~(0~, 9'~), V@)

to--+0+

RE(O, 9', s(0, "Y),V@).

(55)

It is now evident that" It is now evident that lim RF(-O~, ~ , ~(0~, ~ ) , V~) to-+0+

<

inf

- - (/~,9',e(/~,9')) E D o

RE(O, 9', e(O, 9/), V~) - R P

(56)

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Optimal design of gradient fields with applications to electrostatics

and the theorem follows from (53) and (56). 1 With Theorems 5 and 7 in hand it is possible to identify a sequence of configurations specified by X j for which

RP -

lim F(X j, ~(xJ), V~).

(57)

j--+c~

Indeed we consider a minimizing sequence for R P as given by Theorem 7. To each element (0~, ~ , c(0 ~, ~ ) ) of the sequence we can apply Theorem 5 to find a recovery sequence of configurations {X~,n}~=l. In this way we see that R P - lim lim F(X ~'n, ~(X~'n), V~), (58) t~ ----~(x) n--~ (x)

and it follows that we can extract a sequence of configurations {X~J,nJ )~=1 for which R P - lim F(X ~j'~, e(X ~j'nj), V~). (59) j-+cx)

We now establish the following result. We now establish the following result. T h e o r e m 8. - One has

P-

RP

i.e., inf

(X,e(X) )Eado

F(X, e(X), V~) -

inf

(O,')',e(O,q,))E Do

RF(O, 7, e(0, 7), V~).

(60)

Proof. Since ado C Do and from (18) it follows that P >_ RP. Moreover from Theorem 7 and (59), there exist { ( 0 ~ , ~ , c ( t ~ , ~ ) } ~ > 0 E {D~}~>0, such that R P - lim RF(-O ~, 7 ~, e(0 ~, ~ ) , V~) - lim F(X ~j'nj , c(X ~j'~j), V~). (61) ~--+0

j--+c~

On the other hand

F(X ~j'nj, e(X ~j'nj), V~) _> P, for all j.

(62)

Thus (61) and (62) imply that R P >_ P, and we conclude that R P - P. The results presented in this Section provide the way for the identification of a class of minimizing sequences of configurations of the two conductors. Theorem 7 shows how to generate a minimizing sequence of generalized designs coming from discrete problems. Theorem 5 and (39) provide the methodology for constructing an optimizing sequence of configurations based upon the information given in the solution of the generalized design problem. These results give rigorous rules of thumb for the design of two phase conductors. The numerical implementation is given in Section 6.

R. Lipton and A.P. Velo

525

5. A c o m p l e t e c h a r a c t e r i z a t i o n of m i n i m i z i n g s e q u e n c e s In this Section we provide the proof of Theorem 1. We consider a dense class of target fields for which we can account for all oscillations in minimizing sequences of designs. L2(f~) closure of gradients of solutions of (1) with coefficients e(x) given by (3). For X E ado we introduce the set of gradients given by Vu ] u is a W1'2(a) solution of - d i v (e(X)Vu) - f, X E ado.

So -

The strong L2(f~) closure of the set So is denoted by the set So. evident that P -

inf F(X, c(X), Vq3) xCado

=

(63) It is

vuesoinff n [ V u - V ~ 3 [ 2 d x inf VuESo

fiVu-Vq3]2dx. Ja

(64)

In light of (64) and the definition of So we apply Theorem 2 of the introduction to conclude the existence of a G~ subset K of Wo '2 (f~) such that

T h e o r e m 9. - G i v e n a target field ~ E K and a minimizing sequence

{(x

,

n=l E ado

for P then the associated sequence of state variables { ~ n } ~ n--1 solving the equilibrium equation - d i v (e(xn)Vqo n) - f (65) is Cauchy in the W~ '2 (fl) norm given by I[ull2 = f~ [Vu [2 dx.

From the completeness of W~ '2 (f~) there exists a potential q3 E W1'2 (fl) such that limn_,oo qo~ - ~5 strongly in W 1'2 (f~). Passing to subsequences if oo weak LOO(Ft) star converges to a density 0 necessary, the sequence {X }n=l and the compactness property of G-convergence implies that the sequence {e(xn)} ~176 n----1 G-converges to an effective tensor e e where - d i v (eeVq3)- f.

(66)

LFrom the results given in [6] and [11] we have that the set of effective tensors associated with the density 0(x) are all the symmetric 2 x 2 matrices with eigenvalues A1, A2 lying in the set K~ for almost all x in f~. The set K~ is

526

O p t i m a l design o f gradient fields with applications to electrostatics

given by the inequalities 1 k=l AJ -- a

2 k=l

1 ~-Ai

<

1

1

--

hy - a

my - a

< -

1 ~-hy

1 t ~-my"

(67)

On the other hand the work of J. Dvorak, J. Haslinger, and M. Miettinen [5] shows that the strong convergence of the sequence {~n }~=1 delivers the local relation s e v ~ = myV~, a.e. (68) This implies that my is an eigenvalue of ~e. The constraints on the eigenvalues of ~e given by (67) together with (68) allows us to uniquely identify ee as the effective tensor given by e e = R(~)A(-O)RT(~),

(69)

where the angle ~ is chosen according to the requirement given by (68). For this choice of angle we also have the local relation R ( ~ ) H ( - O ) R T ( ~ ) V ~ - 0, a.e.,

(70)

P - Jfa [V~ - V~3[2 dx - R F ( O , ~, e(O, ~/), V~3).

(71)

and we conclude that

In view of Theorem 8 we deduce that the design (0, ~, e(0, ~)) is the optimal design for the problem R P . This establishes parts (1), (2), and (4) of Theorem 1. To proceed we recall the notion of a cluster point (0, r for a sequence of designs associated with a sequence configurations {Xn}n~=l. The definition of a cluster point (0,~ e) implies the existence of a subsequence {XnJ,s(xnJ)}]~ such that {xn/}?=l weak L~176 s t a r converges to 0 and {r G-converges to e e. Arguments identical to those given above show that any cluster point of any minimizing sequence for the problem P is a minimizing design for the problem R P . This establishes the first part of (3) of Theorem 1. The second part of (3) of Theorem 1 follows immediately from the construction of a recovery sequence of configurations based upon Theorems 5 and 7, see equations (59). Part (3) of Theorem 1 together with (68) and (70) point out what kinds of oscillations can occur in minimizing sequences of configurations. In the

R. Lipton and A.P. Velo

527

subregion of ~2 where the minimizing sequence oscillates, i.e., the region where 0 < ~ < 1, we see that the oscillations are in the form of layers of the two conductors. The layers are asymptotically parallel to the optimal gradient V~. This configuration allows for the best effective conductivity properties to be aligned with the direction of the gradient. This is consistent with physical intuition. D

6. Numerical solution and a practical approach to design of graded materials We provide an outline of the method used for the numerical solution of the discrete design problem. For convenience the objective functional is denoted by E (~, -y) and E(O,v)

-

RF(8, "7,e(8, "y), V~)

~

[Vqo- V~I 2 dx

(72)

+ f R(,y(x))H(t~(x))nT(,r(x))V~o 9V~o dx, am

(ra)

where the state variable ~o solves the equilibrium equation (12). For a given partition, the number of subdomains is N(t~) and the design variable (8,-y) is a vector of length 2N(~). The components of (8,-y) are the constant values (8i, %) taken in each subdomain [ti. The components of the design vector are subject to the box constraints: 0 < 8i < 1, i -- 1 . . . , N ( ~ ) , 0 < "yi _< 2rr, i -

1...,N(~).

(74)

We include the resource constraint f~ 8 dx < 0 meas(~) by adding a penalty term

for g >_ 0. The discrete design problem is written

min E(O, 7) + g x ( / (e,-~)

Odx - O meas(~))

(75)

where (6t,7) are subject to the constraints given by (74). The numerical procedure is a straight forward application of the steepest decent method, see [17]. Gradients of the objective E(8, 7) are computed and increments of

528

Optimal design of gradient fields with applications to electrostatics

the design variables ((~0, 5"7) are chosen to insure E(O, "7) >_E(O+50, "7+5"7). The advantage of this procedure is that it is monotone and convergence is assured. We provide numerical examples that illustrate how electrostatic fields can be controlled using functionally graded materials. For all examples the design domain is chosen to be the square centered at the origin given by fl = ( - 1 , 1) x ( - 1 , 1) and we choose the target field to be zero, i.e., V~ - (0, 0). The discrete design is associated with a partition of ft into 20,000 subdomains of diameter on the order of 10 -2. For the first two examples the charge distribution is taken to be uniform in ft and given by f = 1. We choose a = 1 and ~ = 2 and constrain the amount of good dielectric to be 40% of the design domain. The density distribution, O(x), of the better dielectric material in the optimized discrete design is given in Figure la. Here the darkest regions consist of pure ~ dielectric, the white regions are occupied by pure a dielectric and the regions of graded conductivity properties are given by the intermediate shades. The layer normals in the graded parts of the design are given by the arrows in Figure la. The contours are the level lines of the electric potential. Note that the layer normals are tangential to the level lines, hence perpendicular to the electric field. We emphasize that Figure la gives the necessary geometric information for manufacturing graded materials. Indeed, given O(x), '7(x) we can we apply (39) to construct a sequence of graded materials. Because of the continuity expressed by Theorem 5 we are guaranteed that we can construct a two phase configuration thats nearly optimal. For the second example we consider a subdomain D of the design domain Ft. Here we take D = f~\ { ( - 1 / 2 , 1 / 2 ) x ( - 1 / 2 , 1 / 2 ) } . We consider the problem P-

inf ]

xEado

JD

[V~I 2dx.

(76)

The theory presented in this paper easily generalizes to this case and the relaxed problem is

+/D R('7(x))H(~9(x))RT('7(x))V(P " V~ dx} ,

(77)

and P = RP. Here the goal is to screen as much electric field away from the domain D as possible. The good dielectric is constrained to occupy 40% of ft. The density distribution of the good dielectric in the optimal design is given in Figure lb. We point out that we allow the two dielectrics to be placed

R. Lipton and A.P. Velo

Figure 1: a.

529

Figure 1: b.

anywhere in ~, however the algorithm automatically uses the good dielectric only in D. This is consistent with intuition. For the next example we take the charge distribution to be 1 everywhere outside of D and zero inside D. As before we take a - 1 and ~ - 2. The good dielectric is constrained to occupy 15% of the design domain. The density distribution for the optimal design is given in Figure 2a. In Figure 2b we plot the level lines of the potential and the electric field associated with the design. Last we consider the same layout as in Figure 2a but with a = 1 and ~ - 1000 and we plot the electric field for this case in Figure 3. For this layout and choice of ~ we see that the electric field has b4en screened away from D. A c k n o w l e d g m e n t s . This work is supported by AFOSR Grant F4962099-1-0009 and NSF Grants DMS-9700638 and DMS-9403866.

References [1] Bensoussan A., Lions J. L., and Papanicolaou G., Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, 5 North Holland, Amsterdam 1978. [2] Bidaut M., Theorems d'existence et d'existence en g4n~ral d'un contrSle optimal pour des systemes r4gis par des 4quations aux d4riv4es partielles non lin~aires, PhD thesis, Universit4 Paris VI, June 1973.

530

Optimal design of gradient fields with applications to electrostatics

Figure 2" a.

Figure 2: b.

Figure 3: Electric field for f l - 1000.

R. Lipton and A.P. Velo

[3] [4] [5] [6] [7] Is]

[9] [10]

[11] [12]

[13] [14] [15] [16] [17]

531

De Giorgi E. and Spagnolo S. Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. U.M.I. 8 (1973), 391-411. Edelstein M, On nearest points of sets in uniformly convex Banach spaces, J. London Math. Sot., 43 (1968), 375-377. Dvorak J., Haslinger J., and Miettinen M., On the Problem of Optimal Material Distribution, Preprint, (1996). Lurie K. A. and Cherlmev A. V., Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh, 99 A (1984), 71-87. Murat F., Compacit@ par compensation, Ann. Sc. Norm. Sup. Pisa, 5 (1978), 489-507. Murat F. and Tartar L., On the control of coefficients in partial differential equations. In Topics in the mathematical modeling of composite materials, Cherlmev A. and Kohn R. eds., Birkhauser, Boston, (1997), 1-8. Murat F. and Tartar L., H-convergence. In Topics in the mathematical modeling of composite materials, Cherkaev A. and Kohn R. eds, Birkhauser, Boston, (1997), 21-43. Murat F. and Tartar L., Calcul des variations et homog6n6isation. In Les M6thodes de l'homog@n6isation: Th~orie et applications en physique. Collection de la Direction des Etudes et Recherches d'Electricit@ de France, 57, Eyrolles, Paris, (1985), 319-369. Tartar L., Estimations fines de coefficients homog@n@[email protected] Ennio De Giorgi Colloquium, Kree P. ed., Research Notes in Mathematics 125, Pitman, London 1985, 168-187. Tartar L., Compensated compactness and applications to partial differential equations, In Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Volume IV, Knops R.J., ed., Research Notes in Mathematics, 39. Pitman, Boston 1979, 136-212. Tartar L., Remarks on optimal design problems. In Calculus of Variations, Homogenization and Continuum Mech., Buttazzo G., Bouchitte G., and Suquet P. eds., World Scientific, Singapore, (1994), 279-296. Tartar L., H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinb., l15A (1990), 193-230. Pedregal P., Optimal design and constrained quasiconvexity, Preprint, (1998). Pedregal P., Optimization, Relaxation and Young Measures, Bull. Amer. Math. Soc., 36 (1999), 27-58. Pironneau O., Optimal shape design for elliptic systems. Springer Verlag, New York 1984.

532

Optimal design of gradient fields with applications to electrostatics

[18] Spagnolo S., Convergence in energy for elliptic operators. In Proceedings of the third symposium on numerical solutions of partial differential equations, (College Park,1975), Hubbard B. ed. , Academic Press, New York, 496-498. Robert Lipton Department of Mathematics Louisiana State University Baton Rouge, LA 70803-4918 USA E-mail: [email protected] Ani P. Velo Department of Mathematical Sciences United States Military Academy West Point, NY 10996 USA E-mail: [email protected]

Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 24 A BLACKBOX REDUCED-BASIS OUTPUT BOUND METHOD FOR NONCOERCIVE LINEAR PROBLEMS

Y. MADAY,A. T. PATERA AND D. V. ROVAS

1. I n t r o d u c t i o n Reduced-basis methods [2, 5, 9, 19, 20, 24] are very attractive techniques for the efficient prediction of the parametric dependence of outputs - - functionals of solutions - - of partial differential equations. In particular, reduced-basis techniques enjoy a "state-space" optimality property that ensures rapid convergence even in higher dimensional parameter spaces; very good accuracy can thus be obtained with relatively few modes (basis functions), and hence at relatively low cost. In many c a s e s - in which the operator depends affinely on the p a r a m e t e r s - the efficiency can be further enhanced with "blackbox" superposition formulations [12], in which the operation count for the "on-line" stage of the procedure depends only on the dimension of the reduced-basis space and the complexity of the parametric representation. In the limit in which one is interested in rapid evaluations at many points in parameter space (e.g., as in design, optimization, and real-time control), reduced-basis approaches are clearly the most efficient choice. Although there are several a priori error estimates for reduced-basis methods [5, 7, 25] that support the empirically observed exponential convergence rates, there are no sharp and rigorous a posteriori bounds (though see [20] for early efforts in this direction). It is thus difficult, if not impossible, to determine how many empirical basis functions suffice for any given problem, output, and prescribed accuracy; as a result, the application of reduced-basis methods has been rather limited. To remedy the situation, we propose in [12, 17] a blackbox reduced-basis output bound procedure that provides rigorous error bounds for the particular quantities of interest. These methods (as well as an alternative approach to reduced-basis a posteriori error estimation described in [14]) are related to earlier output bound procedures [13, 16, 18, 21], which in turn are based upon classical finite element a posteriori error estimation theory [1, 4, 6, 10].

534

A blackbox reduced-basis output bound method for noncoercive...

In [12, 17] we address either coercive elliptic partial differential equations or very mildly noncoercive equations (in particular, the symmetric positivedefinite eigenvalue problem). In the current paper we consider noncoercive problems; see also [11] for related work within the iterative context. In Section 2 we present the problem formulation; in Section 3 we describe the reduced-basis output approximation and associated error estimator; in Section 4 we prove the optimality of the output approximation as well as the optimality and asymptotic bounding property of the error estimator; in Section 5 we describe the blackbox computational procedure, and present the operation count and storage requirements for each step of the process; and in Section 6 we give numerical results for a particular instantiation, the Helmholtz equation (a future paper will discuss the incompressible Stokes problem).

2. P r o b l e m description 2.1. Nomenclature For a general Hilbert space Z, we denote the associated inner product and induced norm by (-,-)z and I[" [[z, respectively; we identify the corresponding dual space as Z', with norm I1" liz' given by

Ilfilz' = sup f(v) From the Riesz representation theorem we know that for every f E Z' there exists a p~ E Z such that pZI , v ) z = f ( v ) , Vv e z .

It is then readily deduced that

~z

Ilvilz'

and

ilfllz, = iip~ilz, which we will use repeatedly in what follows.

2.2. Problem statement Given a Hilbert space Y of dimension N (possibly infinite), a linear functional s E Y', and a parameter # in a set 79 c ~ P , we look for u(~u) E Y such that a(u(~), ~; ~) = e(v), w e Y, (1)

Y. Maday, A. T. Patera and D. V. Rovas

535

where a(., .;/~) is a bilinear form the assumptions on which are detailed below. We further prescribe an output functional ~o E Y~, in terms of which we can evaluate our output of interest s(/0 as

~(~) = to(~(~)).

(2)

We also introduce a dual, or adjoint, problem associated with go: find r e Y such that

~(v, r

~) = _ t o ( ~ ) , w e Y.

The relevance of this dual problem will become clear in what follows. It is relatively simple to permit p-dependence in I and ~o as well, however for clarity of exposition we do not consider this here. We shall assume (though this is essential for only some of our arguments) that a is symmetric,

a(w, v; 1~) = a(v, w; 1~), Vw, v e y2, v # e z~. We further assume that a(w, v; #) is uniformly continuous,

la(w, v; #)1 < ",'ll~llvllvllv, Vw, v e y2, v/.t e "D, and that a(w, v; #) satisfies a uniform in#sup condition, 0 < &
-

~eY~ev II~llvlivllv

~eY

II~llY

it is classical that these latter two conditions are required for the wellposedness of our primal and dual problems. Finally, we shall make the assumption that our bilinear form a is affine in the parameter # in the sense that, for some finite integer Q, Q

a(w, v; It) = ~ ae(l~)aq(W, v), Vw, v e y2,

(4)

q--1

where the aq are bilinear forms. The assumption (4) permits our blackbox approach; non-blackbox variants of the methods described here - - in which (4) is relaxed - - can also be developed, and will be discussed in a future paper.

2.3. Inf-sup supremizers and infimizers We can rephrase (3) as: for every w e Y, there exists an element T~w in Y, such that #(#)llwllvliT.wllY < a(w, T.w; #), (5)

536

A blackbox reduced-basis output bound method for noncoercive...

where Tgw is the supremizer associated with [la(w, -; #) ][y, . It follows from Section 2.1 that T~,w = Pa(w..;tt) Y that is,

(T.w, v)v = a(w, v; ~), Vv e Y. It is thus clear that Tff Y ~ Y is a linear operator; we can also readily show that T~ is symmetric (since a is symmetric); furthermore, T~ is bounded, since

IIT, wll~ =

a(w. T,w; It)

and hence

<_ 711wllvllT.wllr.

IIT.~llII~llv <_ 7.

(6)

Finally, we can now express our inf-sup parameter in terms of Tg as

#(#) = inf ~eY

IIT~wllY= IIT~x(g)IIY IlwllY

where X(~t) = arg inf

wEY

IIx(~)llY '

(r)

IIT,.,~IIII~llY

is our infimizer; we thus also have

#(#) =

a(x(#),T.x(#))

IIx(~) II Y FIT~----~-~rl Y

It will be useful in the subsequent analysis to recognize that ~/(#) and X(#) are related to the minimum eigenvalue and associated eigenfunction of a symmetric positive-definite eigenproblem. In particular we look for (0, A) e (Y x R ) solution of A(O(/.t), v;/.t) = A(/.t)(O(#), v)y, Vv E Y, and IIO(#)llr = 1,

(8)

A(w, v; #) = (T~,w, T~,v)y, Vw, v e y2;

(9)

where we denote the resulting eigenpairs as (Oi(/~), Ai(#)), i - 1 , . . . , with ~min ----)h __ A2 _< ---. It follows immediately from Rayleigh quotient arguments that A(w w; ~) [[T, wll~ ~2 Amin(#) = min ' = min = (#) .omv ( w , ~ ) v .omr I1~11~Y

and thus ~(#) = r

(#) and 0min (#) --" 01 (~) = )[~(#).

Y. Maday, A. T. Patera and D. V. Rovas

537

To further understand the relationship between the infimizers and supremizers, we consider a second symmetric positive-definite eigenproblem: find (T x w) e (Y x IF/) such that 2 7 ( v ( u ) , v ) r = ~(u)B(:r(u), v; u),

(10)

where B ( ~ , ~; u) = 27(~, ~ ) r - ~(w, ~; u);

note that/3 is symmetric and coercive. By the usual arguments (and appropriate normalization), 27(Ti, Tj)y = wiB(Ti, Tj) = wi6ij, with 0 < wt <_ w2 < ---; here 6ij is the Kroneeker delta symbol, and (Ti,wi) refers to the i th eigenpair associated with (10). We can then write (T.~. v ) r = 2.y(w. v ) r - B(w. v; ~).

expand Af w = E ciTi, i=1

and exploit orthogonality to deduce that

x Tuw = E diTi i--1

for di = 27ci(wi- 1)/wi. Thus .N"

49'2 E IIT.~II~ _ II~II~ -

i--I

wi(#)- 1 w,(u)

2

Af i--1

again by orthogonality. We conclude that

~(u) =

2~

wi.(u)(#)

-

I

and X(#) = ci*(t~)Ti*(~), where i*(z) = a~g

min ie{~ .....g}

wi(#)- 1 ~(~)

(11)

538

A blackbox reduced-basis output bound method for noncoercive...

We thus observe that

TuX =

27

wi.(,)(#)

X,

which states that Tux and X are collinear; in general, Tuw and w will be linearly dependent only if w is proportional to a single eigenfunction T i.

3. Reduced-basis output bound f o r m u l a t i o n 3.1.Approximation space We select M points in our parameter set/9, #m E D, m = 1 , . . . , M , the collection of which we denote

s'= We then introduce three "Lagrangian" spaces [25], W M = span{u(#m), rn = 1 , . . . , M},

(12)

W~ t = span{r

m = 1 , . . . , M},

(13)

WM = s p a n { x ( ~ m ) , m - 1 , . . . , M } ,

(14)

and associated with our primal solutions, dual solutions, and infimizers, respectively. We shall consider two approximation spaces W N. In the first, we set N = 2M and choose W 2v = WoN, where

WoN =

wy+w2

=

span{u(pl),r162

--

span{Cx,...,Qv}.

(15)

In the second ease we set N = 3 M and choose W N = W ~ , where

w:'

=

wy + wy +

=

span{u(u*),O(Ul),X(#I),...,u(uM),r

--

s p a n { ~ l , . . . , ~'N}.

(16)

(Obviously the ~N - - the reduced-basis functions - - take different meanings in the two cases.) The role of each of the components of the W N shall become clear later in our development.

Y. Maday, A. T. Patera and D. V. Rovas

539

Remark. Compliance. In the case in which go = ~, then r = -u(~); we thus redefine WoN = W M (with N = M) and W N = W M + WxM (with N = 2M). This will of course result in computational savings. Note that if ~o is close to ~ then WoN of (15) and W N of (16) can lead to ill-conditioned systems; we shall not address this in the current paper (or the more general issue of ill-conditioning arising in the reduced-basis context). I 3.2. O u t p u t P r e d i c t i o n We shall see shortly that the W N will play the role of the infimizing space. We also need supremizing spaces. To that end, we introduce V N C_ Y, with (-, ")vN = (', ")Y and hence 1[-IIvN = [[" [[Y. We shall consider two spaces in particular- V N = W N (= WoN or WN), and hence of dimension N; and V N = Y, and hence of dimension Af. We next define, for all w N E W N and for all ~oN E W N, the primal and dual residuals, Rpr(-; wN; I~) E Y~ and Rdu(.; ~Og; l~) E Y', respectively: R p~ (v; wN; ~) Ra"(v; ~oN; ~)

-

e(v) - a ( w ~v, v; ~), Vv e Y,

=_ - e ~ (v) - a(v, ~oN ; #), Vv E Y.

It follows from our primal and dual problem statements that R p~(v; wN; ~)

= a(u - w ~ , v; ~),

(17)

= a(v, r - ~,N; ~,), which is the standard residual-error relation evoked in most a posteriori frameworks. We then look for uN(#) e W N, c N (#) E W iv, such that

~N(#)

=

arg

=

arg

=

arg

inf

wNEW N

IIRP"(.;wN;~)II(v~),

R p~(v; wn;

#)

inf

sup

inf

[[Rd~(';~0N;#)[[(V~),

(18)

and

r

~oN E W N

=

arg

inf

sup

Rdu(v; ~oN; l~)

(19)

which is a minimum-residual (or least-squares) projection; see also [5] for discussion of various projections within the reduced-basis context. Our output approximation is then given by:

A blackbox reduced-basis output bound method for noncoercive...

540

8N(it) ---- t O ( u N)

-

-

RPr(r

uN;

It);

(20)

the additional adjoint terms will improve the accuracy [8, 15, 23]. It will be convenient to express our minimum-residual approximation in terms of affine supremizing operators p N. Wiv -+ VN,DN~ : Wiv ~ Viv, defined by

p Nw N

VN

--- p a p r ( . ; w N ;~) ,

VN ~ N w N = PRdu( ;wN;.)'

that is

(P~wiv, v)v = RPr(v; wiv; it), Vv 6 Viv,

(21)

( D iv , w iv , v ) v = RdU(v ; wiv ;it), V v 6 Viv ,

(22)

for any wiv 6 Wiv. In particular, it follows from Section 2.1 that we can now write

u N (it) = arg wNinfewN IIPNw N IIv, cN (it) = arg ~oNinfeWN IIDt~ ivcpiv IIv; (23) the # dependence is through Pff and D ~ . 3.3. E r r o r b o u n d p r e d i c t i o n We first define ~iv (it) 6 Hi~ as inf

sup

~(w N u; ~) '

DN(~) = ,,,~ew~,,ev,,, IIwNIIvllvllY

=

II~(wN; ";~)ll(v~), . w'ew" IIwNIIv inf

(24)

We can rephrase (24) as: for any wiv 6 Wiv, there exists an element T N W iv in Viv, such that

/~r(~)IIwNIIYIITNwNII Y < a(wN, T~w~V; ~),

(25)

where ~TNwN is the supremizer associated with [[a(wN,.;It)ll(vN),. It fo1VN

lows from Section 2.1 that T N- Wiv -+ Viv is given by pa(wN,.;~), or more explicitly,

(TNw N, v)y = a(wiv, v; It), Vv 6 V N, for any w N 6 W N. We can now express ~iv (it) as

f~N(#) =

inf

I[T~NwIv [[v

IITy~(~)IIY IIxN(~)II,"

Y. Maday, A. T. Patera and D. V. Rovas

541

where

XN(~) =

arg

inf

~ew"

IITNwNIIY IlwNIIY

is our infimizer over w N; we thus also have

a(x~(#),Tfx~(,)) BN(~)--IlxN(~) IYIIT~x.N(~)IIY Then, given u N (p), c N (~u), and a real constant a, 0 < a < 1, we compute

1 [Inpr('; A N ( p ) -- a~N(D)

uN ;ll)[]y,l[ndu('; r

;/~)[[Y',

(26)

which will serve as our a posteriori error bound for [ ( s - sN)(/~)[. Remark. Output Bounds. We can of course translate our error bound A N (p) into (symmetric) upper and lower bounds for s, sN(p) -- s g ( p ) + A N(#), sN_(#) = sN(#) -- AN(#). In our earlier work [17, 23] we focus on these output bounds, s N, rather that on the error bound (or bound gap), AN, since for coercive problems the output bounds are in fact nonsymmetric due to a shift which also effectively reduces the bound gap by a factor of two relative to the noncoercive case. II

4. Error analysis In Section 4.1 we analyze the accuracy of our reduced-basis output prediction s N, and in Section 4.2 we consider the properties of our error estimator A N . In both Section 4.1 and 4.2 we make certain hypotheses about fiN(#) that we then p r o v e - at least for the case V N = Y, W g = W g m in Section 4.3. Note that, for convenience of exposition, we shall not always explicitly indicate the # dependence of all quantities.

4.1. A priori theory We first prove that our discrete approximation is well-defined, as summarized in L e m m a 1 - I f fiN(D ) >_ ~o > O, V# e D, then the discrete problems (18) and (19) are well-posed. I f furthermore V g = W N, the minimum-residual statement is equivalent to standard Ga/erkin approximation: u n ( p ) = uN'GaI(#), where uN'Vat(p) e W N satisfies

RP~(v;uN'G"~(~);~) = O, Vv e W N.

(27)

A blackbox reduced-basis output bound method for noncoercive...

542

An analogous result applies/'or the dual. Proof. We consider the primal problem (18); analysis of the dual problem (19) is similar. To begin, we recall t h a t Pty N 6. V/V satisfies

(p[~, v)r = t(v), Vv e v N. It thus follows that, for any w N 6- W N,

P ~ wiv = Ptv ~ _ TNwN; from our minimum-residual statement (23) we then know t h a t u N 6- W N satisfies

( T ~ u N, T y v ) v = t(T~v), Vv e W ~.

(28)

We now choose v = u N in (28) and note that, since T ~ u N is the supremizer over V N associated with u ~r,

( T ~ u N , TNu~r)y = a(u N, T f uN; #) _> and thus

1

BNII~IIYIITy~NIIY,

1

Ilu~rllY _< ~--~-Ilellv, < ~lltllv,. We have thus proven stability; uniqueness follows in the usual way by considering two candidate solutions. Finally, we consider V Iv - w N : from standard arguments we know that, for/~N (/t) > /~o > 0, the Galerkin approximation (27) admits a unique solution u g,vat. But since [[pNuN'aal[[y -- O, UN'Gal must be the (unique) residual minimizer, and hence u N = u N'~al. m We can then prove that u/v, C g are optimal. Indeed, we have Lemma

2 - IT ~lv (t~) >_ ~o > O, V# 6 T), then

min Ilu(~) - w NIIv, Ilu(~) - u N(#)IIY -< 1 + ~2 7 ) w"~w" with an analogous result for the dual. Proof. Since for any w N E W Iv, w N - u N is an element of W Iv, we have from (25) t h a t

~NIIwN -- UNIIYIITN(w N -- UN)IIY

Y. Maday, A. T. Patera and D. V. Rovas

u ~, T ~ ( w N

uN); #)

=

a(w ~

<

I~(~" - ~, T."(w N - ~");,)I + I~(~ - ~", T f ( ~ "

= <

<_

-

u + u

-

-

543

- ~");,I

IR~(T~(~" -~");~";#)I + IR~(T."(~" -~");~";,)I

(IIP.-"Vw"IIY +

IIP~uNIIY)

IIT~(w"

-

uN)llv

N - uN)llY,

211P~wNIIvllT~(w

(29)

where the last three steps follow from (17), (21), and (23), respectively. We now take v = p N w N e V N in (21) and apply (17) and continuity to obtain

(30)

IIPflw NIl" -< ~11~ - w N llv , which then yields, with (29),

27

I1~'~ - u ~ IIv < ~

(31)

I1~ - ~,N IIv, Vw N e W ~.

The desired result then follows by expressing I l u - uNIIv as I l u - w ~ + w N -ulV[[y and applying the triangle inequality, (31), and our hypothesis on ~N (#).

II

Finally, we can prove that our output prediction is optimal in the following result:

T h e o r e m 1 - / f fiN(#) >/~o > 0, V# e 7:), I(~ - ~N)(#)I _< "yll~ - ~NIIvllr -- CNIIv;

if furthermore V N D_W N, which is always satisfied for our choices,

1(8- sN)(~)l <_ ~

1 + ~47 )

inf inf I1r ,o,,ew,, I1~- w NIIY ~oNEW N

~Niiy.

Proof. We have that I(~- ~)(#)!

=

le~

=

I - "(" - . N , r ~) + a(u - u ~ , r

=

la(u - u ~ , r - r

<

~'llu- u N I I v l l r - CNIIY,

which proves the first result.

- tO(u N) + t ( r ~ ) - . ( . ~ , r ~)I

~)I

(32)

544

A b l a c k b o x reduced-basis o u t p u t b o u n d m e t h o d for n o n c o e r c i v e . . .

We also know that, for all qoN E W ~ , w N E W N, la(u - u N, r - ON; ~)1 = la(u - u N, r - ~ N + ~ N _ CN;#)[ <_

la(u - u N, r - ~N;#)[ + [a(u - u N, ~ N _ cN;#)[

<

~llu - uNIIrllr -- ~ONIIv + IRP"(,,o N - CN; UN; #)1

_< 0' 1 +

Ilu-wNIIvllr

+11r r - r

sup vev,,

IIv

RP"(v;uN;~'), Ilvllv

(33)

where we have evoked continuity, (17), Lemma 2, and W N C_ V N. But from (21), (23), (30), and the dual version of (31) sup vEY N

R p~(v; uN; ~) il~N Ilvllv

CN

_<

lC

li

20' lir 20'

<_ ~ l l u - w ~ l l v ~ l l r which with (32) and (33) proves the second result.

- ~N

~N

IIY, II

Theorem 1 indicates in what sense reduced-basis methods yield optimal interpolations in parameter space. We could of course predict s(~) at some new value of # as some interpolant or fit to the s(~um), m - 1 , . . . , M; however, it is not clear how to choose, or whether one has chosen, the best combination of the s(#m), in particular in higher dimensional parameter spaces. In contrast, Theorem 1 states that, by predicting s(~u) via a state space ( w N ) , and by ensuring stability (~0 > 0 independent of N), we obtain in some sense a best a p p r o x i m a t i o n - the correct weights for each of the reduced-basis components. With sufficient smoothness, this best approximation will converge very rapidly with increasing N [7, 25] (see also Section 4.3). Note the importance of W ~ in W N in ensuring that infvNew~ [ [ r ~oN[[y is small - - had we included only WM in W N, this would not be the case, since reduced-basis spaces have no general approximation properties (that is, for arbitrary functions in Y). Of course, Theorem 1 only tells us that we are doing as well as possible; it does not tell us how well we are doing ~ our a posteriori indicators are required for that purpose. 4.2. A posteriori t h e o r y We can directly show that, under certain hypotheses, our error estimators are indeed error bounds.

Y. Maday, A. T. Patera and D. V. Rovas

545

T h e o r e m 2 I [ ~ N ~ fl as N -4 oo, then there exists an N*(#) such that, V N > N*(lz), -

I(~ - ~N)(.)I < ~ ( ~ ) , for AN(I~ ) as given in (26). Proof. We first note, evoking symmetry and our inf-sup condition (5), that

~(~)11r - cNIIvlIT,(r - CN)IIv

< =

~(T.(r - CN). r _ CN; ~) Rd~(T.(r _ CN). CN; ~)

< IIRa~(';r

IIT.(r162

,

or

I1r -

1 Rdu('; cN ;,)llw. CNIIY <- Z(.)II

We then write, from (32) of Theorem 1,

I(~- ~)(,)1

= = <

la(u - ~N, r _ CN; g)l I R ~ ( r - r =~; g)l IIR~(.;uN; ~)llY'il~- ~NIIY 1

_< ~(~) IIRP~('; uN; ~)llY' IIRa~('; ON; g)llY'. The result then directly follows: for a < 1, we have from our hypothesis on ~N that a~ N (#) < /~(#) for N sufficiently large, say N > N*(#), and thus

I ( ~ - 8N)(~)i

1

<- ~(~) IIRP~(';uN;~)IIY'IIRa=(';r 1

< ,r~N('-----~llRP"(';uN;tt)llv,llRa~'(';c'N;~)llr', = ~N(~). for N > N* (#).

m

It is not only important to determine that AN(D ) is a bound for the error, but also that it is a good bound. As a measure of bound quality, we introduce the usual a posteriori effectivity,

AN(p) 0 ~ ( ' ) = I~(') -- ~N(~)I"

(34)

Under the hypothesis of Theorem 2 we know that r/N(/*) > 1 as N --+ oo, providing us with the desired bounds; to ensure that the bound is tight,

546

A blackbox reduced-basis output bound method for noncoercive...

we would also like to verify that ~,]N(~) <_ Const (independent of N) as N -~ c~. We have no proof for this result, but it is certainly plausible given the demonstration of Theorem 2, and is in fact confirmed by the numerical experiments of Section 6.2.

4.3. The discrete inf-sup parameter It should be clear that good behavior of the discrete inf-sup parameter is the essential hypothesis of Theorems 1 and 2. If/~N vanishes, or becomes very small relative to/~, Theorems 1 and 2 indicate we risk that [ ( s - s N ) ( # ) [ and A N will both become very large: accuracy of our predictions thus requires/~N > /~0 > 0. However, too much stability is not desirable, either. If/3 N is large compared to /~ as N ~ oo, Theorem 2 indicates we risk that AN(p) will not bound [(s - sN)(#)[: certainty in our predictions thus requires/~N close to /3. It is clear that the best behavior is /~N _~ /~ from above as N -+ cx~. We now discuss several possible choices for V N, W N, and the extent to which each - - either provably or intuitively - - meets our desiderata.

4.3.1. The choice V N = Y, W N = W1N. M e t h o d 1 It is simple in this case to prove stability:

L e m m a 3 - For V N = Y (and any space W N C Y), ZN(~) > Z(~) > Z0 > 0, for all # E :D.

Proof. We have

Z~(~)

liT"-"~ ll:~

_ inf -~-',,~w~

IIT.wNIIII,.,.,NIIv

=

inf ~-',,~w.',,

> -

inf liT.wilY = Z(~) > Z0 > 0, ~v I1~11~

IIw~ll.

as desired.

I

Thus, for V N = Y, the hypothesis of Theorem 1 is satisfied with/~0 =/~0; we are guaranteed stability. To ensure accuracy of the inf-sup parameter - - and hence asymptotic error bounds from Theorem 2 - - we shall first need

L e m m a 4 - If ~M iS C]lOSell such t h a t sup inf [[#_ #m[[ _.+ 0 ~,~z~ ,,,e{1 ..... M}

547

Y. Maday, A. T. Patera and D. V. Rovas

as M --+ oo, and if X(#) is sufficiently smooth in the sense that

II sup IIV~xll IIv < ~ , DE/)

then

inf

w~r E W ~

(35)

IIx(~) - w NllY -~ o, v~ e ~ ,

as M (and hence N)--r c~. Note [l" 11refers to the usual Euclidean norm. Proof. Recalling that X(#), the infimizer, is defined in (7), we next introduce

~N(~) e w y as ~N(#) = X(#m" 0,)), m* (#) = arg

min

me{1 ..... M }

l# - #m I.

Thus

IIx(~,) - ~NCi,)llr

< (

inf

me(1 ..... M }

< (sup --"

I1~- ~11)II sup IiV~xll IIY

inf

DEW me{1.....M}

#E'D

I1~ - ~mll)ii sup II%xll IIv, v~ e z~, #ET)

and therefore for all # E :D, inf

wNEW N

IIx(#)--wNIIv

<

IIx(~) - ~N(~)llv

<

(sup

inf

pET) me{1..... M }

I1~- ~mll)II sup IIV, xII IIY, pET)

which tends to zero as M (and hence N) tends to infinity from our hypotheses o n (.~M and the smoothness of X(#). II Clearly, with sufficient smoothness, we can develop higher order interpolants [25], suggesting correspondingly higher rates of convergence. For our purposes here, (35) suffices; the method itself will choose a best approximation, typically much closer to X than our simple candidate above. The essential point is the inclusion of W M in W ~ , which provides the necessary approximation properties within our reduced-basis space. We can now prove that, for V N = Y, W N = WxN, fin (#) is an accurate approximation to fl(#). T h e o r e m 3 - For V N = Y, W iv = WxN, flN(#m) = ~(#m), m = 1 , . . . , M .

(36)

Furthermore, under the hypotheses of L e m m a 4, there exists a C independent of N and an N**(#) such that

inf [ [ X ( # ) - w N [ [ ~ , , V N > N ** I ~ ( ~ ) - ~N(#)I <_ c - - - 72 2Z(~) ~N~w~ '

(37)

548

A blackbox reduced-basis output bound m e t h o d for noncoercive...

and thus from L e m m a 4,

(38) Proof. To prove (36), we note that, since X(# m) e W N,

inf

IIT"~wNIIY<

~N(u~)=~W~

IIwNIIr

--

IIT'~x(U~)IIY = ~(U~); IIx(u~)IIY

but BN(#m) k B(#m) from Lemma 3, and thus BN(# m) = B(l~m). To prove (37), we introduce the discrete eigenproblem analogous to (8)" find (0N, AN) e (W~ x ~/) such that

~t(o N(~), v; ~) = AN(~)(o N(~), v)y, Vv e w ~ ,

II0N(~)IIY = 1;

by arguments identical to those of Section 2.3 it is simple to show that = X/Amin(#). We can now apply the standard theory for Galerkin approximation of symmetric positive-definite eigenproblems. To wit, from Theorem 9 of [3] and (35) of our Lemma 4, there exists an N**(#) such that, VN > N** (#), [Amin(#) -- ~min N (~)[ ~---C A ( 0 m i n - w N, 0min -- w N ; # ) ,

VW N ~. W N

for C independent of N. (One can in fact show that C may be taken as (1 + 2~3)~.) But from (6) and (9) of Section 2.3, we know that jd[(0mi n -- w N , 0min -- wN;#) --IlTu(Omin -

wN)II~ _< 72110min -

The result (37) then follows by recalling that (/~)2 = )~min, and noting that

0min

-"

X,

(/~N)2

wNII~. _.

)kmi nN ,

I(~N) ~ - Z21 = I(~ N - ~)11(~ N + ~)1 _> I(~ N - B)I2B since ~N >_ ~ from Lemma 3.

II

The hypothesis of Theorem 2 is thus verified for the case V N = Y, W N W N. The quadratic convergence of ~N is very important" it suggests an accurate prediction for /~.w and hence bounds - - even if W N is rather marginal. 4.3.2. T h e C h o i c e V N = Y, W N = W~ v

Method 2

In this case the X ( # m ) , m = 1 , . . . , M , are no longer members of W N. We see that Lemma 3 still obtains, and thus the method is stable - - in

Y. Maday, A. T. Patera and D. V. Rovas

549

fact, always at least as stable as W N = W N . Furthermore, since WoN still contains W M and W~ r, we expect i l u - u N IIY and 0]r c g ilY to be small, and hence from Theorem 1 I ( s - sN)(#)I should also be small. There is no difficulty at the level of stability or accuracy of our output. However, Lemma 4 can no longer be proven. Thus not only is (36) of Theorem 3 obviously not applicable, but - - and even more importantly u (38) no longer obtains: we can not expect Bg (#) to tend to B(~u) as N --+ oo. In short, the scheme may be too stable, fiN may be too large, and hence for any fixed a < 1 we may not obtain bounds even as N -+ oo. In short, in contrast to the choice W N = W N , the choice W N = WoN no longer ensures that ~N(#) is sufficiently accurate. In practice, however, BN (#) may be sufficiently close to fl(#) that a ~ g (#) < ~(#) for some suitably small a. To understand why, we observe that, in terms of our eigenpairs (Ti, wi) of Section 2.3, ~(Ti) wi(# m) - 1 Ti.

u(#rn) =

(39)

o___

For "generic" ~, u(# m) will thus contain a significant component of T i-(~) and hence x ( # m ) . It is possible to construct ~ such that s = 0, and hence we cannot in general count on X(# m) being predominantly present in WoN; however, in practice, ~ will typically be broadband, and thus W - WoN may sometimes be sufficient. Obviously, for greater certainty that our error bound is, indeed, a bound, W = W N is unambiguously preferred over W = WoN. 4.3.3. T h e C h o i c e

V N = W N,

W N = W N ~

Method 3

We know from Lemma 1 that this case corresponds to Galerkin approximation, but with W M present in our spaces. We first note that not only does Lemma 3 not apply, but unfortunately we can prove that ~N(#m) < ~(#m), m = 1 , . . . , M : -> -

),

sup inf

sup

>

a ( w , v; flwilYfl

sup _

) (40)

ilr

since X(# m) E W N c Y. Stability and accuracy of the output could thus be an issue, though not necessarily so if fin (#) is close to ~(/~). As regards the accuracy of ~N(#), Lemma 4 still applies, however (36), (37) (and hence (38)) of Theorem 3 can no longer be readily proven.

550

A blackbox reduced-basis output bound method for noncoercive...

Nevertheless, in practice, ~N may be quite close to/~. To understand why, we recall from Section 2.3 that X(# m) is not only our infimizer, but also proportional to T~-X(#m). It follows that if X(# ~) is the most dangerous mode in the sense that sup

~ ( x ( , ~ ) , ~ ; , ~)

< sup

~(w, v; ~ ) ,

~ew~ IIx(#")llrllvllv - ~ew." Ilwllrllvllr

w e w~,

(41)

then

BN(#') = sup .(x(t.'). ,,; #")

-(x(#"). T..~ x(#"); #m)

since both X(# m) and Turn(X(#m)) are in WN; note that (41) is a conjecture, since the supremizing space here is W[ v, not Y as in Section 2.3. Under our assumption (41), we thus conclude from (40) that ~lv(#m) = ~(#m).

(42)

By similar arguments we might expect/~N (#) to be quite accurate even for general # E D, as both X(#) and TuX are well represented in the basis. (From this discussion we infer that for nonsymmetric problems a PetrovGalerkin formulation is desirable.) The above arguments are clearly speculative. In order to more rigorously guide our choice of V ;v, we can prove an illustrative relationship between the Galerkin V iv = W N, W ;v = W ~ (superscript "Gal")and minimum residual V/v = Y, W N = W ~ (superscript "MR") approximations: T h e o r e m 4 - For 831 # E D,

~XN,Ma(~) < A~,G~,(~),

(43)

where A N,MR and A N,Gal refer to (26) for the minimum-residuM and Galerkin cases, respectively. Proof. We first note that

B N'Gal =

inf sup a(w,v;#) < inf sup a(w,v;#) ,o~w~ ~ewr Ilwllvllvllv -weW,,, ~ev Ilwllvllvllv

__ ~ N , M R

(44) for all # E 2). We then note from (23) that A N , MR

~_

1

~N.MR IIR'" (" uN'M~; ~')IIY'I1Rd"(', cN, MR;u) IIY' 1 or~N, MR

1

IIP~uN, MRIIv lID.N r ~,MR IIv

,r~N.MRIIP~w~IIvlIDN.~'NIIv. VW~v e Wx~, V~oN e W~r,

551

Y . M a d a y , A . T. P a t e r a a n d D. V. R o v a s

where P ~ - W N --~ Y and D N" WIN --+ Y are here defined for V N = Y. Thus A N , MR

<-

N.MR1 IIPffuN.G,IIYIInN./,N.G,._. 1

=

~-

)IIY'

o.~N, MFI 1 N,Gal II

IIY

uN'Gal;)llY'llnd (" CN,G l; )llv'

where the last step follows from (44).

=

AN'Gal, m

We thus see that, in general, V N = Y will provide sharper error estimates: minimum residual is in fact equivalent to minimum error bound. Conversely, we might expect the Galerkin approximation to be more conservative, providing bounds even when the minimum-residuaJ approach may not (e.g. for N very small). The Galerkin method with W g = W N thus has some redeeming features. However, there is the possibility of a loss of accuracy in both s N a n d A N , reflected in (43) and (44) of Theorem 4. 4.3.4.

The Choice V g = W0N, W N = WoN

Method 4

This case corresponds to Galerkin approximation, but now with W M absent. Here (40) no longer applies: /~N(#) may be greater or less than /~(p). Furthermore, accuracy of/~N (#) now relies on two fortuitous events m the "selective amplification" of (39) and the "most dangerous mode" of (41). Again, in practice, the method may perform well, but it is now more likely that either/~N will be too small and hence ( s - s N ) ( p ) and A N too large, or/~N will be too large and hence ~N < 1 (no bounds). We are able to prove a result analogous to Theorem 4, but now comparing V g - ii, W g - WoN to V N -" WoN , W N "- WoN: t h e / ~ v (respectively A N ) for the former will be larger (respectively smaller) than the corresponding quantities for the latter. We thus expect that V N - W0N, W N - WoN will yield rather poor accuracy.

5. Computational procedure For clarity we shall present the detail for the most "rigorous" and general scheme, V N = Y , W N = W f (Method 1 of Section 4.3.1). As we proceed, we indicate simplifications for the other schemes, and at the conclusion we give a comparison of computational complexity.

A blackbox reduced-basis output bound method [or noncoercive...

552

5.1. A n a l g e b r a i c

representation

5.1.1 . P r e l i m i n a r i e s We assume here that Y is finite dimensional, with associated basis ~i, i = 1 , . . . ,Af. We also recall t h a t W g can be expressed as span{(i, i = 1 , . . . , N } ; we implicitly assume independence of the reduced-basis functions. A m e m b e r w E Y is expressed as wt~, w E ~N'; a m e m b e r w E W N is expressed as w ~(, w E ~ g . Here t denotes the transpose. We next introduce the matrices A Y'Y E j~Arxjv', A w N , w N E ~ N x N , AY, W N ~ j~Ar• BY, Y E KIAt• B Ws'WN E KI N• given by

Y, Y, , A i j t#) wN,w N Aij (It) AY, W N

=

a(fj,fi;#),l

< i , j <_AZ,

=

a((j,(/;#),l

< i , j < N,

~,~ (u) = BY, ~,~V (u) =

a((j,fi;#),l


wN,w N Bij (It)

< N,

(~j,~i)v, 1 <_ i , j < A/',

=

(r

1 < i , j <_ N.

matrices we can derive four further matrices, Z Y'Y z w N , y ~ .~Nx.A[', ~___.Y,Y~.. j~A[xA/', ~.swN,wN E .~N• as

From these

j~.A/'x.~,

Z_v,v(#) z_W~,V(#) S__v,v(#) ~_w',w~ (u)

=

Av, Y(~)(Br, Y)-~

=

(AY, W "(~))~(By,Y)-~ (AV,V(~))~ (Br, V)-~Ar, V(~) (A__v'w'(u))~(Bv'v)-~A_ v'wN(u),

=

=

where t denotes m a t r i x transpose. These matrices are representations m in terms of our bases m of the operators introduced earlier. For example, if w E Y and v E Y are expressed as wt~ and vt~, then w t B Y'Y v is (w, v)y; ((z)Y'Y(#)) ~ is our representation m

m

m

of Tu; and for w E Y expressed as wt~, w ~ sY'Y(It)w is (T~,w,T~,w)y. It follows t h a t (fl(#))2 =

wt SV, v (#)w

min --. w_.e~R.v" wt B Y, Y w

(45)

Similarly, for w E W N expressed as wt~_, w ~ s w N ' W N w is (TNw, T N w ) , and

w~SW''w" (~)w (fiN(#))2

=

min

wE]RN wtBWN,WN w

.

(46)

Y. Maday, A. T. P a t e r a a n d D. V. R o v a s

553

8 WN'WN (~_~)represents the normal equations associated with the leastsquares approach, and /~N is the "generalized" smallest singular value of AY, W N ; see [5] for an earlier discussion of singular values and stability in _ the reduced-basis context. Finally, we shall need the vectors L_pr'Y 6_ I R N , L__pr'WN 6_ _I~N,L du'Y 6_ K I ~ f , L d~'WN 6_ Kl N, defined by Note

Lpr, i Y LPr, W N

=

/(r

< i <.N',

i

=

e(r

< i <__N,

L dU,V

_

eO(~i) l < i <./kf,

L du,W N

_

~O(~i) 1 < i < N ,

which are the obvious representations of our primal and dual linear functionals. 5.1.2.

Reduced

basis

We first find, for m = 1 , . . . , M, u_m E K/j~, r A Y,Y (#m)Um

_A

=

=

(or s _Y ' Y (#m )u m - zY'YL__.Pr'Y),

L_._pr'Y

-Ld

E -~J~, solution of

'v

(or S v ' v (/~m)r

= _zY, YLdu,Y).

(47) (48)

We further obtain (Xm, ~mmin)as the first eigenpair (0_,~) e ( ~ N X ~ ) of the symmetric positive-definite problem S Y'Y (#re)O__ : ~B____Y'Yo__, OtBY'Yo : 1.

(49)

(Note that in some cases it may be preferable to find X m by considering the (-,-)y - B(-,-; #) eigenproblem of Section 2.3; an inverse iteration with shift (of unity, initially) may work well.) It can then readily be shown that u(#m), r X(#m), m = 1 , . . . , M, of (12), (13), and (14) are given by Ac i:l H j:l X j:l

554

A blackbox reduced-basis output bound m e t h o d for noncoercive...

where this last result can be readily motivated from (45); furthermore, ~m~i, = (/~(#m))2, though we shall not have direct need of this result in the construction of W N. Note that for W N = WoN (Methods 2 and 4) we may omit (49); this constitutes significant "off-line" savings (see Section 5.2.2 below). 5.1.3. O u t p u t p r e d i c t i o n We first find, for given # E 2), u N (#) E ]R N, r

(#) E 2/?N, solution of

s_wN, WN (#)u N (#)

=

z WN, v L.Zr,v

s_WN,W N (#)r

=

_ z W " , Y Ld,,,v ;

(#)

(5o)

it is readily shown that uN(#) and cN(#) of (18) and (19) of Section 3 are given by N

E

j=l N

r

j--1

We can then evaluate s N of (20) as 8 N ( ~ ) = (r

(f~))~s

N _ (~.N(/.~))~ (s

N _ AwN,w

N (/.$)~N(~[~)). (51)

Note that for the Galerkin formulations, V N = W N, we may replace (50) with AWN'WNuN -- L_.P"'W~; but since we will need S_WN'WN in the error prediction step below, this is not a significant simplification. 5.1.4. E r r o r b o u n d p r e d i c t i o n We first calculate fiN(#) = J~;mNin(~), w h e r e /~mNin(#) is the eigenvalue associated with the first eigenpair (~_N(#), ~N (#)) e (JlqN X ~I~) of

_sw~'w~ (~)e "(~) = ~"(~)BW~,W~e_" (~),

(e_"(~))~B w~'w~ e"(~) = 1,

as motivated by (46) above. Note in this "integrated formulation" that the same reduced-basis matrix, S_WN'WN , serves to determine u N, c N , and fiN. We next compute _Tpr(#) E ~Ar, rflu(#) E ~N', solution of B Y ' Y T_Pr(#)

=

L._p''Y - A Y'WN (#)uN(#)

(52)

BY'Yrd"(~)

=

- L d"'Y - A Y'w~ (~)_r

(53)

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It can be readily shown that P ~ u N(D) , D ,N r g (p) as defined by (21) and (22) of Section 3 (for V N = Y) are given by Ar

j=l DNcN(,)

=

H ~--~TdU(p)~j--(r_du(p))*~_,

j--1 respectively. Note that these calculations (52), (53) are required even for the Galerkin approach: we must compute the Y~ norm of the residual to estimate the error. Lastly, it then follows that A N (p) of (26) can be evaluated as

AN (#) --___ ~1 ( (T_.Pr(#) ) t BY, Y T.Pr(#))1/2 ( ( TdU(#) ) ~BY, Y TdU (#) )1/2 ,

(54)

which completes the procedure. 5.2. Blackbox approach 5.2.1. P r e l i m i n a r i e s The blackbox approach comprises two separate stages [12, 22]. The first stage, or off-line stage, or pre-processing stage, invokes various solutions and evaluations associated with the space Y; it will typically be expensive, since for many problems Af is very large. The quantities produced by this off-line stage are passed to the second stage, or on-line stage, which serves to predict sN(#) and AN(p) for any new value of #; this stage invokes only low-dimensional entities - - that is, the computational complexity for each new value of p is independent of Af, and depends only on N, the dimension of the reduced-basis space, and Q (defined by (4)), a measure of the "complexity" of the parametric dependence. Our "affine" assumption of (4) plays a critical role in permitting this two-stage decomposition and associated linear superposition. It follows that for N and Q small, as is often the case, the on-line reduced-basis predictions for slY(p) will be much less costly than direct appeal to u(#) in (1) and subsequently s(#) in (2). (In fact, very often A r ' r has structure and sparsity not shared by S_WN'WN , which will typically be dense; nevertheless, for N << Af, as is usually the case, the on-line reducedbasis prediction is still much faster than direct appeal to u(#) and s(p).) Reduced-basis methods are thus very advantageous in situations such as control, in which real-time response is required, and in situations such as

556

A blackbox reduced-basis o u t p u t b o u n d m e t h o d for n o n c o e r c i v e . . .

design or optimization, in which many parameter values are visited (such that the off-line effort can be amortized). Critical to this argument is the ability to assess the error in the reduced-basis predictions - - through A N (#) - - so that the minimal number of modes N may be (safely) retained. To describe the blackbox procedure, and demonstrate the N-independence of the on-line stage, we shall need a few additional definitions. First, we recognize that the ~i, i = 1 , . . . , N, can be represented in terms of the ~j, which we express as ~i = _ where z i E K~~f, i - 1 , . . . , N . Second, we need to introduce the matrices -- .hf • AqY , Y E~R q=l,. ,Q, givenby (AY, . . = aq(~j, ~i), 1 < i, j < Af, q Y~,,,a

where the aq(-,-) are defined (implicitly) in (4). We shall summarize the computational effort in the off-line stage in terms of A Y'Y-solves, _SY'Y-eigensolves, B Y'Y-solves, A Y'Y-actions (matrix vector products), BY'Y-actions, and Y-inner products (inner products between two N-vectors). Note for simplicity we assume that A_qY'Y-actions are roughly equivalent to AY'Y-actions. In many problems of interest, in particular in which there is underlying sparsity in A Y'Y, S_-eigensolves will be the most expensive, then A_-solves, then BY'Y-solves (less costly because the equations are symmetric positive-definite), and then the "actions" (often only O(Af)) and Y-inner products. We shall summarize the on-line computational effort directly in terms of N and Q (albeit somewhat imprecisely, sometimes considering a multiplication and an addition as a single operation). Note that in the on-line stage we are not compelled to exploit all N basis functions computed in the off-line stage, and thus N in the on-line stage may be replaced with Nused(~), with the error bound A N (#) guiding the choice of minimal Nused [26]; this can significantly reduce the cost of the on-line predictions. As regards storage, we shall report, in the off-line stage, both Temporary Storage (required just during the off-line stage) and Permanent Storage (quantities passed by the off-line stage to the on-line stage). All quantities stored in the on-line stage originate in the off-line stage. (We note that the on-line code is completely segregated from the original solution method on Y - - the link is only the relatively few stored quantities - - and furthermore comprises only very elementary (and very few) operations that can be simply implemented and shared.) The simplifications to the procedure in the case of "compliance" should be clear.

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5.2.2. Off-line stage

1. Compute reduced-basis vectors: u_m E ~Af, Cm E ~Af, Xm E j/~Af,m = 1 , . . . , M, from (47), (48), and (49). Recall that the Xm are not needed for = w2'.

Complexity: 2M _AY'Y-solves, M_SY'Y-eigensolves. 2. Compute B WN'WN E ~ N x N

as

wN,w N

B ,j

=

or

_t nY, Y

wN,w N

Bi, j

= z__iD__ z_j.

Complexity: NB__Y'Y-actions, N 2 Y-inner products. Temporary Storage: NAf. Permanent Storage: N 2. 3. Compute lgqi E Kl~ , 1 < q < Q, 1 < i < N, where (l,~qi) k = aq (~i, ~k), 1 ~_ k ~_ J~f, or

~-~-qi = A Y'Y z-i

Complexity: N Q Ag'Y-actions. Temporary Storage: N Q N . 4. Compute V_V_q~E ~ X , 1 <_ q <_ Q, 1 <_ i <_ N, and ~_~" ~_ ~ x , ~F_~,~~_ ~ x , solutions of

B___Y'Y V__qi= Ltqi , B__Y'Y ~_~r = L_p",Y, B__Y,Y V_.du o

= L_L_du, Y .

Complexity: (NQ + 2) B__Y'Y-solves. Temporary Storage: (NQ + 2)Af. 5. Compute

FqyW(--

Fq'qi'i),

1 g q, q~ < Q, 1 <_i,r < N, as rqq, ii,

t

= Llqi~__q,i, .

Complexity: N2Q 2 Y-inner products. Permanent Storage: N2Q 2.

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A blackbox reduced-basis output bound method for noncoercive...

6. Compute A~, Ad~', 1 _ q _< Q, 1 _ i <_ N, as -~-q i "--O , A q i

-

-~--q i "--o

9

Complexity: 2 N Q Y-inner products. Permanent Storage: 2NQ. 7. Compute ~

e ig, Co d" e Ig, as

Complexity: 2 Y-inner products. Permanent Storage: 2. 8. Compute L__pr'WN E KIN,L_ du'WN E Kt N, as

LPr'WN -- (LPr'Y)tzi L --idu'wN -- (Ldu'Y)tz_i ' i = 1 Complexity. 2N Y-inner products. Permanent Storage: 2N. 9. Compute EqU, , 1 <_ q <_ Q, 1 <_ i s N, as "~'qii'

aq ( ~i, ~i' )

-=

Complexity: N2Q Y-inner products. Permanent Storage: N2Q. 5.2.3. O n - l i n e s t a g e

Given a new value of the parameter # E 7): 1. Form S_WN'WN (#) E Kl NxN as Q Q Si,i W'N'WN = E Z aq(p)aq,(tt)Fqq, ii,, 1 <_ i,i' < N. q=l q'=l

Complexity: N2Q 2. 2. Form the necessary "right-hand" sides: x WN'YLP'r'Y ~-~Z~~_,__,j -3

=

j--1

j--1

1 < i < N, - -

q--1

~f

wN,Y Ld.u,Y ~" Z _ ,j -3

Q Zaq(#)A~' Q

--

Z

q--1

Olq(#) Aqi du ,

1 < - i < - N.

N.

Y. Maday, A. T. P a t e r a and D. V. Rovas

559

Complexity: 2NQ. 3. Find ug(#) 6 KlN, cN (#) 6 KlN, ~g(~) G ~ solution of (47), (48), and

(5o). Complexity: 0 (N 3). 4. Compute s N(#) of (51) as

~g (~) __ (s

(p)

_

Q N

N

q:l i:1 i':l

Complexity: 2N + N2Q. 5. Compute AN(p) of (54) as Q or~ N

(~)

q=l i--1

Q +

N

Q

E

N

E

q:l u

q

N

E E aq(P)aq'(p)u~(p)uN(p)Fqq'ii'] 1/2• i:l i':I

N

q : l i=I

Q

Q

N

N

+E E E E Olq(.)Olu(.)r (.)r (.)rqq,ii,]1/2. q : l q ' : l / : I i'--I

Complexity: 2(N2Q 2 + NQ + 1). We now briefly discuss the computational complexity of the different schemes. The first comparison is between minimum residual (V iv - Y) and Galerkin (V N - W N) approaches. The important point to note is t h a t the q u a n t i t y - Fqq,ii, - - required by Method 1 (V r~ = Y, W N = WIN) to form the projection matrix S_WN'WN (p) is the same quantity required by all the methods to compute the error bound A N (#); in both capacities, Fqq,ii, represents the calculation of the necessary Y~ norm. In this sense (see Theorem 4) the arguably better scheme V N = Y, and somewhat riskier scheme V N = W Iv, have similar complexity, and we contend t h a t V ~v = Y is thus

560

A b l a c k b o x reduced-basis o u t p u t b o u n d m e t h o d for n o n c o e r c i v e . . .

preferred. (We note, however, that faster blackbox schemes and related nonblackbox variants require a Galerkin approach, and we thus retain Galerkin projection as an important alternative.) The second comparison is between WoN and W ~ . For the on-line component, the difference is not large m N - 3 M vs. N - 2M; however, for the off-line component, the calculations of X(~U'n) can indeed be onerous, and its omission thus welcome. However, there is a corresponding rather significant loss of security, since the accuracy of BN is no longer controlled (See Sections 4.3.2 and 4.3.4).

6. The Helmholtz problem 6.1. Formulation We take here Y = H~(n), where n is a suitably smooth domain in ~ d , d = 1, 2, or 3, and H~ (12) is the space of functions which are square integrable over n, which have square integrable (distributional) first derivatives over n, and which vanish on the boundary of n, On. For our inner product we take

(w, v)y =/~ Vw-Vv +

wv

~

which induces the norm

IlwllY -- (/n Vw. Vw + ww) It is important to remark that we may substitute for (., .)y any inner product which induces a norm equivalent to the H l ( n ) - n o r m for example, a good preconditioner. The latter will of course greatly reduce the off-line computational cost, as BY'Y-solves will now be much less expensive. For our bilinear form we take

a(w, v; ~) = f~ V w . V v - g(x; ~ ) w v , where we assume that g(x; I~) satisfies

[g(x;~)l__
n

(55)

=

q--1

Y. Maday, A. T. Patera and D. V. Rovas

561

m

where Gq E L~176 q = 1 , . . . , Q. The difficult case, on which we focus here, is of course when g(x; ~) is positive, as in the reduced-wave (Helmholtz) equation. The decomposition (55) is, in fact, reasonably general. We shall consider the situation in which P = 2 with parameter # = (kl, k2), Q = 2, FI(~) = k 2, F 2 ( ~ ) = k 2, and a~(x)=

1 0

xEFh xefh

G2(x) =

0 1

xEf~ x e f12 '

'

where ~ = ~1 t2 ~2; this represents variable "frequencies" in two subdomains. It can be shown that the regularity of X required in Lemma 4 follows from the smoothness of the Fq(#) and the interpretation of X as 0rain of (8). It is simple to see that our requirement (4) is readily satisfied for Q = 3: we choose aq(#) = Fq(#),q = 1, 2, and a3(#) = 1, with aq(w,v) = -f~ Gg(x)wv, q = 1, 2, and a3(w, v) : ff~ V w . Vv. It is similarly easy to show that a is symmetric, and also uniformly continuous with (say) 7 = 1 + gmax- The inf-sup condition will be satisfied so long as we exclude from l) neighborhoods of points # for which there exists a w such that a(w, v; #) -0, Vv E Y. In general, if the inf-sup condition (3) is thus satisfied, and and t o are any bounded linear functionals, then our theoretical results of Section 4 will obtain. We make two points of a more practical nature. First, in practice, we will of course not know where resonances occur, and thus we will typically posit a parameter domain which does indeed contain several points at which the inf-sup condition does not hold. However, unless driven to such a point by a design or optimization process, it is unlikely that a particular # will coincide exactly with an eigenvalue, and thus for some sufficiently small /~0 our hypotheses will be "in practice" satisfied. (Obviously the physical model may also be made more elaborate, for example by including damping that will regularize the resonances.) Second, in practice, we choose not Y = H~ (~), but rather Y = Xjv, a suitably fine (say finite element) approximation of finite (albeit very large) dimension A/'. As we are more and more conservative in defining this "truth" approximation, that is, as we increase Af, the off-line computational effort will of course increase; however, thanks to the blackbox formulation, the on-line computational effort is independent of the dimension Af.

6.2. Numerical Results We take here d = 1 and gt = ]0, 1[ (though obviously the computational savings provided by the reduced-basis approach will only be realized for

562

A b l a c k b o x r e d u c e d - b a s i s o u t p u t b o u n d m e t h o d for n o n c o e r c i v e . . .

more complicated multidimensional (d > 1) problems). Our truth space X~f is a linear finite element approximation with 200 elements. We consider the two-parameter Helmholtz equation defined in Section 6.1, with n l - ]0, 0.5[ and fl2 = ]0.5, 1.0[. For simplicity, we present a "compliance" case in which

t(v) = t~

= fo.55

V,

,]0.45

corresponding to an imposed (oscillatory) distributed force for the input and an associated average displacement amplitude measurement for the output. In the below we shall consider the four methods associated with the four choices of spaces of Sections 4.3.1, 4.3.2, 4.3.3, and 4.3.4: Method 1 refers to the choice of Section 4.3.1, V N = Y, W N = W N, and will be denoted by the line pattern (--) in all plots; Method 2 refers to the choice of Section 4.3.2, V N = Y, W N = WoN, and will be denoted by the line pattern (---); Method 3 refers to the choice of Section 4.3.3, V N = W N , W N = W N , and will be denoted by the line pattern ( - - ) ; and Method 4 refers to the choice of Section 4.3.4, V N = WoN, W N = WoN, and will be denoted by the line pattern (..-). In general, subscripts, unless otherwise indicated, refer to the Method index above. Throughout this section we take a - (1.1)-1: it follows from Theorem 2 that a sufficient (though not necessary) condition for bounds is that /~N be within 10% of/~. For most of the results of this section, we choose an effectively one-dimensional parameter space D which is the subset of Z)' = ]11, 11[ x ]1,20[ in which neighborhoods of the two resonance points p - (kl, k2) = (11, 7.5) and p - (kl, k2) = (11,14.4) have been excised such that fl0 = 0.02. (Of course, in practice, we would not know the location of these resonance points, and we would thus consider 7) - 7:)' - - which would only satisfy our inf-sup stability condition, "in practice," as discussed in the previous section.) To begin, we fix M - 3, and hence N - 2M - 6 since we are in "compliance," with #1 = (11,2), #2 = (11,8) and thus #3 = (11,14), and thus 8 M = {#1,#2,#3}; we shall denote this the "M = 3" case. We first investigate the behavior of the discrete inf-sup parameter, the accuracy of which is critical for both the accuracy and bounding properties of our output prediction. In Figures 1 and 2 we plot the discrete inf-sup parameter ~N, i = 1 , . . . , 4, and the ratio ~ N / ~ , i = 1 , . . . , 4, respectively, as a function of k2 for fixed kl (see Section 6.1); recall that the index i refers to the method under consideration. We first confirm those aspects of the behavior t h a t we have previously demonstrated. First, f/N and BN axe never less than t~N, as shown in Lemma 3 and Section 4.3.2, respectively; and ~N > ~N, as must be the case since the inf space is smaller. The choice V N - Y ensures stability. Second, we see t h a t fin _> ]~3N and ~2N _ ~N, as demonstrated in

Y. Maday, A. T. P a t e r a a n d D. V. R o v a s

563

(44) and Section 4.3.4 respectively; the methods with smaller supremizing spaces are perforce less stable. Third, we see (by closer inspection of the numerical values) that BN is never greater than B at the sample points, consistent with (40); in fact, we observe that equality obtains at the sample points, (42), and hence at least in this particular case the conjecture (41) appears valid. Fourth, we notice that ~ can be either below or above ~, and is clearly the least "controlled" of the four approximations. (Indeed, for other parameter values we observe near zero values of BN at points quite far away from the true resonances of the system.)

Figure 1: The discrete inf-sup parameter for Methods 1, 2, 3, and 4 as a function of k2 (see text for legend). The symbol x denotes the exact value of ~.

Figure 2: The ratio of the discrete inf-sup p a r a m e t e r to the exact inf-sup parameter for Methods 1, 2, 3, and 4, as a function of k2 (see text for legend). The thick line denotes the "sufficient" limit: if f~N < 1.1~, bounds are guaranteed.

It is clear from Figures 1 and 2 t h a t B/v is indeed a very accurate predictor of f~ over most of D for Methods 1 and 3; we anticipated this result in Theorem 3 and the discussion of Section 4.3.3. We now study the convergence of BN to B as N increases. For this test we consider a sample s M = {#m, m = 1 , . . . , M}, with the #m randomly drawn from l:); the particular parameter points selected are given in the second column of Table 1. (Note t h a t for a given M, indicated in the first column of Table 1, S M con= sists of all #m, m = 1 , . . . , M.) We present in Table 1 the vaJues of f~N _ fl for Methods i =1, 2, 3, and 4 for k2 = 11 (and hence D = (11, 11)). We note that, indeed, fin converges very rapidly for i = 1 and i - 3 - - the two methods in which we include the infimizers in V n - - whereas for i - 2 and i = 4 we do not obtain c o n v e r g e n c e - not surprising given the discussion

564

A blackbox reduced-basis output bound m e t h o d for n o n c o e r d v e . . .

Table 1:

M

1 2 3 4 5 6

The error B~v - B for Methods i =1, 2, 3, and 4, for k2 = 11, as a function of M. #M

(11, 4.7351) (11,19.0928) (11,11.4848) (11,13.6038) (11, 1.4975) (11, 2.6998)

i=1 1.81e 1.70e 3.52e 9.25e 6.57e 1.90e -

01 01 04 06 09 11

i-2 2.92e + 4.28e 2.76e 5.76e 4.91e 4.81e -

00 01 01 02 02 02

i=3 -1.88e -2.03e -1.24e 3.99e 2.43e 4.49e

-

01 01 04 06 09 08

i=4 1.08e + 4.14e -9.85e 5.31e 4.13e 3.83e -

00 01 02 02 02 02

of Section 4 . 3 . Note also t h a t whereas the convergence of M e t h o d 1 is (and m u s t be) monotonic, this is not necessarily the case for M e t h o d 3. We conclude t h a t Method 2 a n d in particular M e t h o d 4 are not very reliable: we can certainly not guarantee asymptotic bounds for any given fixed a < 1; for this reason we do not recommend these techniques, and we focus primarily on Methods 1 and 3 in the remainder of this section. However, in practice, all four m e t h o d s may perform reasonably well for some smaller a, in particular since the accuracy of the inf-sup p a r a m e t e r is only a sufficient a n d not a necessary condition for bounds. Indeed, for our M = 3 case of Figures 1 and 2, Methods 1, 2, and 4 produce bounds for all k2 less t h a n approximately 18 and Method 3 in fact produces bounds for all k2 in D; consistent with T h e o r e m 2, bounds are always obtained for all m e t h o d s so long as aB N <_ B. T h e breakdown of bounds for Method 1 (which in fact directly correlates with aB g > B) is due to the poor infimizer a p p r o x i m a t i o n properties of W N for larger k2; if we include an additional sample point, ~u4 = (11, 20), we recover bounds for all D. (In fact, even for lower k2 the infimizer approximation is not overly good; but thanks to the quadratic convergence proven in T h e o r e m 3 the inf-sup p a r a m e t e r remains quite accurate.) T h e fact t h a t M e t h o d 3 produces bounds over the entire range is consistent with the "less stable" a r g u m e n t s of Section 4.3.3. However, by these same arguments, in particular T h e o r e m 4, we expect t h a t the bound gap the controllable error in the o u t p u t prediction - - will be larger for M e t h o d 3 t h a n for M e t h o d 1. To d e m o n s t r a t e this empirically, we plot in Figure 3 A N / I s l , i = 1 and i = 3, as a function of k2, for the M = 3 case of Figures 1 and 2. We observe that, indeed, the bound gap is significantly smaller for M e t h o d 1 t h a n for Method 3. Note also t h a t the normalized bound gap is quite large for the k2 at which we no longer obtain bounds for M e t h o d 1; no doubt these predictions would be rejected as overly inaccurate and

Y. Maday, A. T. Patera and D. V. Rowas

565

Figure 3: The normalized bound gap A~/Is I for Methods i =1 and i =3 as a function of k2.

requiring further expansion of the reduced-basis space (thus also recovering the inf-sup parameter accuracy and hence bounds). As regards the convergence of the bound gap, we present in Table 2 convergence results for A ~ , i = 1 and i = 3, for k2 = 11 (and hence # = (11,11)), as a function of M (analogous to Table 1 for the inf-sup parameter). (Note for Methods 2 and 4 the convergence is slower, with the bound gap typically an order of magnitude larger than for Methods 1 and 3; this suggests that the inclusion of the infimizers can, in fact, reduce the approximation e r r o r - as might be anticipated from (39).) We observe that the differences between Methods 1 and 3 become smaller as M increases; however it is precisely for smaller M that reduced-basis methods are most interesting. "We conclude - - given that the two methods are of comparable c o s t - that Method 1 is perhaps preferred, in particular because we can also better guarantee the behavior of the inf-sup parameter. Note that the difference in the true error for Methods 1 and 3 is much smaller than the difference in the error bound for the two methods; this is expected, since the inf-sup parameters do not differ appreciably. It follows that the effectivity (defined in (34)) of Method 1 is lower (and hence better) than the effectivity of Method 3; this is not surprising, since for Method 1 the approximation is designed to minimize the bound gap. We close by considering a second set of numerical results included to demonstrate the rapid convergence of the reduced-basis prediction as N increases even in higher dimensional parameter spaces: we now consider 7:) = ]1, 20[ x ]1, 20[ (without excision of resonances, and hence satisfying our infsup condition only "in practice"). In particular we repeat, the convergence

A blackbox reduced-basis output bound method for noncoercive...

566

Table 2:

Table 3:

M 1 2 3 4 5

6 7

The bound gap for Methods i =1 and i = 3, for k2 = 11, as a function of M.

M

#M

I 2 3 4 5 6

(Ii, 4.7351) (II, 19.0928) (II, 11.4848) (11, 13.6038) (11, 1.4975) (11, 2.6998)

i=1 2.23e 2.13e 5.37e 4.01e 6.43e 1.65e-

04 04 06 08 11 14

i=3 1.73e 6.21e2.68e 5.80e 6.50e1.62e-

01 01 05 08 11 14

The b o u n d gap and effectivity at # = (11,17), as a function of M , for Methods i = 1 and i - 3, for the two-dimensional p a r a m e t e r space Z~ - ] 1 , 20[ x ]1, 20[. #M (7.5388, 14.2564) (2.9571, 7.1526) (4.2387, 9.0533) (17.7486, 15.9503) (9.7456, 14.0523)

(3.8279,16.3388) (11.2113,

8.0970)

1.30e 1.22e 5.82e 1.10e 2.99e 2.81e 5.40e-

04 04 05 05 08 08 12

24.20 3.51 1.08 2.56 3.47 3.71 3.78

6.09e 3.05e 8.77e 1.24e 3.07e2.88e 5.44e -

04 04 05 05 08 08 12

12.18 11.88 1.49 3.42 3.71 3.97 3.85

scenario of Table 2, but now choose our r a n d o m sample from the t w o dimensional space l) = ]1,20[ x ]1, 20[; we present, in Table 3, the bound gap and effectivity (defined in (34)) for Methods 1 and 3 for a particular "new" p a r a m e t e r point # -- (11, 17). We observe, first, t h a t we obtain bounds in all cases (~/N > 1) m indicative of an accurate inf-sup parameter prediction; second, t h a t the error (true and estimated) tends to zero very rapidly with increasing M , even in this two-dimensional p a r a m e t e r space; and third, t h a t Method 1 again provides smaller bound gaps (and lower effectivities) t h a n Method 3, consistent with Theorem 4 - - though the difference is only significant for very small M. Note t h a t u and the o u t p u t s are order 10 -3, so the relative errors are roughly 1000 times larger than the absolute errors in the table. Results similar to those reported in Table 3 are also obtained if we consider the error over a r a n d o m ensemble of test points # rather t h a n a single test point.

Acknowledgements.

We would like to t h a n k Dr.

Luc Machiels of

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567

Lawrence Livermore National Laboratories and Prof. Jaume Peraire of M.I.T. for numerous helpful discussions, and for suggestions leading to simpler and more illustrative proofs. This work was supported by the Singapore-MIT Alliance, by AFOSR Grant F49620-97-1-0052, and by NASA Grant NAGl-1978.

References [1] M. Ainsworth and J. T. Oden. A posteriori error estimation in finite element analysis. Comp. Meth. Appl. Mech. Engrg., 142:1-88, 1997. [2] B. O. Almroth, P. Stern, and F. A. Brogan. Automatic choice of global shape functions in structural analysis. AIAA Journal, 16:525-528, May 1978. [3] I. Babuska and J. Osborn. Eigenvalue problems. In Handbook of numerical analysis, volume II, pages 641-787. Elsevier, 1991. [4] It. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial differential equations. Math. Comput., 44(170):283-301, 1985. [5] A. Barret and G. Reddien. On the reduced basis method. Z. Angew. Math. Mech., 75:543-549, 1995. [6] R. Becket and R. Rannacher. A feedback approach to error control in finite element method: Basic analysis and examples. East - West J. Numer. Math., 4:237-264, 1996. [7] J. P. Fink and W. C. Rheinboldt. On the error behaviour of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech., 63:21-28, 1983. [8] M. B. Giles and N. A. Pierce. Superconvergent lift estimates through adjoint error analysis. Technical report, Oxford University Computing Laboratory, 1998. [9] M. D. Gunzburger. Finite element methods for viscous incompressible flows. Academic Press, 1989. [10] P. Ladeveze and D. Leguillon. Error estimation procedures in the finite element method and applications. SIAM J. Numer. Anal., 20:485-509, 1983. [11] L. Machiels. Output bounds for iterative solutions of linear partial differential equations. Preprint. [12] L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera, and D. V. Rovas. Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Sdrie I, to appear. [13] L. Machiels, Y. Maday, and A. T. Patera. A "flux-free" nodal Neumann

568

[14] [15]

[16]

[ls]

[19] [2o] [21] [22]

[23] [24] [25]

A blackbox reduced-basis output bound method for noncoercive... subproblem approach to output bounds for partial differential equations. C. R. Acad. Sci. Paris, Sdrie I, 330(3):249-254, Feb 1 2000. L. Machiels, Y. Maday, and A. T. Patera. Output bounds for reducedorder approximations of elliptic partial differential equations. Comp. Meth. Appl. Mech. Engrg., to appear. L. Machiels, A. T. Patera, J. Peraire, and Y. Maday. A general framework for finite element a posteriori error control: Application to linear and nonlinear convection-dominated problems. In ICFD Conference on numerical methods for fluid dynamics, Oxford, England, 1998. L. Machiels, J. Peraire, and A. T. Patera. A posteriori finite element output bounds for the incompressible navier-stokes equations; application to a natural convection problem. Journal o.f Computational Physics, submitted. Y. Maday, L. Machiels, A. T. Patera, and D. V. Rovas. Blackbox reduced-basis output bound methods for shape optimization. In Proceedings 12th International Domain Decomposition Conference, Japan, 2000. To appear. Y. Maday, A. T. Patera, and J. Peraire. A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem. C. R. Acad. Sci. Paris, Sdrie I, 328:823-828, 1999. D. A. Nagy. Modal representation of geometrically nonlinear behaviour by the finite element method. Computers and Structures, 10:683-688, 1979. A. K. Noor and J. M. Peters. Reduced basis technique for nonlinear analysis of structures. AIAA Journal, 18(4):455-462, April 1980. M. Paraschivoiu and A. T. Patera. A hierarchical duality approach to bounds for the outputs of partial differential equations. Comp. Meth. Appl. Mech. Engrg., 158(3-4):389-407, June 1998. M. Paraschivoiu, J. Peraire, Y. Maday, and A. T. Patera. Fast bounds for outputs of partial differential equations. In J. Borgaard, J. Burns, E. Cliff, and S. Schreck, editors, Computational methods for optimal design and control, pages 323-360. Birkh~iuser, 1998. A. T. Patera and E. M. Ronquist. A general output bound result: application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci., 2000. To appear. J. S. Peterson. The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Star. Comput., 10(4):777-786, July 1989. T. A. Porsching. Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation, 45(172):487-496, October 1985.

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[26] R. von Kaenel. Reduced basis methods and output bounds for partial differential equations. Dipl. thesis, MIT, EPFL, 2000.

Yvon Maday Laboratoire Jacques-Louis Lions Universit~ Pierre et Marie Curie Bo~te courrier 187 75252 Paris Cedex 05 France E-mail: [email protected] Anthony T. Patera Massachusetts Institute of Technology Department of Mechanical Engineering Room: 3-266 Cambridge, MA 02139-4307 U.S.A E-mail: [email protected] Dimitrios V. Rovas Massachusetts Institute of Technology Department of Mechanical Engineering Room: 3-264 Cambridge, MA 02139-4307 U.S.A E-mail: rovas~mit.edu

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Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 25 SIMULATION OF FLOW IN A GLASS T A N K

V. NEFEDOV AND R. M. M. MATTHEIJ

I. Introduction The manufacturing of glass is a complicated and expensive process. The glass is produced in a so-called glass oven or tank. The raw materials such as soda and sand are dumped on one side of the oven, which is heated from above by gas burners. Ovens are constructed in such a way that the glass stays inside for some time, and after about 20 hours flows via feeders to the production lines on the other side. There are, therefore, several processes involved, such as flow and heat transfer and various chemical reactions. In order to study how different oven configurations affect the production, numerical simulation is required, as full scale experimental studies are too expensive to carry out. Since the complexity of the problem is quite appalling, existing codes for tackling this problem are in for improvement and further sophistication, like adaptivity in gridding, to make the simulations more feasible (with respect to memory and computing time). In [6] a local refinement procedure was combined with a glass oven model. The method proposed was based on staggered grid finite volumes [7]. In this paper we study a different way to simulate the glas flow, based on locally uniform grids. The next section describes the mathematical model, namely the equations we are dealing with, the boundary conditions and the specific physical parameters. In the third section we discuss the discretisation procedure. In section 4 the solution method is described, including some implementation details. Section 5 deals with a local refinement technique called local defect correction (LDC). Its application to the glass flow is discussed in the two last sections, viz. a stirred flow in section 6 and bubbling in section 7.

2. Mathematical model The oven in fact consists of an oven proper (melting tank) and a second part, connected by a small channel, from which the glass is going to a feeder

Simulation of flow in a glass tank

572

channel for further processing, leading to the actual product eventually, see Fig.1. The mathematical model consists of equations, describing physical processes occuring in the glass flow, complemented by suitable boundary conditions. In order to restrict this still general model to our particular case of interest we perform a dimension analysis based on glass glass flow properties.

2.1. Main equations Since melted glass can be considered a viscous Newtonian fluid, we can model it by the (incompressible) Navier-Stokes equations

(p(x)u, Vu) = F V - ( p ( x ) u ) = 0.

-

Vp + V - ( # ( x ) V u ) ,

(1)

Here p, p are the viscosity and the density of the glass respectively, and F = (0, O, _pg)T is a gravitational force. Since the glass flow exibits steady state behaviour for large time scales, these equations are written in a time independent form. The motion of the glass is caused by temperature differences; thus the set of equations is incomplete without the energy equation. For a steady lowvelocity flow with negligible dissipation we can derive the energy equation in the following form

V . (Cpp(x)uT) = V - ( k ( x ) V T ) ,

(2)

where k and Cp are the conductivity and the heat capacity, respectively.

2.2. Boundary conditions A typical glass oven configuration is depicted in Figure 1. We can define the boundary values for this geometry as follows: the velocity has Dirichlet values on the inflow, outflow and, assuming no-slip, on the walls too. The top layer is modelled as a symmetry plane, that is the normal component is zero; for the rest we prescribe homogeneous Neumann boundary conditions. For the temperature the situation is different, since the heat loss from the walls and heat influx from the top layer are not known a priori, but can be expressed via a Robin-type boundary condition. In particular we can write -(u,n)lr,~1,~

= uo,

-(u,n)lro~,1,~

= ul,

V. Nefedov and R. M. M. Mattheij

573

Figure 1: Sketch of the glass tank (horizontal and vertical cross sections)

(u, T)lr,~,,o~ -- (u, T)lro~,,o~ -- O, ulr~,~ = 0, (u, n)lr,o~ = 0,

0(u,r) On

Ir,o~ = 0,

OT TIr,~,,o~ = To, ~-~nlro~,,o~ = 0,

0TIr~o = H ( T ) ' OT On ~--~nlr~o,, = Q(T).

2.3. Physical parameters The above described equations, together with the boundary conditions, are still too general, since they might as well represent any viscous flow in a tank. In order to resctrict the model to the particular case of the glass flow we are interested in, we need to specify the parameters of the equation, in particualr the viscosity, the density, the heat conductivity and the heat capacity. The most important of these coefficients are the viscosity and the density. The viscosity does significantly affect the flow pattern; it decays exponentially as the temperature grows (Vogel Fucher Tamman law)

574

S i m u l a t i o n o f flow in a glass t a n k

5

Viscosityof glass

x 104

1 9

|

10(~ 1200 1400 1600_ 1800 2{~ tem-/Yeratuf~K

Figure 2: Viscosity of tv-glass

# g l a , s ( X ) -- # 9 1 a s , ( T ( x )

-- a ~ e b~'/(T-c~').

(3)

The coefficients a t , bz, c~, in (3) are specific for the glass type (tv-glass, window-glass etc). The most significant factor in a glass flow computation is the density. It may be modelled as a linear function of the temperature, more precisely Pgla,,(T) = ap(1 - bp(T - Cp)),

(4)

and changes by only about 10%; nevertheless it drives the flow via the convective term and the gravitation. 2.4. D i m e n s i o n a l

analysis

In order to examine the behaviour of the momentum and the energy equations, in particular to determine whether diffusion or convection prevails, we make the equations dimensionless. Let us rewrite them as new variables 1

1

"2.=

T m Tmin

AT

Here X, U are characteristic length and velocity respectively and AT -T,~ax - Train is a maximum possible temperature difference. The gradient

and the divergence operators in the old and the new variables are related a~

-

v-=

1

:=

1

V. Nefedov and R. M. M. Mattheij

2550

575

Densityofglass

.~2500 E

~.2450 (n q)

2400

'%

23~0~ 12'00 14()0 16'00 18'00 2000 temperature. K

Figure 3: Density of tv-glass

A tilde over the gradient and the divergence thus indicates that it is taken with respect to the new variables. After subsitution the Navier-Stokes and the energy equations will look like U2 -~-(Off, Vfi) = F -

1 U x g p + ~--~9-(#9fl),

(5)

v . (p~) = 0, ATU ~~ AT X Cp(pfi, VT) = ~ V - ( k ~ T ) .

(6)

Since the density p changes by only about 10% we assume p to be constant. After dividing both parts of the first equation of (5) by p U 2 / X we obtain an equation in dimensionless variables

( ~ , 9 ~ ) = P --9 ~ + 9 . ( ~ Here

y=X

1 Vfl) .

1 P=~P"

pU ~F,

By analogy we divide both parts of (6) by A T U C p / X to obtain - ~ (~, VT) = 9 " (R~.1 Pr

9T).

Here the Reynolds (Re) and Prandtl (Pe) numbers are defined by Re :=

pUX #

,

Pr :=

~c~ k

Simulation of flow in a glass tank

576

IN

IN

gAN

gAN

gN

A

gN

^

A

gN

A

gN

g"

9

9

~

'

9

gN

IN

IN

gAN

~N

~N

gN

A

A

A

A

gN

A

gN

~

9

~

A

9

9

9

IN

A

IN

A

IN

A

IN

A

IN

A

A

A

A

A

q

A

9

ab

ab

ab

~k

4

Figure 4: Staggered (left) and collocated (right) grids

The Reynolds number expresses the ratio between convection and diffusion, while the product of Reynolds and Prandtl numbers indicates correspondence between radiative and convective heat transfer. In the case of the glass tank (ommiting physical units) X-20,

U-0.01,

p.~2490, 2 5 _ p < 2 5 0 ,

1.2
Cp-1000.

That is 0.2 < Re < 20,

7812.5 < R e P r <<_416667.

3. Discretisation The next step towards the solution is a discretisation of the continuous problem. We have opted for the finite volume method, because it ensures conservation of a number of important properties, such as momentum and mass. After the discretisation procedure has been fixed, we need to choose the grid on which we will discretise the equations. 3.1. Collocated grid versus staggered grid The natural choice of the grid for a finite volume method is a staggered grid [7]. It is called staggered because the velocity components are staggered with respect to the pressure, which is placed in the center of the cells. The staggered grid ensures that the resulting discrete system will not be singular. In some sense it plays the same role as the LBB or Inf-Sup condition in the finite element method [8]. Despite the fact that using a staggered grid we

V. Nefedov and R. M. M. Mattheij

577

always end up with a non-singular system, the programming process using such a grid is not a triviality, especially in 3D. We will use another type of grid, namely a collocated grid, that is the grid where all variables are computed at the same locations. It is far more convenient for programming, but requires a special discretisation procedure to ensure non-singularity of the system.

3.2. Discretisation of the momentum equations Let us consider the momentum equation corresponding to the horizontal component of the velocity

Op

- V - ( # ( T ) V u ) + (pu, Vu) + Oxx = 0.

(7)

For the sake of simplicity we consider here the two dimensional case. Momentum and body force are discretised according to a standard finite volume procedure [3]. Let us restrict our attention to the pressure gradient. Integrating the first component over a control volume, with node C say,

IN

W

w\

e

1" T Figure 5: Pressure stencil

f ~x OPdv " h~(.pe - Pw) - h~(pE -- PC). y~

(8)

We use downstream values of the pressure in (8) instead of midvalues. It leads to first order approximation instead of a possibly second order but,

578

S i m u l a t i o n o f flow in a glass t a n k

as it will be explained later, this will guarantee the non-singularity of the system.

3.3. Discretisation of the continuity equation Since we are using a collocated grid, the continuity equation is integrated over the same control volumes as the momentum equations.

/(

0

) +

cgy

d V " h~[(pu)e - ( p u ) w ] + hx[(pv)n - (pv),] "

v~ hv[(pu)c - (pu)w] + hx[(pv)c - (pv)s].

(9)

Here the upstream values of the velocity are used for the same reason as in the pressure gradient approximation.

4. P I S O m e t h o d A solution method for the Navier-Stokes equations coupled with the energy equation was suggested by Issa [4]. It is an operator splitting approach, which is closely related to the well-known family of S I M P L E / S I M P L E R algorithms by Patankar [7]. Let us assume now that after discretisation the system has the following form

BUp(T) 0

.. 0 0

0 )

0 E(u, T)

=

(.) 0 0

,

(10)

Here M ( u , T) is a discrete momentum operator, E ( u , T) - - discrete energy operator, both depending non-linearly on the velocity and the temperature. B p and B u are the discrete gradient and the discrete divergence, respectively.

4.1. Description o f the algorithm First we append to the momentum equations a time dependent term. Iterants u n, T n and pn are assumed to be known. After solving the prediction step p ( T n)u* - p(T n)u n = F - M ( u n, T n ) u * - BPp n (11) At we find the velocity field u* which is our first guess. Since it might not satisfy the continuity equation we correct it in the following way

P(Tn)u** - P(Tn)un = F - M(u* T n ) u * - B P p *, At

(12)

V. Nefedov and R. M. M. Mattheij

579

where pu** should satisfy the continuity equation. The new value of the pressure p* is not known yet, but we can get around this problem by taking the divergence of (12), and applying the continuity equation to the new velocity field u**. The equation for the new pressure will look like - B U B P p * = - B u [ F - M(u*, T n ) u * + p(Tn)un]At "

(13)

The matrix BUB p is a Laplace-type operator with Neumann boundary conditions. We shall examine it further in the next subsection. After the new pressure has been found we can apply (12) to correct the velocity field. At the next step we compute the new temperature

Cpp(Tn)Tn+ 1 - Cpp(Tn)T n At

: - E ( u * * , T n ) T n+l .

(14)

In his paper Issa showed that two correction steps give the optimal new value of the velocity for the time-dependent computations. Although we are dealing with steady-state computations, numerical tests show that two correction steps are also more efficient in the steady-state case. The second correction step will then look like p(T n + l ) u n+l _ p(Tn)u n At

= F - M ( u * * , T n+a)u** - B P p n+1 ,

and the corresponding pressure system is - B u B P p n+l = - B u I F - M(u**, T n ) u ** §

p(T")u"] At

The PISO method can be now formulated as follows: 0.

Let u n, T n, pn be g i v e n

I.

If t h e r e s i d u a l s r u := F - M ( u n, T n ) u n - B P p n,

r T .= - E ( u n, T n ) T n a r e not small enough proceed to s t e p 2. 2. Compute u* from p ( T n )u* - p ( T n )u n At 3.

= F - M(u

n, T n ) u * - B P p n

Solve

- B U B P p * = - B u IF - M(u* ' Tn)u* +

p(T")u"] At

Simulation of flow in a glass tank

580

4.

Compute u** from

p ( T n )u** - p ( T n )u n

= F - M(u*, Tn)u * - BPp *

At 5.

Compute T n+x from

Cpp(Tn ) T n+l - C p p ( T n ) T n

= - E ( u * * , T n ) T n+l

At 0

Solve

- B U B P p n+l : - B 7.

Compute

U n+l

u IF-

M(u**, T ~ ) u ** +

p(Tn )Un

from

p ( T n + l)un+ 1 -

p(T n)un

= F - M(u**, T ~+x)u** - B P p n+l

At 8.

Increase n; return t o step 1

4.2. T h e s y s t e m for t h e p r e s s u r e

In the previous subsection we derived the system for the pressure, the matrix of this system being a product of two matrices B ~ - discrete divergence and B p - discrete gradient. Let us determine how this matrix acts on a pressure vector in the two-dimensional case. Define B p :=

Bp

,

where the block matrices Bp, Byp are the approximations of the components of the gradient operator. Bearing this in mind we have ( - B U B P p ) c = hu[(BPp)w - ( B P p ) c ]

+ h z [ ( B ~ p ) s - (B~p)c] =

hu hx h-~[(Pc - Pw) + (Pc - PE)] + ~--~[(Pc -- PS) + (Pc -- PN)] = hx

hu

h, pE

hx hz -~'~PS -- -~'~PN

This is a standard 5-point finite volume (or which is the same in this case, finite element) discretisation of the Laplace operator. It can be shown that the discrete pressure system differs from the 5-point approximation of the Laplace operator only at the boundary nodes. Connectivity graphs of the pressure matrix and the discrete Laplacian are shown in Figure 6.

V. Nefedov and R. M. M. Mattheij

581

Figure 6" Connectivity graph of the pressure matrix (left) and discrete Laplacian (right)

4.3. Compatibility condition for the pressure system By construction, the pressure matrix is symmetric, positive-semidefinite, since the diagonal entries are positive, the off-diagonal entries are negative and the row sums are equal to zero. Furthermore this matrix is singular, and the kernel is formed by the constant vectors. Thus before solving the system we need to check whether it is compatible. The compatibility condition can be derived by summing up all equations of the pressure system

E ( - B U B P ) p ) c = E ( - B U [ F - M(u, T)u + C

p(T)u

])c.

(15)

C

Since the matrix is symmetric and all row sums are zero, the left-hand side of the equality is zero. At the boundary point C we have (

p(T)u~

F-M(u,T)u+

At Jc

= (p(T)u)

At

C'

whence p(T)u C

lAt IE

hu(p(Tlulc - E hu(p(Tlulc

+ ~ h,r(p(T)v)c - ~ hx (p(T)v)c]. F,

F,~

This expression should be equal to zero. After multiplying both sides byAt we obtain the compatibility condition

E hu(p(T)u)c r.

-

E hu(p(Tlulc + E h=(p(Tlvlc r~

r,

Simulation of flow in a glass tank

582 "

I

.: .:

n 9

9

9

9

9

llllllll

I I

I I I

9 -" 9

9

9

9

9

9

mm|m|

1~

i~ I

n m d a i m i m l i u m | $ m u l i b l m o d | u i m ~ | !

Figure 7: Boundary points involved in the compatibility condition

Z hx(p(T)v)c = 0.

(16)

Summation is done over four sets of the boundary points, as is shown in Figure 7. The compatibility condition can be interpeted as an approximation of

f

(p(T)u, n)dF = 0,

on

which in turn means that no mass is generated inside the domain. A consequence of (16) is the following. Let us assume that the exact solution is known and satisfies the continuity equation. We use it to obtain the boundary values; but since they also should satisfy (16) it might happen that we cannot use the values of the exact solution at the boundary points. To put it another way, even though the continuous solution satisfies the continuity equation, a test problem constructed from it might not have a solution. The same is true for any solution method for the Stokes/Navier-Stokes equations which makes use of the pressure matrix, though the compatibility condition will perhaps look less "exotic" than for the method considered. One of the reasons why (16) (or its variants) is sometimes ignored, is the correction of the pressure system. The pressure system is singular and since the pressure is determined up to a constant, we can fix the pressure at one point, which means that the solution is fully determined despite the fact that the original system might be incompatible. The result of that will be the new velocity field that does not satisfy the continuity equation. The last remark about the compatibility condition is that it is not known a priori, but follows from the discrete momentum and the continuity equations.

V. Nefedov and R. M. M. Mattheij

583

5. L o c a l d e f e c t c o r r e c t i o n

1

i

'[-1-TT"

E,] r

r

"

o |

0

i g2 H'h

0

1 s

1

.

Y2 ..........

. : l i l t

.

.

.

.

.

.

.

! | i

..

0

.

H

1

0

i i !

'/1

1

Figure 8: The uniform grids n H, n h and n H, and the composite grids ~H,h for H = 1/6, a = 2 and "h = "y2 = 1/2.

Due to the local nature of some processes in the oven, we need a higher resolution in areas where such processes take place. The method we use for local refinement is called Local Defect Correction (LDC) [2]. LDC is an iterative procedure which is able to accurately combine solutions on finer local and coarser global grids. In order to describe the method we consider a model boundary value problem Lu

-

f in fl = (0,1) x (0,1),

(17)

u

=

~o on 0 ~ .

(18)

Here L is an arbitrary elliptic operator, f is a given function on n, ~o a given function on the boundary 0ft. The model composite grid ~H,h (see Fig.8)is composed of a global coarse grid and a local fine grid. The global coarse grid n H is a uniform grid with

Simulation of flow in a glass tank

584

grid size H

n H={(xi,yj)l

x~=iH,

yj=jH,

O
O<j
We suppose that 1 / H E N. The local fine grid nho~ is a uniform grid with grid size h < H, covering the region of local refinement ~loc = (0, ~1) • c n

12hoc = {(xi,Yj)l x ~ = i h , yj - jh, O < i < l / h , O < j < l/h}. We assume V1/H E N, 72/H E N and H / h E N. The interface F is defined as the part of the boundary 0nlo~ that lies inside n,

F = O~to~ N n. We assume also that the high activity region of the boundary value problem lies inside the subregion nloc. The composite grid ~'~H,h is defined by ~ H , h _ ~-~H U ~'~hoc.

The refinement factor a is defined as the ratio of the coarse grid size H and the fine grid size h, a = H/h. Besides the uniform coarse grid n H and the uniform local grid n loc' h there is one more important grid, namely the local coarse grid 121Hocwhich is defined by In the LDC method one starts by discretising the boundary value problem on the global coarse grid n H. This yields the basic discretisation

LHuoH _ f H on ~2H.

(19)

The grid function Uo H is an approximation of the solution of the boundary value problem (17), (18). This grid function is used for discretising the boundary value problem on the local fine grid nlHoc. Boundary conditions for this problem are obtained from u0H by means of interpolation. The discrete problem on the local fine grid is denoted by

" Llo~Uto~, o = ho~(uHo ) on

(20)

The dependence of flaoc on the aproximation Uo H is denoted explicitly in (20). The approximations u0H and U~o~,o are used to define a composite grid approximation u~ 'a

{

xe

n,"o

c}H,h \ c}h

"

V. Nefedov and R. M. M. Mattheij

585

In the local defect correction method the fine grid approximation uho~,o is used to correct the basic discretisation (19) in the following way. The global coarse grid approximation u H and the local fine approximation uhoc,O are combined to define the global coarse grid function w H,

{

o(,)

x E nlHo~ z ~ ~"\~t:o ~

Substituting this function into the basic discretisation yields a residual grid function or defect, d H = LHw H _ fH.

The values of this defect at grid points inside ~loc are used to update the right hand side f H of the basic discretisation,

fH (x) "I-dH (x) x E N~Hc fH (X)

ill(x) : -

The updated coarse grid problem reads (21) Equation (21) yields an approximation u H of the solution of the boundary value problem (17), (18) on the global coarse grid. Like the approximation Uo H, the approximation u H is used to define artificial Dirichlet boundary values on the interface. The related discrete problem on the local fine grid reads h h h [uH~ LlocUtoc,1 = caloc~, 1 I

on

~hoc"

The approximations u H and u 4,1 h are used to define a composite grid approximation u H'h of the solution of the boundary value problem

~"'"(~) = { u~'(~)u~~

X E nhoc

z e nH'h'\n~o~

The LDC method can be summarised as the following scheme: 1. Solve L H unH = f n H on ~ H , f H = f H .

2. Solve h

h

h

H

L~oculoc, n = f loc,n (Un ) on ~oc. (Boundary conditions for this problem are obtained from u H n by means of interpolation.) 3. Construct the composite approximation uH'h(x)

~',"(~) = { ~'~

x E f~o~,

z e n ~,h\n~o~

Simulation of flow in a glass tank

586

4. Construct the global coarse grid function w H XE~

=

ug(

H

loc~

)

5. Compute the defect d H d H _ LHw H _ fH.

6. Update the right-hand part of the global problem

H fn+l(X)-

{ f H ( x ) + d H(x) _

f f f (X)

x e 12tHoc, X E

~H\~l~ c "

7. Check convergence; return to step 1, if not accurate enough.

6. Stirred flow One of the situations where we can use the advantages of LDC is the modelling of a stirrer. Usually the stirrer is located in a so-called doghouse, a small tank from which glass flows to production lines. We model the stirrer as a small rectangular solid piece, with velocity vectors prescribed in such a way that they move the flow around. The problem with computations involving a stirrer is that the size of the stirrer is normally much smaller than the characteristic length. The standard way to resolve the problem is to refine globally in the vicinity of the stirrer. Results of this approach are shown in Figure 9 (since original computations are carried out in 3D, we plot the results for a fixed vertical coordinate) This approach gives the

Figure 9: Solution obtained by means of non-uniform global refinement magnified velocity field (left), absolute value of velocity (right) (horizontal crossection)

V. Nes

and R. M. M. Mattheij

587

best possible accuracy, but has some strong disadvantages. First, by refining globally we obtain a considerable amount of grid points, in which much the information is irrelevant. In 3D this leads to a significant and superfluous increase in memory and computational time. Secondly, the condition number of the discrete system depends on the mesh size ratio. We can avoid this problems by refining locally, and use LDC. The result is depicted in Figure 10. One of the main advantages of LDC is that we don't

Figure 11: Solution after zero iteration - magnified velocity field (left), absolute value of velocity (right) (horizontal cross section)

results from the global refinement fi in order to estimate convergence and accuracy of LDC. U H ' h denotes the solution on the composite grid (see Table

1).

588

Simulation of flow in a glass tank

II~-u",~ll~

# L D C iter.

7.90-10 -2 3.79.10 -2 2.15-10 -2 1.26-10 -2 Table 1: LDC iteration results

Another specific feature of LDC is the use of the defect. Via the defect local pertubations affect the solution globally. If we would stop the procedure after the initial step, that is without updating the global problem, the results would not be accurate enough, see Figure 11

7'. Bubbl|ng Another situation in the glass oven where we can successfully use LDC, is a bubbling process. The air bubbles are injected into the glass tank from the bottom, and while moving to the top, they attract other bubbles, thus removing the air from the glass.

I

O.

I--

t--t.:~[:-}-:ttt-t-t:ttlttttttttt

Ill I IIII-l_I-I I-I

IE3 b---

o_~,o..,,

-

.

!!i! !i: ~i=::=: i~i!:il

:

:

.

.

2

.

.

:

.

.

.

.

:

:

:

.... ii :

:"

+ 0#---

tJ:]. t t l-tf-t

t--l.-Itt-ttf

t

t-t

;

'~.

.

-i

+~l_t+

-t_ t+t--[::t t t t-+f -t t t ~- -~-[~

-o.~_.

Figure 12: Global velocity field (without LDC) (vertical cross section)

Suppose we have with an initial volume per bubble V0. Since the hydrostatic pressure decreases going from the bottom to the top, the bubble diameter changes to Vb = Yo

1 + pgH/po 1 + p g ( H - z)/po'

where H is a total glass height, z the glass height and P0 the athmospheric pressure. Suppose the total bubbling volume flow is Q. Then the distance between the bubbles is d - vVo/Q, where v is the bubble rising velocity,

V. Nefedov and R. M. M. Mattheij

589

Figure 14: Absolute value of the velocity (vertical cross section)

i.e. the sum of the undisturbed vertical glass velocity and the relative glassbubble velocity. From one bubble, a force acting on the glass is Fb = --Vbpg. Since per vertical meter, there are on the average 1/d bubbles pushing on the glass, the average force per meter height is: Fa = Fb/d = _Vbpg Q.

vVo

After that we correct the right-hand side of the vertical momentum equation as follows: Fc = hz Fa - hx huhzp(Tc)g The computations on a coarse grid appear to be not accurate enough, see Fig.12. In order to improve the solution we use LDC, see Fig.13. The absolute value of the composite solution and the velocity field in the region of the local refinement are depicted in Figure 14 and 15 respectively.

590

S i m u l a t i o n o f flow in a glass t a n k

,plt ,It ,,

. . . . . . . . . . . . . / I l l .

. . . . . . . .

llll

II

, , J l t l !

.,~//I11

Ill/l,

.......

.,l/Ill! .... IIIII II I II//, I// ._.,/t/lll ._.,//IIII I l I / / ~_...i/llll t I i / / . . . . t//lll III// . . . . ....... /,,//llllll/, -" ....... //llllllt~ . . . . 0~8

2

22

24

26

28

3

32

Figure 15: Local velocity field (vertical cross section)

References [1] G. K. Batchelor, A n introduction to fluid dynamics, Cambridge University Press, (1970). [2] P. J. J. Ferket and A. A. Reusken, Further analysis of the local defect correction method, Computing, 56 (1996), 117-139. [3] J. H. Ferziger, M. Peric, Computational methods for fluid dynamics, Springer (1996). [4] R. I. Issa, Solution of the Implicitly Discretised Fluid Flow Equations by Operator-Splitting, Journal of Comput. Physics, 62 (1985), 40-65. [5] P. K. Kundu, Fluid Mechanics, Academic Press, San Diego (1990). [6] S. Nefedov, Simulation of glass flow in an oven, in Progress in Industrial Mathematics at ECMI98, B. G. Teubner, (1999), 106-113. [7] S. V. Patankar, Numerical heat transfer and fluid flow, Hemisphere Publishing Corporation (1980). [8] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Springer (1994). V.Nefedov and R.M.M.Mattheij Scientific Computing group Department of Mathematics and Computing Science Eindhoven University of Technology, PO BOX 513 5600MB Eindhoven The Netherlands E-mail: [email protected]; E-mail: [email protected]

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 26 CONTROL LOCALIZED ON THIN STRUCTURES FOR SEMILINEAR PARABOLIC EQUATIONS

P. A. NGUYEN AND J . - P .

RAYMOND

1. Introduction Let us consider the heat equation with Stefan-Boltzmann boundary conditions

Oy O'-"t- A y = 0 in Q,

Oy + lyl3y = ub.y on E, O--n

y(0) = y0 in ~,

where ~t is a bounded open subset in R N (N >_2) with a regular b o u n d a r y F, Q - ~tx]0, T[, Z - F x]0, T[, ~/is a regular manifold in F of dimension D <_ N - 2, 57 denotes the Dirac measure supported by V, and the control variable u belongs to a closed convex subset g v of Lq(O, T; L"(~/)). For example i f N - 3 and D - 1, 7 is a heating wire. This example is a particular case of equations studied in this paper. We address the two following control problems: 9 Find a control u E Ku such that (y~, u) minimize the functional

I(y, u) - CQ /Q I V y - Vdl'~dxdt + C a / ~ ly(T) - Ydl~ +C~ j~oT(~ luladr } dt. (y~ is the solution of the above equation associated with u, and CQ, C~, C~ are nonnegative constants.) W h a t are the necessary optimality conditions for the solutions (y~, u) ? 9 Let us suppose that D - 1, that is ~/is a curve, and that u is a control only depending on the time variable. Let y~,~ be the solution of above

592

Control localized on thin structures for semilinear parabolic...

equation associated with (u, ~/). What is the derivative of the functional

J(~/, u ) = / ~ ly~,~(T) - ydledx with respect to a field of deformations of -y ? This question is connected with the characterization of the optimal shape of the curve -y in a set of admissible curves. More generally, we are concerned with control problems for parabolic equations of the form:

Oy Oy O--t + Ay + ~(., y) - 0 in Q, ~nA + ~('' y) -- ub~ on ~, (1.1)

y(0) = y0 in ~,

where A is a second order elliptic operator, (I) is a Carath~odory function from Q x R into R, 9 is a Carath~odory function from ~ x R into R. We suppose that ~ C F, which corresponds to the most difficult problem, because the boundary condition is not necessarily linear. But the case where c ~ may also be considered. We study the control problem

(P1) inf{I(y,u) l (y,u) e L~(O,T; WI'~(~)) x g v , (y,u) satisfies (1.1)}. There is an important literature on pointwise control problems, they correspond to D -- 0 ([3], [4], [9], [12], [13], [20], [22]). Some problems in three dimensional domain (N = 3), with a control localized on a curve (D - 1), have recently been studied for the linear heat equation in [17] (see also [16], [19]). In our knowledge the case of nonlinear equations is not yet studied in the literature. For simplicity, first consider the following typical situation

(~(., y) = ly]tt--ly,

~(., y) __ lylu--ly.

E x i s t e n c e of s o l u t i o n s for e q u a t i o n (1.1). Before studying the control problem (/)1), we have to clarify the conditions on (a, q, N, D, #, u) for which equation (1.1) admits solutions. Parabolic equations with measure as data have been studied in the case of homogeneous Dirichlet boundary conditions in [7], [6]. In particular Baras and Pierre [6] have proved that the equation

Oy -~+Ay+lylt'-iy=AinQ,

y=0onE,

y(0)=y0ingt,

(1.2)

with A E .A4b(Q), and Y0 E L~(~), admits a solution if, and only if, the measure A does not charge the sets of C2,1,t,,-capacity zero. For example if

P. A. Nguyen and J. -P. Raymond

593

is of the form A = u5~, with u E LI(0, T; L ~ (-y)), and -y c gt is a manifold of dimension D <_ N - 2, then equation (1.2) admits a solution if, and only if, # < N-D-2N-D. In the case of equation (1.1), when 6 p ( . , y ) - lyl~-ly, and t~(., y) = ly[U-ly, we prove the existence of solutions when # < N -ND- D- 2 and (see Remark 3.1). From the analysis of equations with measures l] . ( N N -- D D -- 12 as data, we know that the solution y to (1.1) belongs to L~(O,T; wl'd(f~)) for every } + N > g + 89 It can also be proved that y(T) belongs to Ll(~t) ([24]). These regularity results are not sufficient to attack the control problem (P1). With more restrictive conditions on #, u and Y0 (assumptions (A5) and (A6) stated below), we can improve these results (Theorem 3.1), and we are able to prove the existence of solutions for (P1) for some 0 > 1 and ~ > 1 (Theorem 4.1). O p t i m a l i t y c o n d i t i o n s for (P1). The obtention of optimality conditions is more delicate. Indeed we have to prove that the adjoint state associated with a solution of (P1) has a trace on -y• T[, which belongs to L q' (0, T; L ~' (-y)). The adjoint state p can be written in the form p - p~ +p0, where p~ corresponds to the contribution of the term f q l V y - Vdl~dxdt in the adjoint equation, and po corresponds to the contribution of the term f~ ly(T) - ydl~ The trace of p~ on ~/• T[ belongs to L q' (0, T; L ~' ('y)) only if ~ satisfies some conditions depending on a, q, D, #, u, and p. These conditions are stated in assumptions (A10)-(A12). The corresponding regularity results for p~ are stated in Theorems 4.2 and 4.3. The conditions on 0 for which the trace of po belongs to L q' (0, T; L ~' ('y)) are stated in assumption (A13). Regularity results for po are stated in Theorem 4.4. Some examples satisfying the technical assumptions (All)-(A13) are given at the end of Section 4.3. Finally the derivative of J with respect to deformation of ~/is obtained in Theorem 5.2.

2. P r e l i m i n a r y

estimates for s o m e l i n e a r equations

2.1. Notations and a s s u m p t i o n s We denote by T~ (respectively Tr, T.y• TE) the trace mapping on '7 (respectively on F, ~/• E). Throughout the paper, we denote by C, Ci, K, Ki for i c N, various positive constants depending on known quantities. The same letter may be used for different constants. ( A 1 ) - The elliptic operator A is defined by Ay = - ~i,N=i D i ( a i j ( x ) D j y ) + ao(x)y. The coefficient a0 is positive and belongs to C(gt), the coefficients

594

Control localized on thin structures for semilinear parabolic...

aij belong to Cl'u(~) with 0 < u <_ 1, aij = aji, and they satisfy N

a,j(x)~,~j > mo[~l~ for every ~ E R N and every x E ~, with m0 > 0. i,j=l

( A 2 ) - F is of class C ~, with k = max(2, [g~,D] + 1), and -y is a submanifold of F of class C ~, of dimension D <_ N - 2. (A3) - (I) is a Carath~odory function from Q x R into R. For almost all (x, t) E Q, (I)(x, t, .) is of class C 1, and the following estimates hold:

['~(x,t,y)l

<_ C(1 + lyl'~),

!

o _ ,I,~,(x,t, y) ~ c(1 + ly[~-l).

For simplicity in the writing of equations we also suppose t h a t (I)(x, t, 0) = 0. Still for simplicity, we shall write (I)(y) in place of (I)(x, t, y). (A4) - ~ is a Carath~odory function from E x R into R. For almost all (s, t) e E, ~(s, t, .) is of class C 1, and we have

[~(s,t, Y)l-< C(1 + lyl~), 0 ~ ~ ( s , t , y) ~ C(1 + lyl~-X), As for (I), we suppose that @(s,t, 0) = 0, and we write ~(y) in place of

t~(s,t, y). ( A 5 ) - We make the following assumption on # and u" 1 < tz < inf~u f

N+2 __

D

~-~

'

N D N--gr-2

'

N-D+2 N-D--~

2'

N-D N-D-2

/j,

'

(2.1)

and 1

<

I/

<

inf~N f

N+I D ,,,

N-1 ~2 ' N - ~D - 2

,

N-D+I N-D-~

N-D-1 N-D-2

}"

(2.2)

( A 6 ) - The initial condition y0 belongs to LP(f~) with p > max (~2, N - ~N- ~

)'

K u is a closed convex subset in Lq(O, T; L ~ (-~)), and 2 < a < q < oc.

2.2. Estimates for linear equations First recall some results for analytic semigroups. We denote by A the operator defined by

{

Z)(A)=

yeC~(~)l~=oonr

}

, .4y=Ay.

595

P. A. Nguyen and J. -P. Raymond

For 1 < t~ < c~, we denote by Ae the closure of A in Le(f~). The operator -A~ is the generator of a strongly continuous analytic semigroup Se(t)t>o in Le(~t) [2]. For 1 < t~ < oc the domain of A~ is Oy T)(Ae) = {y E W2'~(gt)] 0--~A = 0 on F}.

For g = 1, T)(A1) is the set of functions y in Ll(f~) such that there exists z e LI(Ft) satisfying fn z ( x ) v ( x ) d x = fn y ( x ) A v ( x ) d x for all v e D(A). For any 1 < g < c~, 0 belongs to the resolvent of - A e and there exists 5 > 0 such Re a(Ae) _> 5 (it is a consequence of (A1) and of the fact that a(Ae) is independent of g). Therefore, for a > 0, there exists a constant K = K(g, a) such that

JlA~Se(t)~JJL~(~) ~_ Kt-~ll~[[Lt(a), Henry, [23], A~ is the a-power of Ae). Thanks to this result the following lemma can be established. L e m m a 2.1 [2], [26] - For every 1 < ~ < A < oc with ~ < oc, there exists a constant K1 = K1 (A, g) such that

(2.3)

[LS~(t)~IIL~(a) <_ Klt--~(~--})[l~llL~(a )

for every ~ E L~(~) and every t > O. For every I <_ ~ <_ A <_ oc with ~ < c~, and every a > O, there exists a constant K2 = K2(A, g., a) such that IIA~S~(t)~IIL~(a) <_ K2t-~(~-~)-"]l~]]L~(n

)

(2.4)

for every ~ E L~(Ft) and every t > O.

L e m m a 2.2 ([1], Theorem 7.58) - I f assumption (A2) is satisfied, then T~ is a continuous linear operator from Wm'P(~) into Lq('y) for all ( m , p , q ) such that 0 < m < k, 0 < N - mp < D, p < q < Dp --

--

--

N

-

m

p

"

Throughout the sequel, O denotes a diffeomorphism of class C ~, ( ] - 1, ID N - D - 1 onto an open set O of F, such that J O (the matrix of O) and H O (the Hessian matrix of O) are uniformly on V x ( ] - 1, 1[) N - D - l , and such that O(~, 0) - ~ for every ~ shortening we write ] - 1 , 1 [ N-D-1 for ( ] - 1, 1[) N - D - 1 .

]-- 1, 1[N - D - l , we denote by % the manifold 0('7, ~).

from "7 x Jacobian bounded e "i'. For

For each ; e

Control localized on thin structures for semilinear parabolic...

596

L e m m a 2.3 - There exists a constant C such that

II~(O(., ~))IILo' (~) -< cIl~llw~,o,(~),

(2.5)

Vw E W k'a'(~t), and all ~ E ] - 1, 1[N - D - l , where N~,D < k < [NsD] + 1 . Proof. The proof may be performed with a system of local charts, in order to use Theorem 7.58 in [1]. I Let J o be the Jacobian of O, that is the determinant of JO. We introduce an approximation of uS~, denoted by u~f,~, and defined as follows. Set

OnO(~/x]II[N-D-I) = --'n -n ,

(2)N-D-1 I J o o 1O - '

fn=

IX~

where Xo~ is the characteristic function of ON. We define un on On by 1 ~I[N-D-1 and un(O(r ~), t) = ~(o(r o), t) = u(r t), vr e 7, v~ E] - ~, Yt E [0,T]. L e m m a 2.4 - Let Un and fn be defined as above. There exists a constant C such that 1

i

un(s, t)fn(s)w(s) ds I <_ C

lu(r t)lad~

Ilwilwk,~, (f~),

for almost everyt e]0, T[, for allw e W k'~' (f~), with N~,D < k < [N~D]+l. Proof. We conclude with Lemma 2.3, since from the definition of un and fn, one has

(~•177 ~ [ ~ - ~ - ~ ) I J o ( O - l ( ~ ) ) l

--<

(n)

N-D-1

II

n ~

Un (0(r

•

~), t)w(O(r ~) )dCd~

lu(r

x

•177 • 1

x

]w(O(r ~)) I"' d~dg 1

1 [N--D--1

X]--n, n

(2)N-D-I(2)N-D-I(I <

_

)~ lu(f,t)l~dr

n

•

sup

qE]--l,l[ N-D-1

IIw(O(., ~))II Lo' (~).

• m

P. A. Nguyen and J. -P. Raymond

597

2.5 - Let Un and fn be defined as above. The sequence (Unfn)n converges to uS~ in the weak-star topology of L q(0, T; M L(-))"

Lemma

Proof. We prove that I f [ fr Unfn~dsdt - f [ f7 u(~, t)~(r t)d~dt I when n --~ c~, for all ~ belonging to L q' (0, T; C(F)). We have

(

o

I n

~(~,t)

--" [ 2

(,),X]__I I[N--D--1 [ J o ( 0 -1

)

,~

(~))1

~0

~(s,t)dsdt

- f o T ~ u(" t)T((' t)dCdtl = [I

u(r t)~({9(r ~), t)dCdcdt

--~0 T

--

fT f --

Jo J~ lu(r

f u(~, t)~(O(~, O), t)d~dt I

sup

e]_~,~t,,_D_1

]~(O(C, ~), t) -- ~(O(C, 0), t)ldCdt --~

m

as n --, oc. The proof is complete. Proposition

0

2.1 - Let r be in T)(f~), and w be the solution of the Cauchy

problem: Ow 0---~+ Aw = 0 in Q,

Ow OnA = 0 on E,

w(0) = A~,r in fi,

(2.6)

where 0 < a < 1. The mapping that associates w with r is continuous from L d'(f~) into Li(0, T; Wk,J(f~)), for all (a,j,i,k,d) satisfying: 2>k>0,

i>l,

j>d',

k

N

1

N

a+~+x-~"<-+2--='zza'~3

(2.7)

Consequently, the mapping that associates T~xlO,T[(W) with r is continuous from L d' (f~) into Li(O,T;LJ(7)) for all (a,j,i,d) satisfying: i>l,

j>d',

N 1 D a+x-~,<-+x-=.. za" ~ z3

(2.8)

598

Control localized on thin structures for semilinear parabolic...

Proof. Let (~, k, j, i, d) satisfy (2.7). Since the operator (--Ad,) is the infinitesimal generator of an analytic semlgroup " (S d, (t))t>_o m9 L d' (f~), we have: w(t) = Sd,(t)A~,r = A~,Sd,(t)r Observe that if/3 obeys k < 2/~, then D(A~) r wk'J(f~). Moreover, if d' < j < oc, with (2.3) and (2.4) in Lemma 2.1, we have:

IIw(t)llw~,j(a) = IIA~,Sd,(t)r -<

<_ CIIA~+~Sd,(t)r

ct-~-~-~(~-})llr

'(n).

1 Due to (2.7), we can choose t3 > k/2 to have a + ~ + g ( ~ , _ y) < 1. Since

the mapping t ~-, t - ~ - ~ - ~ ( ~ - ~ ) belongs to Li(O,T) for all i satisfying ~ + ~ + ~ ( ~ , - ~ ) 1 < ~1 _< 1 , the mapping r ~ w is continuous from L d' (f~) into Li(0, T; W k'j (ft)). Let (a,j, i,d) satisfying (2.8). There exists k obeying (2.7), such that kj > N - D. Thus T.r(Wk,J(~)) r LJ('T), and the proof is complete, m Proposition

2.2

-

Let f be in/)(Q), and z be the solution to the equation:

Oz -~ + Az = f in Q,

Oz OnA = 0 on E,

z(O) = 0 in f~.

(2.9)

The mapping that associates z with f is continuous from L~(O,T;Ln(~)) into L~(0, T; w~'d(f~)) for all (~, ~, y, 5, d) satisfying: 0
1<~<5,

1
1

~+~+~-~

g

1

g

< ~+~-~+1.

(2.10)

Proof. Due to (2.10), there exists a such that
1 5

1 N N ~) + 1 - 2-~ + 2-d"

(2.11)

w~'d(~). Let w be the solution of the Cauchy problem

fa A~z(x, t)r =

~ ' 2
dx = ~ A~,r

~(

~(~,t -

t) dx r)z(~,

~-)a~)a,-

fot ~ (Aw(x,t -- T)Z(X T) -- W(X t -- T)Az(x, T))dxdT +

/o/o

~(~, t - ~-)f(~, ~-)a~aT.

P. A. Nguyen and J. -P. Raymond

599

Hence A~ z(x, t)r

dx =

w(x, t - T) f (x, T)dxdT

= ~ot ~ A ~ , S d , ( t - T ) r

7")dxdT.

Using (2.4) in Lemma 2.1 with 1 < d' < y' < co, d' < co, we have" [[z(t)[[w~,d(a)<<_C sup { [ / ~ A~z(t)r -- C sup

I


A~,Sd,(t - T)r f(T)dxdT,

]]r

-"

[lr

d, (~) ---

1} 1

}

(t-r)-~-~(~-~)l[I(r)llL,(a)dr.

1 The mapping r ~4 [If ( r ) tlL, (a) belongs to L~(O,T). We set -}. = 1 + ~1 ~. 1 Thus the Due to (2.10) and (2.11), we have i >_ 1 and c~ + ~ ( ~1 - ~) < 7"

mapping t ~ t - ~ - ~ ( ~ -~t) belongs to Li(O, T). Therefore, using estimates on convolution, we have: (

(

(t-

T)-a--~( 88

~ ~_ cmlfmmL~(O,T;L,(f~)),

and the proof is complete,

m

2.3. Linear equations with measure We first consider the equation Oy O---t+ Ay = 0 in Q,

Oy = uS~ on E, OnA

y(0) = 0 in gt,

(2.12)

where u belongs to Lq (0, T; L ~ (~/)). P r o p o s i t i o n 2.3 - Equation (2.12) admits a unique solution y~ in the space LI(0, T; Wl'l(fl)). The mapping that associates yu with u is continuous from Lq(O,T; L('(-)')) into L~(O,T; L"(~)) for every (~, r) satisfying: q <_ ~ <_ oc, q <_ ~ <_ oc,

N-D[_D 1 2~ 1 2 ~-~ ~- ~ < + ~ +1, N-D if a < N-D-2 ' N-D N -2D ~ "q i < N2---rD+ -~ 1 + 1, 1 < r < N-D-2,

a <_ r <

N

N_D_2,

if (T > N - D -- N - D - 2

"

(2.13)

600

Control localized on thin structures for semilinear parabolic...

The mapping u H y= from Lq(O,T;La(7)) into LZl(O,T; W I ' d l ( ~ ) ) is also continuous for every (~1, dl) satisfying a _

q<--~l,

[-" ~Dg + : ,.1 1

N--D

N-D ifor < N_D_ q <__ (~l,

1 < dl < N - -DD- I --

(2.1a)

l < "~l + N -d ~- "q

N-1 N__p.r_2,

a<s<

q < ~ < oc,

s

+

N-D N

-

D

-

1

"

The mapping u ~-~ TE(yu) from Lq(O,T;L~(7)) into tinuous for every (~, s) satisfying: q<~
+

1 ,

N- D

i f cr >

< ~1 + 2~-1

N-D

2

N-D-1 i f (T < N - D - 2 N-D-1 g-D < N-D-2' 2 i f (7 > N - D - 1 -- N - D - 2

D

1

L~(O,T;L~(F)) is

~-~g+~<~+

1

~sl

, ~_ 1 1 N-D-1 q < ~ -~ 2s

+ 1

con-

+1, (2.15) '

"

Proof. 1 - Let us prove t h a t the mapping u H y~ is continuous from Lq(O,T;L~(7)) into L~I(O,T;WI,dl(~)). For this we proceed by duality. Let w be the solution of the Cauchy problem (2.6) with a - 1, d -- dl. By a straightforward calculation, we can prove t h a t

A~I y(x, t)r

dx =

u(~, T)w(~, t -- T)d~dT.

Therefore we have

]]y(t)Jiw,,.~(~) < C sup c sup

I

U(T)W(r

1

~_ CIBA~lY(t)IILdl(~ )

t -- T ) d C d T I ,

I]r

{/oI t Ilu(T)IIL < )IIw(t-- T)IIL '< )dTI, IIr

Observe t h a t if g ~ D < k < 2/~, then TT(D(A~,)) r

(n)

=1} -1}

TT(wk'~'(~)) r

L~' (7). By using estimate (2.4) in L e m m a 2.1, with 1 < d~ < a ' < cr oc, and a = 1, we obtain:

~0 t (t - T) N (1; _ ~ ) _ 8 9 il ( )llLor liy(t)llWl.~l (a) < _ C

dT.

d~ <

P. A. Nguyen and J. -P. Raymond

601

D I f a < g N-D - D - l ' due to (2.14), we can choose/~ > N~D to have g2D + 2~ N 1 1 From estimates for convolution ~-j~+ 89< N ( ; 1- - ~ 1 ) + 1 + / ~ < 1 + (~1 q" we deduce

<-- CllulIL (O,T;Lo( ,)). The case when a > NN- D- D- 1 may be deduced from the previous one. 2 - Let us prove t h a t the mapping t h a t associates y~, the solution to equation (2.12), with u is continuous from Lq(O,T;L~(~/))into Le(O,T;L~(~)). Let w be the solution of the Cauchy problem (2.6) with a - 0, d - r. If D O" < N-D-2N-D, due to (2.13), we can choose fl > N~D to h a v e N 2 D 2t-~--~--N < N(~21 _ 7)1 + / ~ < 1 ~ ~1 ql. We can conclude with the same arguments as in Step 1. The continuity of the mapping u H Tr.(y~) m a y be obtained in the same manner. 1 -

-

C o r o l l a r y 2.1 - Let y be the solution to equation (2.12).

The mapping that associates y with u, is continuous from Lq(O,T;L~(7)) into L~(Q) for every r satisfying 1 _< r < inf { ( N

N+2 q,

N

N-D+2 (N-D-~)

~,) ( N - 2 - ~ )

N-D N-D-2

I

(2.16)

The mapping that associates yl~ with u, is continuous from Lq(o, T; L a ('7)) into Ls(E) for every s satisfying N-1

N+I

_

l<s
N-D+1 N-D-1 ( N - D - -~) N - D - 2

~-T) ( N - 2 - 7 )

i

" (2.17)

P r o o f . To prove the first result, we distinguish four cases. -r)'N-D+2 then q < (N-D-2)N-D and a < 1 - I f q < (N_2N~) and i f a < (N-D-2q ( N N-D -D-2)

q _< r, 2 - If q

"

Due to Proposition 2.3, y belongs to L"(Q) for all r satisfying

D 1 < N + 7 -1+ - l , , i.e., s u p { q , a } _< r < (N_7_DT) N2~2 a _< r, N2D-~-~-j~-~ " > D - - ( N , 2L ~--r)

and if a < inf{ ( NN-D+2 - D - - q-r2) '

(Ng - D--D 2)}'

then u belongs to

Lr(O,T;L~('y)) for all r < (g_2N_ ~)" D Due to Proposition 2.3 y belongs to N -2D + D < N + 1, t h a t is equivalent to L r (Q) for all r satisfying a _ r, --

N ( N - 2 - a--r D )"

3 - If q < inf{

(N:2:N m-2)} and if a ~D) , (NN-D -

to

Lq(O,T;L~('~))for all r

<

N-D-F2 then u belongs > (N_D_q3r),

N-D+2 Due to (N-D-~)"

to L ~(Q) for all r satisfying q _ r, equivalent to q _ r < ( gN-D+2 _ D _ ~)"

N--D

2

Proposition 2.3, y belongs

+ D + ~1 < 7I + g + 1, t h a t is

Control localized on thin structures for semifinear parabofic...

602

4 - If q > -- ( NN-D - D - 2 ) and if a > ( NN-D - D - 2 ) ' then u belongs to L r ( 0 , T ; Lr(~/)) N D for all r < (N-D-2)" Due to Proposition 2.3, y belongs to L~(Q) for all r 1 -~- N _jr_1, t h a t is e q u i v a l e n t t o 1 _ r < ( NN- D - D- 2 ) " s a t i s f y i n g N 2 D -Jr-D _jr.r1 < -~ The second result for the trace of y on E can be obtained in the same manner, m 2.4 - Let u,~fn be the approximation of uS~ defined in Section 2.2. Let yn be the solution to the equation:

Proposition

Oy O--t + Ay = 0 in Q,

Oy OnA = Unfn on E,

y(O) = 0 in f~.

(2.18)

There exist constants C(~, r), C(~, s), C((~l, dl), independent of n, such that

IIY IIL (O.T;L ( )) (

<---

r)IIUlIL~(O,T;L~(~))

) sati /yi g (2. lS), b]Y~I~I]L~(O,T;L~(r)) <---C(~, s)mmulmL~(O,T;L~(~))

for every (~, s) satisfying (2.15), IlYnlmL~, (O,T;W,,a, (~)) ~_ C ( 5 1 , dl )lmullLq(O,T;L~(.y))

for every (51,dl) satisfying (2.1~). Proof. Let w be the solution of (2.6). Let us prove the estimate in the space L ~1(0, T; W 1'dl (~)). With Lemma 2.4 and the identity: A~lyn(x,t)r

=

un(s,T)fn(s)w(s,t-r)dsdT

for a = ~,

we can write:

< C sup

=1},

]lu(v)]lL,(~)llw(t- T)l]Wk,~,(a)dv I IIr

where k > N~D. The rest of the proof is similar to t h a t of the previous proposition, m Proposition

Oy

--

Ot

+ Ay

2.5 - Let y be the solution to the equation -

0 in Q,

Oy COnA

-

0 on E,

y(O) -

yo in

f~.

(2.19)

P. A. Nguyen and J. -P. Raymond

603

The mapping yo ~ y is continuous from LP(~) into L e ( O , T ; L r ( ~ ) ) N L~*(O, T; wl'd4(fl)), for every (~, r) satisfying (2.13), and for every (54, d4) satisfying 1<54<2,

N

p < d a < g _Np p,

1

N

1

1 + ~p < ~4 + -~4 + ~ "

(2.20)

Consequently, y belongs to Lr(Q) for all r satisfying (2.16). Moreover, the mapping Yo H YI~ is continuous from LP(f~) into L~(O,T;L~(F)) for all (~,s) satisfying (2.15), such that ~ < 2p. Consequently, YlE belongs to atl s satisfying (2.17). Proof. 1 - To prove t h a t y E L~4(0, T; w l ' d 4 ( ~ ) ) for every ((~4, d4) satisfying (2.20), it is sufficient to use Proposition 2.1 with a - 0 and k - 1. g-D and if r > p, 2 - Let (~,r) be a pair satisfying (2.13). If a < N-D-2 then 1 + ~ -< 1 + g2 D 2~' q'1 < g + ~1 + 1. From Proposition 2.1 with k = 0, it follows t h a t y belongs to Le(O,T;Lr(fl)).

If r < p, then

y e Lcc(O,T;LP(~)) r L~(O,T;Lr(f~)). 3 Let (~,s) be a pair satisfying (2 15), such t h a t ~ < 2p. If a < ~__D-1 D-2 and if s > 1+ N - - p, then we have 1 + ~ -< 2 2D a' qt' < -N_~ - 1 + ~1 + 1. From Proposition 2.1, it follows t h a t y[~ belongs to L~(0, T; LS(F)). If s < p, we have 1 + ~p < - ~ + ~1 + 1 (because ~ < 2p). Thus y]:~ belongs to -

9

L~(O,T;LP(F)) ~ L~(O,T;L~(F)). 4 - Since 1 + 2~ < N_~__A+ p1 -t- 1, the trace of y on E belongs to LP(E). If s satisfies (2.17), then s <_ p, and the trace of y on E belongs to Ls(E). m

2.4. Linearized equation Let 5 be the solution to equation (1.1). T h e n ~ is also a solution to the linear equation

Oy Oy O---t+ Ay + ay - O in Q' ~nA + by = ub~ ~ E' y ( 0 ) = y 0 i n f l ,

(2.21)

where a = fo ~'(x~)dx and b = f~ t~'(x~)d X. (We have supposed t h a t (I)(0) = 0 and ~(0) = 0, see (A3) and (A4).) Observe t h a t , in the case vchere u _> 0 and y0 _> 0, with a comparison principle, we can prove t h a t 0_<~<~

a. e. i n Q ,

and 0

~[~<_~[:~ a. e. i n E ,

where ~ is the solution of (2.21) corresponding to a - 0 and b - 0. Hence it is natural to look for a solution ~ to (1.1), belonging to L e ( 0 , T ; Lr(f~)) for every (~,r) satisfying (2.13), and such t h a t YI~ belongs to L~(O,T;L~(F))

604

Control localized on thin structures for semilinear parabolic...

for every (~, s) satisfying (2.15) and ~ < 2p. In this case, we can verify that the coefficient a belongs to Lk(0,T; Lk(~)) for every (k, k) satisfying q ~-1

<~g, --

a ~


N ,

D

if q

D 1 -[- 2"a -[- q <

2

< k, 1 < k,

a

<

1 -[- ( ~ - l ) k

+I

N-D N-D-2' 1

N-D

-+-~ <

N2D

N 2(~-l)k

2(~-l)k

(2.22)

1

-[- (~-I)Tr -[- 1

N-D

if a > N - D - 2 , and b belongs to Li(0,T; Lt(F)) for every (~,g) satisfying q <~< v--1 - -

2p v--l'

a <~, v--1 - -

N-D 2

if a < ~

v-I

--

2p

_}_ D _}_ 1 < 2"~ q

+ 1

N-D-1 N-D-2' 1

'

N-1 _[_ 1 2(v--1)t (v--1)~

N-D-1

(2.23)

1

--

ifa>

N-D-1

-- N-D-2"

P r o p o s i t i o n 2.6 - Assume that a is a nonnegative function belonging to Lk(O,T;Lk(f~)) for every (k,k) satisfying (2.22), and b is a nonnegative function belonging to L~(O, T; L~(r)) for every (~, e) satisfying (2.2~). Equation (2.21) admits a unique solution y in LI(0, T; Wl,l(f~)). This solution belongs to Le(0, T; Lr(f~)) for every (~,r) satisfying (2.13), and there exists a constant C(~, r), not depending on the functions a and b, such that

IlYlIL~(O,T;L~(~)) ~ Cff, r)(IlulIL~(O,T;L~<~)) + IlYolIL~(~)). In particular for ~ = oo and for every 1 < r < inf{

N-D

N- D-

(2.24) N

2u ' N - ~ D- ~

}, y

belongs to C([0, T]; L r ( ~ ) ) and there exists a constant C(r), not depending on the functions a and b, such that

IlYlIL~eO,T;L~(~)) ~ C(r)(IlulIL~eO,T;L~<~)) + IlYOIIL~(~)).

(2.25)

(C([O,T]; L ~ ( ~ ) ) denotes the space of continuous functions from [0, T] into Lr(~) endowed with its weak topology.) Moreover, the trace YI~. belongs to L~(O,T;L~(F)) for every (~,s) satisfying (2.15) and g < 2p. There exists a constant C(~, s), not depending on the functions a and b, such that

IIYI~IIL~(O,T;L~(r)) ~ C(~,S)(IlUlIL~(O,T;L~(~)) + IlYolIL~(~)).

(2.26)

Proof. 1 - Let ~ be the solution to (2.21) corresponding to a, b - 0. For ~), estimates (2.24), (2.25) and (2.26) follow from Propositions 2.3 and 2.5.

P. A. Nguyen and J. -P. Raymond

605

2 - Since a and b are nonnegative functions, with a comparison principle we can prove that ]]ylIL~(O,T;Lr(a)) < 2]]~IKL~(O,T;Lr(a)), IDYHKL~(O,T;L~(a)) <_ 21i~IIL~(O,T;L~(a)), and liYH2ilL~(O,T;Ls(r)) <- 2[lYlr=IOL~(O,T;Ls(r)). 3 - In particular, when ~ = c~, we can proceed as in the proof of Proposition 2.1 in [13], to conclude that y belongs to C([0, T]; L~(~)). i Now, we want to prove that the solution of equation (2.21) belongs to the space L zl (0, T; W i'd1 ( ~ ) ) "3t- L ~2(0, T; W i'd2 ( ~ ) ) nt- L ~3(0, T; W 1,d3(~-~)) -}LZ4(0, T; w l ' d ' (~)), where (~1, all)obeys (2.14), (54, d4)obeys (2.20), (52, d2) and (53, d3) satisfy: qtt <--5 2 ,

~ _< d 2 ,

~(N

~D +~D g+~ 1

if q < 52

1

< d2

a

q- < 53 < 2-e-

q < 53,

if

1

N-D

tiN(N-D-2)

2(N-D)

a >

N-D

-- N - D - 2

N

< ~

1

-~- ~ -~-

(2.27)

1

'

D =~_ 1 _ 1 )

z < d3 ff 1 < d3,

~1+ ~ + ~N,

< N-D-2'

~q~if

1)<

<

1 +

N

N-D-1

a<-W='_-D-=~ , v

v(N-1)(N-D-2)

2(N-D-I) 0"> N-D-1 -- N - D - 2 " -'4-

N

< ~

I

(2.28)

'f- ~ss

P r o p o s i t i o n 2.7 - Assume that a and b satisfy the assumptions of Proposition 2.6. Let y be the solution of equation (2.21). Denote by ~1 the solution of equation (2.12), by ~4 the solution of equation (2.19). Let ~2 and ~3 be the solutions to the equations

0J_22 _+_A~2 = - a y in Q,

0~2 = 0 on E,

Ot

COnA

0J_33 + A~3 = 0 in Q, cot

0~3 = - b y on E,

~2(0) -- 0 in gt, ~3(0) = 0 in gt .

OnA

Then y = ~1 -}- ~2 -}- ~3 -Jr-~4, and the following estimates hold for some (k, k) satisfying (2.22), and some (~,~) satisfying (2.23):

II~2ILL~(O,T;W',~(~)) <---CIIaIIL~(O,T;L~(~))(IlUlILq(O,T;L~(~)) + IlYolIL~(~)),

Proof. Due to Proposition 2.6, y belongs to L~(O,T;L~(~)) for every (~,r) satisfying (2.13), and YiE belongs to L~(0, T; L~ (F)) for every (~, s) satisfying (2.15) and ~ < 2p.

Control localized on thin structures for semilinear parabolic...

606

Let (62, d2) be a pair satisfying (2.27). We take/r = 62#', k = d2#', ~ = 62#, r = d2p. Then (k,k) satisfies (2.22), and (~,r) satisfies (2.13). Moreover, we have 1

k

I

1

1 ---- ,

1

4

1

1 =--,

1 1 and = + - +

N

(

1

1 +-)<

1

+

N

1 +xz

Due to Proposition 2.2, it follows that

11~21IL~(O,T;WI,~=(~))<--CI[alILk(O,T;L~(~))IlYlIL~(O,T;L~(~)). Now, let ((~3,d3) be a pair satisfying (2.28). We take g = 63v', k = d3v', fi = 63v, r - - d 3 v . Then (g, g) satisfies (2.23), and (~, s) satisfies (2.15) and g < 2p. Moreover, we have 1 1 1 - ~ =--, g g (~3

1 1 1 ~ =~, g s d3

1 1 and = + - + g g

N-11 1 ( +-)< 2 ~ s

1 N -4---. ~33 2d3

Using the same arguments as in the proof of Proposition 3.2 in [27] (see also L e m m a 4.3), we obtain

]]~3]ILS3(O,T;WI,da(~)) <__C]]b]]#(O,T;L~(p))]IY]~]]L~(O,T;L~(P)),

m

and the proof is complete.

3. State equation Definition 3.1 We say that y E L~(0, T; WI'I(~)) is a weak solution of equation (1.1) if ~(., y(.)) belongs to LI(Q), ~(., y(.)) belongs to i l ( ~ ) , and -

+

fQ y -0r~

dxdt

(I)(y)r dxdt +

N +/Q ~ aq(x)DjyDir dxdt + / Q a o y r i,j--1 ~ ( y ) r dsdt =

for every r E C 1(-Q) such that r

ur d~ dt +

= 0 on -~.

3.1. Compactness results To study equation (1.1), we need the following lemma:

r

dxdt

dx

R A. Nguyen and J.-P. Raymond

607 m

L e m m a 3.1 - Let A be a measure belonging to A/Ib(Q \ (~ • {T})). Let

yx be the solution to the equation: Oy O---t+ Ay = AQ in Q,

Oy = A2 on ~, OnA

y(O) = A-5 in -~

(3.1)

where AQ (respectively AE, A-5) denotes the restriction of A to Q (respectively E, ~ • {0}). The mapping A 9 A H (y~, y~[E) is a compact linear operator from A/Ib(-Q \ (-~ • {T})) into LI(Q) • L I ( E ) .

Proof Let m and n be such that m > g + 1 and n > N + 1. The operator h is continuous from A/Ib(Q\ ( ~ x {T})) into L m' (Q) x L ~' (E) (see [24]). To prove the compactness we proceed by transposition. Let z be the solution to

Oz - - - + Az = r in Q, Ot

Oz = r on E, OnA

z(T) = 0 in f~.

With a Green formula (see Theorem 4.1 in [24]), we have

zdA = /Q yC dxdt + f

yC dsdt

for all A 6 A4b(Q \ (~ x {T})), all r e Lm(Q), all r e Ln(~).

Set L(r r - z. Using the same arguments as in [25], proof of Proposition 3.2, we prove that L is continuous from Lm(Q) x Ln(~) into C ~ Co(Q \ (f~ x {T})), for some a > 0. So, L is a compact operator from Lm(Q) x Ln(~) into C0(Q \ (~ x {T})). Since h - L*, the proof is complete, m

3.2. Nonlinear equation L e m m a 3.2 - If the function a belongs to Lk(O,T;Lk(~)) for every (it, k)

satisfying (2.22), and if y belongs to L~(Q) for every r satisfying (2.16), then ay belongs to La(Q) for every a such that

1 < a < inf { #(N

N+2 D a t

N 2 ~ ~-,) # ( N - ~ vD- 2 )

N-D+2 ~

p(N-D-~)

N-D 2

p(N-D-2)

}" (3.2)

Proof. As in the proof of Corollary 2.1, we can prove t h a t a belongs to L ~ (Q) for every r satisfying (2.16). Thus ay belongs to L~(Q) for all a such that s( ~ complete,

t~-I + i?" -- - ~~ ' for some r obeying (2.16) 7"

The proof is m

608

Control localized on thin structures for semilinear parabolic...

L e m m a 3.3 - If the function b belongs to L~(O,T;Lt(Y)) for every ({,t) satisfying (2.23), and if y[z belongs to Ls(E) for every s satisfying (2.17), then by]z belongs to L~(E) for every ~ such that 1
N+I D 0.!

N-1 N-D+1 ~,) 2 , v(N - D~7- 2 ) ' v ( N - D - o )

N-D-1 2 ' v(N-D-2)

i

y" (3.3)

Proof. Using the second part of Corollary 2.1, the proof is similar to the previous one. 9 3 . 8 - Assume that a and b satisfy the assumptions of Proposition 2.6. Let Unfn be the approximation of uS~ defined in Section 2.2. Let Yn be the solution of the equation

Proposition

Oy Oy - ~ + Ay + ay = O in Q, ~nA + by - u'~fn o n E ,

y(O) - yo in ~. (3.4)

Then the following estimates hold:

liY~IIL~), for all (~,r) satisfying (2.13);

]]Y~I~.I)L~
m

T h e o r e m 3.1 - The state equation (1.1) admits a unique solution, it belongs to Le(O,T;Lr(fl)) for every (~,r) satisfying (2.13) and we have

IJV(IL~(O,T;L~(~)) --< C(IlulIL.(O,r;L~ inf{

+ IIYoIIL~(~)) 9

The solution of (1.1) also belongs to C([0, T]; L~(~)) for every 1 < r < g-D N }, and we have

llVlIL~(0,V;L~(~)) --< C(llulIL~(0,V;L~

IIV0IJL~(~))"

The solution y of (1.1) also belongs to L~I(0, T; wl'dl(~-~))-~-L~2(0,T; wI,d2(~)) + L~3(O,T; wI,d3(~)) + L~4(O,T; w I , d ' ( ~ ) ) , for all (51, dl) obeying (2.1~), all (52,d2) obeying (2.27), all (53, d3) obeying (2.28), and all (54, d4) obeying (2. 20).

609

P. A. Nguyen and J. -P. Raymond

Moreover, we have

[[y -- ~4[[L~1(0,T;WI,4I (~t))+L~2(O,T;WI,~2(Ft))+L$3(O,T;WI,d3(~t)) -FI[~4[[L84(O,T;Wl,a4(ft)) ~_ C([[U[[Lq(O,T;L~(w)) + I[UI[~Lq(O,T;L~(.r))

+llullL
0~4 Ot

--

0~4 OnA

-F A~4 = 0 i n Q ,

= 0 o n Y],

~4(0)

-- YO

in ~.

Proof. We approximate uh~ by the sequence of functions (u~f~)~ defined in Section 2.2. We denote by y~ the solution of (1.1) corresponding to unf~. 1- Estimate in Le(O,T;L~(f~)). By setting an = f~ ~'(Xyn)dX and bn = f~ ~'(Xyn)dX, from Proposition 3.8, it follows that (Y~)n is bounded in Le(O,T;L~(~t)) for all (~,r) satisfying (2.13), (y~[E)~ is bounded in L~(O,T;L~(F)) for all (~,s) satisfying (2.15) and ~ < 2p. Therefore, with assumption (A4), the sequence (an)~ is bounded in Lk(0, T; Lk(~t)) for all (k, k) satisfying (2.22), and the sequence (b~)~ is bounded in L~(O,T;L~(F)) for all ([,g) satisfying (2.23). Due to Lemmas 3.2 and 3.3, the sequence (a~yn)n is bounded in L~(Q) for all a satisfying (3.2), and the sequence (bnYn[E) n is bounded in L~(E) for all /3 satisfying (3.3). From Lemma 3.1, it follows that the sequence (y~),~ is relatively compact in LI(Q), and the sequence (Yn E)n is relatively compact in L I(E). 4 Lhj (0, T; w l , d j (gt)) . Let ~ln ~ ~2n ~3n be the solutions 2 - Estimate in )-~j=l of the following equations ~O~ln - ~-A~ln = 0 in Q, Ot O~2n Ot

+ A~2~ = - a ~ y ~

0~3~

Ot + A~3n = O in Q ,

O~ln = unf~ on E, OnA

in Q,

~1~(0) = 0 in ~t,

O~2n = 0 on E, OnA

~2,(0) = 0 in f~,

on E,

~3~ (0) = 0 in ~t.

0~3~ _ -bnYn On A --

Then we have y,~ = ~1~ + ~2n + ~3~ + ~4. From Propositions 2.4,we have

]I~I~[]L~I(O,T;WI,~I(a)) <_ C(51,d1)IlUl]Lq(O,T;L~(~)), for all

((~1, dl)

satisfying (2.14).

610

Control locMized on thin structures for semilinear parabolic...

Let (52,d2) satisfy (2.27). satisfying (2.22) such that

As in Proposition 2.7, we can find (k,k)

liEi2nllL, (o,r;w,, ( )) <_ Clla.IIL (O,T;L ( ))(JlUlIL (O,T;L ( )) + IlY011L.( )). From assumption (A4), and due to Proposition 3.8, we can write

+ILUJfL(O,T;Lo( )) + ItY0tl'L
--< C(liuiii.(o,r;io( ))+ ilY011L ( ) +iluiiL(o,r;L ( )) + liyotlL(a)), for all (53, d3) satisfying (2.28), and all n. Due to Proposition 2.5, we also have ll~4[i/~,(0,r;W'.",(n)) <-- CilYoOKL,(~),

for all (54, d4) satisfying (2.20).

3 - Passage to the limit. There exists a subsequence, still indexed by n, such that (Yn)n converges to some y in LI(Q) and almost everywhere in Q, (ynl~)~ converges to some z in LI(E) and almost everywhere in E. It is clear that z = YI~. Due to Lemma 4.1 in [13], we can also prove that (y,~)n converges to y in L ~(Q) for all r satisfying (2.16), and (y~ I~)~ converges to YHE in L~(E) for all s satisfying (2.17). Thus (any,-,)n converges to (I)(y) in L'~(Q) for all a satisfying (3.2), and (bnyn)n converges to ~ ( y ) i n LZ(E) for all ~ satisfying (3.3). Now by passing to the limit in the variational formulation satisfied by yn, we can show that y is a solution of (1.1). We also obtain estimates of y in L ~1(0, T; W l'd~ (~))+ L ~2(0, T; W 1'd2(~))+ L~3(0, T; w l ' d 3 ( a ) ) + L~4(0, T; w l ' d ' (gt)), for all (51, dl) obeying (2.14), all (52, d2) obeying (2.27), all (53, d3) obeying (2.28), and all (a4,da)obeying (2.20), by passing to the limit in Step 2. Remark 3.1. Due to Theorem 3.1, if u belongs to L~(0, T; L~176 if # < g-D and/2 < NN -- DD -- 21 ~ then equation (1 "1) admits a unique weak solution. N-D-2 N-D and Now consider the case when u belongs to LI(O,T;LI('7)), # < N-D-2 /2 < N-D-2"N-D-1 Set fin : - max (min (n, u), - n ) , and denote by yn the solution of (1.1) corresponding to fin. As in [8], using monotonicity arguments, we can prove that (Yn)n converges to some function y in LI(Q), (Yni:~)n converges to y[n in LI(E), y E Lt'(Q), y E L~'(E), and y is the weak solution of (1.1) corresponding to u. Therefore, we have proved the existence of a unique solution even if # and /2 do not satisfy (2.1) and (2.2), but only -D-1 < N-D-2N-D and /2 < NN-D-2" However the regularity results stated in Theorem 3.1 are not true in this case.

P. A. Nguyen and J. -P. Raymond

611

4. Control problem (P1) In all the sequel, we suppose that the control set K u is a closed convex subset of Lq (0, T; L ~ (7)). 4.1. Existence of solutions to problem (P1)

Theorem 4.1 - Assume that hypotheses (A1) to (A6) are fulfilled. Suppose that Yd belongs to Le(f~), and Vd belongs to L~(0, T; (L~(f~))N). Suppose that there exist ((~l,dl) satisfying (2.1~), (52, d2) satisfying (2.27), (53, d3) satisfying (2.28) and (54, d4) satisfying (2.20), such that I < t~ < 5i and ~ < di for i = 1, ..., 4. Suppose in addition that 1 < 0 < p, and that either K u is bounded in Lq(O, T; L a (7)), or C~ > O. Then the control problem (P1) admits solutions. Proof. Let (u,,)n be a minimizing sequence in K u . Then (Un)n is bounded in Lq(0,T; La(7)). We can suppose that (un), converges to some u for the weak-star topology of Lq(o, T; La(7)). Since K u is convex and closed in Lq(O,T;La(7)), then u e Ku. Let yn be the solution of (1.1) corresponding to u,. Due to Theorem 3.1, the sequence ( y , ) , is bounded in L~(0,T; WI'K;(~"~)), and the sequence (yn(T))n is bounded in L~ Let y, be the solution of (1.1) corresponding to u. Using the same arguments as in the proof of Theorem 3.1, we can prove that ( y , ) , converges to y, for the weak topology of L~(0,T; Wl,~(f~)), and (yn(T))n converges to yu(T) for the weak topology of L~ By classical arguments, we can prove that u is a solution of (P1). m

4.2. Regularity results for the adjoint equation First, to study the adjoint equation for the control problem (P1), we have to suppose that there exist (k, k) obeying (2.22), and (~,g) obeying (2.23), such that 1 N 1 N-1 1 ~+~<1, 2~<:. (4.1)

-~+

Setting k = ~q-1 and ~ = v-'~-:, the above condition will be satisfied if assumption (A7), stated as below, holds. (AT) - In addition to conditions (2.1) and (2.2), we suppose that # and u satisfy

I_1> # ( N 2 D -4- 2~a -[-- -~ D

< N ~ D + :0 "4- -~:

1 __ 1 ) <

N~D

D

1

if

N-D O" < N - D - 2

N-D

0" > 0"-- N - D - 2

'

(4.2)

Control localized on thin structures for semilinear parabolic...

612

and

D -}- -~ 1 V ( N 2 D -}- "T~

__ 1)_

<

D Jr "~ 1 -[- ~'~

a < N-D-2'

if

D l](N2 D +~g~x + ~ 1- 1 )

if

N o..D

N-D-1

NoD ..

<

D 1 -[- ~-~-1 -} q

N-D-1

N-D-2

a_>al=

(4.3)

9

Next, to simplify the analysis of Problem (P1), we suppose that #, u, q, a and p satisfy the following additionnal conditions. ( A 8 ) - I f NN_~_i ~ a < ~ , then /z( N - D

2a' If

N-D

< a <

N-D-1

--

N-D-1 N-D-2

~

1

1

1

(4.4)

q')< q+2"

then

N-D v( 2a'

1

1

q' ) < -. q

(Ag) - We suppose that p > max ( N -

(4.5)

D , ~N 2, N - D7-~ ). The function Vd

N--D

belongs to Lq(O,T; (LN-D-1 (~t)) N) + (L2(0, T; (Lp(~))N). The function Yd -D N belongs to Le(~), where ~ = inf{ ( g _N D _ ~ ) , (g-~D _~)}" Moreover, to study the control problem (Pc), we look for solution to equation (1.1) belonging to L~(0, T; Wx'~(~)). Therefore we must have ~_ 5i, ~ <_ di for 1 _ i < 4, where (bi, di) satisfy the conditions stated in Section 2. Observe that if (51,dl) satisfies (2.14), then 2 < q < 51, and dl < g-D-1g-D _< 2 (because D _< N - 2). This simple observation leads to simplifications stated in Proposition 4.1. P r o p o s i t i o n 4.1 - Assume that (AT) and (A8) hold. If q > 2, if 51 = q, and (51, dl) obeys (2.14), then there exist (52, d2) obeying (2.27), and (53, d3) obeying (2.28), such that 52 = 53 = q,

d2 > dl,

d3 > ds.

Proof. We only prove the existence of (52, d2), the rest of the proof can be performed in a similar manner. If a < N-D-1,N-D then dl < N-D-1N . We set 52 -- 51 = q, d2 - dl. Then 52_> q > ~, u andd2 =dl have #(

N-D 2

_> a > ~. Moreover, with assumption (4.2) we

D 1 N-D +~a +--1)<~+q 2

D 1 N 1 1 ~aa + - < q ~ 2 + q + 2 "

613

P. A. Nguyen and J. -P. R a y m o n d

Thus the pair (52 d2) obeys (2.27) If N - D < a < N-D then we have dl < N-D-IN-D. We choose 52 = 51 - q. Due to assumptions (4.2) and (4.4) there exists d2 _> dl such that '

#( and

"

N-D 2

#(N-

D 2

N - D - I

--

N - D - 2 '

D 1 +~aa + q

N 1 1 N 1 1 1)< ~-;~ + ~ + ~ - <~ - - ~ + q + - 2'

D 1 +~aa+ q

N 1 1)< ~+q+2-

1

N/z 1 1 < 2a + - + 2 " q

Then (52 d2) satisfies (2.27) If a > N - D then dl < N - D 52 --51 --q and d2 - d l . With assumption (4.2), we have "

N-D #("

2

D

--

1

+~aa + q

N - D - 2 ~

N-D 1 ) <- # ( ~

+

2

N-D < 2

+

N - D - I "

D(N-

D-

2)

We choose

1 q

+--1)

2 ( N - D) D - 2) 1 U 1 1 2(U - D) + -q < ~ + -q + 2"

D(N-

Thus (2.27) is satisfied by (52,d2).

m

Due to Proposition 4.1 with assumption (A9), we notice that N - D < 2 < a, and that the solution to equation (1.1) belongs to Lq(O,T; wl'd(~)) -[L~4(0, T; WI'P(~)), for all d < NN- -DD - 1 and all 54 < 2. The assumption (A10) below is needed in the proof of Theorems 4.2, 4.3, 4.4. (A10) - The exponents a, #, u satisfy N

1
a < q'

N-1 ~7 N

1 < u < 1 + a'(N-

D

1 < I~ < 1 + a ' ( N

D

v
-

D

-

I

/~
--

--

(4.6)

N-D-1 _

2)

if

a < N-

D-

2 '

(4.7)

2)

if

N-D a < N- D- 2 "

(4.8)

Let u be a solution of (P1), Y~ be the solution of equation (1.1) corresponding to u, and set a = ~'(y~),

b = @'(y~),

h = ~CQIVyu -- Vdl

p r = OC~ly~(T) - ydl~

~ y,

Vd)

- yd).

The adjoint equation for (P1) associated with (y~, u) is _ 0190t+ A p + ap = - div h in Q, p ( T ) = PT in f~,

Op ~ n A + bp = f~. g on E,

(4.9)

614

Control localized on thin structures for semifinear parabolic...

where ~ is the unit normal to F outward gt. When f~. ~ is not defined, equation (4.9) is a formal writing for the variational equation N

+ E

a+~DjpD~y + copy + apy) dxdt +

bpy dsdt

i,j--1

=

/++

h. Vydxdt +

]:

y(T)pTdx,

for all y e e l ( Q ) such that y ( 0 ) = 0. Lemma

4.1 - One has

[Vy~-VdI~-2(Vy~_Vd)eL~-~-~ (O,T; (L~-~-, (~))N)+ L~A~-,(O,T; (L~-~-~(~))N) for all d < NN-D - D - l , and all 54 < 2. P r o o f . We know that (Vy~ - Vd) belongs to Lq(0, T; (Ld(~)) N) + L~4(0, T; g-D and all 54 < 2. To prove the Lemma, we ver(Lp(~))N), for all d < N-D-I, ify that [Vy~-Vd[ ~-1 belongs to L ~--~-,(0, T; L ~--~-(gt))+L ~-~ - (0, T; L ~-~-,(~t)). Set Vyu - Vd -- gl + ~2, where ffl belongs to Lq(O, T; (Ld(~))g), and if2 belongs to Ls4(0,T; (LP(~))N). Since [ V y ~ - Vd[~-1 <_ C([tTl[ ~-1 + [ff2[~-1), we can prove that the mapping f H fQ [Vy~ - Vd[~-lf dxdt belongs to

((L~-~-, (O, T; L~-~-, (~)))' n (L~--~, (O, T; L~-~-~(~)))')'. But (see [30], Page 69) ((L~-~-, (O, T; L~-~- (~)))'N(L~-I (O, T; L~-~-, ')' - (L~-~-,(O,T;L~-~-,(~))+L~--~,(O,T;L~-~-,(~))), the function y~ ~ [ V y ~ Vd[ ~-1 belongs to L ~--~-,(0,T; L ~ - , (gt)) + L ~A~ -, (0, T;L ~--~-,(gt)), and the proof is complete, m The equation (4.9) is linear, we can set p = p~ + pc, where p~ is the solution to

_ Op +Ap+ap = - div f~ in Q, Ot

op ~+bp OnA

= h.g on E,

p(T) - 0 in 12,

and p0 is the solution to

0p - O-t + A p + ap - 0 in Q ,

0p On A ~ bp - 0 on E,

p(T)

PT in ft.

Due to L e m m a 4.1, f~ - ~[Vy~ - Vdl~-2(Vy~ -- Vd) -- f~l + f~2, where f~l belongs to L~--~-~(0, T; (L~---T-,(~))N) for all d < NN- -DD- l , and f~2 belongs to

P. A. Nguyen and J. -P. Raymond

615

L~-~-(0, T; (L~-4"f-(~)) N) for all 54 < 2. We can write p~ -- p l -b p2, where p l is the solution corresponding to f~l, and p2 is the solution corresponding to ~2. To establish optimality conditions for (P1), we have to prove that the trace of p on -yx]0, T[ belongs to Lq' (O, T; L"' (~) ). In assumptions (A10) and ( A l l ) , we state sufficient conditions on a so t h a t the trace of p l on -yx]0, T[ exists and belongs to L q' (0, T; L a' ('y)) (Theorem 4.2). A similar result for p2 is established in Theorem 4.3 when assumptions (A10) and (A12) are fulfilled. The trace of po on -yx]0, T[ is studied in Theorem 4.4 under assumptions (A10) and (A13). ( A l l ) - Assumptions needed in the proof of Theorem 4.2. ( A l l a ) - Assumptions needed to estimate the term bp. ,~-1 ~ (,~-I)(N-D-1) q 2 ( N - D)

1 < q'

u-1 q

I 1 2 "

(4.10)

If a < N-D-1 N--D--2 ' then

N(a-1)(N-D-1) 2 ( N - D) v 1 g

1

q'

q

N(,~- I ) ( N - D - 1 ) If a ->- N -ND -- -D - - 1

,~-1 q < 1 q'

( v - 1)(N

20"' <

2 ( N - D)

1

N 1 2a' --

2)

D

2 ( v - 1)(N 2

--

1

"2 ' D

~7

--

2)

1 + = . 2

(4.11)

(4.12)

then

I

N(,~- I ) ( N - D - 1 ) 2 ( N - D) v- 1 ~ N- 1 - ( v - 1 ) ( N - 1 ) ( N - D - 2) + ~1, (4.13) q 2 ( N - D - 1) 2 ( N - n - 1)

N ( , ~ - 1 ) ( N - D - 1) 2 ( N - D)

N-1

(v-1)(N-1)(N-D-2)

< 2(N-D-I)(Allb)

2(N-D-l)

1 -+2"

(4.14)

- Assumptions needed to estimate the term ap. ,r q

<

1 q'

~-I q

i

1 2

(4.15)

616

Control localized on thin structures for semilinear parabolic...

If a <

N-D

N--D--2'

then ~-1

N(a-1)(N-D-1)

q

2 ( N - D) 1

<

/z -

q'

1

I

q

N(a-1)(N-D-1)

--

g-D

N - - D - - 2

(~ -

- ~TD _ 2)

I)(N

1

~2'

2

(/~

N

2 ( N - D) If a >

g

2a I

1)(N 2

< 2a'

D

2)

§

(4.16)

1

(4.17)

then N(a-1)(N-D-1)

2 ( N - D) 1

/z-

q'

1 -~

q

g

(/~- 1)g(g-

D-

2)

2 ( g - D)

g-D

N ( a - 1 ) ( N - D - 1) N 2 ( N - D) < N- D

1

(4.18)

+2 '

(/z- 1 ) N ( N - D - 2) 1 2 ( N - D) + ~ .

(4.19)

L e m m a 4.2 - Let f~ be in L0(0, T; (L'(f2))g), where ~ > 1 and ~ > 1. We consider the equation: _ 019+Ap=_ cot

divfzinQ,

Op = f ~ . ~ o n E , On A

p(T)=O inf'.

(4.20)

The solution p to equation (t.20) belongs to L I ( O , T ; W I ' I ( D ) ) A Le(0,T; Lr(~)) for all (~,r) satisfying:

7)<~,_

r/
The trace of p on E belongs to

~<~, -

~<s, -

N 1 N 1 2 - ~ + - < r / ~rr §

1

(4.21)

L~(O,T;L~(F)) for all N 1 ~-+-< zr/

N-1

rl

2s

(~,s) satisfying:

1 1 ~-+-. s

2

(4.22)

If moreover 71 > N - D, the trace of p on 7 x ]0, T[ belongs to L ~ (0, T; L ~ (~/))

fo~ aZl (~, ~) ~ati~fy~g: ~<~, -

~<~, -

N

1

D

~

,1

~

~+-<

1

1

~

2

+-+-.

(4.23)

P. A. Nguyen and J. -P. Raymond

617

Proof. 1 - We know t h a t p belongs to LI(0, T; Wl'l(~)) (see [31]). 2 - We want to prove that p belongs to Le(O,T;Lr(f'l)). Let w be the solution to the equation

cow + Aw = O in Q, Ot

Ow =0onE, i)nA

w(T) = r in f~.

(4.24)

Then

fo

=

Vw(x, T + t -

. h(x,

)dxd

.

Using (2.4) in L e m m a 2.1 with 1 < r' _ rl' _< oc, r' < oo, and a = 1, we have:

IIp(t)IIL~(~) <_ C =

C sup{[

/t /o

sup{[ Ja p(x,t)r

V w ( x , r + t - T). h(T)dxdT[,

Due to (4.21), i > 1 and __

--1

= 1}

[lr

= 1}

ILf~(~)ll(L.(a))~ belongs to L'~(O, T) " We set

The mapping T ~

t_l_N~_ (~1

I[r

89+ N ( ~1_

1 = 1-t ~1 0" 1 7 1 < 7" 1 Thus the mapping t ~-, 7)

) belongs to Li(O,T). Therefore, with estimates on convolution,

we have

1 1 ( fo T ( jft T (T + t - T) _ 89 ~(-~--;)[[fZ(T)[I(L,(a))NdT )edt) ~1 < C[[h[[L~(O,T;(L,(n))N), and p belongs to L~(O,T;L~(f~)). 3 - We want to prove t h a t the trace of p on 2 belongs to L~(0, T; LS(F)). Due to (4.22), there exists/3 such that 1

2---~
/3< -

1

1

1

N

~

rl

2

2,/

~,

J

N

2s

.

(4.25)

Thus Tr(D(A~)) ~ L~(F). Let ~ be the solution of the equation -

0---t- + A ~ = 0 in Q,

OnA

= 0 on E, ~ ( T ) = A~,r in s

(4.26)

Control localized on thin structures for semilinear parabolic...

618

Then

A~p(x, t ) r

V ~ ( x , T + t - T) . h(x, T)dxdT.

=

Using (2.4) in Lemma 2.1 with 1 < s' < rf _< co, s' < co, and a = have

IIp(t)lrilL.(r) <_C sup {I/~ A~p(x,t)r = C sup

V ~ ( x , T + t - v). h(x, r)dxdrl,

I

<_ C

IIr

jft T (T

89 we

1} IIr

= 1

+ t - r)-- 89188

-* 1 The mapping T ~-~ []h(r)ll(L,(~))belongs to L~(O,T). W e s e t 71 = 1 + 71 ~" Due to (4.22) and (4.25), i >_ I and I + / ~ + N ( ~ _1 ;1) < 7"1 Thus the mapping

t ~-, t - 8 9 -.!) belongs to Li(O,T). convolution, we have

(

(

Therefore, using estimates on

(T + t - T)- 89189 ~_ C[IhIIL~(O,T;(L,(~))N),

and the proof is complete. 4 - We can proceed in the same manner to prove t h a t the trace of p on ~• belongs to L~(O,T;L~(~/)). m L e m m a 4.3 - Let g be in L$(O,T; L~(F)), where ~ > 1 and ~ > 1. We

consider the equation:

_

Op-~- Ap = 0 in Q,

Ot

Op = g on E,

OnA

p(T) = 0 in ~.

(4.27)

The solution p of the equation (4.27) belongs to Le(0, T; L r ( ~ ) ) for all (~,r) satisfying: N-1 1 N 1 1

~<~'- ~<~'- 2 - - T + ~ < ~ + - + - ' ~2

(4.281

P. A. Nguyen and J. -P. Raymond

619

The solution p of the equation (~.27) belongs to L~(0, T; wl'd(f~)) for all (~, d) satisfying: ~<~,

~
-

N-1

-

1

N

2----fi- + ~ < ~

1

+ ~

(4.29)

The trace of p on E belongs to L~(O,T;L~(F)) for all (~,s) satisfying:

< ~'

~ < ~'

If moreover ~ > N - D - l , for all (~, T) satisfying:

N-1 1 N-1 1 1 2---Y- + -5 < 2---7- + -~ + ~

(4.30)

the trace of p on-rx]O, T[ belongs to L~(0, T; L~('r))

~

N-1

1

~ -

-~ <

D

1

1

+ - + ~.

(4.31)

Proof. We use the same arguments as in the proof of Proposition 2.3.

I

L e m m a 4.4 - Let f~ be in L~(0, T; (LV(12))g), where ~ > 1 and • > 1.

Let b be a nonnegative function belonging to L~(O, T; L~(r)) for some (~, ~) satisfying (2.23) and (~.1). Consider the equation --

_

-~

Op

OPot ~-Ap - - div h in Q, ~nA +bp - h. ~ on E, p(T) - 0 in f~, (4.32)

The solution p to equation (~.32) belongs to LI(o, T; WI'I(f~))NLe(O,T; Lr(f~)) for every (~, r) satisfying (~.21). Moreover, the trace of p on E belongs to L~(0,T; n'(r)) for every (~,s) satisfying ~

~ <. ~ ,

_

. ~ ' < ~. ,

~. < s ,

N

~'<s,

1

N-1

~ - ~ + - < ~ + - +2s 7/

1

1

s

~.

(4.33)

Remark. In Lemma 4.4, we implicitly assume t h a t the set of pairs (~, s) satisfying (4.33)is non empty. This assumption must be checked every time that we use Lemma 4.4. Proof. 1 - Let (~, s) be a pair obeying (4.33). By a fixed point argument, we prove that the equation

_Op + Ap = - div f~ in ~ x ] T - t, T[, 0t

Op + bp = f~. ~ on r x ] T - E,T[, p(T) = 0 in f~ OnA

(4.34)

admits a solution for t small enough. Set

V ( T - { , T ) : = { p e L ~ ( T - { , T ; W~'~(~2)) I pir•

L~(T - t, T; L~(F))}.

Control localized on thin structures [or semilinear parabolic...

620

It is clear that V ( T -

t, T) is a Banach space for the n o r m IIPlIV(T-~,T) :-IIPlILI(T-~,T;WI,~(a)) + IlPlr• 9 Let ~ belong to V ( T -

t, T), and p~ be the solution of the equation"

ap

---+Ap:-divf~ Ot

Op OnA

:f~.g

in f t x ] T - t , T [ ,

-b(on

Fx]T-t,T[,

p(T):Oinft.

(4.35)

We have p~ = Pl + P2, where pl is the solution to

Op }-Ap = - div f~ in gtx ] T - [ , T[, cot

Op : f~.g on F x ] T - < T[,

On A

p(T) = 0 in ~t, and p2 is the solution to

_ Op + Ap - 0 in f t • Ot

t, T[,

Op = - b ~

COnA

on F x ] T - t , T [ ,

p(T) = 0 in ~. Using Lemma 4.2, from condition (4.33), it follows that pl belongs to belongs to L~(T - t, T; L~(r)). Ob-

L I ( T - t, T; W1'1 (~)), and Pllr• serve that b~]r•

belongs to L777 (T - t, T; L +To-;(F)) ' where t-~ + ~

>

--

1, ~ _> 1. Using Lemma 4.3, p: belongs to Lt-'T7(T - t, T; WI'~-~. (n)). With condition (4.1), and still from Lemma 4.3, P21r• belongs to L~(T - E, T; L~(F)). Thus p belongs to V ( T - ~, T). Let ~1 and ~2 belong to V ( T - t, T), let p~ and p~ be the solutions of equation (4.35) corresponding to ~1 and ~2. Still from condition (4.1), and using Lemma 4.3, we have: [[Pr -- Pr

+ II(Pr -- P~2)Ir•

~-- C]IblILi(T-{,T;L~(F))II(~I -- ~2)IF• where C can be chosen depending on T, but independent of t. The mapping t H C ~ fTT_t lib(s, T)II~L~(r)dT is absolutely continuous, then there exists > 0 such that C(f:ax{t_~,0} lib(., T)ll~Lt(r)dT)~i < C -- 1 for all t e [0, T]. Thus the mapping ~ F-, pc is a contraction in the Banach space V ( T - t, T), and admits a unique fixed point. Therefore equation (4.32) admits a local solution p.

P. A. Nguyen and J. -P. R a y m o n d

621

2 - Existence of a global solution. We prove t h a t a solution exists in the space V ( T - 2t, T), by repeating the above process. Let (~1, ~2) belong to {7r e V ( T - 2t, T) ] ~r = p on IT - t, T[}. We still denote by P~I and P~2 the solutions to equation (4.35) on ( T - 2t, T) corresponding to ~1 and ~2. As in Step 1, we can prove t h a t pc1 and pe~ belong to V ( T - 2t, T). Moreover, we have

liPs1 -- P~2 IILI(T--2t, T;W"I(fl)) -Jr-[I(P~ C[[b[ILg(T_2{,T_{;L*(F)) ([[~1 -[-[](~1

--

P~2)IFxIT-2t, T[IIL~(T-2t, T;L~(F))

-

-

-

~2[[L'(T-2{,T;W',I(f~))

~2)[FxIT--2LT[IIL,(T_2~,T;L.(r)))

1 < -([[~1 -- ~2[[L~(T--2LT;W',~(a)) + I](~1 -- ~2)]Fx]T--2LT[IIL~(T_2~,T;L~(r)))" --2 Thus the m a p p i n g ~ H pe admits a unique fixed point in the metric space {Tr e V ( T - 2t, T) [ zr = p on l T - t, T[} (the distance is defined by the norm of V ( T - 2t, T)). After a finite number of iterations, we prove t h a t equation (4.32) admits a global solution in Y(0, T). 3 - Uniqueness. If we consider the equation

Op - - - + Ap = 0 in Q,

Ot

op ~ + bp = 0 on E,

p(T) = 0 in ~t,

OnA

we can apply the above fixed point m e t h o d to prove t h a t p - 0 is the unique solution to this equation in the space {p E LI(0, T; W l ' l ( ~ t ) ) I P[E E

L~' (O,T; Lt' (F) ) }. 4 - We see t h a t p is the solution of the equation

_ Op + Ap = - div f~ in Q,

Op = ft. ~ - bp on E,

cot

p(T) = 0 in ~t.

COnA

Let (~, r) be a pair obeying (4.21). Then there exists (~, s) satisfying (4.33) and such t h a t

~ < ~, ~+ ~ -

ts

< r,

~+ s -

1 ~

~

N2~

1

1 NF- + ~ s

2s

1

1 N 1 < - + + -~. r

Therefore bpl~ e Lz-47 (O,T; L~-;~ (F)), and p e L~(O,T; Lr(f~)).

~r

B

Let f be in L a ( O , T ; L ~ ( ~ ) ) , where & >_ 1 and a >_ 1. Let b be a nonnegative function belonging to Lt(O, T; L~(F)) for some (~, e) satisfying (2.23) and (4.1). Consider the equation

Lemma

4.5 -

_ OP + A p = f in Q,

Ot

Op + bp = O o n E ,

OnA

p(T) = O in f~. (4.36)

622

Control localized on thin structures for semilinear parabolic...

The solution p to equation (~.36) belongs to Le(O,T;Lr(f~)) for every (~,r) satisfying N 1 N 1 5_< ~, a _< r, ~ +-a < ~r +-~ +1" The solution p to equation (~.36) belongs to L~(O,T;WI,d(ft)) for every (5, d) satisfying

5<(f,_

a
N 1 2da + - < a

N 1 1 2-d + ~ + 2 "

I f m o r e o v e r a > N~D, the trace of p on 7x]O,T[ belongs to L~(O,T;L~(7)) for all (~', T) satisfying: N &<~,_

a
1

D

1

~a + - < a ~TT+--+I'T

Proof. The proof can be established as in Lemma 4.2.

I

L e m m a 4.6 Let fz be in L~(0,T; (LV(f~))N), where r > 1 and rl > 1. Let b be a nonnegative function belonging to Le(O, T; Le(f~)) for some (~, ~) satisfying (2.23) and (4.1). Let a E be a nonnegative function belonging to Lk(O,T;ik(f~)) for some (k,k) satisfying (2.22) and (4.1). Consider the equation -

Op O--t + Ap + ap = - div h in Q, Op + bp = f~ . ~ on E, p(T) = 0 in f~. OnA

(4.37)

Its solution p E LI(0, T; WI'I(f~))MLe(O,T; Lr(ft)) for every (~, r) satisfying ~ .< ~, . [c' <. ~,

.rl < r ,

The trace orB on ~ belongs to

k' < r,

N 1 N 1 1 ~-~+-<~7 ~rr + - + " r 2

(4.38)

L~(O,T;L~(F)), V(~,s) satisfying (4.33).

Remark. In Lemma 4.6, we implicitly assume that the set of pairs (g, s) satisfying (4.33) and the set of pairs (?, r) satisfying (4.38) are non empty. This assumption must be checked every time that we use Lemma 4.6. Proof. 1 - Let (?, r) be a pair obeying (4.38). By a fixed point argument, we prove that the following equation admits a solution for t small enough:

Op Ot

+ Ap + ap = - div f~ i n f / •

Op

OnA

+ bp = ~. ~ on r •

E, T[,

p(T) = 0 in f~

(4.39)

P. A. Nguyen and J. -P. Raymond

623

Let ~ belong to LI(T - t, T; W I ' I ( ~ ' ~ ) ) I ' 1 L ~ ( T - t, T; L~(~)), and pc be the solution of the equation

op 0t

+Ap=-divft-a~

in f ~ x ] T - t , T [ ,

Op + bp = f~. ~ on r x ] T - ~, T[,

p(T) = 0 in f~.

OnA

(4.40)

We have p~ = Pl + P2, where Pl is the solution to

0p

---+Ap=-divft Ot Op

~-t-bp=ft.~ OnA

in f ~ x ] T - t , T [ , on

rx]T-{,T[,

p(T)=0inf~,

and p2 is the solution to

Op

---

0t

Op OnA

+ Ap = -a~

in ~ x ]T - t-, T[,

+ bp = 0 on F x ] T - t-, T[,

p(T) - 0 in f~.

Using Lemma 4.4, Pl belongs to LI(T - t, T; WI'I(~))CILe(T- t, T; L~(gt)). We know that a~ e L-~'~ (T - t, T; Lr';-;(f~)), where ~k +-e- > > 1. -- 1 ' -~From Lemma 4.5, and under condition (4.1), it follows that p2 belongs to

L%=-i(T- t,T; wl'r~-; (D)) n L e ( T - f, T; L~(f~)). Then pe e LI(T - t,T; WI'I(g/)) n Le(T - t-, T; Lr(12)). Let ~1 and ~2 belong to L I ( T - t,T; W I ' I ( f ~ ) ) n L e ( T - E, T; Lr(f~)), ar, d let pc, and PC2 be the solutions of equation (4.40) corresponding to ~1 and ~2. Still from Lemma 4.5 and under condition (4.1), we have: [[Pr -P~2 I[L 1 (0,3; W 1,1 (f~))NL e (T-t,T;L" (f't) ) __~ C [ l a [ I L ~ ( T _ ~ , T ; L k ( n ) ) 1 1 ~ I - - ~ 2 [ [ L I ( T _ ~ , T ; W I , , ( 1 2 ) ) n L e ' ( T _ ~ , T ; L , ' ( ~ ) ) ,

where C depends on T, but independent of t. The mapping t C~ fT-t T Ila(X'T)llki~(a) dT is absolutely continuous,

t then 3t- > 0 such that C (fmax{t-~,0} [la(., T)llki~(f~)dT)~ <_ C = 1 for all t 6 [0, T]. Thus ( H pe is a contraction in the Banach space LI(T - {, T; WI'I (f~)) L e ( T - {, T; Lr(f~)), and it admits a unique fixed point, so equation (4.37) admits a l o c i solution p.

Control localized on thin structures for semilinear parabolic...

624

As in Step 2 in the proof of L e m m a 4.4, we can iterate the process on ] T - 2t, T[ to obtain a fixed point in {Tr e L I ( T - 2t, T; W l ' l ( f ~ ) ) N L ~ ( T - 2t, T; Lr(f~)) [ 7r = p on ]T - t, T[}. After a finite number of iterations, we prove t h a t (4.37) admits a global solution in LI(0, T; W l ' l ( f ~ ) ) N L~(O,T; Lr(f~)). 2 - Uniqueness. If we consider the equation

Op

Op

- - - + A p + ap = 0 in Q, Ot

-+ bp = 0 on E, OnA

p(T) = 0 in gt,

we can apply the above fixed point method to prove t h a t p _= 0 is the unique solution to this equation in LI(0, T; WX'X(f~)) N L ~' (O,T;L k' (f~)). 3 - We see t h a t p is the solution of the equation

Op

_OP + Ap = _ div h _ ap i n Q , . ~nA + bp = fz " g ~ E' p(T) = O in f~ Ot Let (~, s) be a pair obeying (4.33). Under condition (4.1), there exists (~, r ) s a t i s f y i n g (4.38) and such t h a t

k~

kr

< ~,

1

< s,

+

N

Therefore a p e Lrg'i (O, T; nr4-; (12)) and

1 +-+

N

1 N<--+

plr~ e

L~(O, T; LS(F)).

2s

1

t-1. 1

T h e o r e m 4.2 - Assume that a and b satisfy the assumptions of Proposition 2.6. Let p be the solution to equation (~.37), where h belongs to - D the space L~-~-~(0, T; (L~--'~-~(f~))N) for all d < NN- D -l"

Under assumptions

(A1)-(A11), the trace of p on ~/x]0,T[ belongs to Lq'(O,T; L~'(~)). Proof. 1 - We first prove t h a t there exists (~,~) obeying (2.23) and (4.1), such t h a t the set of pairs (~, s) satisfying ~_>

q a-l'

~>~',

s_>

d

1

_g,

a-l' 1

1

=+- <-s ~ - q"

s> 1

-+ s

1

N

, (a-1)(q+

1

1

~-

<-a"

1

-+ s

1

~

1 )<~+

<

1

N-D-I'

N-1 2s

1 (4.41) +2' (4.42)

is non e m p t y for all d < Ng--DD- l , big enough. We distinguish two cases. First case: a < N - D - 12 , choose ~ = u - q- 1 . Due to (4.3), (4.7) ' (4.11) and (4.12), we can choose ~ such t h a t a ' < g < ( u - 1 ) ((N-l) gD --2) ' u-1 q

+

N-1 2~

1 < 2'

(4.43)

P. A. Nguyen and J. -P. R a y m o n d

-1

t N(a-1)(N-D-1)

q

1

2 ( N - D)

and

625

N-1

u-1

N-1

1

< ~7 -t

2a'

q

2s

~- ~,

<

N-1 2a'

N(n-1)(N-D-1) 2 ( N - D)

N-1 2t~

~

1 . 2

(4.44) (4.45)

i n f { ~ -qI ' q 1' 1 Observe t h a t ~1 - ~ 1 > 0 b e c a u s e u < q . Now we set ~1 7}' 1 - 2}" 1 Let us prove t h a t the pair (~, s) obeys and 71 = i n f { ( ~ - l ) N( N- D- D - 1 ) , ~-7 - 1 ~ N(,~- l ) ( N - D - 1 ) q 2 ( N - D) If~l ._ q'l

~1 a n d

tion (4.10).

I f ~1

71 __ .!.n--1)(N--D--!)N_D , c o n d i t i o n q,1

=

71 a n d

71

=

~,1

1

g-1

s

2s

< - +

1

+ ~.

(4.46)

(4.46) f o l l o w s f r o m a s s u m p (4.46) f o l l o w s f r o m

1 condition e,

(4.44). If ~1 __-- ,~--lq and ;1 __ (n-1)(N-D-1)N_D , condition (4.46) follows from l condition (4.46) follows from a s s u m p t i o n (4.6). I f ~1 -_- n-1 q and s1 -__- a1' e, (4.45). - D- 1, big enough, so t h a t (4.41) and Due to (4.46), we can choose d < NN- D (4.42) be satisfied. due Second case: a _> N-D-2'N-D-1 choose ~~ = v-l"q Since /2 < NN -- D D -- 12 ' to (4.3), (4.13) and (4.14), we can choose g satisfying (4.43), such t h a t g-

D-q 1 t

D-1

1 < g < (v-i~-g-D-2), N(~-I)(N-D-1) 2 ( N - D)

1 g-1 < ~+2(ND-

1)

u-1 q

g-1 2s

1 (4.47) t-~,

1

(4.48)

and N(~-

1)(N-

D-

2(N-D)

1)

N-

1

< 2(N-D-l)

N-

2g

1

+2"

1 Die Observe t h a t ~1 - ~1 > 0 because u < q, and set ~i _ i n f { ~ -q1 , q,I 7}. distinguish two subcases. When (,~-I)(N-D-1)N_D < ~ N - D - 1 1 -- ~'1 we set -~1 __ (~-I)(N-D-1)N_D . Let us prove

t h a t the pair (~, s) obeys (4.46). If ]1 -__ q,1

1

condition (4.46) follows from

a s s u m p t i o n (4.10). If ~l = trq .... , condition (4.46) follows from a s s u m p t i o n

(4.6). W h e n ( ~ - I )N( -ND- D ' I )

> --

1

N-D-I

_ !~ due to (4.47) and (4.48) (depending on

1 we can choose s such that 0 < 7 1 < N-D i -1 the value of ]),

1 a n d such

t h a t (~, s) satisfy (4.46). big enough, so t h a t (4.41) and Due to (4.46), we can choose d < N g- D- D- I ' (4.42) be satisfied. 2 - In the same way, using a s s u m p t i o n s (4.2), (A10) and ( A l l b ) , we can

626

Control localized on thin structures for semilinear parabolic...

prove t h a t there exists (fr k) obeying (2.22) and (4.1), such t h a t the set of pairs (~, r) satisfying ~> q -a-l'

~ > k', r > d -a-l' 1 + =" 1 < 1 -z

r

1 N (a-1)(q+~-~)<

r_>k', -1+

k--~'

1 < - -1 -k-a"

r

-1+

r

1 N 1 (4.49) ~+~rr+~,

1 < 2 -k N - D '

big enough. is non e m p t y for all d < N N- D- D- I ' 3 - From L e m m a 4.6, it follows t h a t the trace of the solution p to equation (4.37) belongs to L~(O,T;L~(F)), where (g,s) is the pair defined in Step 1. 1 1 We set ~1 -- ~1 + 7' ~ -- ~1 § ~' then bp E L~(O, T; L~(r)). Let 7r be the solution to the equation 07r 07r . Ot. F ATr . . 0 in. Q,. Onm

bp on ~

7r(T)-0

in ~

(4.50)

N-DD -- 21) ' then/~ _> q~, a n d / 3 > a~ > N - D - 1 If a < N - D - 21 ( r e s p . a > N N -- D (resp. ~ > N - D - 1 _ a~). Using Lemma 4.3, we deduce t h a t the trace of r on 7x]0, T[ exists, and belongs to Lq' (0, T; L ~' (7)). 4 - From L e m m a 4.6, it follows t h a t the solution p to equation (4.37) belongs to Le(0, T; Lr(f~)), where (~, r) is defined in Step 2. Let ~r be the solution to the equation -

0# - O---t-F A ~ r -

-

- a p in Q,

0# OnA - - 0

on ~,

7 7 ( T ) - 0 in f~,

(4.51)

we can prove t h a t the trace of # on 7x]0, T[ belongs to L q' (0, T; L ~' (7)). 5 - Let # be the solution to the equation - --+ at

A# - - div h in Q,

0# OnA

=h.gonE,

#(T)=0inf~.

N-D

(4.52)

> NSince ~ < 1 + N - D1 - I ' for d < N - D - 1 big enough, we have From L e m m a 4.2, setting ~ = ~ = ~ in (4.23), it follows t h a t #]~• belongs to L ~--br-(0, T; L~--~-(7)). From (4.6), we have ~

D.

> q'. Since ~-1 >

N - D > 2 > a', the trace of # on 7x]0, T[ belongs to L g'(0,T; L ~'(7)). Notice that p -- zr + # + #. The proof is complete, m ( A 1 2 ) - Assumptions needed in the proof of Theorem 4.3. ( A 1 2 a ) - Assumptions needed to estimate the term bp. Jr 2

+

~r 2p

<

1 ql

v-1 q

~

1 . 2

(4.53)

P. A. Nguyen and J. -P. Raymond

627

N-D-1 then If a < N-D-2, N(tr . 1) . 1. v 1 N 1 ( v - 1 ) ( N - D--2) < t 2p q' q 24' 2 N ( a - 11 N- 1 (v-ll(N~vD--2) 1 2p < 24' 2 + 2 " If a > N-D-1 then -- N-~=~, . 1 2

t

- 1

N ( ~ - 1)

1

2p

q'

2

N ( a - 1) 2p

N- 1 2 ( N - D - 1)

1 +~,

N- 1 2 ( N - D - 1) ( v - 1 ) ( N - I ) ( N - D - 2) 1 + ~ , 2 ( N - D - 1)

(4.54) (4.55)

v- 1

q

1 ( v - 1 ) ( N - 1 ) ( N - D - 2) 42, 2 ( N - D - 1)

(4.56) (4.57)

( A 1 2 b ) - Assumptions needed to estimate the t e r m ap. ~-1 2

1 q'

<

#-1 q

~

1 . 2

(4.~s)

N-D then If a < N-D-2, ~-1 2

If a >

--

N (tt 24' (it 1)(N

N(~-I) < 1 #-1 2p q' q N ( ~ - 1) < N 2p 2a' N-D then

1)(N D 2) 1 + ~ , 2 D _ 2) 1

2

+~"

(4.59) (4.60)

N--D--2

- 1 § N ( ~ - 1) < 1 2 2p qt N ( s : - 1) 2p

/z- 1 q N

N ( # - 1 ) N ( N - D - 2) 1 (4.61) N-D 2(N-D) +2 ' ( / z - 1 ) N ( N - D - 2) 1 + ~ . (4.62) 2 ( g - D)

N-D

4.3 - A s s u m e that a and b satisfy the assumptionsof Proposition 2.6. Let p be the solution to the equation (~.37), where h belongs to _L4_ L~-I (O,T; L~-~-~(f~)) for every 1 < 54 < 2. Under assumptions (A1)-(A10), (A12), the trace of p on ~/x]0,T[ belongs to Lq'(O,T;L~'(~/)).

Theorem

Proof. 1 - We first prove t h a t there exists (g,g) obeying (2.23) and (4.1), such t h a t the set of pairs (g, s) satisfying ~> -

54 tr

(~-1)(

~>[',

s>

-

N 1 +~pp)<-+s

-

P tr

N-1 2S

s>_g', 1 +2'

(4.63)

628

Control localized on thin structures for semilinear parabolic... 1 1 1 -=< ~+g-~'

-1+ s

1 < - -1 -[-a"

-1+ s

1 <

1

-d

(4.64)

N-D-I'

is non e m p t y for all 64 < 2, big enough. We distinguish two cases. -D-1 We choose g = u -ql " 9 First case: a < NN-D-2" and (4.55), we can choose g satisfying (4.43),

(N-l) 1)(N

al
2

N(a-

~

1)

1

2p

N-

< q 7q

1

Due to (4.3), (4.7), (4.54)

D

2)'

v-

2o"

1

N-

q

1

2g

1

(4.65)

t 2'

and N ( a - 1) N-1 2p < 2a ~

N-1 2g

1 + 2 "

(4.66)

1},

1 - 71 > 0 because u < q. Now we set ]1 = inf{ tr 2 , q, 1 Observe t h a t ~v 1 ____ i n f { ~ i 1 1 and 7 p , ~, e }. Let us prove t h a t the pair (~, s) obeys

- 1

N(a-

2 If ~q , _-1 1 If ~q , 1-_

~1 1

t

1)

2p

1

<-s +

N-

1

2s

1

+~.

(4.67)

and 7=1 a - l p , condition (4.67) follows from a s s u m p t i o n (4.53). ~1

and 71 __ a~1

~'1

condition (4.67) follows from (4.65).

If

and 7 - _1 ~--lp, condition (4.67) follows from the fact t h a t ( ~ - 1) < ~, condition (4.67) follows from I
~r

9 Second case: a ->- NN - - D D- -21 " W e choose g~ - v - lq" Since v < (4.3), (4.56) and (4.57), we can choose g satisfying (4.43),

2 and

N(a-

}

1)

2p

1

due t o

N-D-1 (u-1)(N-D-2)'

N-D-I
N -- D D -- 2 1 ' N

N-

1

u-

< ~7 + 2 ( N - D - l )

N ( a - 1) N-1 < 2p 2(N- D-

1

N-

q

1)

N-1 2g

2g ~

1

1

(4.68)

I 2 '

1 . 2

Observe t h a t ~1_ - ~1 > 0 because u < q. We set ]1 = i n f { ~ - I 2 , q'1 consider two subcases.

(4.69) 1 ~}, and

P. A. Nguyen and J. -P. Raymond

629

When < N - D 1- 1 t1' we set s1 = ~ ~ X . Let us prove t h a t (~,s) obeys x - 7, x condition (4.67) follows from assumption (4.53). If (4.67). If ~x = ~7 = ~-x ~ condition (4.67) follows from the fact t h a t ( ~ - 1 ) < 1 < N - D <_ p. 8 2 When ~-~ -> N - D - 1 1 e'X due to (4.68) and (4.69) (depending on the value 1 x and such t h a t (g, s) of ~), we can choose s such t h a t 0 < ;x < N - Dx- X ~' satisfy (4.67). Due to (4.67), it follows that we can choose 1 < 54 < 2, close enough to 2, so that (4.63) and (4.64) are satisfied. 2 - In the same way, using assumptions (4.2), (A10) and (A12b), we can prove t h a t there exists (k, k) obeying (2.22) and (4.1), such t h a t the set of pairs (~, r) satisfying

r>k'

, r >- ~ - 1 ' - 1' N < ~1+ ~ +N 1 ( n - l ) ( ~4 + zp~-'7-)

-

:1+

r

1 < 1

~-~'

-1+

r

1 < ~1

-1+

-k-a"

r

(4.70) 1 <

-k

2

N-D'

is non e m p t y for 3 - From L e m m a (4.37) belongs to 1 1 = ~1 + ~, ~ =

all 54 < 2, big enough. 4.6, it follows t h a t the trace of the solution p to equation L~(0, T; L~(r)) for the pair (~, s) chosen in Step 1. We set 71 + ~" Then bp e L~(O, T; L~(F)). Let 7r be the solution -D-1 (resp. a -> NN -- DD--12 ) ' _ q, , and to equation (4.50). If a < NN-D-2 then/3 > 13 > a' > N - D - 1 (resp. /3 > N - D - 1 > a'). Using L e m m a 4.3, we deduce t h a t the trace of 7r on "yx ]0, T[ exists, and belongs to L q' (0, T; L ~' (7)). 4 - From L e m m a 4.6, it follows t h a t the solution p to equation (4.37) belongs to Le(O,T;Lr(gt)), where (~,r) is chosen in Step 2. Let r be the solution to equation (4.51), we can prove t h a t the trace of r on 3,x]0, T[ belongs to

Lq' (O, T; La' (,./)). 5 - Let # be the solution to equation (4.52). Since ~

> N-

D, from

L e m m a 4.2, it follows t h a t #i~• belongs to L--~ (0, T;L~-~-r-~(7)). We have a' < 2 < p < ~-~-l" Moreover, we can choose 54 < 2 big enough, to obtain q' < 2 < ~-~_~. Thus #l~xl0,X[ belongs to L q' (0, T ; L ~' (7)). Notice t h a t p = 7r + 7? + ~. The proof is complete, m -D In the sequel, we set # = rain (a, N -ND -~

).

( A 1 3 ) - Assumptions needed in the proof of Theorem 4.4. N 0 < inf{1 + ( N -

D + 2 ~"

q.~2p

D) (N

a,D

.-7 0

q,)2 ' 1 + (N

~,D

~)2 }.

(4.71)

630

Control localized on thin structures for semifinear parabolic...

( A 1 3 a ) - A s s u m p t i o n s needed to estimate the t e r m bp.

( 0 - 1)(N

D

a'

2N If a <

N-D--1 N--D-

'

( 0 - 1)(N

v--1 < 1. q

(4.72)

then

D ~'

2 7) <1

v-1 q

2 If a ->- N - N- D- -- D- - - 1

2

if)

(v --1) (N -- ~D -- 2) + N-1 2(N- D2

1)' (4.73)

then

( 0 - 1)(N

D ~,

~2 )

2

v- 1 ( v - 1 ) ( N - 1 ) ( N - D - 2) q 2 ( g - D - 1) N-1 2 ( N - D - 1) '

<1

(4.74)

( A 1 3 b ) - A s s u m p t i o n s needed to estimate the t e r m ap. then If a < N -ND- -D- 2 '

(8- 1)(N- ~_~)D 2 < 1 2 If a ->-

N-D

N-D--2

(tt --1) (N -- ~D -- 2) + N 2 ( N - D)"

~- 1 q

( ~ - 1 ) N ( N - D - 2) N + ~ (4.76) 2 ( N - D) ( N - D)"

(4.75)

then

(9- 1)N(N- D2 ( N - D) Lemma

it-1 q

2

~)

<1

4 . 7 - Let p be the solution to the equation

Op

-- O-'t + Ap = 0 in Q,

Op

OnA - - 0 on E,

p(T)-

PT in ~t,

(4.77)

where PT belongs to L ~(~) for some N2 D < e < ( N - ~ - ~N) ( e - 1 ) " Then Pl-yxl0,T[ belongs to L~(O,T;L~(~)) for all ( ~ , r ) s a t i s f y i n g ~ >_1,

r >

N 1 ~e < -+--r

D

2r

Proof. This is a straightforward consequence of P r o p o s i t i o n 2.1. Lemma

4.8 -

m

Let PT be in L~(~) for some e < (N-7-D N~)(0--1)" Let b

be a nonnegative function belonging to Le(0, T; Le(~t)) for some (g,i) satisfying (2.23) and (4.1). Let a be a nonnegative function belonging to

P. A. Nguyen and J. -P. Raymond

631

Lk(O, T; Lk(~t)) for some (k, k) satisfying (2.22) and (~.1). Consider the equation

Op

---+Ap+ap-O

in Q,

Ot

Op OnA

+ bp = 0 on E,

p(T) = PT in ft.

(4.78)

The solution p to equation (4.78) belongs to LI(0, T; W1'1 (g2))nLe(0, T; L~(f~)) for every (~, r) satisfying -

N

k' <_~, e<_r, The trace of p on E belongs to ~'<_~,

e<s,

k' <_r,

N

L~(O,T;L~(F)) /or every k'<_s,

1

2ee<~rr +='r

(4.79)

(~,s) satisfying

N N- 1 1 7e< 2s ~ ~"

Proof. We can proceed as in the proof of Proposition 2.6.

(4.80) m

T h e o r e m 4.4 - Assume that a and b satisfy the assumptions of Proposition 2.6. Let p be the solution to equation (~. 78), where PT belongs to N2 . Under assumptions (A1)-(AIO), (A13), L~(~) for all e < (N-D~-V)(o-i)

the trace of p on ~x]0, T[ belongs to L q' (0, T; L ~' (~)). Proof. 1 - Let p be the solution corresponvding to a - 0 and b -- 0. Due to condition (4.71), there exists e < ( g - VD- 7 ) ( 0 , 1 ) such that ~g < ~71 _~_~D and ~g < ~ + ~ . 1D

g < ~1 + ~; D the trace of p on If a' _< e, due to T

x ]0, T[ belongs to Lq' (0, T; L ~(~/)) ~ Lq' (0, T; L ~' (~)). If a' > e, due to the trace of p on ~/x]0, T[ belongs to L q' (0, T', L ~' (~/)). 2 - Now suppose that a and b are not necessarily equal to zero. 2 a - We first prove that there exists (~,g) satisfying (2.23) and (4.1), such that the set of pairs (~, s) obeying 2_~
~>[',

s>e,

s>~',

1+1 s ~<1,

1+1 1 N 1 N-1 s ~ < N- D- l'2e < s + ~'2s

(4.81)

is non empty for all e < (N-~---r--q-~)(0--1)"D N big enough. We distinguish two cases. First case: a <

N N -- D D -- 12 "

We choose ~ -- u--l" q From (4.7), we have u <

Control localized on thin structures for semilinear parabolic...

632

1 1 + ~ ' ( NN-- D 2) < 1 + a--r --

N-1 (N-D-1)(N-

D-2)

"

Therefore, from (4.3) and (4.73),

we can choose g satisfying (4.43), such t h a t N - D - 1 < ~ <

(N-l) D ~2) ~ (v--I)(N-

and (0-

D ~,

1)(N

2

Now

2 ~)

1 __ I set ~

we

1 ~

< 1

v-I q

and

1 _ s (0_I)(N_

(4.72), it follows t h a t 8 > (g, s) obeys (tg-1)(N-

N-I 2g

N-1 § 2(N-D-l)

inf{(0-1)(N-N 0

D

.

2

- ~r ) , N - D -11

(4.82)

1 -- ~ }" F r o m

2

2N T - y )

~D- 7 ) 2

2

>

. Let us prove t h a t the pair

1 N-1 < - + ~ . s 2s

(4.83) D

If ;1 __ N-D-11 --7,1 condition (4.83) follows from (4.82). If -sl __ ( 0 - 1 ) ( N - N v - ~) condition (4.83) follows from a s s u m p t i o n (4.72). Then, due to (4.83), we can choose e < ( g _ 7-~g ~) (0" 1)' big enough, so t h a t (4.81) is satisfied N --DD- 1- 2 ' due to Second case: a _> N-D-2"N-D-1W e choose g~ = v-l"q Since v < N (4.3) and (4.74), we can choose g satisfying (4.43), such t h a t N - D - 1 < N- D- 1 , and ~ satisfies (4.82). ~" < ( v - l ) ( g - D - 2 )

1

We set ~1 = 1 - 7"1 Due to (4.72), we have ~ > 1

N-D-I

__

1~ < -1

O<s
(N-

(0-1)(N-

~D -r -- ~-~r) 2N

"

If

D -- ~2-r)(0-- 1)

~r s

t h a n k s to (4.82), we can choose s such t h a t 1 1 (0-1)(N~D- ~ r ) and s satisfies (4.83). If N - D - 1 ~ > N ' N

we set 1s = ( 0 - 1 ) ( gN- ~ - ~ r ) . Condition (4.83) follows from (4.72). Due to (4.83), we can choose e < ( g - ~ - ~ g) ( 0 - 1 ) ~ big enough, so t h a t (4.81) is satisfied. 2 b - In the same way, using assumptions (4.2), (A10) and (A13b), we can prove t h a t there exists (]r k) satisfies (2.22) and (4.1), such t h a t the set of pairs (~, r) satisfying

>k', -

r>e, -

r>k' -

'

-1+1 _ r

~ < 1,

is non e m p t y for all e < (N-~-r-q - - rN) (2O - - 1 ) D

1

-r +

1 -k

<

2 N - D

,

N

1

N

-2e - < -~ + ~ r r '

(4.84)

big enough.

2c - From L e m m a 4.8, it follows t h a t the trace of the solution p to equation (4.78) belongs to L~(O,T;L'(F)), where (g,s) is chosen in Step 1. We set 1

1 = ~1 + ~, ~ = 71 + 89 Then bp e L 3 (0, T; L ~ (F)) w i t h / 3 > Y - D - 1. Let

P. A. Nguyen and J. -P. Raymond

633

7r be the solution to e q u a t i o n (4.50). Using L e m m a 4.3, we deduce t h a t t h e trace of 7r on ~x]0, T[ exists in LI(O,T;LI(9/)). 2 d - F r o m L e m m a 4.8, it follows t h a t the solution p to e q u a t i o n (4.37) belongs to Le(O,T;L"(~)), where (~,r) is chosen in Step 2. Let # be the solution to equation (4.51), we can prove t h a t the t r a c e of # on -yx]0, T[

belongs to LI(0, T; Ll(q,)). 3 - From Steps 1, 2b and 2c, it follows t h a t the trace P[~x]0,T[ of t h e solution to e q u a t i o n (4.78) belongs to LI(0, T; L1 (7)). By a c o m p a r i s o n principle and using Step 1, we can prove t h a t it belongs to L q' (0, T; L ~' (9/')). E1 E x a m p l e s . Let us give e x a m p l e s in the three dimensional case, for which (A5), ( A 7 ) - ( A 1 3 ) a r e satisfied. 9 Suppose t h a t N = 3, D = 1, q -

a = 2. We have

N - D N - D - I

---

2 < --

N - D

N m - 2 = C~ a n d 2 -<- a < N D -- 2I N -- D = OC. Conditions (4.2) and # < q correspond to # < 2. Condition (4.3) is satisfied if v < 5 Conditions (4.11) and (4.16) on ~ are satisfied if m < 9 Condition (4.71) on 0 is satisfied if 5 ~ < ~9 a n d p > 2, 0 < 2. If p > 2, (A9) is satisfied. Thus, if # < 2, v < ~, all t h e conditions are satisfied.

(7 <

-

9 Suppose t h a t N = 3, D = 1, and t h a t K u is b o u n d e d in L ~ ( T x ]0, T D. We do not set q - a - cc since the regularity results in Section 2 are s t a t e d in the case w h e n q and a are finite, b u t we can take q and a as big as we want. We have a < N N- -DD - 2 - - " (:X:)and a < . ~ _ _ D0--1- 2 = c~. C o n d i t i o n s (2.1), (4.2), (4.4), (4.6) and ( 4 . 8 ) c o r r e s p o n d to # < co. C o n d i t i o n s (2.2), (4.3), (4.5), (4.6) and (4.7) are satisfied if v < co. Conditions (4.6) and (4.12) on a are satisfied if a < 2. Condition (A13) on 0 is satisfied if 0 < co. If p = co, (A9) is satisfied. Thus, if # < co, v < co, a < 2, 0 < cc a n d p = co, all the conditions are satisfied. 9 Suppose t h a t N = 3, D = 0, q = a = 2. We have N -D-1 N-D--2 N - D N - D - 2

---

_

--

2 < a < --

3 " Conditions (4.2) and # < q correspond to # < 2 Condition

3 Condition (4.6) on m is satisfied if m < 3 (4.3) is satisfied if v < ~. Condition (4.71) on 0 is satisfied if 0 < 3. If p > 3, (A9) is satisfied. Thus, if # < 2, v < 3, a < 3, 0 < 2 a n d p > 3, all the conditions are satisfied.

9 Suppose t h a t N = 3, D = 0, and t h a t K u is b o u n d e d in L ~ (7 x ]0, T D. Therefore we can take a and q as big as we want. We have a ->- N N -- D D - 2 - - - 3 and a ->- NN -- DD -- 12 __ 2. Conditions (2.1) and (4.2) are satisfied if # < 3. Conditions (2.2) and (4.3) are satisfied if v < 2. Conditions (4.6), (4.14) and (4.19) on a correspond to a < 3. Conditions (4.71), (4.74) a n d (4.76) on 0 correspond to 0 < 3. If p = oc, (A9) is satisfied. T h u s , if # < 3, v < 2, a < 3, 0 < 3 and p - co, all the conditions are satisfied.

634

Control localized on thin structures for semifinear parabofic...

4.1. O p t i m a l i t y conditions for (P1) L e m m a 4.9 - Let (an)n be a sequence of nonnegative functions converging to a in Lk(0, T; Lk(~t)) for every (k,k) satisfying (2.22). Let (b,~)n be a sequence of nonnegative functions converging to b in Lt(0, T; Lt(r)) for every (~,~) satisfying (2.23). Let zn be the solution of Oz

+ A z + anz = 0 in Q,

Oz OnA F bnz

u ~ on E,

z(O) = 0 in ~,

and let z be the solution of Oz O---t+ A z + az = O in Q,

Oz OnA + bz = us

onE,

z(O) = O i n n .

(4.85)

Then (Zn)n converges to z in Le(0, T; Lr(12)) for all (~, r) satisfying (2.13), and in Lq(O,T; wl'd(fl)) for all d < N - D - 1 to z(T) for the weak topology of Lr(12) for all 1 <_ r < ( g - 7N -7)D 2 (where -

N - O

a = min (a, (N_O_~q_r)

)).

Proof. Due to Proposition 2.6, we have

IIz. IIL+(O,T;L:(m) + IIZ. I~:IIL:(O,T;L=(r)) <-- CllU[IL~
~W

--+Aw+aw=(a-an)z,~ in Q, Ot Ow ~+bw=(b-bn)z, on E, w ( 0 ) = 0 OnA

in f~.

(a.s6)

Due to Propositions 2.6, 2.7, and 4.1, there exist (/r k) satisfying (2.22) and ({, e) satisfying (2.23), such that" IIz.

-

ZlIL+
IIz.

--

ZIIL~
<_ c(}la. - alli~(O,T;i~<m)llz.lli+
Thus the convergence results for (zn)n are established.

635

P. A. Nguyen and J. -P. Raymond

N Since (zn(T))n is bounded in L~(~) for all 1 < r < (N--~--~)'D 2 we can

suppose that (z,~(T)),~ converges to some ~ for the weak topology of L~(~) g for all 1 < r < (g-~--~)D 9By passing to the limit in the Green formula

(-z,~-O-/- + y~ ~jDjz,~D~r + ~oznr + ~nZ,~r

+

bnz.r

i,j=l

we obtain fn r = fa r solution of (4.85). Thus ~ = z(T).

for every r e C 1(Q), where z is the m

T h e o r e m 4.5 - Let z be the solution to the equation (~.85). Let h = ,~ f~l+f~2, where f~l belongs to L~-~-~(0, T; (L~-~-~ (fl))g) for all 1 < d < NN- -D D- l ,

and It2 belongs to L .-5r - , (0,T; (L~z~-~ (~'~))N) for all 1 < 5 4 < 2 . Let p be the solution to equation (4.9) corresponding to f~. If assumptions (AT)-(A13) are satisfied, then we have the Green formula o

p(s, t)u(s, t) dsdt =

V z . f~ dxdt +

pT(x)z(x, T) dx.

(4.87)

Proof. Observe that z(T) belongs to L~(fl) for all 1 < r < g - ~ N -~"

With

(4.71), we can find e satisfying e < ( g - ~D N~)(0-1) such that z(T) belongs to L~ ~' (g/)). Thus pTz(T) belongs to Ll(12). Let (hlm)m (respectively (f~2m)m) be a sequence of regular functions converging to f~l (respectively f~2) in L~-~-~(O,T;L~d-~-'(~)) for all 1 < d <

N-D (respectively L~-~-~(O,T;L~-z~-~(~)) for all 1 < 54 < 2) Let (p~)m N-D-1 be a sequence of regular functions converging to PT in the space L~(~) for N2 -'1 + f~2 and let pm be the solution of a l l e < (N_D 9 Let hm - hm a--r-- q-r)(0-- 1)

the equation

_OP+Ap+ap___divf~m Ot Op + bp - f~m " ~ on ~,

in Q,

p(T) - p~ in ~.

(4.88)

OnA

Using Theorem 4.2 in [24] for equations (4.85) and (4.88), we obtain:

pm(~,t)u(s,t)d~dt

=

/o

V z . ~,,d~dt +

/o

p~(~)z(~,T)d~.

(4.S9)

636

Control localized on thin structures for semilinear parabolic...

Due to estimates established in the proofs of Theorems 4.2, 4.3 and 4.4, we can prove that (Pmlv• converges to PI~• for the weak-star topology of Lq' (0, T ; L ~' (~)). Therefore the Green formula (4.87) may be obtained by passing to the limit in (4.89). Theorem

4.6 - If u is a solution of (P1), then

p(v -- u)d~dt + qC7

(

lul~dr )

lul~-2u(v-

~(

u)d~) dt > 0

for every v ~ K u , where p is the solution of

op

-O---t + Ap + ap = - a C Q div (IVy~ - Vdl~-2(Vyu -- Vd)) in Q,

Op On---~ + bp =

~Cq(IVy~

- Ydl~-2(Vy~

p(T) = #Cnly~(T) - ydl~

- Yd)). ~

on r~,

- Yd) in ~,

with a = (I)'(y~), b - ~'(y~), and y~ is the solution of (1.1) corresponding to u. Proof. 1 - Let v be in K u , and A > 0. Denote by y~ the solution of (1.1) corresponding to u + A(v - u). Let us set w~ - y~ - y~, then w~ is the solution of the equation: Ow --+Aw+a~w Ot

=0

in Q,

(~W

~ n A + b ~ w = A(v-u)5~ on E,

w(0) = 0 in ~t,

where a~ = f~ ~'(y~ + zg(y,x - y~))dvg, and b~ = f : ~'(y~ + O(y~ - y~))dO. Applying Proposition 2.6 to the above equation, we obtain

IJwxlrL~(O,T;L~(~)) + IIWxlE]]L+(O,T;L+(F)) ~

C~llv -

UlILq(O,T;L~(~))

for every (~, r) satisfying (2.13), and every (~,s) satisfying (2.15) and ~ < 2p. Therefore, when A tends to zero, the sequence (y~)~ tends to y, in Le(0, T; L r ( ~ ) ) for all (~, r) satisfying (2.13), and the sequence (Y~lz)~ tends to Y~,lz in L~(O,T;L~(F)) for all (~,s) satisfying (2.15) and ~ < 2p. Thus (ax)~ tends to a in Lk(0, T; Lk(~)) for every (k,k) satisfying (2.22). And (bx)~ tends to b in L~(0, T; Le(r)) for every ({,g) satisfying (2.23). For all l _ < r < ( N - ~N- 7 )2~ , we have the estimate"

[Jwx(T)lIi.(~)

<_ [[W~,IIL=(O,T;L~(~))<--CA]Iv-

U[[Lq(O,T;L,,(,~)).

Thus (y~(T))~ tends to y~(T) in Lr(~t) for all 1 < r < --

~g - 7 )

(N_D

2

9

P. A. Nguyen and J. -P. Raymond

637

Due to the previous convergence results for (a~)~, (b~)~, and (y~)~, using L e m m a 4.9, we can prove t h a t (y~)~ tends to y~ in Lq(O,T; w l ' d ( f ~ ) ) + N-D and all ~4 < 2. L~4(0, T; WI'P(f~)) for all d < N-D-1 2 - Now we set z~ = (y~ - yu)/A, and we denote by z the solution of

Oz O--t+ Az + az = 0 in Q,

Oz OnA 4-bz = (v - u)57 on E, _

z(0) = 0 in f~.

Due to L e m m a 4.9, we can prove t h a t the sequence (z~)~ tends to z in Lq(O, T; wI'd(f~)) for every d < NN-D - D - l , and (z~(T))~ tends to z(T) for the N weak topology of Lr(f~), for all 1 _< r < iN- ~_~)D 2 9

3 - We want to calculate the gradient of the cost functional. From the convexity of the mapping y H fa [ y - yd[edx + fQ I V Y - Vd[~dxdt, we have 0 C n / n lye(T) - ydl~

-- yd)zA(T)dx

+nCQ /Q LVy~ - Vd[~-2(Vy~ -- Vd)Vz~dx < ~ ( I ( u + A(v - u)) - I(u)) <_ OCn / a lye(T) - YdI~

+ cq fQIVy -

-- yd)z~(T)dx VdI~-2(VyA - Vd)VzAdx.

Let us set F(u) - I(yu, u). Due to the previous convergence results on (y~)~, and (z~)~, by passing to the limit in the above inequalities, we obtain f

F'(u)(v - u) = ./o OlY~(T) - YdlO-2(y~(T) -- yd)z(T)dx

-[-~ / o Ivyu -- Ydl~-2(Vyu -- Yd)Vzdx

+qC~

(

lul~dr )

(

lu

Finally we conclude with the Green formula (4.87).

2u(v - u)dr dr. m

5. Derivative of the cost functional with respect to deformations of ~/ In this section we consider the case when ~/is a curve, t h a t is D - 1. Since we suppose t h a t N >_ D + 2, we only consider the case where N > 3.

Control localized on thin structures for semilinear parabofic...

638

We calculate the derivative of the cost functional with respect to fields of deformation of the curve 7. For simplicity we suppose that the control u is only depending on the time variable. Let I be a bounded interval in R, and let (g(~))~ex be a regular parametrization of 7. More precisely we suppose that g is a mapping of class C k from I to R N such that 7 = (g(O))o~i, and such that ]g'(0)[, the Euclidean norm of g'(O), is different from zero for all 0 E I. Let (G(r162 be a regular field of deformation of 7 such that G is a mapping of class C ~ from 7 into R N, and G(~) is a tangent vector to F at r We suppose in addition that there exists a family (gA)xe]0,1[ of mappings from I into F, such that 7~ = (g~(O))~eI is a curve of class C ~, and

and where lim~_.0 II~(A,')IILoc(I;RN) = 0, and lim~_~0 II~I(A,')IILo~(I;RN) -- 0. Let y~,~ (resp. Yu,~) be the solution of (1.1) corresponding to u and 7~ (resp. 7~). Recall that u only depends on t. Set F(7)

J (y,~,.y, u)

We want to calculate limA_~0

= L lYu"r(T)- YdlOdx "

g ( 7 ~ ) - F(7)

For this calculation, we suppose that the assumptions (A1), (A3) are satisfied. The initial condition y0 belongs to LP(Et) for p _> ~ - 3 " Some additional assumptions stated below are needed. (AS')-

-

o.

( A 9 ' ) - The control variable u belongs to L~176 For the regularity of the adjoint state, we make the following assumptions on 0 and p: 8<1+

N-1 ( m - 3)N '

#<1+

N-1 ( N - 3)N "

(Notice that we can use the regularity results in Section 2.2 with D = 1, q and a are as big as we want.)

5.1. Regularity results of the adjoint equation We consider the following terminal boundary value problem

Op

- 0-7 + Ap + ap = 0 in Q,

0P - - 0 on E, OnA

p(T) = p T in ~, (5.1)

P. A. Nguyen and J. -P. Raymond

639

where a is a nonnegative function belonging to Lk (0, T; Lk (gt)) for every (/r k) satisfying N- 1 (5.2) 1r k< (N-3)(#-l)' N-1 and where PT belongs to L~(~) for all 1 _ e < (N-3)(0-1)" If ~ is the weak solution of (1.1) corresponding to u and -y, if we set a = (I)'(~), PT -0]~(T) - ydle-2(~(T) -- Yd), then equation (5.1) corresponds to the adjoint equation for J associated with (-y, ~).

5 . 1 - Let p be the weak solution to equation (5.1), where PT N-1 the mapping that associates belongs to L~(~). For all 1 _< e < (N-3)(0-1), p with PT is continuous from L~(~) into L ~ ( O , T ; L m ( ~ ) ) for all ( ~ , m ) satisfying N 1 N rh>l, m_>e, ~e < - m + - ' 2 m (5.3) Theorem

We have the estimate

where C(Cn, m, e) does not depend on a. Moreover the solution p to equation (5.1) belongs to LI(0,T; C1,~(~)) for some 0 < a < 1, and

IlpilL,(o,r;c ,o( )) --- c(1 + ilaliL (o,r;L (a)))iiPrilL ( ), for some ]r < oc, k < (N-3)(~-I),N-1 and e

N-1

< (N'3)(0-1)

Proof. 1 - When a - 0, the estimate of p in L~(O,T;Lm(f~)) can be obtained in a classical manner. When a ~ 0, this result can be obtained with a comparison principle. 2 - Let lr be the solution to 07r . Ot ~. A r = .- a p in. Q,

07r OnA

0 on E,

7r(T)-0

in f~.

(5.4)

Due to (A9'), we can find a pair (~l,jl), 1 < ~1 < 2, j l > N, satisfying sup { N ( N - 3 ) ( 0 - 1) N ( N - 3 ) ( # - 1) N 2 ( N - 1) ' 2 ( N - 1) } < 1 -~ 2jl

~1 < 1 2 2"

N-1 and k < (N-3)(ttN-1 1), such t h a t Next, we can choose e < (N-3)(o-1) N

N

0 < 2ee < 1 - ~ 2jl

~1

2'

(5.5)

Control localized on thin structures for semilinear parabolic...

640

sup { N ( N - 3 ) ( # - 1) N 2(N-l) '2jl

N N N 2e } < ~ <1-~ 2jl

~1 2"

Observe that N < 1 + 2jl -gx2 < 89 and N < 1 + -2jl g- e > N and k > N. N and 1 + 2j~ N 2~

< 2+

-~ 2 < 89 Thus g ~ < 1 < l _ g N Since 1 + 2j--7- 2 ~ ~ < 1+ N 2 2~, N

~2< 2 q

N - ~2 2j---7

g - g! N we also have ~kk < 1 + N2 2T 2 - 27,

2~N and

N Thus there exists rh > 1 such that - 2~"

sup {-0, N 2e

N

N N

1-

N

2 t 2k'2e

2E'

2-~1 +

N

1

+~-~}<--
Finally, we can find m > e such that sup

{-0 N ' 2e

1 N rh' 2jl

N N N N 2k } < ~ < inf{ 2e' 2

N N 2k' 1 ~ 2jl

~1 2

N 2k }"

Then (rh, m) satisfy (5.3). Due to Step 1, p belongs to L~(0, T; Lm(f~)). Let N N N N 1 _- - •m + ~ ' then r > 1 and ap E LI(0, T; Lr(f~)) Since 2-97, < ~-~ + 2-~ = 2-7, we have jl > r. Moreover, N __-- 2.__~ k g_~_ ~k belongs to LI(0, T; Wr (f~)). Let # be the solution to the equation 0# - - - + A# = 0 in Q,

0#

Ot

OnA

<

1 + 2-j7 - - N ix2, and therefore 7r

#(T) = PT in f~,

= 0 on E,

(5.6)

Under condition (A9'), we can find (~,j) satisfying

0<~<2,

N(N-3)(O-1) 2 ( g - 1)

< l - t . . .N. 2j

~ < - .1 2 2

(5.7)

We can choose e _< j satisfying

N(N-3)(O-1) 2(N-

1)

N

N

< ~ <1+2-j-2

Then # c LI(0, T; W~J(f~)). Since p - 7r + #, we have shown that p belongs to LI(0, T; W~IJ~ (f~)) + LI(0, T; W~J(f~)). 3 - Let (~,j) obey (5.7), and (~1,jl) obey (5.5). Then there exists c~ such that N N l>c~>0, ~>c~+l+__, ~1>c~+1+--. (5.8) 3 jl Thus LI(0, T; W~I'Jl(f~))+LI(O,T; W~'J(Ft)) is included in LI(0, T; Cl'~(~)). The estimate of p in LI(0, T; C1,~(~)) in function of ]IpTIIL~(~) may be deduced from the analysis of Step 1 and Step 2. II

641

P. A. Nguyen and J. -P. Raymond

5.2. D e r i v a t i v e o f t h e c o s t f u n c t i o n n a l First we give a convergence result. Proposition 5.1 - Let yx be the solution of (1.1) corresponding to u and ~ , and ~j be the solution of (1.1) corresponding to u and ~/, then (yx)x converges to ~j in L~(O, T; Lr(Ft)) for every (~, r) obeying

r~< c ~ ,

l_
N-1 N-3

.

(5.9)

Therefore, if we set a~ = f l ~, (~ + X(Yx - Y ) ) d x , the sequence ( a ~ ) ~ tends to a = ~'(~) in L~(O,T;Lk(a)) for every (k,k) satisfying (5.2). Proof. The function w~ = yx - ~j satisfies Ow +-Aw+a~w = 0 in Q, Ot

aw = u ( t ) ( 5 ~ - 5 ~ ) on E, OnA

w(0)-0

in ~t.

Due to Proposition 2.6, the sequence (w~)~ is bounded in Le(O,T;Lr(~)) for every (~, r) obeying (5.9). Since the sequence (u(5~ - 5 ~ ) ) ~ converges to zero for the weak-star topology of A//[(Q \ (@ x {T})), due to Lemma 2.3.1, the sequence (w~)~ (or at least a subsequence) converges to zero in LI(Q). With Lemma 4.1 in [13], we can prove that (w~)~ converges to zero in L~(0, T; Lr(~)) for every (~, r) obeying (5.9). m Proposition 5.2 - Let p be the solution of (5.1) corresponding to a and PT. Let (a~)~ be a sequence of nonnegative functions converging to a in Lk(O,T;Lk(~)), for all (k,k) satisfying (5.2). Let (p~)~ be a sequence of functions converging to PT for the weak topology of Le(~), for all e < g-1 (g-B)(e-1) 9Let p~ be the solution to

Op ~-Ap+a~p = O in Q Ot '

Op __0 on ~, OnA

p(T) = P~T in Ft. (5.10)

Then the sequence (p~) ~ converges to p in L1 (0, T; C 1(-~)). Proof. From Theorem 5.1, we know that the sequence (px)x is bounded in LI(O,T;CI'a(~)) for some 0 < a < 1. The identity mapping from C1,~(~) into C1(~) is compact. The sequence (dd-~t~)~ is bounded in the space LI(0,T; (WI'~(~)) ') for some ~ big enough. From a compactness result ([28], Corollary 4), we deduce that the identity mapping is compact from LI(O,T;CI,~(-~)) n WI'I(0, T; (Wl'~(gt)) ') into LI(O,T;CI(-~)). Therefore the sequence (px)x converges to some function p in LI(0, T; CX(~)). Next, we can easily verify that p is the solution to equation (5.1). m

Control localized on thin structures for semilinear parabolic...

642

L e m m a 5.1 - Let p be the solution to equation (5.1) corresponding to a, pT. Suppose that (a~)~ and (p~)~ satisfy the assumptions of Propositon 5.2. Let p~ be the solution to equation (5.10), then we have 1

T

lim~_~o -A ~o ( i ~ P~d~ - ~ P~d~) udt =

Vp.adCudt +

p(~)v(~)Ta'(~)v(~)ud~dt,

9'(9) is the unit tangent vector to ~ at ~ = g(vg) where T(r = T(g(v~)) = Ig'(O)l Proof. We can write 1 where

1I

J~ = N

p~(g~(e))(Igi(~)l- Ig'(e)l)d~,

J~ = f~ Vp~,(g(O)). (a(g(e)) + e(a, e))lg'(e)lde, J~ =

///ol [

]

Vp~(g(O) + x(g~(O) - g(e))) - Vp~(g(o)) dx x (G(g(e)) + e(A, 0))lg'(0)lde.

First, observe t h a t the sequence (Ig'~l-lg'l )~ converges to ( ~ ) T . G ' ( g ) . g' uniformly on I. Due to Proposition 5.2, the sequence (p~)~ converges to p in LI(O,T;CI(~)). Thus we have the following convergence in L~(0, T): lim J1~ = f p(~)T(~)TG'(~)T(~) )~---, 0 J-y

d~.

Secondly, lim J~ = [ Vp(g(~)) . G(g(~))ig'(O)idv~ -- f Vp. Gd~ . A----,O J~ J! Due to Theorem 5.1, the sequence (p~)~ is bounded in LI(0, T; C1'~(~)) for some 0 < a < 1. Therefore we have

j~0Tu(t)J3~dt


~T lu(t)[ jf/~l Jc(A)dx leA(O) - g(O)l"dOdt

<_ Ciiulin~(o,T)iiVpxlin,(o,T;CO.~(~)) sup ]g~(#) - g(#)]" OEI

~ 0 as A ~ O,

P. A. Nguyen and J. -P. Raymond

643

where

~(A) -- IVP~(g(O) -4- x(g~(O) - g(O))) - Vp~(g(tg))l 19(0) +

-

- g(

)l

The proof is complete, Theorem

m

5.2 - The derivative of the functional F is given by

lim),~o F('/)' ) A- F(7) = ~0 T u(t)

~

Vp. G d~dt + ~0T u(t)

pTT. G'.T d~dt,

where p is the solution to -

Op +Ap+ap=O in Q, oat p(T) = Olfl(T ) - ydl~

op = 0 on E, Ohm -- Yd) in ~,

(5.11)

is the solution of (1.1) corresponding to u and 7, a = ~'(~). Proof. We denote by y~ the solution of (1.1) corresponding to u and 7~. Set a~ = f01 ~'(~-4-O(y~,- ~))dO. Thanks to Lemma 5.1, the sequence (yx)~, converges to ~ in Le(O,T;Lr(~)) for every (~,r) obeying (5.9). Therefore, (a~)~ tends to a = ~'(~) in Lk(0, T; Lk(~t)) for every (k, k) satisfying (5.2). We set z~ = (y~ - ~)/A and we denote by z the solution to Oz O---~+Az+anz = 0 in Q,

Oz = u ( t ) ( 5 ~ - 5 ~ ) / A OnA

on E

From the convexity of the mapping y ~-, fa l Y - Yd] ~

0 In Ill(T) - YdI~

z(0) = 0 in ~t.

it follows t h a t

-- yd)z~(T)dx < - F(7~) A- F(7)

<_ 0 / ~ lye(T) - yd]o-2(y~(T) -- yd)zx(T)dx . Let p~ be the solution to (5.10) when p~(T) = O l y ~ ( T ) - y d l ~ and 15~ be the solution to the equation

Ot

4-AiS~+a~iS~=0

in Q,

~x(T) = OI~(T ) - ydl~

OnA = 0

on E,

-- ya) in ~t.

Using the Green formula (4.87) when h = 0 and b = 0, we obtain T1

dr

F(7~) - F ( 7 )

<_,

,

I u

p~dr

We conclude with Lemma 5.1, by passing to the limit when A -~ 0.

n

644

Control localized on thin structures for semilinear parabolic...

References

[1] [2] [3] [4] [5]

R. A. Adams, Sobolev spaces, Academic Press, New-York, 1975. H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Is. J. Math., Vol. 45 (1983), 225-254. S. Anita, Optimal impulse-control of population dynamics with diffusion, in "Differential equations and control theory, V. Barbu Ed., Longman, Harlow, 1991, 1-6. S. Anita, Optimal control of parameter distributed systems with impulses, App. Math. Optim., Vol. 29 (1994), 93-107. P. Baras, Compacit~ de l'op~rateur f H u solution d'une ~quation non lin~aire f c ( d u / d t ) + Au, C. R. Acad. Sc. Paris, S~rie A, Vol. 286

(197s), 1113-1116. [6] P. Baras, M. Pierre, Probl~mes paraboliques semilin~aires avec donn~es

[7] IS] [9]

[10] [11] [12] [13]

[14] [15] [16]

[17]

mesures, Applicable Anal., Vol. 18 (1984), 111-149. H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., Vol. 62 (1983), 73-97. H. Brezis, W. Strauss, Semilinear Elliptic Equation in L 1, J. Math. Soc. Japan, Vol. 25, 1973, 15-26. M. Berggren, R. Glowinski, J. L. Lions, A Computational Approach to Controllability Issues for Flow-Related Models (Part 1), International Journal of Computational Fluid Dynamics, Vol.7 (1996), 237-252. M. Berggren, R. Glowinski, J. L. Lions, A Computational Approach to Controllability Issues for Flow-Related Models (Part 2), International Journal of Computational Fluid Dynamics, Vol. 6 (1996), 253-247. R. Dautray, J.-L. Lions, Analyse Math~matique et Calcul Num~rique, Tome 8, Masson, Paris, 1987. E. J. Dean and P. Gubernatis, Pointwise Control of Burgers' Equation - A Numerical Approach, Computers and Mathematics with Applications, Vol. 22 (1991), No. 7, 93-100. J. Droniou, J.-P. Raymond, Optimal pointwise control of semilinear parabolic equations, Nonlinear An., T.M.A, Vol. 39 (2000), 135-156. R. Glowinski, J. L. Lions, Exact and approximate controllability for distributed parameter systems (Part 1), Acta Numerica (1994), 269378. R. Glowinski, J. L. Lions, Exact and approximate controllability for distributed parameter systems (Part 2), Acta Numerica (1995), 269378. H. Henrot, W. Horn, J. Sokolowski, Domain optimization problem for stationary heat equation, Appl. Math. and Comp. Sci., vol. 6, No. 2, 1-21. H. Henrot, J. Sokolowski, Shape Optimization Problem for Heat Equation, Rapport de recherche INRIA, Juin 1997.

P. A. Nguyen and J. -P. Raymond

[IS] [19] [20]

[21] [22]

[23] [24]

[25] [26] [27] [2s] [29] [30] [31]

645

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin/Heidelberg/New York, 1981. K. - H. Hoffmann, J. Sokolowski, Interface optimization problems for parabolic equations, Control and Cybernetics, vol. 23 (1994), No. 3. I. Lasiecka, R. Triggiani, Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory, Lecture Notes in Control and Information Sciences, 164, Springer-Verlag, Berlin, 1991. J. -L. Lions, Some Methods in the Mathematical Analysis of Systems and their Control, Science Press, Beijing, Gordon and Breach, 1981. J. -L. Lions, Pointwise control for distributed systems, in "Control and estimation in distributed parameters sytems", H. T. Banks Ed., SIAM, Philadelphia, 1992, 1-39. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin Heidelberg, 1983. J. P. Raymond, Nonlinear boundary control of semilinear parabolic equation, Disc. and Cont. Dyn. Syst., Vol. 3 (1997), 341-370. J. P. Raymond, H. Zidani, Pontryagin's principles for state-constrained control problems governed by parabolic equations with unbounded controls, SIAM J. Control Optim., Vol. 36 (1998), No. 6, 1853-1879. J. P. Raymond, H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations, Appl. Math. Optim., Vol. 39 (1999), 143-177. J. P. Raymond, H. Zidani, Time optimal problems with boundary controls, Diff. Int. Eq., Vol. 13 (2000), 1039-1072. J. Simon, Compact Sets in the Space LP(O,T;B), Ann. Mat. Pura Appl., 196 (1987), 65-96. J. Simon, Caract~risation d'un espace fonctionnel intervenant en contr61e optimal, Ann. Fac. Sciences Toulouse, Vol. 5 (1983), 149-169. H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators", North Holland, Amsterdam New York, Oxford, 1977. V. Vespri, Analytic Semigroups Generated in H -'''p by Elliptic Variational Operators and Applications to Linear Cauchy Problems, Semigroup theory and applications (Clemens et al., ed.), Marcel Dekker, New York, 1989, 419-431. P. A. Nguyen and J.-P. Raymond UMR CNRS MIP UFR MIG Universit~ Paul Sabatier 31062 Toulouse Cedex 4 France E-mail: [email protected], fr, E-mail:[email protected]

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Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 27 STABILITE DES ONDES DE CHOC DE VISCOSITE QUI P E U V E N T ETRE CARACTERISTIQUES

D. SERRE

R ~ s u m ~ . Tout choc satisfaisant la condition d'Oleinik dans un sens strict admet bien stir un profil de viscositY. On donne ici une d~monstration simple de la stabilit~ asymptotique dans L 1(JR) d'un tel profil, qui n'utilise que le principe de contraction, c'est-~-dire une forme pr~cis~e du principe du maximum. Les hypotheses sont naturelles et bien plus faibles que ce qui est connu 9la condition initiale prend ses valeurs entre ses deux limites ua, Ud en i c ~ , tandis que la perturbation est int~grable. En particulier, il n'y a pas d'hypoth~se de petitesse. De plus, le choc (ug, Ud) peut ~tre caract~ristique d'un cot~ ou m~me des deux cot~s. A cause de cette g~n~ralit~, on n'obtient pas d'estimation du tanx de convergence quand t tend vers +c~.

Abstract. Any choc satisfying the condition of Oleinik in a strict sense admits obviously a viscosity profile. We give here a simple proof of the L l ( I R ) - a s y m p t o t i c stability of such a profile, by making use only of the contraction principle, i.e., a precise form of the m a x i m u m principle. The hypotheses we make are natural and are much weaker than those known in the litterature. They asks to the initial condition to take its values between two limits ug, Ud at • while the perturbation has to be integrable. In particular, there is no "smalness" hypothesis. Moreover, the choc (ug, Ud) can be characteristic on one side or even on both sides. Due to this generality, we do not obtain an estimate of the rate of convergence as t --. cx~.

1. Introduction Consid~rons une loi de conservation scalaire

Ou

0

- ~ -~- --~xf ( U) -- 0,

(1)

648

Stabifitd des ondes de choc de v i s c o s i t d . . .

off f est une fonction de classe C 2. Les solutions admissibles de (1) sont classiquement les limites des solutions de l'6quation perturb6e Ou 0 02u O--t -~- -~x f ( u ) -- e Ox----~,

(2)

lorsque e --. 0+. Les solutions de (1) d6veloppent en g6n6ral des discontinuit6s au bout d'un temps fini, m~me lorsque la donn6e initiale est tr~s r6guli~re. L'arch6type d'une discontinuit6 est une solution constante de part et d'autre d'une droite :

u ( x , t ) = ~ Ug, x < st, ( Ztd, X ~ St. Le nombre s est la vitesse de la discontinuit6. Celle-ci n'est admissible que si les conditions suivantes sont remplies : f(Ud)

-- f ( U g )

- - S(Ud -- U g ) ,

(3)

(~d -- u~)f(~) > ( ~ - ~ ) f ( u d ) + (Ud -- ~)f(u~), V~ (~d -- ~ ) ( ~ -

U~) > O.

(4)

La premibre condition est dite de R a n k i n e - H u g o n i o t et exprime que u est une solution au sens des distributions pour (1). La seconde est la condition d'Oleinik et exprime une forme de stabilit6. Lorsqu'elle est satisfaJte avec une in6galit6 stricte, l'6quation diff6rentielle v' - f ( v ) - f(Ug) - s ( v - ug)

(5)

poss~de, h translation pros, une et une seule solution h~t~rocline allant de u avers Ud, appel~e profil de viscositd. A cette solution correspond une onde progressive pour l'~quation perturb~e 9

~'(~, t ) = ~ ( ~ - ~ t ) , E qui tend vers u lorsque e --, 0+. Une question importante est l'uniformit6 de cette convergence par rapport s des perturbations. Une question connexe est, en raison des changements d'6chelles possibles ( ( x , t , e ) ~ (ex, et, 1)), la stabilit6 asymptotique d'un profil de viscositY. En clair, ~tant donn~e une solution r de (5) et une perturbation c~ : IR ~-. IR, disons int~grable, la solution (bien d~finie et r~guli~re) du probl~me de Cauchy Ou 0 O-~ + ~ x f ( U )

=

02u Ox2,

~(~,0) = r

(6)

+ ~(~)

(7)

D. Serre

649

est-elle asymptotique ~ U(x, t) := r dans une norme convenable lorsque t --. +oc ? Si la r6ponse est positive pour la norme de L 1(IR), alors le d6calage x0 est enti~rement d6termin~ par la condition initiale h cause de la conservation de la masse. En effet, u et U 6tant deux solutions ayant les mSmes valeurs en x - =kc~, on a

/

(u(x, t) - U(x, t))dx

=

/~(~(~, o) - u(~, O))d~

=

/

(r

- r

- xo) + ~(~))a~

= ( ~ - ug)~o + f~ a(x)dx. Ainsi,

xo =

a(x)dx.

u d -- Ug

Le premier r~sultat de stabilit~ a ~t~ obtenu par II'in et Oleinik [1] lorsque i n f ~ f " > 0, ce qui s'applique ~ l'~quation de Burgers-Hopf. Comme dans ce cas toutes les discontinuit~s sont non caract6ristiques (on a f'(ud) < s < f'(Ug)), la convergence de u vers U est exponentiellement rapide par rapport au temps. Ceci est li~ ~ une propri6t6 du spectre de l'op6rateur lin~aris~

Lz :=

d2z d dx 2 + ~xx(f'(r

).

Par un changement de variable (x,t) H ( x - st, t), on peut toujours se ramener au cas off s est nul, c'est-~-dire que r est une solution stationnaire. On a alors Lr ~ -- 0 et r > 0, ce qui montre que le spectre de L e s t positif, strictement k l'exception de la valeur propre 0. En ce plaqant dans un espace de type L 2 ~ poids, on peut obtenir que le spectre de L soit discret. Enfin, en se restreignant au sous-espace invariant d~fini par

/

(z - r

-- f

~dx,

le spectre de L devient minor~ par un nombre strictement positif. Le second r~sultat important est celui de Osher et Ralston [2], dans lequel le flux f e s t quelconque. La m~thode employee est la bonne, qui utilise la propri~t~ de contraction dans L 1 du semi-groupe associ~ ~ la r~solution du probl~me de Cauchy pour (6). Mais, curieusement, trop de d~tails techniques en cachent la g6n~ralit~ et les auteurs n'obtiennent la convergence que pour un choc v~rifiant la condition de Lax f'(Ud)

< S < f'(ug),

(S)

650

Stabilitd des ondes de choc de viscositd ...

au lieu d'in~galit4s larges. Ici, une restriction importante a lieu, qui est peut~tre n4c4ssaire : la condition initiale r + c~ est ~ valeurs dans l'intervalle I d'extr4mit~s u 9 et Ud. Enfin, un article r4cent de Matsumura et Nishihara montre la stabilit~ du profil d'un choc caract4ristique (c'est-g-dire qui v4rifie f ' ( u d ) -- s o u f ' ( u g ) -- s). Cependant, la perturbation initiale est suppos4e petite. De plus, la m4thode est bas4e sur des estimations d'~nergie et la perturbation doit donc appartenir g u n espace L 2 g poids, qui n'a pas de sens physique. Nous pr4sentons donc ici un r4sultat plus complet, sans hypoth~se de petitesse sur c~, dans l'espace naturel LI(IR) et sans autre restriction que r + ~(x) E I pour tout x E IR. T h ~ o r ~ m e 1.1 - Soit r un profil de viscositd entre Ug et Ud. Soit a E L I(IR) une perturbation initiale telle que +

e I,

pour presque tout x E IR. Soit xo -- (Ud -- Ug) - 1 f i R (~(x)dx et soit enfin V(x,t) = r - st - xo). Alors lim /iR iu(x , t) - U(x, t)ldx - O.

t--*+c~

2. R a p p e l s ; r ~ d u c t i o n h un cas particulier La th~orie des op~rateurs monotones permet de construire un semigroupe (S(t))t>0 qui r~sout le probl~me de Cauchy pour l'~quation (6) lorsque la donn~e initiale u0 est dans L~(IR). Comme l'~quation (6) satisfait le principe du maximum, ce semi-groupe jouit des propri~t~s suivantes. S G 1 (r~gularit~) Pour tout a E L~176 et born~ sur IR pour tout t > 0.

S ( t ) a est ind~finiment diff~rentiable

S G 2 (principe du maximum) Si a < b presque-partout, alors S ( t ) a < S(t)b pour tout t > 0. S G 3 (conservation de la masse) Si a L 1(IR) et on a pour tout t > 0

/iR(

b E LI(IR), alors S ( t ) a -

S ( t ) a - S ( t ) b ) d x = / i R ( a - b)dx.

S(t)b E

D. Serre

651

S G 4 (contraction) Sous les m(~mes hypotheses qu'en (SG3), l'application

t ~/~t

I S ( t ) a - S(t)bldx

est d4croissante. I1 n'est pas n4cessaire ~ la compr4hension de ce rapport de d4montrer les assertions ci-dessus. Elles sont classiques. Le th4or~me se d4duit ais4ment du lemme suivant 9 L e m m e 2.1 - Soit r un profil de viseositg qui joint ug ~ ua. Soit a une perturbation, fonction mesurable sur IR, telle que r + a soit compris entre deux translatds de r (il existe ~/, ~ E ]It tels que r + "y) <_ r a(x) < r + fl) presque partout). Alors la solution u(t) = S ( t ) ( r a) du probl~me de Cauchy pour (6) satisfait lim f In(t) - r J~t

xo - st)ldx = O,

t-,+c~

o~ xo -- (Ud -- ug) -1 fir a(x)dx. En effet, si a E L 1 est tel que r + a prenne ses valeurs dans I, alors il existe, pour tout ~/> 0, une fonction an E L 1 et des nombres r~els/3,-y tels que +

<

+

<

+

Soit u ~ la solution du probl~me de Cauchy dont la condition initiale est r + an. Alors, pour tout t > 0, on a

~t lun(x, t) - u(x, t)ldx < d'apr~s (SG4). Appliquant le lemme 2.1 h u n lim [ t ---~A- o o

J~t

lun(x, t) - r

9

- st - x~)ldx = O.

De plus, I x ~ - xo[. lUd- Ugl < ~. Finalement, lim sup [ iu(x, t) - r t---* -t-oo J ~ ce qui prouve le th6or~me 1.1.

s t - xo)ldx < 2~1,

652

3. P r e u v e

Stabifitd des ondes de choc de viscositd... du lemme

2.1

On notera II" Ill l& norme de L I(IR). Tout d'abord, quitte k faire le changement de variables (t, x) H (t, x - s t ) , on peut supposer que le choc est stationnaire 9s - 0. Les fonctions (t, x) H r + 7) et (t, x) ~-, r + fl) sont donc des solutions stationnaires de (6). Utilisant le principe du m a x i m u m et l'hypoth~se concernant la perturbation (qui assure que a est borne), on a r + -y) < u(t, x) < r + fl). Notons v(t) = u ( t ) - r Alors v(t) est compris entre deux fonctions int~grables qui ne d~pendent pas du temps et reste donc dans un born~ de L 1(JR). De plus, la propri~t~ de contraction fournit l'in~galit~ Ilv(t, 9+ r ) - v(t)lll = Ilu(t,. + r) - u(t)l[1 ___ [[u(0,. + r) - u(0)l[1 [IC~('-~ r) -- o~][1 qui tend vers z6ro avec r. D'apr~s le th6or~me de compacit6 de Fr~chet-Kolmogorov, la famille (v(t))t>_o est donc relativement compacte dans LI(IR). L'ensemble w-limite A : " r oh Bs est l'adh6rence dans L~(]R) de {v(t); t > s}, est donc non vide puisque A - r est l'intersection d6croissante de compacts non vides. Cet ensemble est celui de toutes les valeurs d'adh6rence, pour la distance d(z, w) = Ilz - will, des sous-suites (u(t~))ne~ off t~ ~ +c~. L'ensemble w-limite est invariant par le semi-groupe S puisque si a E A, oh a = l i m n _ ~ u(tn), alors S(t)a = lim~-~oo u(t + t~). Pour la m~me raison, S(t) est surjectif sur A car on a aussi a = S(t)b oh b est une valeur d'adh6rence de la suite ( u ( t n - t))ne~. La propri6t6 de r~gularit~ [SG1] implique donc que A est inclus dans C ~176 Soit m a i n t e n a n t k E IR. La fonction t --, I l u ( t ) - r k)lll , d6croissante, a d m e t une limite not6e c(k) quand t --, c~. Si a E A, on en d6duit que lid - r k)lll = c(k). Cependant, S(t)a appartient encore h A. I1 s'ensuit que t ~-, I i S ( t ) a - r k)lI1 est constant. Notons provisoirement w(t) S(t)a et z(t) = S ( t ) a - r k). On a :

0 = ~-~[[z(t)l[1 --

ztsgn zdx.

Or zt + ( f ( w ) - f ( r k)))x = zx~, oh on a utilis~ l'~quation de profil pour r Multipliant ceci par sgn z, on en d~duit

Izlt + ((f(w) - f ( r

k)))sgn z)x = zxzsgn z,

ce qui donne apr~s integration sur IR"

d / dt

]zidx = /

zxxsgn z dx.

Finalement,

0 = / ~ zxxsgn z dx

D. Serre

et donc

I

653

'

0 = ./,~ ax~sgn a dx.

(9)

Cependant les estimations a priori faites lors de la construction du semigroupe S m o n t r e n t que wxx est int~grable sur ]R et donc aussi axx. On a donc, d'apr~s le th~or~me de convergence domin~e, 0 = ~-,01im/~a~xj~(a)dx Oh j e ( T ) - - V / e 2 -[- T 2. Int~grant par parties, il vient f

a~3~ 2.,, (a) ax.

O=lim[ ~---*0 Jla

Soit Y0 un point oh a s'annule et soit 5 > 0 tel que 51a~(yo)l < 1. Pour e > 0 assez petit, on a lal < e sur ] y o - 5e, yo + 5el puisque a est continument 1 j ( ~~) avec J(T) = (1 + T2) -3/2 Ainsi diff~rentiable. Or j~'(T) : -~

IR a,3e 2 .,, (a) dx > -1 [yo+~e J(1)a2dx, C. J yo--Se

dont le second m e m b r e tend vers 25J(1)ax(yo) 2 quand e tend vers z~ro. On en d~duit que ax(yo) = O. Finalement nous avons prouv~ que

'Ca e A, Vk e IR, Vx e ]It,

(a(x) = r

---> (a'(x) = r

(10)

Pour conclure, on note d ' a b o r d que a est compris entre r et r comme limite de telles fonctions, donc a prend ses valeurs strictement entre ug et Ud. Aussi la fonction x H k(x) - : x - r o a(x) est-elle bien d~finie et r~guli~re (rappelons que r est strictement monotone). P a r construction, a(x) = r ce qui donne en d~rivant a'(x) = r k'(x)). Vtilisant (10) avec k - k(x), il reste r

- k(x))k'(x) = 0

et donc M(x) - 0 puisque r ne s'annule pas. Finalement, k est une constante et a - r k). Cependant, les ~l~ments de A satisfont

/

(a - r - a)dx = O,

ce qui fixe, on l'a vu, la valeur de k : on a k - - y o . On a donc prouv~ que l'ensemble w-limite est r~duit t~ un seul ~l~ment (au moins un puisqu'il n'est pas vide et seulement celui-l~) : r yo). Puisque

654

S t a b i l i t d des o n d e s de c h o c d e viscositd . . .

la famille (v(t))t>_o est relativement compacte dans LI(IR) et comme elle n'a qu'une seule valeur d'adh~rence quand t --, +co, elle est convergente, c'est-~-dire que lim Ilu(t) - r Y0)lli -- 0. t--,~c~

Ceci ach~ve la preuve du lemme et donc celle du th~or~me.

Remerciements Je remercie Heinrich Freistiihler pour l'int~r~t qu'il a port~ ~ ce travail, les discussions fructueuses et pour avoir attir~ mon attention sur des travaux ant~rieurs. Je remercie aussi J.-L. Lions et ses collaborateurs pour leurs constants encouragements. References [1] A. M. II'in, O. A. Oleinik, Behaviour of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of time. AMS Translations, 42 (1964), 19-23. [2] S. Osher, J. Ralston, L 1 stability of travelling waves with application to convective porous media flow. Comm. Pure and Applied Maths., 35 (1982), 737-751. [3] A. Matsumura, K. Nishihara, Asymptotic stability of travelling waves for scalar viscous conservation laws with non-convex nonlinearity. Comm. Math. Physics, 165 (1994), 83-96.

Denis Serre Unit~ de Math~matiques Pures et Appliqu~es CNRS UMR 128 ENS Lyon 46, All~e d'Italie 69364 Lyon France E-mail: [email protected], fr