Patterns and Waves: Qualitative Analysis of Nonlinear Differential Equations

Patterns and Waves -Qualitative Analysis of Nonlinear Differential Equations-

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 18

Editors : J. L. LIONS, Paris G. PAPANICOLAOU, New York H. FUJITA, Tokyo H. B. KELLER, Pasadena

KIN0KUNIYA:COMPANY LTD.-TOKYO JAPAN

NORTH-HOLLAND-AMSTERDAM

NEW YORK

OXFORD TOKYO

PATTERNS AND WAVES QUALITATIVE ANALYSIS OF NONLINEAR DIFFERENTIAL EQUATIONS

Edited by TAKAAKI NISHIDA MASAYASU MIMURA HIROSHI FUJI1

KINOKUNIYA COMPANY LTD.-TOKYO JAPAN

NORTH-HOLLAND-AMSTERDAM

-

-

NEW YORK OXFORD TOKYO

0Kinokuniya Company LTD., 1986

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN: 0 444 70144 3 Publishers KINOKUNIYA COMPANY LTD. TOKYO JAPAN

ELSEVIER SCIENCE PUBLISHERS B. V. P. 0. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U.S. A. and Canada ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK. N. Y. 10017 Distributed in Japan by KINOKUNIYA COMPANY LTD. Distributed outside Japan by ELSEVIER SCIENCE PUBLISHERS B. V. (NORTH-HOLLAND)

PRINTED IN JAPAN By The International Academic Printing Co., Ltd.

In recognition of the stimulating inspiration by his works, this book is dedicated to Professor Masaya Yamaguti.

Preface

This volume is dedicated to Professor Masaya Yamaguti in celebration of his 60th birthday. It is a compilation of contributions from mathematicians who have without a doubt been under the great influence of Professor Yamaguti, and from those people who have directly or indirectly during their scientific career been inspired by Professor Yamaguti. The title “Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations-” was chosen to symbolically reflect the subjects and topics contained within. Also, the title does not necessarily imply that the contents are restricted to nonlinear differential equations. These papers, in fact, deal with nonlinear problems related to the following such fields: General analysis, fluid dynamics, mathematical biology and computer sciences, and their underlying mathematical structures including nonlinear waves and propagations, bifurcation phenomena, chaotic phenomena, fractals, and so on. This volume is arranged into two parts: Part I contains papers which survey the developments in the analysis of nonlinear phenomena in our country in the past decade. Part I1 consists of up-to-date original papers concerning qualitative theories and their applications. Professor Yamaguti has left his great footprints on many fields of mathematics as well as mathematical sciences. Among other things, his contributions include his early works on the behavior of solutions of ODE’s (Mem. Col. Sci. Univ. Kyoto, 28 (1953) 87-96), characterization of strong and weak hyperbolicity of the Cauchy problem for linear PDE’s (Mem. Col. Sci. Univ. Kyoto, 32 (1959) 121-151, 33 (1960) 1-23), works on finite difference schemes for hyperbolic systems and pseudo-difference schemes (Math. Comp. AMS, 21 (1967) 611-619, Proc. Inter. Conf. Func. Anal. Related Topics (1969) 161170), mathematical treatment of pattern formation in biology (Adv. Biophy. 15 (1982) 19-65) as well as his recent studies in chaos related to the discretization of ODE’s and a new treatment of fractal theory which has a deep insight into the relationship between chaos, fractals and singular functions such as the Takagi function and the Weierstrass function (Physica 3D (1981) 618626, Japan J. Appl. Math., 1 (1984) 183-199).

vii

It is widely recognized that as a n “organizing center” of nonlinear sciences in Japan, he has inspired and given enormous impact upon many people through active and intensive discussions as well as his informal way of communicating and expressing his attitude, beliefs and approach to mathematical sciences. In this sense, Professor Yamaguti’s achievements and influence go far beyond the conventional notion of mathematics. He is known to have many friends in physics, biology, economy and philosophy, and a series of publications are planned to be released in the near future in collaboration with these people. It should be emphasized that this present volume reflects only a “mathematical projection” of his broad activities. The editors wish to express their gratitude to Professor Peter D . Lax, who presented us with a message of greetings. Professor Lax is a long-time friend of Professor Yamaguti since their meeting during Professor Yamaguti’s visit to Courant Institute of Mathematical Sciences in 1965-66. His personal relationship with Professor Lax has long been one of the incentives to deepen Professor Yamaguti’s research in nonlinear PDE’s and numerical analysis. Finally the editors wish to thank the authors to this volume for their cooperation in its preparation and production. Takaaki NISHIDA Masayasu MIMURA Hiroshi FUJII Editors

Message of Greetings

The Soul of Mathematics In the middle of our century, a time of resurgence of the human spirit following the conclusion of the second World War, a fierce fight broke out for the soul of mathematics. The purists, championed by Bourbaki, claimed that so many branches are budding and blooming today that mathematics will develop autonomously, with no need for input from the real world; applied mathematicians thought otherwise; their view was represented most forcefully by v. Neumann. He concluded his article “The Mathematician”*, with these words: “I think that it is a relatively good approximation to truth-which is much too complicated to allow anything but approximations-that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to a n empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more if it is a second and third generation only indirectly inspired by ideas coming from “reality”, it is beset with very grave dangers. It becomes more and more purely aestheticizing more and more purely I’art pour I’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections or if the discipline is under the influence of men with a n exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical. In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I a m convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.”

*

Reprinted in J. R. Neurnan’s “The World of Mathematics”.

ix

V. Neumann had very definite ideas where many of these empirical ideas might be coming from. I n a talk at Montreal delivered in 1946**, when fast electronic computers were merely figments of his imagination, he said: “We could, of course, continue to mention still other examples to justify our contention that many branches of both pure and applied mathematics are in great need of computing instruments to break the present stalemate created by the failure of the purely analytical approach to non-linear probelms. Instead we conclude by remarking that really efficient high-speed computing devices may, in the field of nonlinear partial differential equations as well as in many other fields which are now difficult or entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress. In the specific case of fluid dynamics these hints have not been forthcoming for the last two generations from the pure intuition of mathematicians, although a great deal of first-class mathematical effort has been expended in attempts to break the deadlock in that field. To the extent to which such hints arose at all (and that was much less than one might d,esire), they originated in a type of physical experimentation which is really computing. We can now make computing so much more efficient,fast and flexible that it should be possible to use the new computers to supply the needed heuristic hints. This should ultimately lead to important analytical advances”. It was a great personal tragedy and a great loss for science, that v. Neumann did not live to see his prophecy fulfilled. But many scientists in many countries followed the road he marked out. One of the leaders in Japan is Masaya Yamaguti; this volume of articles, contributed by his former students and others influenced by him, are a n eloquent testimony to his scientific interests and style. On the occasion of his 60th birthday we express our admiration and affection for him, and transmit to him our best wishes for the future.

Peter D. Lax

** Contained in Vol. V of his Collected Works under the title “On the Principles of Large Scale Computing Machines”.

Contents

vi

Preface Message of Greetings The Soul of Mathematics Peter D. LAX

viii

Part I

Hyperbolic Partial Differential Equations On Weakly Hyperbolic Equations with Constant Multiplicities Sigeru MIZOHATA Strongly Hyperbolic Equations and their Applications Nobuhisa IWASAKI

1 11

Flow Problems Solutions of the Boltzmann Equation Seiji UKAI

37

Equations of Motion of Compressible Viscous Fluids Takaaki NISHIDA

97

Reaction-DiJFusionEquations Predation-Mediated Coexistence and Segregation Structures Masayasu MIMURA and Yukio KAN-ON

129

On the Structure of Multiple Existence of Stable Stationary Solutions in Systems of Reaction-Diffusion Equations Hiroshi FUJII,Yasumasa NISHIURA and Yuzo HOSONO

157

Chaos and Fractals Chaotic Phenomena and Fractal Objects in Numerical Analysis Shigehiro USHIKI

221

Fractals in Mathematics Masayoshi HATA

259

xi Numerical Analysis & Computations Parallel Computation Tatsuo NOGI

279

A Theoretical and Computational Study of Upwind-Type Finite Element Methods Masahisa TABATA

319

Part I1

Positive Solutions to Some Semilinear Elliptic Equations in L'(R") Masaharu ARAIand Akira NAKAOKA

357

On the Vlasov-Poisson Limit of the Vlasov-Maxwell Equation Kiyoshi ASANOand Seiji UKAI

369

A Discrete Model for Spatially Aggregating Phenomena Tsutomu IKEDA

385

A Note on the Blowing-up Problem of a Certain System of Nonlinear Parabolic Equations Nobutoshi ITAYA

419

L'Bquation de Kadomtsev-Petviashvili approchant les ondes longues d e surface de I'eau en Bcoulement trois-dimensionnel Tadayoshi KANO

431

Application of a n Iteration Scheme to the Analysis of Incompressible or Nearly Incompressible Media Fumio KIKUCHI

445

On a Local Existence Theorem for the Evolution Equation of Gaseous Stars Teru MAKINO

459

Fundamental Solution of the Linearized System for the Exterior Stationary Problem of Compressible Viscous Flow Akitaka MATSUMURA

481

Nonlocal Advection Effect on Bistable Reaction-Diffusion Equations Masayasu MIMURA, David TERMAN and Tohru TSUJIKAWA

507

On Small Data Scattering for Some Nonlinear Wave Equations Kiyoshi MOCHIZUKI and Takahiro MOTAI

543

xii

The Near-field Finite Difference Approximation for Wave Propagation Problems in Infinite Media Tomoyasu-Taguti NAKAGAWA

561

Energy Decay for Nonlinear Wave Equations with Degenerate Dissipative Terms Mitsuhiro NAKAO

583

On the Stochastic Integral Equation of Fredholm Type Shigeyoshi OGAWA An Approach to Constrained Equations and Strange Attractors Hiroe OKAand Hiroshi KOKUBO On the Existence of Progressive Waves in the Flow of Perfect Fluid axound a Circle Hisashi OKAMOTO and Mayumi S H ~ J I On Laminar Boundary Layers with Suction Taira SHIROTA Initial Value Problem for Kac’s Model of the Boltzmann Equation Yasushi SHIZUTA and Hideko NISHIYAMA The Initial Value Problem for the Equations of the Motion of Compressible Viscous Fluid with Some Slip Boundary Condition Atusi TANI

Index

597 607

631 645 663

675 685

This Page Intentionally Left Blank

Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equationspp. 1-10 (1986)

On Weakly Hyperbolic Equations with Constant Mu1tiplicities By Sigeru MIZOHATA Abstract. Let P(x, t , D,, D,) be a formally hyperbolic differential operator with constant multiplicity. This article gives a brief survey of the development of the research on the Cauchy problem for P. In the C"-case, the necessary and sufficient condition for the C"-wellposedness (Levi condition) is given in an explicit form by using the perfect factorization of P . Next, when the Levi condition is violated, the wellposedness in the spaces of Gevrey class is discussed. The best possible class of Gevrey is given also through the perfect factorization of P . Key words: Cauchy problem, hyperbolic equation, C" wellposed, Gevrey wellposed, constant multiplicity

S 1.

Introduction

We trace briefly the development of research on weekly hyperbolic equations, and next we summarize the results obtained hitherto from our viewpoint. Since a n article by N. Iwasaki appears in this issue, we restrict ourselves to equations of constant multiplicities. We are concerned with the equation of the form: P ( x , t ; D , D t ) = P m ( x ,t ; D ,D , ) + P , - , ( x , t ; D , D,)+

...

where P,(x, t ; E, r ) is a polynomial of homogeneous degree j in (5, r ) with and ( x , t )E R " x [ 0 , TI; D = -i(a/ax), D,=-i(a/at). coefficients in eZ,,, We assume always P,(x, t ; 0, T)=?', and PAX, t ; E, r ) =

ii (r-A,(x,

j=

1

t ; 5))"'

,

+m,=m, where , l j ( x , t ; E ) are real and distinct, and m, are constant. mi+ More precisely we assume l u x , t ; E)-J,(x, t ; 812 6 l E l ,

for all ( x , t ; E ) , where 6 ( > O ) is a constant. Received May 4, 1985.

S. MIZOHATA

2

In 1956 the work of A. Lax [9] appeared. Although it treats only the case n= 1 (one space dimension), this work elucidated the fundamental properties of this problem, and it has its own beauty. It presents two equivalent conditions. Condition (A).

Put d , = D t - I , ( x , t ) D z . P can be expressed in the form

...

a l ml a 2 m2

akmk+

2 a l ( m l - z ) +...

ak(llZk-')+

121

&f 1(x, I;D , Dt) 9

where M l are (arbitrary) differential operators satisfying order

(a,(ml-z)+ ... a k ( m b - z ) +1 M )<m-z.

(Let us note t h a t j + = m a x ( j , O).) The next equivalent condition, suggested by R. Courant, is more maniable. Let us consider the basis of the form almi-li

.. . a b m k - l a

---

where m,-Z,20, I,+ +Z,>l. Let S be the set of all differential operators which can be spanned, with coefficients in B ' functions, on the above basis. Then. Condition (C). P can be expressed as p=almt

.-.

a k m k + ~ ,

NES.

It is proved that (1) when P is of constant coefficient, (A) is necessary and sufficient for P-wellposedness, (2) (A) and (C) are equivalent, and (3) the Cauchy problem is C"-wellposed under the assumption (A) (or (C)). In 1957, A. P. Calderon-A. Zygmund [l], and in 1958 A. P. Calderon [2] appeared. The author began trying to retreat the works of I. G . Petrowsky [21], J. Leray [lo], L. Girding [6] from the view-point of singular integral operators. M. Yamaguti also showed keen interest in this subject, and we spent a good deal of time in enthusiastic discussions. Both M. Yamaguti and the author considered the elementary hyperbolic operator D , - I ( x , t ; D) b,(x, t ; D ) as a fundamental operator (see [14]). M. Yamaguti, however, aimed at a more delicate problem: the general extension of A. Lax's work. He succeeded with a publication in 1959 [22]. He replaces

in higher dimensional x , and extends the module S by all pseudodifferential operators of the form:

Weakly Hyperbolic Equation

3

where C n ( x ,t ; D)are pseudodifferential operators of the order 10. He showed that, under the assumption (C), the Cauchy problem for P is H"-wellposed. In 1968 S. Mizohata-Y. Ohya [18] appeared. This article takes a slightly different view-point from the previous one. It is a n extension of E. E. Levi [12], assuming the multiplicities at most double. It presents a necessary and sufficient condition for C"-wellposedness under the form of vanishing of the subprincipal symbol on the characteristic set of double multiplicity (see S 2). In 1971 H . Flaschke-G. Strang [ 5 ] appeared. It presents a necessary condition for C"-wellposedness in the case of general multiplicities under the form of (L') in S2, which they called the Lax-Levi condition. Finally J. Chazarain [30] showed that this condition is sufficient.

S 2.

C"-wellposedness in the Case of Constant Multiplicities We can summarize the above results.

We use perfect factorization. Let

P ( x , t ; D,D,)=Pm+P,,-l+

*..

,

where P , is the homogeneous part of degree j of P. Let k

P,(x, t ; E,

TI=

II (T--Aj(X,t ; El)"' ,

j=1

where Aj(x, t ; E) are real and distinct. We know that P can be factorized in the following way: P = P k * P k - ,- * - P , + R ,

-

where denotes the operator product and P , ( l < j < k ) is a pseudodifferential operator whose principal symbol is

(T-Ay(X, t ; 0 ) " j

.

To be precise,

P,=(D,-A,(x, t ; D))"'+al,,(x, t ; D ) ( D , - A ~ ( Xt ;, ~ ) ) " j - ' + +a,, ,(D,-A,(X, t ; D))"'-8+ * * fa,,,, ,

-

with order a,,,(x, t ; E)<s-l.

We remark that

.

(D,-A,(X, t ; D))k=(D,-A,(x, I ; D ) ) * ( D , - I , ( x ,t ; D)) * (D,-A,(X, t ; D)). k The operator R is a regularizing operator, which means the following:

R=

m

C r j ( x , t ; D)D,"-3 , j=1

S . MIZOHATA

4

where Y,(x, t ; E ) ( l < j < r n ) are null symbols. Then the Levi condition, namely the necessary and sufficient condition for C”-wellposedness, can be formulated as follows. (L)

order a,,,(x, t ; 0 1 0

for all s (l<J
(I-’)

as P tends to infinity, where $,(x, t ) is any phase function corresponding to 2,. Next, we consider the case where one of A,, say A,, is a double root of the characteristic equation. Denoting 1, by 1, put

P,(x,

(2.1)

I;E,

r)=(r--R)2Q,-Ax, ti E,

.

Let us show that (L) is equivalent to (2.2)

f’L-I(x7 t ; E , r)lr=l =P?,,-I(X,

i:

t ; E, ~)+(i/2)((32/ataT)+

(az/axjaEj))PmIr=i

,=1

=o. First, observe that, denoting P I

=(D,-1)* (D,--R) +a,(D,--) + b , ( x , t ; E ) ,

the condition (L) claims that bp(x, t ; E ) = O , for all (x, t ; E), by being the principal symbol i.e. the part of homogeneous degree 1 in E of b,. First, we rewrite P, in usual form:

we see

where 6: is the homogeneous part degree 1 of

6,. Now,

Weakly Hyperbolic Equation

5

In view of (2.3),

which shows our assertion because Qm-2+0. Remark that the condition (2.2) is the one which Levi proposed in [12] as a sufficient condition in the case n = l . The sufficiency of (L) is seen in the following way. For simplicity, we consider the case k = 2 : (2.4)

P=P,.P,+R.

Put

Observe D:U=CjUO+cj-,UI+

- * -

+c:u,-,+u, ,

(OIjlrn-1),

where order c:(x, t ; D ) < k , Uj=d$+dj-iD,U+

where order d;(x, t ; D ) < k .

(O<j
+d:Di-'u+Diu,

In view of these, if we denote U=YUo,

u1,

..*,% - I ) ,

(2.4) is equivalent to

1

0

(2.5)

D,U=

21

1

U+Bo(x, t ; D ) U + F

12

*

-

9

2%

.

--

where Bo(x, t ; D ) belongs to So, and F=,(O, 0, f). Now we use the following fundamental lemma.

Lemma. Let k be an arbitrary real number and consider the equation a,u=i;i(x, t ; D)u+bo(x, t ; D)u+f , where 2(x, t ; E ) is a real-valued and homogeneous degree 1 in

c, and bo belongs

S . MIZOHATA

6

to So. Then, f o r all uo(x)E Hk, and f(-,t ) G Co([O, T I ; Hk), there exists a unique Moreover, we have the folsolution u ( x , t ) E Co([O,TI; Hk) n C1([O, TI; €I--'). lowing estimate:

This lemma shows that (2.5) is solved by successive approximations, and moreover the uniqueness of the solutions is derived from the above inequality. In fact let us designate (2.5) by D,U=DU+B,U+F. We solve this assuming UI,=,=O. D,Uo= D U o + F , D,U,=DU,+B,U,-,

Let UoI,=,=O ,

9

V,l,=,=O,

(j2l).

We see easily that

IlU,(., t ) l l k l M ( C r ) ~ + c ' / ( j + l ) ! , ( O < t < T ) , j r l

.

The uniqueness is derived in elementary way. The necessary part of the condition is a little delicate. A detailed account is given in [17]. From the view point of principle, it amounts to considering the following simple case: Let (a,-U(x, t ; D ) - c ( x , t ; D))u=f , uI,=,,,=u0(x) , Oito
t € [to,T ]

where l ( x , t ; E ) belongs to Si,oand c ( x , t ; E ) to Sf,,, with O < p < 1, and suppose that c ( x , t ; E ) = c " ( x , t ; C)+E(x, t ; E )

9

where co is homogeneous of degree p, and order ?LO. Then Theorem 1.

Zf there exists ( x o ,t o ; t osuch ) thaf Re c o ( x o ,t o ;Co) > O ,

then the above Cauchy problem is never H"-wellposed.

S 3.

Gevrey Wellposedness in the Case of Constant Multiplicities

If the Levi condition is violated, the Cauchy problem in the C"-space is no longer wellposed. However, we know the following classical result. Let us consider the following Cauchy problem for heat equation.

7

Weakly Hyperbolic Equation

are C"-functions in t. We know that this equation is not C"where Qo and wellposed. However, if &(t) satisfy the following condition: there exist A and C such that

then when s 1 2 , there exists a unique solution u ( x , t ) in a neighborhood of

x=O, and when s<2, then u ( x , t ) exists globally, namely for x of R . This is easily seen by considering formal solutions. We say that $t belong to 7 ( $ ) (Gevrey class s) if (3.1) is fulfilled. More generally, we say that a function $(x) € C" belongs to 7(", with s ~ 1if, it satisfies (3A. 3C1), l$(@(X)l

lACj"'a!S,

a20

i

Hereafter, for simplicity we assume s > l . We assume, besides the assumptions made in § 1, all the coefficients a,,,(x, t ) of P belong to 7 2 ) . Then the Cauchy problem in 7(s) means:

Pu=f € C!([O, T I ; 72)) (Oijlrn-1)

a:ul,=,,=U,(X)€7:)

,

where t o € [ O , TI, and the solution u ( x , t ) € C y ( [ t oT, I ; 72)). However, if we assume a,,,(x, t ) € 72,)', then it is natural to seek a solution in 72,)'. Systematic treatment of the Cauchy problem in Gevrey class has been done by Y. Ohya 1191 and J . Leray-Y. Ohya [ l l ] . [19] proved its wellposedness for l<s
and [ l l ] showed the same result for s
where r is the maximum multiplicity of P, and rn-r+q order of

p-

ii a&,

,=I

( q 2 0 )is the minimum

t ; D, D,)7

where a j is defined as a n arbitrary strictly hyperbolic operator. starts with

This article

8

S. MIZOHATA r

PJx, t ; 6, r)=

JJ a;(,, j=1

t ; 5, r )

.

where a; are homogeneous strictly hyperbolic polynomials. S. Matsuura [ 131 proved that under the assumption of constant multiplicities, the hyperbolic homogeneous polynomial P,(x, r ; 5, T) can be factored by polynomials, p,=

I?

/= 1

q j b , t ; 5,

.rIdj

,

where q, q, is a strictly hyperbolic polynomial. J. C . De Paris [4] has shown a kind of factorization by differential polynomials and defined “bien dkomposable” operators which are now known to be equivalent to satisfying the Levi condition. H. Komatsu [S], analyzing the above works, showed a general factorization theorem, and defined the notion of irregularity through this factorization. Now we summarize a part of their results by using the factorization explained in S 2 . Since we assume the Gevrey property only in x , we don’t use perfect factorization. By taking only finite terms of formal symbols there, we can assume i) all pseudodifferential operators ai,,(x, t ; D ) are of Gevrey calss s in x , ii) in the expression

R = C rj(x,t ; 0 ) D T - j , 1

r j ( x , t ; D ) € S.$;

and they belong to rp) in ’ x. Then, instead of the Levi condition (L), we consider

We can show that when p j > O , p j (hence p) is invariant with respect to the order of factorization. Then Theorem 2. In order that P be r(s”-wellposed, it is necessary and sufficient that l/s>p.

When l / s= p , only the local solution exists. In other words, when l/s
This theorem is proved in V. Ya. Ivrii [7] in slightly different form, and sufficient part is given also by H . Komatsu [8]. A fairly detailed account of this theorem is given in [15], [16].

Weakly Hyperbolic Equation

9

References

171

191

1131 [141 I151 I161

[I91 I201 [21]

A. P. Calder6n and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 902-921. A. P. Calderbn, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16-36. J. Chazarain, Optrateurs hyperboliques a caractkristiques de multiplicitt constante, Ann. Inst. Fourier, 24 (1974), 173-202. J.-C. De Paris, Probleme de Cauchy analytique a donntes singulieres pour un optrateur difftrential bien dtcomposable, J. Math. Pures Appl., 51 (1972), 465488. H. Flaschka and G. Strang, The correctness of the Cauchy problems, Adv. in Math., 6 (1971), 347-379. L. Ghrding, Solution directe du probleme de Cauchy pour les tquations hyperboliques, Colloque sur les Cquations aux dtrivtes partielles du C.N.R.S., 1956, 71-90. V. Ya. Ivril, Conditions for correctness in Gevrey class of the Cauchy problem for weakly hyperbolic equations, Siberian J. Math., 17 (1976), 422-435 (English Translation). H. Komatsu, Linear hyperbolic equations with Gevrey coefficients, J. Math. Pures Appl., 59 (1980), 145-185. A. Lax, On the Cauchy problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math., 9 (1956), 135-169. J. Leray, Hyperbolic equations, (Princeton Lecture Note), 1954. J. Leray and Y. Ohya, Systitmes lintaires, hyperboliques nonstricts, Deuxieme Colloque sur I’analyse fonctionnelle a Liege, 1964. E. E. Levi, Caratteristiche multiple e problema di Cauchy, Ann. Math. Pura Appl., 16 (1909), 109-127. S. Matsuura, On non-strict hyperbolicity, Proc. Internat. Conf. on Funct. Anal. and Related Topics, 1969, Univ. Tokyo Press, Tokyo, 1970, 171-176. S. Mizohata, Systkmes hyperboliques, J. Math. SOC.Japan, 11 (1959), 205-233. -, On the Cauchy problem, (Lectures at the Wuhan Univ.), to appear in Science Press, Beijing. -, Sur I’indice de Gevrey, Skminaire de Vaillant, (Propagation des singularitts et opkrateurs difftrentiels), 106-120, Hermann, 1985. -, On the Cauchy problem for hyperbolic equations and related problems, to appear in Proc. of the Taniguchi Symposium on hyperbolic equations and related topics, Kinokuniya Publisher, Tokyo. S. Mizohata and Y. Ohya, Sur la condition de E. E. Levi concernant des tquations hyperboliques, Pub]. R.I.M.S., Kyoto Univ., 4 (1968), 511-526; Sur la condition d’hyperbolicitk pour les Cquations a caracttristiques multiples 11, Japan. J. Math., 40 (1971), 63-104. Y. Ohya, Le problkme de Cauchy pour les Cquations hyperboliques a caracttristique multiple, J. Math. SOC.Japan, 16 (1964), 268-286. -, Le problitme de Cauchy a caractkristiques multiples, Cours a I’Univ. Paris VI, 1981. I. G. Petrowsky, ‘iSber das Cauchysche Problem fur ein System von partieller

10

s. MIZOHATA

Differentialgleichungen, Mat. Sb., 44 (1937), 815-868. [22] M. Yamaguti, Le probltme de Cauchy et les opkrateurs d’integrale singuliere, Mem. Coll. Sci., Kyoto Univ. Ser. A Math., 32 (1959), 121-151. Department of Mathematics Faculty of Sciences Kyoto University Kyoto 606, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 11-36 (1986)

Strongly Hyperbolic Equations and their Applications By Nobuhisa

IWASAKI

Abstract. Recent results on the effectively hyperbolic equations are summarized. It will be found that the effective hyperbolicity, more general than the strict hyperbolicity, is an important sufficient condition for the strong hyperbolicity. Some examples are given with respect to the real MongeAmpere equation, the nonlinear wave equation and the compressible Euler equation. Key words: Cauchy problem, effectively hyperbolic equation, NashMoser implicit function theorem, Monge-Ampere equation, nonlinear wave equation, compressible Euler equation

S 0. Introduction Our recent study proved that any effectively hyperbolic equation was well posed with n o particularly restrictive condition. It means that we may replace assuming the effective hyperbolicity for having done the strict hyperbolicity as far as we are interested in the unique existence of local solutions for the Cauchy problem of hyperbolic equations. In this article we shall explain the background on the effectively hyperbolic equations and our results extended to the general nonlinear systems by means of the Nash-Moser implicit function theorem. We shall also show a respective simple example of applications to the hyperbolic Monge-Amp6re equation, one of the nonlinear wave equations, and the Euler equation of fluid dynamics. We are going to make a brief historical remark on the strong hyperbolicity in this section. Two types of partial differential equations, strictly hyperbolic and symmetric hyperbolic equations (uniformly and smoothly symmetrizable if unifying), are well known and basic when attempting to check the Cauchy problem to have a smooth local solution uniquely for any smooth datum (in other words, to be well posed) in view of applications. One of the reasons why they are important is that many basic equations in applications are included, while the conditions are stated in terms of the principal symbols of partial differential operators and are in a general situation of hyperbolic equations. In fact they Received May 4, 1985.

12

N. IWASAKI

are always well posed without any restriction of lower terms except for smoothness (said strongly hyperbolic). This strong hyperbolicity is useful because we know n o more than the principal symbol (terms with the highest order) and the subprinciplal symbol (roughly corresponding ones with the next order) for geometrical representations of general partial differential operators into handy invariances under the change of variables. These types of equations are historically credited to I. G . Petrowsky [36] and K. 0. Friedrichs [6] (also J. Leray [22] and L. Gsrding [S]), although the easier and clearer present understanding is the result of a clarification of the notion of pseudodifferential operators (S. Mizohata [32], M. Yamaguti [38], K . 0. Friedrichs [7] and the recent books for partial differential equations). After L. Ggrding [9] completed the theory for equations with constant coefficients, they were characterized by K . Kasahara and M. Yamaguti [19] as uniformly symmetrizable equations equivalent to strongly hyperbolic equations. But this was for equations with constant coefficients, and there was a counter example in general. On the other hand, P. D. Lax [21] and S. Mizohata [31] showed that the principal symbols had to be hyperbolic for the Cauchy problem to be well posed also in the case of equations with variable coefficients. This was the first success of the theoretical approach for the characterization of general well posed equations. We note that S. Mizohata’s lecture [30] will offer us a vivid story of this period. The next fruitful step was begun just before 1970 by S. Mizohata and Y. Ohya [28, 291. They obtained a necessary and sufficient condition for lower terms (called the Levi condition in connection with E. E. Levi’s work [23]) when a single equation had the characteristics with double constant multiplicity, that is, the condition that the subprincipal symbol vanishes identically on the double characteristic points (at the points with double multiplicity). This relation was extended in succession by H. Flaschka and G . Strang [5] (necessity) and J. Chazarain [2] (sufficiency) for equations having the characteristics with higher constant multiplicities (multi-products of distinct strictly hyperbolic operators, S. Matsuura [26]). S. Mizohata’s exposition of this issue provides a thorough understanding of these cases. It reminds us that the part of sufficiency is a reaffirmation of M. Yamaguti [38] (a generalization of A. Lax [20]). 0. A. Olejnik [34] in parallel with them gave a sufficient condition for a single equation of the second order to be well posed. The condition was general in the sense that it did not require the smoothness of characteristics and that it was essentially valid for a kind of equations never appearing in ones with constant coefficients. Especially, it picked out a class of strongly hyperbolic equations which was neither strictly nor symmetric hyperbolic. This is what we now call effectively hyperbolic. And V. Ya. Ivrii and V. M. Petkov [13] claimed that it was intrinsic for the strong hyperbolicity of single equations by studying the necessity of the Levi condition for general single equations with n o restriction to characteristics. Their results were refined by L. Hormander [12]

Strongly Hyperbolic Equations

13

to come to give a necessary condition on double characteristic points following the results by P. D. Lax and S. Mizohata. A corollary is deduced from this result. A single equation should be effectively hyperbolic if it is strongly hyperbolic. Our recent result claims that the converse is also true. Therefore the effective hyperbolicity is equivalent to the strong hyperbolicity for single equations. A decade after these studies, many scholars are trying to extend these results, to refine them or to discover a new condition with respect to a n equation of the higher order, a system of equations or one with more singlular characteristics. We shall comment on a few of them in the latter half of the next section, where we shall extract the main part of results by V . Ya. Ivrii, V. M. Petkov and L. Hormander.

S 1. Fundamental Matters Let p be a single partial differential operator of the order m on an open set 9 of R n t l . The non characteristic Cauchy problem is to find a solution of the equation p u = f on 9 satisfying the initial data on a given hypersurface of the derivatives up to m-1 of u to the conormal direction of the surface. So it is also called the initial value problem. It belongs to our consentaneous comprehension after J. Hadamard [lo] that, when the Cauchy problem is called solvable, what it means consists of two parts of the local existence of solutions and the local uniqueness. There are some variants exactly to realize it. We here use one of somewhat strong definitions. Definition 1.1. The Cauchy problem for p is said to be well posed at a point x - with respect to a non characteristic direction O f 0 if there exist a neighborhood D of x - and a n infinitely differentiable function $ satisfying $(x-)=O and d@(x-)=O such that the following statements (E), and (U), hold for any small t .

(E), For every f belonging to C,"(9) there is a distribution u belonging to W ( Q )and satisfying the equation p u = f on 9,, where 9, is the set of x in 9 such that $ ( x ) < t . (U), If u belonging to W ( 9 )satisfies pu=O on Q,, then u vanishes identically on D,. By P. D. Lax and S. Mizohata, we have known that the principal part p m of p is hyperbolic if the Cauchy problem for p is well posed. (It is proved under weaker conditions of wellposedness.) Theorem 1.1 (P. D. Lax and S. Mizohata). If the Cauchy problem for p is well posed at x - with respect to the direction Of0 such that p m ( x - , O)#0, then Pm@, ClO+E)=O

N. IWASAKI

14

has only real roots in 5 f o r any real vector E and f o r any x of a neighborhood of x - , that is, p m is hyperbolic.

Remark. We also assume in the rest that the principal symbol p m is real valued when we call it hyperbolic. More precise information to be well posed may be described by relations at characteristics (a set {pm=O})between the principal symbol and the lower terms. At simple characteristics (p,=O and Vpm#O), we have and need n o other information about such relations. In fact if characteristics are simple at all points, p m has to be strictly hyperbolic (see after Definition 1.2), and p is always well posed. At multiple characteristics (pm=Vpm=O: critical points) the fundamental matrix 3 gives a relation. 3- is defined by the Hessian of p m at critical points. Let Q be a quadratic form defined by the Hessian H , of pm. Then

~ ( xr ,W = Q(W , where X = ( x , E ) and u is the canonical two form

s=(-:%m;

C dE,r\dx,.

-a,aep, acqpln)

More directly,

.

The hyperbolicity yields the properties of the fundamental matrix 9that the eigenvalues of F a r e only on the purely real and purely imaginary axes. The non zero real eigenvalues, if they exist, are only one by one on the positive part and the negative part of the real axis, respectively. If we define the positive trace Tr-d-of .3 by Tr-57=2-'

C IIm R,I ,

where 2, are all eigenvalues of .F,then we are able to conclude the following.

Remark. We denote the subprincipal symbol by p m - l . If p k - l stands for the usual symbol of the order m - 1 by means of the usual notation, then Pm-1=PLL-1-(2i)-1

i:(a/axc)(a/att)pm.

1=0

Theorem 1.2 (V. Ya. Ivrii, V. M. Petkov and L. Hormander). cipal part p m - , should be real and bounded as

The subprin-

-2-l T r - ~ - < p ~ - ~ < 2T- r~- 9 by the positive trace of the fundamental matrix 9on the singular points of the characteristic set where the fundamental matrix had no non zero real eigenvalue

Strongly Hyperbolic Equations

15

if the Cauchy problem for p were well posed. We may also call the inequality of the above theorem the Levi condition. In case of double constant multiplicity, it means that the subprincipal symbol should vanish a t characteristics. (Refer to S. Mizohata’s exposition of this issue.) Now we introduce two notations. Definition 1.2. 1) Let p , be hyperbolic with respect to the direction 8#0. p , is said to be effectively hyperbolic (with respect to the direction 8+0) if the fundamental matrix at any critical point of p , has non zero real eigenvalues. 2 ) p , is said to be strongly hyperbolic (with respect to the direction 8+0) if the Cauchy problem for p with any arbitrary lower order term is well posed (with respect to the direction 8 f O ) .

Remark. Naturally, p m is effectively hyperbolic if it is strictly hyperbolic, that is, if the characteristic polynomial p,(rO+e) in r has only real distinct roots for any real 5 independent of 8.

Corollary 1.3 ( V . Ya. Ivrii and V . M. Petkov). If p , is strongly hyperbolic, then p , is eflectively hyperbolic. Examples. 1)

-Eoz+xo2~E12+E22)+a2x12~E12+E22)+b2E~2 on

R3

is a n effectively hyperbolic operator. (Roughly speaking, to be effectively hyperbolic means generally to include the term -EO2+xo2jEl2.) 2)

on

-Eo2+a2x,2(E,2+f22)+b2E12

R3

and -Eo2+2EoE1+~12(E~2+E32~+a2x22(Ez2+E32)+b2E22 on

R4

need the Levi conditions bounded by 2-l T r - F = lab1 at Eo=bE1=ax,=O and at Eo=E,= bez = x l =ax,=O, respectively.

3)

- E o Z + ~ x o E 2 + E 1 ~ Z + ~ ~ , 2 E 2on 2 R3

is effectively hyperbolic if O < v < l and of the same types as 2) if ~ I Y . 4) Let a(xo,x’, E)=Cn,,=la,,(xo, x’)EiEl be a symbol of elliptic operators in x of the second order and f be infinitely differentiable. -EO2++(xo, x’)a(xo,x’, E )

on R”+‘

is effectively hyperbolic if f 2 0 and if aZfst.0 at the points where f vanishes. 5 ) Let us put z=Vf/lVfl ( V f + O ) . -e02-ZoEo
E>+

1EI2-
o2

16

N. IWASAKI

is strictly hyperbolic if a, f f O and effectively hyperbolic if linearly independent at the points where dof =O.

Vf and V a ,f are

The Olejnik's result is applicable to 1) of the above example, and the usual method supported by the GArding-Melin's inequality is to 2) with lower terms satisfying the pure inequality at the Levi condition, if we apply the general results to these examples in order to show their wellposedness. The several were improving the Olejnik's result or the other cases to extend them to the equations of the higher order and the systems of them. (For examples, R. Berzin and J. Vaillant, Y. Demay, D. Gourdin, K. Igari, V. Ya. Ivrii, K. Kitagawa and T. Sadamatsu, A . Menikoff, T. Nishitani, Y. Ohya, R. Sakamoto, S . Tarama, H. Uryu, and M. Zeman, see the references of N. Iwasaki [17].) When the characteristics consist of a union of smooth surfaces, it is natural that we are able to know sharper results than at the general standpoint. From this point of view, some of them were arranged to the t-involutive cases by K. Yamamoto [40]when the characteristics consisted of smooth characteristic roots (as a matter of fact, all such types were not covered yet), and the typified cases were verified in view of necessity by T. Mandai [24]. However, the situation for the system is very complicated. V. M. Petkov [35], H. Yamahara [39] and K. Kajitani [18] tried to extend the case of constant multiplicity for single equations, especially the results by H. Flaschka and G . Strang, to the case for systems under some rank condition, but there are yet some gaps between the sufficiency and the necessity. Moreover, without such rank condition, W. Matsumoto 1251 noted, the peculiar difficulty related to infinitely differentiable functions as well as a non well posed example for a single equation by F. Colombini and S. Spagnolo [3], already occurred. It is the same even with the strong hyperbolicity. We note this by examples. Let us consider a 2 x 2 system of the first order on R2,p = p l + p o , where the order of p o is 0 and

We assume it is strongly hyperbolic. For simplicity, we further suppose the wellposedness is the stronger one, that all solutions for infinitely differentiable data are also infinitely differentiable, and that it is satisfied to the negative direction in xo. Then the formal cofactor operator p l C ois also strongly hyperbolic so that p c o p is well posed, where p c o = p l e o + p o - . If the offdiagonal parts of p1 do not vanish, then we have the freedom to choose the lower terms p o and p o l as one of the offdiagonal parts of pcop identically vanishes. In this case the wellposedness of peop is equivalent to one of the diagonal parts of p"p. Since the lower terms of the diagonal parts remain sufficient freedom, the principal symbol of the diagonal parts, which is the determinant of the symbol of pI, has to be effectively hyperbolic. Therefore the symbol of p 1 is sym-

Strongly Hyperbolic Equations

17

metric or effectively hyperbolic if it is strongly hyperbolic (under the stronger wellposedness). However it does not mean that the symbol of p 1 is symmetric on a n open set. In fact there exists a series of similar types of operators between the effectively hyperbolic one and the symmetric (uniformly symmetrizable) one. For example,

are strongly hyperbolic. We are going to explain the strong hyperbolicity of the effectively hyperbolic equations in the next section, but the following one of the second order is not included in them though it is strongly hyperbolic at the neighborhood of the origin. - xO2A ,

$j2.

The Effectively Hyperbolic Equation

Here we consider an Z X Z system of linear partial differential operators of the higher order with a single diagonal principal symbol. The principal symbol is of the order m 2 2 and is denoted by pm. The lower term Q= (q&,,=l,...,Lhas the following type. There exists a set of integer indices a, (j=l,-..,I) such that order q t , < a t - a j + m - l

.

The system we consider is p=p,Z+

Q

These systems are considered on a suitable open set SZ of Rn+'with the variable (xo, x)=(xo, xl, xn). It may be assumed that SZ is a neighborhood of the origin and the initial hypersurface is {xo=O}. The Cauchy problem is to solve the equation that

-..,

PK=f, do~ul,o=o=q5j

(j=O,. - - m-1) ,

,

where q5, and f are given functions and u is unknown. Almost all of the Cauchy problem for hyperbolic equations, which has been known up to the present to be well pos'ed on the space of infinitely differentiable functions, have stronger properties than Definition 1.1. One of them is that the solution has the regularity in proportion to one of the data and the coefficients of equation. The other is the finite propagation speed (the existence of a sharp region for the

N. IWASAKI

18

uniqueness). The case in this section also has these properties, while its special feature is that it may happen that the grade of loss in the regularity grows in proportion to the size of the subprincipal symbol. This feature is different from that found in strictly hyperbolic equations and symmetric hyperbolic equations, and has made difficult the obtaining of wellposedness.

Theorem 2.1. Let P be a system of partial differential operators of the order m as (2.1). If the principal symbol p m is effectively hyperbolic with respect to d x , at a neighborhood of the origin, then P is well posed at the origin with respect to d x , for any smooth lower term Q . Moreover there exists a lens neighborhood Q of the origin contained in a neighborhood given beforehand such that f o r any infinitely differentiable function u on Q supported on x o 2 Q ,it holds the estimates

I l U l l s C,(IIPull*+L+ ~ Ilalls+LllP~IIJ

(2.2)

for any real positive s and f o r a fixed I , where a is coefficients of P and where 11. ] I l stand for the Sobolev norms on the lens domain

(2.3)

Q={(x,, x ) :

--E<X,<&-~X12}

.

Q, the constants C, and I are uniform as f a r as the principal symbol p m being hyperbolic, and coefficients a of P belong to a small neighborhood of a fixed effectively hyperbolic one and a bounded set in the space of infinitely differentiable functions, respectively. A Simple Example.

On R3,

Q-(c:i: : En+[, )+lower ‘2

terms

.

Then, P= QaQ(QQco,resp.)= (~02-[12-xo2~22)I+lower terms

.

It should be noted that many systems which were formally more general, could be reduced to the system (2.1), for example, by taking the product with the formal dual operator. One of them is as follows. Let us consider the 1x 1 system

(2.4)

P=(p,,)

9

where there exist two sets of indices of real numbers { a J }and { P I } such that C:=,(aj+,8,)=m and the order of p L jdoes not exceed aa+Pr. We denote by p p r , l jthe principal symbol of pzi just with the order az+/3, and call the system of symbols Ppr={ppr,tl} the principal symbol of P . It is clear that detP,, is independent of the choice (ai, P I ) for fixed m while P,, depends on.

Strongly Hyperbolic Equations

19

Theorem 2.2. Let the principal part of det P,, be efectively hyperbolic of the order lm with respect to dx, at the origin. Then it holds for a neighborhood Q of the origin that there exists the unique infinitely differentiable extension u onto Q of uo for any infinitely differentiable u, on Q n {x,
S 3.

The Extension to Nonlinear Systems

It may be better directly to clarify each nonlinear equation in order to get fine results even if the method of proof is as similar as one for linearized cases. However it is not a standpoint of the general theory. In fact many efforts have been made to normalize the method of adapting the results of linear cases to ones of nonlinear cases, and they come to bearing the NashMoser implicit function theorem. This naturally includes some techniques obtained in the clarification of the Cauchy problem. We here translate the Nash-Moser implicit function theorem to a theorem by means of words of the Cauchy problem in the infinitely differentiable class for partial differential equations. This makes it possible not only to extend the results in the previous section but also to apply many known results which we do not mention in this article. Moreover, a new result for linear cases can be extended to a theorem for nonlinear cases as soon as it is established. Let us consider an 1x1 system similar to (2.4) (3.1) where u= ( u l ,

-

*

-,u,)

is a real vector valued unknown function and

P,=P,(x, a"u,(x)l ia.lGr,t,9,) ( = p 2 ( x ,a w ) is a nonlinear partial differential operator with the order a,+/3, at most in u,. That is, p t ( x ,?,,J is a function in ( x , ~ 3 , a l l n , ~ a z t p j : j=1, -,I ) and p L ( x ,P u )

.-

N. IWASAKI

20

is a function in which aauj(x) is substituted for of P in u is denoted by DP, that is,

?,,a.

The Fr6chet derivative

(3.2) P,, stands for the principal symbol of D P (and also called the one of P), following the definition after (2.4). We require a stronger definition for the solvability of the Cauchy problem than Definition 1.1 at Section 1. We should note that it was satisfied by the conclusion of Theorem 2.2 and also by almost all already known cases. Let Q, be domains such that

(3.3)

a,,, ={x:

-EExo-yoIE-

Ix’-y’(2}

.

Definition 3.1 (Strongly well posed). Let 9 be a set of linear partial differential operators P with the type (3.1), that is, Pu=DPu. 9(and also its element P) is said to be strongly well posed at the origin with respect to the direction dx, when there exists a neighborhood U of the origin and a positive E , such that any P of 9satisfies uniformly the following l), 2) and 3). 1) P is well posed on U with respect to the direction dx, in the sense of Definition 1.1. 2) For anyfbelonging to Cm(Qncr,) and vanishing at { x o i y o } there , exists a unique solution u of Pu=f belonging to C-(Q(,,) and vanishing at {xo
(3.4) holds for the above solution u and for any s 2 0 , where ll-llS is the Sobolev norm or the Holder norm of a,,, and “a” stands for coefficients of P. It is a different, essential and important point from the previous definition that the estimate (3.4) holds, namely, the gaps r of regularities between data (including coefficients) and solutions are always constant. We prepare next a word for a nonlinear operator to simplify the description of theorems. The word will mean that the linearizations almost satisfy the above definition a n d exact at a solution of the nonlinear equation. The Nash-Moser theorem requires usually the existence of (right and/or left) inverses for linearizations in appearance. However, it needs actually only the existence of parametrices with suitable properties if obtaining a solution for a specified datum is required. Definition 3.2 (Strongly well posed for nonlinear systems). We call a nonlinear operator P strongly well posed with respect to the direction dx, a t (x-, u-, h-) if there exist linear partial differential operators Q and R with the type (2.4) or (3.1-2), which are a parametrix and its remainder of the lineariza-

Strongly Hyperbolic Equarions

21

tion DP in the sense that the following 1)-5) hold for all u belonging to a neighborhood U of u-. 1) The coefficients of Q and R are functions in ( x , v8)and in ( x , vp, Cp), respectively, where substituting for vp a finite number of unknown functions u and their derivatives apu, and where doing for Ca a finite number of parameter functions h and their derivatives Ph. 2) The order indices (a‘, P j ) for Q and R are taken independently of the variance of unknown functions u and parameter functions h. 3) D P = Q + R with h=Pu-h-. 4) @={Q: for all u belonging to U } is strongly well posed at x - with respect to the direction dx, in the sense of Definition 3.1. 5) R r O at h-0.

Remark. 1) The essential points are 3)-5). 2) We call P strongly well posed at a set K of (u-, h-) if there exists a common U such that 4) of Definition 3.2 holds for the union of @ with respect to elements of K . It is naturally assumed that all Q and R have a common bound of their derivatives of coefficients and other. 3) We also call P XXX (some other name), if Q satisfies at 4) the condition named XXX, instead of “strongly well posed”. For example, P is effectively hyperbolic if Q is so. Theorem 3.1 (Unique extension). Let a nonlinear system P be strongly well posed with respect to dx, at ( x - , u-, h-). I f there exists a neighborhood Q, of x such that u- is a solution of Pu-=h- at {X,<X,-} n Q, then there exists a unique solution u of Pu=h- on a neighborhood 9 of x - such that u=u- on {x,<x,-} n Q. (It is assumed that u-, h- and also u are suitably smooth. The uniqueness holds under such smoothness. Zf all data are infinitely diferentiable, then the solution u is so.) The existence theorem in the usual sense for the Cauchy problem is deduced by assuming a stronger condition.

Corollary 3.2 (Unique existence). Let P be strongly well posed with respect to dx, at ( x - , u , Pu) for all u in a neighborhood of u-. If u- is a formal solution of Pu-=O at x,=x,-, then there exists a unique solution u of Pu=O at a neighborhood of x - such that u-u- is flat at xo=xo-. In fact, we define a function f as f = O at x , 2 0 and f=Pu- a t x,
Remark. The assumption of Corollary 3.2 is satisfied if {DP} is strongly well posed. Put Q=DP and R=O. Corollary 3.3. Theorem 3.1 and Corollary 3.2 hold if P is efectively hyperbolic, symmetric hyperbolic or a hyperbolic system satisfying a suitable Levi condi-

N. IWASAKI

22

tion at a proper position attached to Theorem 3.1 and Corollary 3.2, respectively.

S 4.

Applications

We try to apply the previous results to some concrete examples. We shall start from the Monge-Ampere equation as well as R. Courant and D. Hilbert, Appendix 1 to Chapter V [4].

S 4.1.

The real Monge-AmpZre equation This is one of the typical nonlinear equations closely related to geometry; It appears for examples in the local construction of a piece of hypersurface with a given Gauss curvature, the local embedding on R3 of a 2-dimensional manifold with given a smooth Riemannian metric, and also a similar problem of locally connectable isometric surfaces by a bending (the problem of E. E. Levi). (Refer to M. Spivak [37] vol. 5 and S. T. Yau [41] problem section no. 54.) The n + l dimensional Monge-AmpBre equation is as follows because we consider local problems.

(4.1.1)

@(u)=det( P u + C ( x , u, B u ) ) - f ( x , u, Bu)

=o, where u is a n unknown real valued function, v u = ( ( m x , ) u ) , = , , . ., n 9 P u = ((a?/ax,ax,)u) j , *=, ,..., , C ( x , u, w ) is a n (n+ 1)x ( n + 1) real symmetric matrix valued infinitely differentiable function on Rntl x R x R n t l , and f ( x , v, w ) is a real valued infinitely differentiable function on RZntS. For example, the Gauss curvature of a hypersurface of Rn+' denoted by K satisfies the equation

det ( P u )=K(x)( 1

+ I Bul*) +'"' (n

if a hypersurface of Rntl is given by a function y = u ( x ) on R". Let us put it as 4 u ) = ( a i j ) t , j = o. , , = V 2 u f C , and denote minor matrices of n x n with respect to a,, and the cofactor matrix by A,j=(akl)k,lt*.

and

j

Strongly Hyperbolic Equations

23

So the equation is written as di=det A - f . The FrCchet derivative DO$ of di in u has a form DO$ = T r (AcoV2q5) +lower terms =A""(a)$+ * * . , where a second order partial differential operator defined by a quadratic form H is denoted by H(a)$=Tr ( H e r 2 $ ) . We assume that the equation (4.1.1)is kowalevskian so that the cofactor det Aoo=aooshould not vanish, because we investigate the non characteristic Cauchy problem with the initial surface {xo=O}. The initial condition

(4.1.2)

(u, (a/~~o)u)l,o=o=(go, Sl)

has also to be restricted, bacause we treat the hyperbolic cases. We assume that the matrix Aoo(u)l,O=o, which is determined by the initial datum (go,g J , is strictly positive,

(4.1.3)

Aoo(u)l,o=o~O ,

and that the function f is non positive,

fl0.

(4.1.4)

It is easy to check that these conditions limit the type of the linearization if a solution of the equation exists. Proposition 4.1. The Frkchet derivative Ddi(u)$ of @(u) in u is a linear hyperbolic partial diferential operator of the order 2 in $ with respect to the direction d x , at a neighborhood of the origin under the condition (4.1.4),if u is a solution of (4.1.1)and (4.1.2)at a neighborhood of the origin and if the initial datum ( g o ,g l ) satisfies the condition (4.1.3). Moreover, iff is negative, then Ddi is strictly hyperbolic. And also if

(4.1.5)

(-w"f (x,u, Vu)l
at the origin, when f ( x , u, Vu) vanishes there, then D@(u)is effectively hyperbolic. where 9is a vector field defined by .P=

2 a;?(a/ax,)

>=O

N. IWASAKI

24

The purpose is to deduce the existence of a solution from the conditions (4.1.2-5). The linearizations are not always hyperbolic except for the case where f is negative, though these conditions are stable under a small change of unkown functions. The linearizations are however effectively hyperbolic in the sense of Definition 3.2 and Remark. So we get a solution of (4.1.1-2) through the unique extension theorem and a lemma backing it up. In fact, we can show that the linearization DO has the following form. The condition (4.1.3) implies that O is kowalevskian because @=(det A,,)u,,+the

terms not including uoo.

Therefore we get a formal solution u- of @ = O a t xo=O satisfying (4.1.2), that is, aa@(u-)=O at xo=O for all a. We consider the linearizations near this formal solution u-. Let us denote the eigenvalues and the projections to their eigenspaces of A""by 0, and p , ( j = O , -,n), respectively. They are connected with the eigenvalues A, of A as

--

and the eigenspaces of B j and A, are the same p,. (4.1.3) implies the existence A,= of n positive eigenvalues A j > O ( j = l , -,n) of A . Since det A = l , f l 0 at u=u- and x,=O, so A,
.-

where E =

C,"=,A,-'pj.

- +

The principal parts of linearizations are written as

where and

Since E vanishes at the eigenvector vo of Oo(Ao) and is positive except for it, namely, since E(d) is elliptic on the transversal directions to v,, so it is easy to check under the condition (4.1.4-5) that Q,(a) is effectively hyperbolic with

Strongly Hyperbolic Equations

25

respect to the direction vo so d x , because Aao(dxo) >O. We can conclude under the condition (4.1.2-5) that the linearizations Dd,of d, are effectively hyperbolic with respect to d x , at ( x , u, d,)=(O, u-, 0 ) in the sense of Definition 3.2 and Remark if we put R ( x , h, u)=hE(a) and Q ( x , u)=Dd,-d,E(a) with the principal part Qo(a). Therefore it is strongly well posed by Theorem 2.1. Theorem 4.1. Let u- be a formal solution at x,=O of the Monge-Ampire equation (4.1.1) d,(u)=O satisfying the conditions (4.1.2-3). If the conditions (4.1.4-5) hold at a neighborhood of ( x , u)=(O, u - ) , then there exists a unique solution of d,(u)=O on a neighborhood of the origin x=O such that u-u- is flat at x,=O (aa(u-u-)~,,=o=O f o r any a ) . Example. Let M be a two dimensional Riemannian manifold. If the curvature K of M is non positive at a neighborhood of a point x and if the Hessian V2Kof K does not absolutely vanish at points where K vanishes, then some neighborhood of x is isometrically embeddable into R3. (This is equivalent to finding a solution of the Darboux equation. See M . Spivak.) We need a simple lemma to prove the above theorem as a corollary of the unique extension theorem. Lemma. Let $ be an infinitely di'erentiable function being flat at xo=O. Then there exists a non negative infinitely di'erentiable $- in xo (independent of other variables) also being flat at xo=O such that $+$- is non negative. By using this lemma we can find a n infinitely differentiable Q(xo),flat at x,=O, such that d,(u-+$)gO

at x,IO

and also it is flat at x,=O. If we put P=@-g, where g=O at x o 2 0 and g= O(u-+$) at xoO keeping P(u)=@(u)=O at xo>O. Such a solution u is able to extend again to x,
S 4.2. A non-linear wave equation Let us consider two fucctions H ( x , v, u j , u) and G ( x , v, u j , u) which have the relation H,+vG,=O

.

We use the notations that f(,) =(a/ax,)f, f,=(a/av)j, f,=(a/au,)f andf,=(a/au)f. We are going to find by the formal calculation a canonical equation of the Hamiltonian

26

N. IWASAKI

s

%= H(x,, x‘, v , Vu, u)dx‘

under the canonical one form

=s

@(q)

G(xo,x’,

V,

VU, u)qdx’

.

where

Z? in ( v , u) is The FrCchet derivatives of S

Then the Hamilton vector field F is a ( F :P, 4)=6=(P,

4)

i

Putting q = O in it, we get

so that

+ 5 G,V,F(q)+ 5 V,(G,F(q))= -5 V,H,+ H ,

G,FP)

j=1

j=1

j=1

Considering that H,+vG,=O so that H,,+vG,,=O, we get

a

27

Strongly Hyperbolic Equations

Therefore the canonical equation a,u= F(q) and aov= F ( p ) is

2

(4.2.1)

5 2GjV,d,u

P ( U ) = ~ ~ ~ U + G , - ’ [(H,,+d,uG,,)B,V,u+ J ,k = 1

+2(Hj,+&~Gj,)V,u+ f=1

J=l

J=1

(H,,,, +&uGj,j))-HuI

=o, where (&u, d p ) are substituted for (v,u f ) of H , G and their derivatives. The principal part of this equation is

(4.2.2)

pz(u)=Eo2+G,-’[J , k = l (Hj,+&uGf,)E,E,+

5 2GfE,EoI

3=l

Now we put

(4.2.3)

ajr=G,-2G,G,-G,-1(H,,+vGj,)

and assume that

(4.2.4)

G,
and

(4.2.5)

{a,x},,* = I . . . . , , 2 0

as functions in ( x , v, uf,u). Then P is hyperbolic. Theorem 4.2. Let u- be a formal solution of P(u)=O (4.2.1) at x=O satisfying an initial datum (u-, dou-)I,o=o=(~o, ul). We assume (4.2.4-5) for a,, (4.2.3), (namely, p z (4.2.2) is hyperbolic) and also that p2(u-)is efectively hyperbolic. Then there exists a unique smooth local solution of P(u)=O at a neighborhood of the origin satisfying the initial datum. Remark. We assume that H and Cj”=, H,,,, -H, are functions in ( x , , x’) with compact support a t (v,u,, u)=O, and that the initial datum (u,, u,) is with a compact support in XI. If the assumptions of Theorem 4.2 are satisfied for

N. IWASAKI

28

all (0, x’), then there exists a function with a compact support such that u is a solution of P(u)=O near xo=O. Therefore it satisfies

5

ff(X0,

X’,

a&, =

ajU,

J:”

U)dX’-

s

H(0, X’,

a&,

a,U, U ) d X ’

J

d x , H o , ( x o ,x’, aou, aju, u)dx’

.

Example 1. Let H and G be

where a,,=a,,(x), c j = c j ( x ) and f = . f ( x , u ) , and

G=v. Then

P(u ) =ao%- 2 (a, -C j ) ( U j k (a,u+

c,u)} +fu =0

.

3,k=l

This is semilinear.

So if the principal part

I.k=l

is effectively hyperbolic, then it has a solution for any smooth initial datum.

Example 2 . H=2-’v2+(

f:u,”f(u)”)”2 .

,=1

G=v. P(u)=2,2u--

f:a,( v-la,U)+

3=1

where

Then

where

We assume VfO

.

V-tffu=0 ,

Strongly Hyperbolic Equations

29

Then p z is hyperbolic. If f + O , then 1b,1< 1, so that p z is strictly hyperbolic. At f=O, p z has double characteristics. Now we put h , = a , [ f ( u ( x ) ) l=f,(u(x))&lu(x)

and h j = u o j ( x ) = d o a j u ( x ):

j=1,

,It.

If the inequality ho2+

5 h,2-(5

3=1

bjhj)2>0

J=1

holds at f ( u ( x ) ) = O , then p z is effectively hyperbolic. Example 3 .

and

Here we have assumed

5

j=1

(ajU)2+fZ#0

.

The principal part is

This is hyperbolic. If f ( u ) f O , then it is strictly hyperbolic as well as the previous example. If at f ( u )=0,

N. IWASAKI

30 aof

(4=f,(uPoufO

7

then it is effectively hyperbolic.

S 4.3.

The compressible Euler equation We consider the most known case, namely, the compressible, irrotational and barotropic fluid. The equations may be written as

(4.3.1)

1

(a,+v. V ) p + p V . v = o

(a,+v - V ) v+p-'Vp

=0

Bxv=O.

We here give the equation of state by (4.3.2)

p=F(p)

*

The equation (4.3.1)is reduced to a hyperbolic system, if (d/dp)F>O,especially, to a strictly hyperbolic system if (d/dp)F>O. We consider the case that (d/dp)Fmay vanish somewhere, namely, a minor phase transition may arise at some value of density p. An example is the isothermal state of water with the temperature not less than a critical value 374°C. The equation of state at 374°C is critical at the point of pressure 218 atmospheres and density about 0.4gm/cms. (See G . K. Batchelor [ l ] ,S 1 . 8 . )

Theorem 4.3. W e assume (d/dp)F>O, and ( d i d ~ ) ~ F >at O p such that (d/dp)F=O. If a smooth initial datum (po, v o ) , satisfying po>O and VxvO=O, keeps the relation that V . v o f O at x such that (d/dp)F(po(x))=O,then there exists a unique local smooth solution of (4.3.1-2) satisfying the initial datum. A n example. Let us consider the van der Waals' equation of state, p=Rp7'/(1-bp)-ap2,

( O < p < l / b (=cc if b = O ) ) ,

where T>O is the temperature, R>O is the gas constant, and a , b T 0 are also constants for a given gas. If b R T 2 ( 2 / 3 ) 3 a ,then it satisfies the conditions for p = F ( p ) at O
(a,+v * B)$+

Bv =0

.

We may replace p - V p by V f , where f is a function of $. The velocity v is written by the velocity potential $ as

v=V$

Strongly HyperboIic Equations

because v is rotation free.

Then the equation of motion implies

.

V((a,$+2-11V$12)+f)=o

(4.3.4)

31

Therefore (8, +V$. V)2$-g = -(a,+~$-~'lf(qN

= -f(1Y$)(4+V$*V)$

7

where g is a function depending only on t . We use the notation that f (d/d$)rf. By using (4.3.3), we get

(a,+ V$* V $ -f'"($)d$ =g

(j)=

.

We calculate again (4.3.3), operated by (a,+V$.V), by means of two relations,

(a,+ V$* ma,+ V$* V)#= (a,+V$* 8)2$-

f ( l )

IP$12

and by (4.3.4)

(a,+V$ .B)d$ = -f ( ' ) d $-f ( 2 )I V$ 12- 18%)12 . Then we obtain two equations

(a,+

B$ * V ) 2 $ - - f'"d$- (f'1

+f

(2')

IV$I'IP"l~

=o

and

(a,+

V$* V $ -f(1Y$)4 =g

It is clear that we can attain the conclusion independent of g if we find a solution of this system under a n initial condition satisfying on a n initial surface

. +

a =(a,+V$ V)$h A$ =0

because these equations mean

and

+

+

(8, V$ * V ) 6 Vz$ * 6- F ( f (')($)u)=0 ,

and because the uniqueness of this linear system is also assured. The independence of g follows the uniqueness and the fact that, if $h and $ are sohtions for g=O, then $ and $+/I for g=(d/dt)%.

N. IWASAKI

32

We take the linearization of this system.

=~,~ where the lower terms C = { C ~ , } , , , are

+

c,, = -f @ ) A $ - ( f ' , ) f

IV$lZ- 2 ( f ' " cl2=2(a,v$*V+v$. VZ$. 8)-2VZ+. vz , c Z ~ = - f tz)A+ , (3))

+f ' 2 ) ) V $ . v ,

cz2=2(a,v+-v+v+. V2$. V ) . cI1and cZ2are differential operators of the order 1, c,, of the order 2 and czl of the order 0, respectively. If f c l ) > O , then this system is strictly hyperbolic. If f c l ) > O and if

(a,+v+.

v ) z j - ( l=) p 3 )

at f " ) ( $ ) = O ,

Appendix.

l(a,+v+.v)$y

( = f ( 3 )

I A + I Z )> 0 ,

then it is effectively hyperbolic.

The Nash-Moser Implicit Function Theorem

The one we have used is different in appearance from the usual one. So we mention it in this appendix. However, n o essential change is required in the proof. It follows L. Hormander's appendix [ l l ] . This theorem also has its own history. But we note nothing about it. Let 3,(j=O, 1) be two Banach scales. There exist sequences of Banach spaces {H,s} s 2 0 (j=O, 1) with a real parameter s. Hj8 is continuously imo is dense in HjS. We also assume mersed in H,$ for s > t . And H f m = n s SH,8 that these scales are interpolation spaces and have the smoothing operators. It means the following.

1) There exists a sequence of norms the logarithmic convexity that ll4l2a+&2)t.

5~

II.IJS with

respect to HjS,which has

ll~ll~ll4l~-~

for O
then the integral

,

If it satisfies for a =

Strongly Hyperbolic Equations

33

existing in Hjo ( H I a for a < a ) , belongs to H I a and satisfies the estimate llulla I C(at -a-)(a- a-)-l(a, - a ) - ' M

.

3) There exists a sequence of continuous linear and strongly differentiable in 0 operators & from H I o to H,", satisfying the estimates (i) (iv) (smoothing operators). ( i 1 lls8ullb
---

Remark. The constants C in the above estimates are uniform as far as a and b are in a bounded interval. Let 0 be a closed nonlinear operator from Hornto H,O for some m>O, which is defined on a bounded neighborhood V, of the origin of Horn. We assume the following properties for 0. a) For some fixed p ( O < p I o o ) @ ( V , n H o ~ t mbelongs ) to H f , and @ satisfies the estimates i). b) @ has first and second degree FrBchet derivatives @ ( l ) ( u : v )and @(2)(u:v1, u,) on V,n HOPtm, satisfying the estimates ii) and iii). c) The first derivative has a right (or left) parametrix F(u:g)as iv). i) For all u of V,fl H O P + m and for all a: O
I I @ ~ ~ ) l l a l ~ ~ ( I I ~ l l 1) a t .m + ii) For all u of VonHoPtm,for all v of Hoptmand for all a : O < a l p ,

ll@(l)(u: iii) For all u of V,n

ll@(z)(u'

'17

'2)

v)IIRICR(llullRtmllvIImf

Hoptm,

/la

ca(ll '11

IIvllR+?II)

*

for all vl, vz of HOP+m and for all a: O
11 m +

llvl I l R t 9 ? l l l v z

119llf

llvl llmIIvzIlat7n) '

iv) A right parametrix F(u:g)is a continuous linear operator from H f t z m to Hoptmfor all u of V,n Hornsuch that

@'"?#-=I-@, where Q = @ ( @ ( u ) u, : g ) is a n operator from HIPtzmto HIP. For all u of V,n H,", for all g of HIPtzmand for all a : O g a l p ,

l '(K:

g)11a 2 ' R (

Ilglldtm+

11sI l m l l u l l a + m )

and

ll@(' ( u)*

': 9)I l R

ca(llg I I R t m l l @ ( u )

Ilm

f llgllmll@(u) I I a t m + llgllnlIl'(u)ll

71LIIullrZt9ft)

*

34

N. IWASAKI

(In case of a left parametrix, change a n d f r o m HOPtzm t o HOP.)

I a n d 0 to

one s from HOPtzm to HIP

Theorem (Nash-Moser). (existence) W e assume the existence of a right parametrix. Let parameters satisfy a relation pfm>r>2a>6m. there exists Then for any open set V , of V , and for any bounded ball B, of HOT, a positive E such that it holds that for all f of @(Vl n B, n H,")

Ilf

Ill(a-m>

2~

or that an element u of V,n Ho(@exists and satisfies @(u)=O, where Ho(p)= H,B. (uniqueness) W e assume the existence of a left parametrix. Let parameters satisfy a relation

na.P

p+m>a>3m. For any bounded ball B, of HOa, there exists a positive Ilu-vllo
E

such that it holds that,

if

References

131

t71 t81

t 101

G . K . Batchelor, An introduction to fluid dynamics, Cambridge, 1967. J. Chazarain, Optrateurs hyperboliques a caracttristiques de multiplicitt constant, Ann. Inst. Fourier, 24 (1974), 173-202. F. Colornbini and S . Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in C", Acta Math., 148 (1982), 243-253. R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 11, Interscience, New York, 1962. H. Flaschka and G. Strang, The correctness of the Cauchy problem, Advances in Math., 6 (1971), 347-379. K. 0. Friedrichs, Symmetric hyperbolic linear equations, Comm. Pure Appl. Math., 7 (1954), 354-392. -, Lectures on pseudo-differential operators, New York Univ., 1968. L. Girding, Cauchy's problem for hyperbolic equations, Lecture Note in Chicago Univ., 1958. -, Linear hyperbolic partial differential equations with constant coefficients, Acta Math., 85 (1950), 1-62. J. Hadamard, Le Probltme de Cauchy et les Equations aux Dtrivtes Partielles Lintaires Hyperboliques, Herrnann, Paris, 1932. L. Horrnander, The boundary problems of physical geodesy, Arch, Rational Mech. Anal., 62 (1976), 1-52. -, The Cauchy problem for differential equations with double characteristics, J. Analyse Math., 32 (1977), 118-196.

Strongly Hyperbolic Equations

35

[13] V. Ya. Ivrii and V. M. Petkov, Necessary conditions for the Cauchy problem for nonstrictly hyperbolic equations to be well-posed, Uspehi Mat. Nauk, 29 (1974), 3-70. (Russian Math. Surveys, 29 (1974), 1-70.) [14] N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (a special case), J. Math. Kyoto Univ., 23 (1983), 503-562; (a standard type), Publ. RIMS, Kyoto Univ., 20 (1984), 551-592. [15] -, The Cauchy problem for effectively hyperbolic equations (general cases), J. Math. Kyoto Univ., 25 (1985), 727-743; (Remarks), Proc. of the Taniguchi Symposium on hyperbolic equations and related topics, Kinokuniya Publisher, Tokyo, 1986, 99-110. [16] --, Applications of Nash-Moser theory to nonlinear Cauchy problems, to appear in Proc. Sym. Pure Math., 45 (1986). [17] -, The Cauchy problem for hyperbolic equations with double characteristics, Publ. RIMS, Kyoto Univ., 19 (1983), 927-942. [18] K. Kajitani, Cauchy problem for non-strictly hyperbolic systems, Publ. RIMS, Kyoto Univ., 15 (1979), 519-550. [19] K. Kasahara and M. Yamaguti, Strongly hyperbolic system of linear partial differential equations with constant coefficients, Mem. COIL Sci. Kyoto Univ., Ser A, 33 (1960), 1-23. [20] A. Lax, On Cauchy’s problem for partial differential equations with mulitiple characteristics, Comm. Pure Appl. Math., 9 (1956), 135-169. [21] P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646. [22] J. Leray, Hyperbolic differential equations, The Institute for Advanced Study, Princeton, N.J., 1952. [23] E. E. Levi, Caracteritiche multiple e problema di Cauchy, Ann. Mat. Pura Appl., 16 (1909), 109-127. [24] T. Mandai, A necessary and sufficient condition for the well-posedness of some weakly hyperbolic Cauchy problems, Comm. Partial Differential Equations, 8 (1983), 735-771. [25] W. Matsumoto, On the conditions for hyperbolicity of systems with double characteristic roots, I, 11, J. Math. Kyoto Univ., 21 (1981), 47-84, 251-271. [26] S. Matsuura, On non-strict hyperbolicity, Proc. of Intern. Conf. on Functional Anal., Tokyo, 1969, 171-176. [27] R. Melrose, The Cauchy problem for effectively hyperbolic operators, Hokkaido Math. J., 12 (1983), 371-391. [28] S. Mizohata, and Y . Ohya, Sur la condition de E. E. Levi concernant des Cquations hyperboliques, Publ. RIMS Kyoto Univ., 4 (1968/9), 511-526. [29] -, Sur la condition d’hyperbolicitk pour les tquations a caracttristiques multiples 11, Japan. J. Math., 40 (1971), 63-104. [30] S. Mizohata, On the Cauchy problem, (Lectures at the Wuhan Univ.), to appear in Science Press, Beijing. [31] -, Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 (1961), 109127. Systtmes hyperboliques, J. Math. SOC.Japan, 11 (1959), 205-233. [32] -, [33] T. Nishitani, Local energy integrals for effectively hyperbolic operators I, 11, J. Math. Kyoto Univ., 24 (1984), 623-658, 659-666.

36

N. IWASAKI

[34] 0. A. Olejnik, On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math., 23 (1970), 569-586. [35] V. M. Petkov, The parametrix of the Cauchy problem for non symmetrizable hyperbolic systems with characteristics of constant multiplicity, Trudy Moskov. Mat. ObS6, 37 (1978), 3-47. (Trans. Moscow Math. SOC.,1 (1980), 1-47.) [36] I. G. Petrovskii, u b e r das Cauchysche Problem fur Systeme von partiellen Differentialgleichungen, Mat. Sb., 2 (1937), 815-866. [37] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 5 (second ed.), Publish or Perish, Inc., Berkeley, 1979. [38] M. Yamaguti, Le peoblime de Cauchy et les optrateurs d’intkgrale singulikre, Mem. Coll. Sci. Kyoto Univ., Ser. A, 32 (1959), 121-151. [39] H. Yamahara, On the Cauchy problems for weakly hyperbolic systems, Publ. RIMS, Kyoto Univ., 12 (1976), 493-512. [40] K. Yamamoto, The Cauchy problem for some class of hyperbolic differential operators with variable multiple characteristics, J. Math. SOC.Japan, 31 (1979), 481-502. [41] S.-T. Yau, Seminar on Differential Geometry, Princeton Univ. Press, Princeton, 1982. Research Institute for Mathematical Sciences Kyoto University Kyoto 606, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 37-96 (1986)

Solutions of the Boltzmann Equation By Seiji UKAI Abstract. This note is devoted to the review of the major results obtained in these ten years on the existence problem for the Boltzmann equation. The topics cover the local and global existence theorems for a variety of the Cauchy and initial boundary value problems, the existence and stability of the stationary solution for an exterior problem and the relation between the Boltzmann equation and fluid dynamics. They will be presented with emphasis on mathematical ideas of proofs. Also, important open problems will be pointed out. Key words: Boltzmann equation, Cauchy problem, initial boundary value problem, local solution, global solution, stationary flow, fluid dynamical limit Contents

1. Introduction 2. Local solutions, I. The case of cutoff potentials 3. Local solutions, 11. The case of non-cutoff potentials 4. Global solutions for the Cauchy problem 5. The initial boundary value problem 6 . The stationary flow 7. The Euler limit and the initial layer

37

44 52 57 74 82 89

1. Introduction The Boltzmann equation which describes the motion of a gas has been a fascinating subject for many mathematicians since its discovery by L. Boltzmann in 1872, because its physical significance is well-established and the mathematical problems are various. It is one of the nice nonlinear partial differential equations in mathematical physics, providing many challenging mathematical problems with fruitful results. However, the early development of the equation’s theory was rather slow. No existence theorems had been known until 1932 when Carleman [13] solved the spatially homogeneous case globally in time. The physically more realistic spatially non-homogeneous case was solved much later. The first solution, although local in time, was Received April 9, 1985.

S . UKAI

38

given by Grad [19] in 1965 and the global one by Ukai [37] in 1974. The subsequent progress, however, has been remarkable and is still lively. This note is intended as a review of the major existence theorems and properties of solutions known so far, with emphasis on the presentation of mathematical ideas of proofs. Also, important open problems are pointed out. There are two important topics which are not touched upon here, the Chapman-Enskog expansion and the one-dimensional shock profile solution for which the reader is referred to the review article [ l l ] . In this section, we give a brief explanation of the Boltzmann equation and the formulation of problems associated with it.

The Boltzmann equation The unknown of this equation is a scalar function f = f ( t , x , E ) which describes the number (or probability) density of gas particles having the position x = ( x l , x 2 , -,x , ) € R" and velocity E=(E,, E,, -,E,) € R" at time t € R. The specific phenomenon in a gas is the collision of gas particles. Boltzmann has derived his equation assuming (i) that collisions of more than two particles rarely occur and (ii) that between successive collisions, each particle moves according to Newton's law of motion. It is, 1.1.

-

--

(1.1.1) Here, the sum of the first two terms on the right-hand side describes the rate of change o f f due to (ii) in which . stands for the inner product of R", V , for the gradient (a/&,, ajax,, a/ax,) and similarly for V t , and a ( x , E) denotes the external force acting on particles such as gravity, electric and magnetic fields. The last term Q [ f ] is the rate of change of f d u e to binary collisions (i) and given by

.-

(1.1.2)

a ,

d v , 0) l f ( ~ ) f -f(E)f(S')WE'du (~')

Q[fl=

+

RnxSn-1

where v=15-E'1, cosO=(E-E').w/v, v) etc., with

o E S n - ' (the unit sphere in

R")andf(v)=

f(t, x ,

(1.1.3)

p = ~ - ( cos ~ e)w ,

v ' = ~ ' +cos ( ~ 010,

which denote velocities after a collision of particles having velocities E, E' before the collision and vice versa. The function q(u, 0) is called the collision cross section and determined by the interaction law which is specific to the species of gas particles. Two classical examples are the hard ball gas for which (1.1.4)

q(u, ~ ) = ~ , ~ Ielc ,o s

where uo is the surface area of the ball, and the inverse power law potential

39

Solutions of the Boltzmann Equation

of order s for which ( n = 3 , s > l ) , (1.1.5)

q(v, 8)=u’lcos Si-’‘qo(S)

,

r=l-- 4S ,

2 r’=l+-, S

where qo(S)>O is bounded and does not vanish near 8=lr/2. Extensive literature is available on the derivation of (1.1.1)-( 1.1.9, see e.g. [13, 14, 361, in which the following important properties of Q are also proved. Denote the bilinear symmetric operator induced from the quadratic operator Q by Q[., -1;

Further, define the inner product,

f(E)s(E)dS €= .

(1.1.7)

Theorem 1.1.1.

R”

(i)

Define thefunctions,

Then,for a n y f , g and O < j < n + l , (1.1.9)


Q[f, gl>~=O

*

( i i ) For a n y f 2 0 , (1.1.10) (iii) Q[fo]=O (1.1.11)

< k f , Q[fl>c
if and only i f , f0(E)=p(21rT)-n’z exp ( - 1C--vI2/(2T))

with any p > 0 , v E R”,T>O independent of

6.

The functions h,’s are called collision (summational) invariants, because from (1.1.3) it holds that, for O l j l n f l , (1.1.12)

hj(E)+hj(E’)=hj(p)+hj(q’)

9

which are the conservation laws of the number (j=O), momentum ( l s j l n ) and energy ( j = n + l ) of particles during a collision. The function f o of (1.1.11) is called a Maxwellian and describes the equilibrium state of a gas having the mass density p, flow velocity v and tem-

40

S. UKAI

perature T . If p, v, T are constant in t , x, then f O is called a n absolute Maxwellian, and otherwise, a local Maxwellian. Besides its physical importance, f ois in the special situation that it happens to be a stationary solution to (1.1.1) (e.g. a n absolute Maxwellian for the case a ( x , E ) = O ) , thanks to Theorem 1.1.1 (iii). The statements in Theorem 1 . 1 . 1 are all true, of course, only when relevant integrals converge. Cutoff and non-cutoff potentials The simplest problem associated with (1.1.1) is the local (in t ) existence of solutions to the Cauchy problem. However, this is not a trivial problem because the nonlinear operator Q is unbounded, which comes from two different singular behaviors of q(v, 0). One is the strong singularity at 8=7r/2 in (1.1.5). Since f > l , the integral over 9-' in (1.1.2) does not converge i f f is only bounded, but it does i f f is smooth because p = E , pf=E' at 0=7r/2, so the quantity { - . .} in (1.1.2) then vanishes there, thereby cancelling the singularity of q(v, 8). Thus, Q is well-defined only for smooth f and behaves like a (pseudo-) differential operator, an unbounded operator. In $ 3 we present a method to control this unboundedness, which is analogous to, but much simpler than, that used for the abstract nonlinear Cauchy-Kowalewski theorem of Nishida-Nirenberg-Obsjannikov [30]. However, this method has a limited application, and at present we can only go farther in removing this singularity by cutoff approximations. The most successful cutoff is Grad's angular cutoff [18], with which the local and global existence has been established for a variety of the Cauchy and initial boundary value problems for (1.1.1). This will be seen in $$ 4-7. It is Grad's estimates for Q and related operators with this cutoff that largely promoted the study of (1.1.1). Grad's cutoff is equivalent to replacing q by its angular cutoff, 1.2.

(1.2.1)

q"v, 8 ) = x " M v , 0)

1

where xc(0)=l (l8-~/21 > E ) , = O ( ~ O - R / ~ / < E ) . Obviously, the ambiguity in choosing the value of e arises unless the convergence is established in some way. This will be discussed in S 3.2. Observe that the hard ball gas (1.1.4) has no singularities and is considered as a cutoff potential case. Even with cutoff, Q may be unbounded since q(v, 0)-w ( v - > m ) , as seen in (1.1.4) and (1.1.5) with s > 4 . This unboundedness is handled in $ 2 by a method similar to that of S 3.

1.3. Conservation laws Once local solutions are found, global solutions can be constructed if nice a priori estimates are available. The Boltzmann equation has L' estimates be the inner coming from physically natural conservation laws. Let

<-, ->

Solutions of the Boltzmann Equation

41

product (1.1.7) but over R Z n x R E n . Suppose a ( x , E ) = O and let f be a smooth solution to (1.1.1). Then, by integration by parts and by (1.1.9), we have, (1.3.1)

=, ~ E R ,O < i < n + l ,

which are the conservation laws of the total number (j=O), momentum ( l < j < n ) and energy ( j = n + l ) of particles. In $ 2 . 4 , we see that f ( t ) 2 O provided the initial datum f , = f ( 0 ) 2 0 (and q(v, 0)20), a physically natural conclusion because f denotes the density. Hence, (1.3.1) is L‘ a priori estimates. But it is not useful because it seems difficult to find L1-local solutions by the reason stated in s 2 . 2 (see also [33]), which presents a contrast to the spatially homogeneous case where (1.3.1) is essential in costructing global solutions [2, 31. In $ 4 , we propose a method to find global solutions, which takes advantage of nice asymptotic behaviors of solutions of the linearized problem at the cost of the restrictive condition that the initials are small in some sense.

1.4. Boundary conditions When the gas is contained in a vessel 52, the boundary condition which describes the reflection of gas particles by the wall 852 is imposed on f. Denote the outward normal to 852 at x€i?52 by n ( ~and ) set, (1.4.1)

s*= {(x, E) € a52 x W” I n(x).E2O} ,

(1.4.2)

r’f=fIs.

,

(same signs). are trace operators to S” whose existence will be established in S 5.1. Since particles at (x, E) E S+ (resp. S - ) are particles incident to (resp. reflected by) the wall as, r i f represent the corresponding densities. Therefore, the boundary condition must take the form, 7-f=M r +f

(1.4.3)

t

where M is the boundary operator determined by the reflection law. Classical examples are; (1.4.4) ( i ) (ii) (iii) (iv)

(perfect absorption) M=O, (specular reflection) Mr+f=f(t, x, E-2(n(x)-E)n(x)), (reverse reflection) My+f=f(t, x, -[), (diffuse reflection) n(x)*E’f(t, x, E’)dE’

Mr+f=g,(E)

9

n(z).e’>o

sw(E)=p,(2aTw)-“/2exp ( - lElz/(2Tw)), where T, is the temperature of the wall.

~ , = ( T ~ / ( 2 7 d ) ”, ~

M may be any (convex) linear com-

S. UKAI

42

bination of (i)-(iv). Also, (ii), (iii) may be generalized as ([24]), ( 1.4.5)

My+f=f(r, x , m ( x , E ) ) ,

m(x, E) E S-

,

and (iv) as

dux being the measure on 352, and so on. With these boundary conditions, we are given initial boundary value problems for ( l . l . l ) ,the subject of SS 5-6.

1.5. Large initial data In solving the Cauchy and initial boundary value problems globally in t , a n unpleasant restriction is required for initial data f , . Whereas the local existence can be established for arbitrarily large f , (SS 2,3,5), the global one requires the smallness condition on f,, as stated at the end of S 1.3. More precisely, f , should be near a n equilibrium described by a n absolute Maxwellian go (§§ 4-6). Of course, this is a technical restriction specific to our method of $ 4 . 1 , but at present, n o other methods are available. Thus, the global existence for f , far from go is a big open problem. For such f,, it is expected that the shock appears after a finite time, however smooth f , may be. Notice that all known solutions, local or global, are classical (smooth) ones, and so d o not present the shock. Therefore, the shock solution, if any, should be a weak one, but up to the present no definitions of weak solutions have been found which promise the global existence. Recently, Arkeryd [4] introduced a new concept to this problem, constructing global ‘non-standard’ solutions for arbitrary L1 initials f,. More precisely, his solutions solve (1.1.1) in the sense of the non-standard analysis that the variable t is considered in *R (the set of non-standard real numbers) and x , E in *Rn while the Lebesgue integral $ dE’do over R ” x Sn-’ is replaced by the Loeb integral $ L(*dE’ *do) over *Rnx*Sn-l where L(*d[’*do) is the Loeb measure induced from the Lebesgue measure dt’do. Although this result is remarkable, its conversion to the standard result seems difficult. For the spatially homogeneous case which will not be discussed in this note, global solutions exist for any f , , with and without cutoff, see [2, 31.

- -.

1.6.

.-

Fluid dynamical limits The motion of a gas can be described not only by the Boltzmann equation but also fluid dynamical equations such as the Euler and Navier-Stokes equations. The former gives a microscopic description in the sense that the gas

Solutions of the Boltzmann Equation

43

is looked at as a many-body system, while the latter gives a macroscopic one because the gas is regarded as a continuum like a fluid. Evidently, the former provides more precise information, but if the number N of gas particles is very large as is usually the case (N=1OZs,Avogadro's number), then the gas will behave much like a fluid, and at the limit N=oo both descriptions are expected to coincide. The Hilbert and Chapman-Enskog expansions reveal this. In 1912 the celebrated Hilbert, the first mathematician to attack the Boltzmann equation as well as many other big problems, gave a solution of (1.1.1) in a formal power series,

(1.6.1)

...

f=f'O'+Ef(1)+E2f'2)+

of € E N - ' (the mean free path). To determine the coefficients, assume a ( x , f ) = O and write ( 1 . 1 . 1 ) as 1

Df=, Q[fl ,

(1.6.2)

D=ac+E*V,,

which is obtained dividing (1.1.1) by N and writing f/N as f (the normalized probability density). Substituting (1.6.1) and equating the coefficients, he then obtains,

(1.6.3)

Q[f(o'l=o,

(1.6.4)

Df")=2 Q[f (O), f'"] ,

( 1.6.5)

D f C r= )

k+l

C Q [ f ( I ) f, ' k ' l - j ) ] ,

k> 1

J=O

.

By Theorem 1.1.1 (iii), f t o ) = f o , the Maxwellian ( l . l . l l ) ,with fluid dynamical quantities p, v, T still undetermined. Once they are known, (1.6.4) is a linear integral equation for f"). Applying his theory of integral equations, and by Theorem 1.1.1 (i), Hilbert has the solvability conditions,

(1.6.6)

E=O,

O<j
.

The integrals can be calculated explicitly, giving a system of nonlinear hyperbolic conservation laws for p, v, T ;

pt+V.(pv)=O, (1.6.7)

+v - ( p 'vv)+v p =0 ,

(pv)t

(pE),fV.(pEv+pv)=O ,

tvv = ( v,vr) , 1 E=e+- 1vI2, 2

supplemented by the equation of state of the ideal gas,

S . UKAI

44

where V=V,, p is the pressure and e the internal energy per unit mass. (1.6.7) is the compressible Euler equation. Similarly, (1.6.5) give inhomogeneous Euler equations which determine higher terms f(". The Chapman-Enskog expansion is not a n asymptotic expansion like (1.6.1); the procedure is complicated and is referred to [ l l ] . The first term of this expansion is again a local Maxwellian, but its p, u, T should solve the compressible Navier-Stokes equation, not the Euler equation (1.6.7). Although the relations thus revealed between the kinetic theory of gases and the fluid dynamics confirm a physical validity of the Boltzmann equation, these expansions are not yet justified fully at rigorous mathematical levels. In particular, the convergence is not known, so they d o not serve as the existence theorem. Only partial justifications have been given [7, 10, 11, 16, 19, 26, 31, 431. In S 7, we will discuss (1.6.1) to the first term. 2.

Local Solutions, I.

The case of cutoff potentials

We begin by solving the Cauchy problem for (1.1.1) locally in t , with a cutoff assumption general enough to include (1.1.4) and (1.1.5) cut by (1.2.1). This is the simplest case in which we meet the unboundedness of Q ($2.1). The main idea to control it is to set the Cauchy problem in a scale of Banach spaces, in order that the integration with respect to t can play a role of a smoothing operator (S 2.3). The same idea will be used in later sections where the problem of unboundedness arises in different situations. Two types of solutions are presented, one in a different function class from that of initials f, ($2.3) and the other in the same class but with further restrictions on q andf, (S 2.4). 2.1.

An estimate for Q We assume that q(v, 0) is measurable in (0, m) x [0,IC]and that,

with a constant q,>_O. For example, this is satisfied if (2.1.2)

M v , ~ ) I I ~ , ( ~ + ~ - ~ + z I ~ ) I c o s ~,~ - ~ O' < p < n ,

O17<2,

f
.

Since 7'>1 in (1.1.5), this may be taken as the weakest cutoff assumption. Here, q is not assumed nonnegative. Let a , /3 E R and set,

, <E>=(I+ I W ) ~,' ~ - c , p = I f = f ( E ) I P a , a f € L"(R")I , Ilf lIa,p=s;P P,,p(E) I f ( E ) I . ,o,,a(t)=e-"(E)2J

Solutions of the Boltzmann Equation

45

The main results of this section is. Theorem 2.1.1. For any a>O, there is a constant C=C,>O, decreasing in a , and 3 0 3 (a-0), -0 (a+m), such that f o r any p > O and f , g € L;,p,

II Qrf, gl l l , , ~ - ~ ~ ~ I I f I I ~ ~ p I I ~ I I ~ , ~

(2.1.3) Proof.

Define the operators Q j , j = 1 , 2 by

(2.1.4) and similarly for Q2 with f ( v ) g ( f )replaced by f(E)g(E’). Then, (2.1.5)

Qrf,

1 gl=l

{Ql[f,

gl+Qi[g, f l - Q z [ . f , ~ l - Q z [ g , f l I

9

so that it suffices to prove (2.1.3) for each Q j . By (1.1.3) or (1.1.12) f o r j = n f 1, < ~ > 2 + < l j ’ > 2 = < E > 2 + < [ ’ > 2 holds, which gives, (2.1.6) for p 2 0 .

Pa,&$Pa,B(?’)

>-Pa,p(E)Pa,o(E’)

9

Hence we have,

I Q i [ f ,gl(E)I ~ ~ ~ t ~ ( E ) ~ l l f l l a . ~ l l ~ l l a , ~ 7

Then, Ir, by (2.1.1) and by (2.1.7)

1

1E-E’[le“(E’’2dE’IC,<E>1

,

R”

which is valid for R > -n. This proves the case j = 1 , and similarly forj-2, though more straight-forward. Remark 2.1.2. Estimates similar to (2.1.3) are possible in the spaces LP,,,, because the transformation ( E , E’)+(?, 7’) by (1.1.3) has the unit Jacobian, [13].

p<

00,

The estimate (2.1.3) says that if r>O in (2.1.1), Q i s unbounded in L:,,. This can be improved partially, under a more restrictive condition. Theorem 2.1.3. a>Oandp>l,

(2.1.8)

Assume (2.1.2) with an additional condition r’=O.

IlQi[f,

~ l I l ~ , p + ~ - r I ~ ~ l l f I l ~ , -p l l ~ l l ~ , p

Then, i f

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46

For the proof, it suffices to proceed as before, using

1

(2.1.9)

Sn-l

-F-fldo< C<E>-fl-l ,

instead of (2.1.6), which is valid for ,8>1 (see the appendix of [19] where f = - 1 is assumed but the result holds for ,821). Note that (2.1.8) is not true for Q z . The linear Cauchy problem The linear part of (1.1.1) can be treated fairly simply in the case of the Cauchy problem. Let us solve the problem,

2.2.

(2.2.1) where k = k ( t , x, E ) and f o = f o ( x ,E ) are given functions. According to the theory of first order partial differential equations, we shall first solve the characteristic equation dX/dt= B ,

(2.2.2)

dE/dt=a(X , 3) ,

under the initial condition (X,E ) = ( x , 5) at t = O . If a ( x , E ) € C1,a unique solution exists at least locally in t near t=O, denoted by ( X , B)= S ,(x, E ) = ( W t , x, El, &t, x, 5 ) )

(2.2.3)

1

and (2.2.1) is solved by Jo This is a classical solution if f o and k are in C', and exists as long as (2.2.3) exists. A simple sufficient condition, convenient for a later purpose, for (2.2.3) to exist globally in t for all initials (x, E ) is (2.2.5)

a(x, E)

has the form

a(x, E)=-V,b(x)+a,(x,

E) ,

where ( i ) b ( x )E C2(Rn)is a scalar function, and a , ( x , E ) € C'(R"x R") is a n nvector function, (ii) b(x)>l, (iii) E . a , ( x , E ) = V e . a l ( x , E ) = O . Here, b ( x ) is the potential and (ii) implies merely that it is lower bounded, because the potential is unique only up to additive constants. An example statisfying (iii) is the magnetic field for which a , = ~ x B ( x ()n = 3 ) .

47

Solutions of the Boltzmann Equation

Theorem 2.2.1. Suppose (2.2.5) be fulfilled. Then the solution (2.2.3) exists for all t E R , for any initial (x, 6). Further, for each fixed t , the mapping S, is (i) C1-diffeomorphicand (ii) measure preserving, on R" x R". Proof. We introduce the one particle energy,

and note that (2.2.2) is Newton's equation of motion. conserved, that is,

In view of (2.2.5), E is

E(S,(X,6))= H x , E )

(2.2.7)

holds, as long as (2.2.3) exists. But, since b ( x ) > l , (2.2.7) provides a n a priori estimate for X and 9,and thereby assures the global existence. Now (i) is obvious and (ii) comes from the fact that div,,€ ( E , a , ( x , E ) ) = O . If a ( x , E ) = O , then S,(x, E)=(x+tE, E ) .

Remark 2.2.2.

Define the operator U(t), t E R , by (2.2.8)

(U(tlfo)(x,E ) =fo(X( - t, x , E l , b(- t, x, E ) ) =fo(S-t(x, E ) ) .

Then (2.2.4) can be rewritten as f ( t ) = U ( t y o + ] ' U(t-s)k(s)ds .

(2.2.9)

0

We shall discuss U ( t ) in function spaces suitable for the nonlinear problem. Let E ( x , f ) be as in (2.2.6) and set pu,&x, f)=eaE(">E)E(x, E)b'2

(2.2.10)

.

Define the space (2.2.11)

X,.,,={f=f(x,

E ) I p a , p f E W R n xR")}

Ilfll..u.a=IIPu,pfll~~

9

.

In view of Theorem 2.2.1 and (2.2.7), we readily have the

Theorem 2.2.3.

For any p E [l, 001 and

IIU(t)fllp,u,p=

llfllP,u,p

7

(Y,

3/ E R , it holds that

t f sR

t

f E

x:.p .

Obviously U ( t ) has the group property U ( t + s ) = U ( t ) U ( s ) ,s, t E R . As for the continuity in t , we can prove the

S . UKAI

48

Theorem 2.2.4. Let p~ [l, 00) and a , P E R . X,”,a. Its generator, denoted by A , is given by

Then U(t) is a C,-group on

Af= -(E.Vz+4x, E ) . V e ) f , whose domain is maximal in X{,,+ sense of distribution.

Here the derivatives are to be taken in the

This theorem is not valid for p = m , but it is if U ( t ) is restricted in a suitable subset of X : , p . For example, we have, Theorem 2.2.5. For p = m , the previous theorem remains true i f X:,, is replaced by

z:,,={fE x,,n where x R = x R ( x , E ) is such that

x Rn)I l l ~ ~ f l l - , ~ , ~ - -( -R>-Om ) } ,

xR=

1 if 1x1

+ IEi > R , and =O

otherwise.

Another example will be obtained by @-operation of [21J, noting that the dual of X2a,-B is X ; , p and the dual of U ( - t ) is U(t).

2.3.

Construction of local solutions We are now in a position to solve the Cauchy problem for (1.1.1);

(2.3.1) Iff is a solution, it must solve the integral equation (2.3.2) which comes from (2.2.9). We shall solve this in the space X:,B of (2.2.10), but need to vary the index LY with time t to control the unboundedness of Q which is seen in the Lemma 2.3.1.

Theorem 2.1.1 remains true i f L:,p is replaced by X : , p

For the proof, it suffices to repeat the proof of Theorem 2.1.1 with the assumption (2.2.5) (ii) taken into account. In spite of Remark 2.1.2, this lemma fails in X,”,, if p < 00. Define the bilinear operator (2.3.3) and the nonlinear operator

Solutions of the Boltzmann Equation

49

Then (2.3.2) can be written as f = N [ f ] , so we shall find a fixed point of N . The simplest but powerful tool for this is the contraction mapping principle. However, if r > O , it does not work if a is kept fixed, because due to the above lemma, N loses the weight ErIz a t each step of iteration. O n the other hand, if a is varied with I , then it does work well since the integration in (2.3.3) with respect to t turns to a smoothing operator. T o be precise, introduce the space (2.3.5)

x=x,,.,p([-T, TI)=If(t, x, 5 ) I e - c l c l E f ( t ) ~ L " ( [ -TI; -,

II If1II = I IIf I II x =

ik!

Ile-"l"'"f(t)

ll-,a.j=

Ilf(t)Il-.a--.ltl ItlST SUP

X , j ) }

,

.j

Lemma 2.3.2. Under the assumption (2.1.1) and f o r any a>O and B 2 0 , there is a constant Co>O such that for any K > O and with T = a / ( 2 r ) , we have, for all f,9€X, (2.3.6)

IIINo[f, sllll

co.-'lllflll

lllglll

.

Proof. Put w(t, s)= IIU(t-s)Q[f(s), g(s)]ll,,a-.lsl,~-r.Using Theorem 2.2.3 and then Lemma 2.3.1, we get w(t, ~ ~ ~ ~ u - K ~ lllglll s ~ l~l Cl fal/ l~ll l l JIllslll, lll for Is1 I T . Hence,

where

The last integral is majorized by

K - ~ ,

so (2.3.5) follows.

The fact that the last integral is bounded although Er/2+cc (IEl+oo)

for

y>O implies that the integration in t plays a role of a smoothing operator.

Now N is a bounded operator.

Further, it is a contraction on the space

Va={f€

x I lllflllla)

if a>O is chosen appropriately. Actually, we first choose d= 1- 4 C 0 K - ~ l l f o l l ~ , ~ , > s 0,

and then set a= (1- 42)/(2C,,K-') ,

which is the smaller root of the quadratic equation C o ~ - ' a ~ - a +Ilfollm.a,e=O

K

so large that

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50

We easily see that a<2/lfollm,m,j.By virtue of Theorem 2.2.3 and Lemma 2.3.2, we have, for any f , g c V,, (2.3.7)

III"fllI12 llfoll-,a,,9+ CoK-'a2=a II I"f1 -"sllII = IllNo[f+s, f-sll II 5 2CoK-'alIIS-Slll 9

7

which shows, since 2COr-'a=1--z/T< 1 , that N is a contraction on V,. Thus N has a unique fixed point f E V,. N o other fixed points exist in any V,, b#a. For, if g € V, be such a fixed point, we have, similar to (2.3.7),

I II f -

gl I I = I I I"f1

so we conclude Illf-glll=O, proved the

- "sll

choosing

K

II

CoK-Ya+b)l

I If-sll I

9

large enough. Summarizing, we have

Theorem 2.3.3. Suppose (2.1.1) and (2.2.5) be fulfilled. Then, for any f o € X z , p , n>O, @20,there are constants K , T>O such that (2.3.2) has a unique solution f € -LK,,d[--T,TI) satisfying Ilf(f)llm.n-I.,l~.B~211follmra,B

Remark 2.3.4. to (2.3.1).

If, in addition, a,,,fo

-

c X ; , B , then f is a classical solution

Remark 2.3.5. Thus the Boltzmann equation can be solved backwards as well as forwards, in t. However, f ( t ) is not in the same function class as f,, and in a worse class as It1 increases. This is the same situation as in the Cauchy-Kowalewski theorem. But, f happens to be within the same class as f , (see Theorem 2.4.3 below). 2.4.

Nonnegativity of solutions In this section, we always assume q, f o 2 0 . We shall prove thatf(t)>O for t 2 0 , a physically plausible property required for the Boltzmann equation because f is the density and q is the cross section. First, we write Q2 of (2.1.4) as

and (2.3.1) as

Given a function g, we define the operator V ( t ,s; g) by

Solutions of the Boltzmann Equation

51

Obviously, f= V(r, s; g)fosolves a linearized equation of (2.4.2), and similarly as before, we can write (2.4.2) as, instead of (2.3.2),

Recall (2.3.5) and set,

X + = X L , = l f € &,,.B([O,

TI) l f ( 4 2 0 ) .

Here, t is in [0, T I , not in [ - T , T I , and similarly for Then, @>O and hence,

111.111.

Suppose f,g €

X+.

for O < s < t < T = a / ( 2 r ) . The latter is proved noting that ~ [ f satisfies ] the linear version of the estimate (2.1.3) and proceeding as in the proof of Lemma 2.3.2. Repeat the proof of (2.3.7), using (2.4.4), to see that a slightly modified version of (2.3.7) holds for N , and for f, g € VanX+. Furthermore, (2.4.5)

N,[f]>O

whenever f 2 0 .

Consequently, N , is contractive on V a nX'. It is clear that the fixed point of N , coincides with that of N . Thus, we proved the Theorem 2.4.1. Let f = f ( t ) , t~ [ - T , TI be the solution given in Theorem 2.3.3. Thenf(t)>Ofor t € [ O ,TI i f q , f o 2 0 . Remark 2.4.2. This is not the case for t € [ - T , 0). For, then (2.4.4) no longer holds and N , is not contractive. Now, we can have a solution belonging to the same class as f,. Theorem 2.4.3. Let f be that in Theorem 2.3.3. Under the situation of Theorem 2.1.3, assume further that y < 1 and q, f o > O . Then, f € L"([O, T I ] ;X;,& with some TI < T . Proof. If y<1, then (2.1.8) shows that Q z is bounded, so by the help of (2.4.4), we can deduce from (2.4.3) that

II If1II'= I IINl[fllI / I < Ilf o II+ CT'II If1I

9

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52

II-II

where

and

111-111’

denote the norms of X ; , p and L”([O, TI]; X ; , , ) =

X,,,,,([O, T’])respectively. If T’ is small, we have lllflll’5~llfoll. Remark 2.4.4. Theorem 2.4.1 is stated in [S, 24, 361 and Theorem 2.4.3 in [l], all as existence theorems. 3.

Local Solutions, 11. The case of non-cutoff potentials

Let us turn to the non-cutoff case, where Q behaves like a pseudodifferential operator: It is well-defined only for smooth functions and induces the loss of smoothness (see S 3.1). Following [40], we show the local existence in the scale of spaces consisting of functions which are analytic in x and of Gevrey class in E. Up to the present, this scale is the only one in which the Cauchy problem can be solved. The existence of less smooth solutions is a n open big problem, a contrast to the spatially homogeneous case [3]. Also, the convergence of the cutoff (1.2.1) will be shown. 3.1.

The operator Q in Gevrey class Throughout this section we assume that q(v, 19) is measurable and satisfies /q(u, ell I q , ( l + v - r + v r ) l

(3.1.1)

cos 01-r‘ ,

with

This involves the inverse power law potential (1.1.5) for s > 2 . Although we deal with smooth functions only, q need not be smooth. Since 7‘2 1, Q cannot be evaluated termwise like (2.1.5)because the integrals in o is divergent, but it can be if suitably combined. To see this, define, Q,,j = 1 , 2, by

QJf,gl=

(3.1.2)

1

RnxSn-1

q(u, e){f(s)-f(E)}s(v’)de’do

and similarly for Q,with {f(v)--f(E)}g(v’) replaced by {f(q’)--f(E’)}g(E).This definition is different from that of Q, in (2.1.4), but (2.1.5) remains to hold. Since 17°C’ at 8=lr/2 by (1.1.3), the quantities in the above vanish and may cancel out the singularity of q, iff, g are smooth enough. T o illustrate this, let p,=exp (a<E>z)and put u=p,f, w=p,g. From (1.1.3) we see that

v=e,

Qi

{ . a * }

[f,gl(E)=p,’(E)

Let llull=sup lu(E)I and let lula=sup Iu(p)-u(E)l/lv-Elg be the Holder quotient. We have

Solutions of the Boltzmann Equation

53

where

~=j

R n xS"-1

m, s)ip,-i(E')is-Eidd~'do .

From (1.1.3), 1~-El=IvcosOl, so we get by (3.1.1),

Z Ic

1

(1 +v-'+Z)r)Dap,-'(E')dE'

R"

1:

lcos 01-r'+dd0

The integral in 0 converges if --y'+6> - 1, whereas that in 5' can be evaluated using (2.1.9). The result is Zr+d. QZcan be estimated similarly. Thus Q,, and hence Q , are well-defined if 6 E (7'- 1, 11. For our purpose, however, this estimate is not convenient, and we shall deduce a n estimate in Gevrey classes re$;, defined as follows. Let pa be as above and define the norm

Definition 3.1.1. Let a , p, ,020 and f = f ( x , E ) E Cm(RnxR") satisfying

X,

v2l.

rg;:b is the set of functions

(3.1.4) where the sum is taken for all k, 1 E N". Remark 3.1.2. ( i ) Endowed with the norm (3.1.4), algebra. This is seen by Leibniz' rule and by

(3.1.5)

k!m!/(k+m)!
is a Banach

for all k, r n E N n ,

, is analytic in x in the strip R"+i{y€R* I l y l < p } , ( i i ) If ~ = 1 fE7Ei$ and similarly in E if Y = 1. (iii) If a'
(3.1.6)

ll~tflla.P,c,

P-0, Y

and similarly for 8, in place of 3,.

(3.1.7)

IC " p Y - l ~ - YIla. l l fp , r , p , u

9

This follows from (3.1.4) and

s,uy (l-ulp)*((s+l)"
where C U = ~ ~ p 1e-$(s+ 2 : a 1)".

S . UKAI

54

Theorem 3.1.3. Suppose (3.1.1) andlet a>O, 6 € ( f - l , 11 and ~ 2 1 . Then there is a constant C 2 0 and for any p, p>O, r > l , /3€(0, a ) , u c ( 0 , p ) and f , g € r:$‘,$, it holds that

IIQrf,

(3.1.8) where

II.II=ll.Ila,p,r,P,”

a)llfll llgll

~ l l I n - ~ . p . ~ , p - a . P, ~ ~ ~ ~ ~ ( P ~9

and a,(

p, a) =/3-

(7

+ +

+al 2{ /3F2 ( 1 p)pyd-lu-yd} .

Moreover, C=C, is decreasing in a , and C,+O (a--,co), -00

(a-0).

This is a modified version of the previous estimate given in terms of the Holder norm. For details of the proof, see [40]. Comparing with (3.1.6), we see that Q is a pseudo-differential operator in E of order 6 € ( f - 1, 11. Construction of solutions In solving the Cauchy problem (2.3.1) under the assumption (3.1.1), we shall restrict ourselves in the case a(x, E ) = O : Thus,

3.2.

fc+E*vzf=Q[fl f Ic=o=fo . 9

(3.2.1)

We rewrite this in the form of the integral equation (2.3.2), where, according to Remark 2.2.2, U ( t )is given by U(t)fo=fo(X--rE, E )

(3.2.2)

*

With this, we recall the definition of the operators N o and N of (2.3.3) and (2.3.4). In order to find a fixed point of N , we shall first study U ( t ) of (3.2.2) in rg;:p. In spite of its simple appearance, and its nice behavior in L p spaces (Theorem 2.2.3), it induces the loss of smoothness in x in our space, as is shown in the Proposition 3.2.1. we have, for I t ] I T ,

Let f o € rg;J:pand assume

V 2 K 2 1 .

II U ( M o l l a , p , . r , p , v i I l f 0 l l a . p . r . p . v p’ = { p - ( pI t I)l’E}K

Then, with T = p / p ,

?

.

For the proof, see [40]. Similarly as in S 2 , we must set the Cauchy problem in a scale of Banach spaces. More precisely, we define the space X = X : y B . p . l , p . a by X 3 f - f = f ( t , x , E ) € C o ( [ - T , T ] C”(RnxRn)), ;

lllflll

Ilf(f)lla-~ltl.,~-llrl.r.P-~,rl.”<

O0

-

55

Solutions of the Boltzmann Equation

In the sequel, T is fixed as

.

T=min (a/2/3, p/21, p/2n)

The essential point is, just as in Lemma 2.3.2, that the integration in t plays a role of a smoothing operator and N o becomes a bounded operator in X . To see this, notice from the last assumption of (3.1.1) that (3-p-f)/2(+ 1) > 1. Choose Y such that 1 < Y < (3-r- f)/2(7’- 1)

(3.2.3) Then, r’-1<(2-7)/(1+2u),

so we can find 6 € ( f - l ,

J- 1 < 0“ < (2- r)/(1 +2Y)

(3.2.4) Lemma 3.2.2. above. Then.

. 11 satisfying

.

Let a, @, p, 1, p, o>O with 1>p.

Let ~ = and l let 8,

Y

be as

Since ~ = and l 1>p, and since O s s i t or t i s I O , it holds that p”ip-1ltlf Hence, on account of Remark 3.1.2 (iii) and then by Theorem 3.1.3. we see rhat

plt-sl
@(s, p*”)I@(s,

p-~lsl)iC,-gltl~o(pl~--sl, p-olsl,

The last two norms are majorized by choice of T , Ca-altlI C,,, and ao(pIt -4,p -Alsl,

Illflll,

~ l ~ - ~ l ) l l f ( ~ ) l l J l l.s ( ~ ) l l s

lllglll for Isl
01 t -sI) I(1 +2( 1 +p)pU8-l)al(p, a)aAIt- sl) ,

where a, is as above and a,(T)= T -

(7+28)/2fT-

(7+d+Zd/Z

.

In view of (3.2.4), (r+26)/2<(y+d+2~6)/2<1, so a, is integrable;

56

S. UKAI

Substituting these estimates into (3.2.6), we conclude the lemma. Apart from the definition of norms, the estimate (3.2.5) differs from (2.3.6) only by the factor l i - I replaced by a,(P, a). Further, a,(/$ o)+O as j?, a+w. Therefore, the proof of (2.3.7) applies to our N , and N becomes contractive for large /3, a. Thus, we get, Theorem 3.2.3. Let a, p, p > O and let Y be as in (3.2.3). Then, for every foE r2G;p,there exist constants T , /3, 1, a > 0 such that (3.2.1) has a unique classical solution f =f ( t )on [ - T , T I , satisfying ( i ) f J2,;,/fJ,P,u? ( i i ) lllf 11112llfOlln.p,l.p.Y. (iii) f ( t ) is analytic in x in the strip R " + i { y € R " I lyl
for O < s < t

and t < s < O .

UI

tl)l/"If -sl1/=}" Ip(s) ,

Rewrite this as

p(s)1/r- p( t)"'

2 (p- a1t I ) 1 / ~ 1I-

SI 11.

,

which shows that p(t)"' is monotone for t 2 0 and for t10, differentiable if li > 1, a contradiction.

but also nowhere

Convergence of cutoff approximation Suppose q(v, 8) satisfy (3.1.1) and consider its angular cutoff qe(v,8) given by (1.2.1). Obviously, qe also fulfill (3.1.1), so Theorem 3.2.3 applies. Furthermore, the constants T , /3, 1, a given there d o not change. Denote the solution thus obtained for qf by f ' . The original solution (i.e. for q) will always be denoted by f. 3.3.

Theorem 3.3.1. Under the situation of Theorem 3.2.3, let f and above, for the same initial f,. Then,

f E

be as

111 f-fq[I
E,

and 6 is as in (3.2.4).

For the proof, see [40]. This theorem establishes the convergence of the cutoff approximation in spaces of Gevrey class. Any similar results are not known for less smooth solutions.

57

Solutions of the Boltzmann Equation

As a simple application of the above theorem, we can prove the nonnegativity off of Theorem 3.2.3. Suppose q(v, B ) > O , in addition to (3.1.1).

Theorem 3.3.2.

Then, f =f (t )20

for t € [ O , TI i f f o 2 0 .

Proof. qf satisfies (2.1.1) so that, according to Theorem 2.4.1, f S ( t ) 2 O for t € [ O , T I ] . In general T , may tend to 0 with E , but for our smooth f,, f‘ satisfies Theorem 3.2.3 (ii) independently of E . Hence we can take T,=T, and going to the limit E+O in the previous theorem, we are done. For the non-cutoff case, we cannot write (3.2.1) in the form of the integral equation (2.4.3) which makes use of the nice splitting of Q . Therefore, without appealing to f‘, a direct proof of the above theorem seems difficult. 4.

Global Solutions for the Cauchy Problem

We now discuss the global existence for the Cauchy problem to (1.1.1). The restrictions are: potentials are angular cutoff, the external force a ( x , E) is absent and initial data are ‘small’. Otherwise, the global existence is still an open question. The smallness of initial data is a condition specific to our method of constructing global solutions described in S 4.1, which requires nice decay estimates as t+m of solutions of the linearized problem. A method of obtaining such estimates is given in S 4.2 and is used in S 4.4 assuming the angular cutoff hard potential and a ( x , E ) = O . Two types of global solutions are presented: one is near non-zero Maxwellian and one near zero. The latter is much simpler to construct but the method employed seems applicable only for the Cauchy problem. The scheme Here we present a method for constructing global solutions. Consider the following equation of evolution in a Banach space X . 4.1.

*=Bu(t)+T[u(t)]

(4.1.1)

dt

where B is a linear operator in X while (4.1.2)

(i)

,

t 2 0 ; u(O)=u, ,

r is nonlinear.

Suppose,

B generates a C,-semigroup erB on X which has the decay estimate, IletBu0II
t20 ,

with constants C,20 and a>O independent of t , u,. ( i i ) r[O]=O and, with some constants a,, C,, r 2 0 ,

Ilrtul -rtv1 II ICI( 1. + Ilv II) r Ilu - II 2,

holds for all u, v € X with norms bounded by a,.

7

S. UKAI

58

Of course II. I I denotes the norm of X . Note that a reptesents the decay rate of solutions of the linearized problem to (4.1.1) and y + l the degree of the nonlinearity of r. Write (4.1.1) as the integral equation,

s:

(4.1.3)

u(t)=etBuo+ e ( t - x ) B[&)Ids f

.

Theorem 4.1.1. Suppose, in addition, a > ( y + l ) - I . Then, there are constants a,, a, > O such that for any uoE X with j[uo[l
Ilu(t)II< d + t ) - "

.

Proof. Set III~III=S~Ptto (1+t)=llu(t)Il and V,=Iu€Co([O, m); X ) 1 I l l u l l l ~ 4 . Define the operator N by writing the right-hand side of (4.1.3) as N [ u ] ( t ) . We shall show that N is a contraction on V , with a suitable choice of a>O. For this, note that if a > (y+ l)-l,

5:

,

(l+t-s)-"(l+s)-u(rtl)dsiC,(l+r)-"

t20

,

holds, which, together with (4.1.2), yields, for all u, v € V,, a l a , , (4.1.4)

II]"I I II 5 coII uo I1+ C,Czar + I III"u1 -"vll I I I ClCz(2a)rII Iu- 41I 9

This implies that if a, a , are chosen small enough and if lluoll
An asymptotic behavior of a semigroup In this section H denotes a Hilbert space with the inner product ( , ) and norm [I 11, and A and K are linear operators in H subject to the condition, (4.2.1) ( i ) A generates a C,-semigroup etA with I[etAll
72ptd

llKR(y+ir, A)II-O

(lrlbm) ,

where 11 I( denotes the operator norm and R ( I , A)=(RZ-A)-' is the resolvent of A which exists for Re I > & due to (i) ([25]). Denote the domain of A by D ( A ) , and the resolvent set and spectrum of A by p ( A ) and u ( A ) , respectively. The set of discrete eigenvalues of A with finite multiplicity is denoted by ud(A) and the essential spectrum ([25, p. 2431) by a,(B). Finally, C,(p)= (A E C I Re ;I2/3}.

Solutions of the Boltzmann Equation

59

Define the operator B by D(B)=D(A).

B=A+K,

By (i) and (ii) in the above, B is a semigroup generator. Our aim is to deduce a n asymptotic behavior of e t B . To this end, we first discuss o(B).

Theorem 4.2.1. ( i ) a ( B ) c C - ( p +llKIl), p(B)3C+(p+IlKll). (ii) a,(~)=o,(~)co(~)cC-(p). (iii) a(B)n C + ( P + S ) is in o,(B) and is a finite set, where 6 is that of (4.2.1) (iii).

Proof. (i) and (ii) follow from [25, Theorems IX. 2.1 and IV. 5.351 respectively. Set G(R)=KR(R, A ) and write the second resolvent equation, R(R, B)=R(R, A ) + R ( I , B)G(I) .

(4.2.2) Solve this as

R ( I , B)=R(R, A)(Z-G(R))-' ,

(4.2.3) and conclude that

(4.2.4)

R € p(A) and I-G(R) is boundedly invertible

3

R E p(B) .

Now, by [25, Theorem VII. 1.91, a(B)n C+(p)is contained in a,(B), with possible accumulation points only on the boundary of C+(p). Put ao=p+6. In view of (4.2.1) (iii), one can find a r,>O such that IlG(R)]I<1/2 for all RELY= (?+(a,) f l {R I IIm 21 >ro}. Therefore, by the Neumann series, Z-G(R) can be boundedly invertible with (4.2.5)

ll(Z-G(I))-1/l<2,

for all R E L Y .

From this and (4.2.4), a(B) does not contain Z. Therefore, by (i) of the present theorem, a,(B)nC+(a,) is a compact set, and thereby, necessarily a finite set. To proceed further, we need the

Lemma 4.2.2. For any r > p, we have,

'j* llR(r+ir,

~)ull2dr<~~K(r--~)-lllull2 ,

-m

Proof. Recall ([21]) that (4.2.6)

R(I, A ) =

'j- e-lte 0

dt,

tA

REC+(p).

~

E

.H

S . UKAI

60

Let ~ + ( be t ) such that ~ + = for 1 t 2 0 and =O for t < O .

Then,

s".,

+

R( y ir, A ) = ( 2 ~ ) - " ~ e-""{ ( 2 7 ~ ) t( " t~)e-rcecA}dt ~ . This is a Fourier transform, and since Parseval' equality is valid for Hilbert space valued functions, one gets,

Using (4.2.1) (i), we conclude the lemma. According to Theorem 4.2.1 (iii), the eigenvalues R , of B in Ct(uO)are finite in number, so they can be enumerated as

-

Re R,2Re d , 2

2 R e R,2u0

.

Denote the multiplicity, eigenprojection and eigennilpotent of R j by m,,P, and Q,,respectively. Write 7

P= 2 P, . j=1

The main result of this section is as follows. Theorem 4.2.3.

Suppose Re R,#uo=P+6.

(ii) IletB(l-P)II
Then,

.

Proof. The inverse Laplace transform of (4.2.6) for B is, [21],

with y sufficiently large. By (4.2.2) and (4.2.3), we have R ( I , B ) = R ( I , A ) + Z(& Z(d)=R(R,A)(Z-G(d))-lKR(R, A ) . Substituting this,

(4.2.7)

1 ecBu=etAu+s-lim-U,,,(t)u , a-m 2111

U,,,(t) =

r+ia

eltZ(R)d;i .

r--ta

The main ingredient is to shift the integration path to the line ReR=u,. Since Z(R) is meromorphic in Ct(uo), and with ro of (4.2.5), we have,

61

Solutions of the Boltzmann Equation 7

Ua,r(t)=2jri C Res e2CZ(I)+Z+Ua,,,,,(t), 1=1 2 = 2 j

for all a > r o , where Res means the residue and

, by [25, p. 1811, we find First, since I j € C + ( , 8 ) c p ( A )and Res eAcZ(I) = Res e"R(I, B ) =eijc(.. 2=2j

2=2j

"'l +2 -Q$). k! tk

k=l

Second, it readily follows from (4.2.1) (i) that (4.2.8)

IIR(I,A)II
Rel>B.

Combine this with (4.2.1) (iii) and (4.2.5) to deduce that llZ(rkiu)ll+O (@-.a) uniformly for y>uo, whence IlZll-0 (a-too). Finally, since I,#u, and by [25, Theorem VII. 1.91, ~ ~ ( Z - G ( u o + i r )
~ ~ ~ ( u , + i r )llR(uo-ir, ull A*)vlldr ,


where A* is adjoint to A. Lemma 4.2.2 is valid also for A*, so by Schwarz' 11011. inequality, the last integral is found to be majorized by Con(ao-,8)-111~11 This implies not only that Ua,o,,(t)converges as a+oo in the weak operator topology, but also that the limit Um,,,,,(f)satisfies

II Um,oo(t) II

Czeaot,

t€R

.

Combining these estimates in (4.2.7) proves the theorem. It is clear that eCB(Z--P)=eCn+(2~i)-~U,,,,,,( t). The above theorem has been proved in [37] in a concrete context of the Boltzmann equation.

4.3.

The linearized Boltzmann operator Our aim is to find a global solution to the Cauchy problem,

(4.3.1) We seek a solution near a n absolute Maxwellian g,#O. Here, go may be any Maxwellian, but can be normalized, without loss of generality, as

62

S. UKAI

go(E)=e-lel*/*.

(4.3.2) Putf=g,+g;% (4.3.3)

Then, (4.3.1) is reduced to u,=--E*V,u+Lu+f[u] ~I,=o=~o

,

7

where (4.3.4)

Lu=2g;'/*Q[g0, g;"~] , f [ u , ~ ] = g ; l / ~ Q [ g ~gA/2~] /~u, .

I'[u]=f"u, u] ,

The linearized Boltzmann operator B is the linear operator appearing in (4.3.3);

B= -f.V,+L

(4.3.5)

.

Then, (4.3.3) takes the form (4.1.1). Therefore, if B and f fulfill the assumption (4.1.2), Theorem 4.1.1 will have given global solutions. In particular, since our r is quadratic, i.e., r=1, it will suffice to deduce the estimate (4.1.2) (i) with a > 1 / 2 . This will be done in the next section. In our situation, however, r is unbounded, so it violates (4.1.2) (ii). In § 4.5, we will give a modified version of Theorem 4.1.1 which ensures the global existence for (4.3.3). In the rest of this section, we state some fundamental properties of L and r. We assume the angular cutoff hard potential proposed by Grad in [18], which includes (1.1.4) and angular cutoff of (1.1.5) for s 2 4 . In [18], the followings are proved. First, (4.3.6)

L=--Y(E)x + K ,

where -Y(E)=Y[~~](E) (see (2.4.1)) and satisfies (4.3.7)

O < Y , I I J ( ~ ~ Y ~ < E ,> ~

with some positive numbers

yo,

IJ,and

<E>=(l+IEIz)'/*

~ 2 0 and ,

where K(E, E') is real measurable on R* x R" satisfying (4.3.9) ( i 1 W E , E')=K(E', E ) ,

,

Solutions of the Boltzmann Equation

63

with constants k,, k,>0 independent of E. Actually (iii) holds only for n=3, and should be modified for n f 3 . Here, we assume it for all n to simplify the argument. Denote by B(X, Y ) the set of all linear bounded operators from a Banach space X into another Banach space Y, and by C ( X , Y ) its subset consisting of compact operators. We write B ( X )=B(X, X ) and similarly for C ( X ) . Set, L2=L2(Rn),

L;={u(E) 1 <E>b(E)

€

L"(R")} .

Proposition 4.3.1. ( i ) K € B(L2)n B(L;, L;+J, p € R. ( i i ) K € B ( L 2 , L;). (iii) K C C ( L z ) . Proof. (i) and (ii) follows from (4.3.9) (ii) and (iii) respectively. Let 1 for IE[ < R and =O for 161 > R . In view of (4.3.9) (ii),

X,(E) be such that X,=

~ ~ ( z - ~ R ) K ~ ~
holds in B(LZ),while, thanks to (4.3.9) (iii), X,K is of Hilbert-Schmit type, i.e., XRK€ C ( L z ) . Then, [25, Theorem 111. 4.71 completes the proof of (iii). Let L be defined in L2 with the maximal domain D ( L ) and similarly for the multiplication operator u(E)x. Then, both are self-adjoint, with D ( L ) = D ( v ( E ) x ) . As for the spectrum, we have the Proposition 4.3.2. ( i ) o,(L)=o,(-vy(E)X)c(--, ( i i ) a ( L ) c ( - - , 01. (iii) a ( L )n (--yo, OIco,(L). (iv) 0 E u,(L) with the eigenspace spanned by

(4.3.10)

$j=go1'21zj ,

j=O, 1,

-vO].

- - -,n + I ,

where h,'s are those of Proposition 1.1.1. Proof. (i) and (iii) can be proved similarly as Theorem 4.2.1, by the aid of (4.3.7) and the previous proposition. (ii) and (iv) come from Propostion 1.1.1. Denote the eigenprojection for 0 € UJL)by P o . For f , we see

Lemma 4.3.3. ( i ) P , r [ u , vI=O. ( i i ) With some constant C20, Ilu(C)-'f[u, v]II
Proof. (i) is Proposition 1.1.1 (i), and (ii) is roughly Theorem 2.1.1, see ~91.

S. UKAI

64

Decay estimates of etB Let us confirm (4.1.2)(i) with a > 1/2 for our B of (4.3.5). However, this is possible only if the norms for etBuoand uo are chosen differently, as will be explained later. To start, we shall use the Fourier transformation in x ;

4.4.

k € R n, i = d q

s z u = f i ( k , E ) = ( ~ R ) - ~ ' ~ e-ik"2u(x, 5)dx , SR.

.

The Fourier transform of (4.3.5) is,

Introduce the auxiliary operator

A ( k ) = -(ik.E+v(E))x

(4.4.2)

i

We look at the dual variable k of x as a parameter and at the operators A ( k ) and B(k) as those in L 2 = L 2 ( R ; )with the maximal domains. Then, from (4.3.6), (4.4.3)

B(k)=A(k)+K,

D ( B ( k ) ) = D ( A ( k ) ) = { uLz € 1 k - f u ,Y(E)u€L'},

the last equality of which comes readily because ik.6 is purely imaginary while is real. The first step is to apply Theorem 4.2.3 to B ( k ) . Thus we shall establish (4.2.1) for A ( k ) and K , taking H = L 2 . We note from (4.4.2) that

v(E)

et"(k'

=,-(ik.E+Y(E))

x

t

which, together with (4.3.7), gives (4.2.1)(i) for A = A ( k ) with j3= -vo, irrespective of k . And (4.2.1) (ii) is just Proposition 4.3.1, so it remains to prove (iii). It is important to show that (iii) holds uniformly for k € R n . Put G ( I , k ) =KR(R, A ( k ) ) . Lemma 4.4.1. For any 6>0, we have, IIG(I, k)ll-O(lkl-co) uniformly for I e C + ( - v o + 6 ) , ( i i ) IIG(o+ir, k ) / l - O ( l r l + ~ ) uniforrnzy for o>-vo+6 and k , I k l I r , f o r each fixed K,,> 0 . (i)

Proof. Let X R be as in the proof of Proposition 4.3.1 and put G,= KXRR(I,A ( k ) ) . By (4.4.2), we have R ( I , A(k))=(R+ik.E+v(E))-'x, so by the aid of (4.3.9) (iii) and Schwarz' inequality,

Solutions of the Boltzmann Equation

65

Denote the last integral by Z=Z(R, k , R ) . Set R=a+ir and

.ZC,={E€ R" I 151< R , Ir+k.El I e l k l } , where e > O .

H2={E€ R" I IEI < R)\Z,,

It is easy to see that mes El CeRn-' ,

mes E2I CR" ,

holds where mes means the measure in Rn and C 2 0 is independent of k, R, E , r. Then, for o 2 - u 0 + 6 ,

I=

L'

+

sz,


Choose ~ = ( R / l k l ) ~and / ~ ,find, llGR[l
On the other hand, put GR'=K(Z-XX,)R(R,A ( k ) ) . Then, by (4.2.8) and (4.3.9), we get I[GR'[lIC(1+R)-'6-' ,

(4.4.4)

Re12-uo+G

.

Now, choosing R=lk12/(3n+"in these two estimates, we conclude (i) of the lemma. T o prove (ii), let Irl 2 2 ~ 8 .Then, Ir+k.El> lrl/2 whenever Ikl I r , and 151I R, so that for a > -u0+S, Z
Choose here and in (4.4.4), R = I T ~ ~ ' ( ~which + ~ ) , is possible for Irl22s0R= 2rolr12/(n+2' or for I r ~ > ( 2 ~ ~ ) ~Thus, + ~ / (ii) ~ . follows. The above lemma assures (4.2.1)(iii) for our A ( k ) and K , uniformly for k € R n and with any d>O. Now that all conditions of (42.1) are confirmed, Theorems 4.2.1 and 4.2.3 apply to B(k) for all k € R n , with ,B=-u0 and for any d>O. R,, P,, Q, possibly depend on k , and will be denoted as R,(k), etc. The next step is to discuss these quantities. The following theorem is due to [16]. Set S 1 [ r ] = { k E R n I I k l S r } , SZ[r]=Rn\S1[r]. Theorem 4.4.2.

There exist positive numbers -,n+ 1, such that ( i ) f o r any k € Sl[~,l,

p j ( r ) € Cm([-ro, r , ] ) ,j = O ,

--

lie,

uo( < y o ) , and functions

u , ( ~ ( kn ) )C + ( ~ o ) = { ~ j ( k ),~ , " = R +, (dW. = P ~ ( I,~ I ) /f,(") = i&)K - p$%2+0

with py)€ R and py)> 0 , and

(I 4 3 )

( llil-0)

,

66

S . UKAI

for j = O ,

-,n+ 1,

where PSo)are orthogonal projections with

c P:"(k) ,

n+l

Po=

j=0

Po being that in Lemma 4.3.3, and P y ) ( k )e B(L2)with uniformly bounded norms, and ( i i ) for any k c S z [ l i o ]u(B(k))nC+(-a,)=Q,. , From this and Theorem 4.3.3, we conclude the

Theorem 4.4.3. There is a constant C>O such that, ( i 1 for any k E S1[liOI, n+l &B(k)

=

2 epj ( ' x' ) t P j ( k)U+( t ,k ) ,

j=O

[lU(t,k)IIICe-"ot,

r20,

( i i ) for any ~ € S , [ K - ,IlecB(k)/IICe-"oc, ], t20. In the above, the constant C can be taken independently of k because we have established (4.2.1) (iii) uniformly in k. Now we can deduce a decay estimate of ere. Let HL(R;) be the L2-Sobolev space, and define, Hl=L2(R;; H1(R;)) ,

Lg,2=L2(R;; Lq(R;))

.

Theorem 4.4.4. For any I E R and qe [1, 21, there is a constant C>_Osuch that, ( i ) IIecBull IC(1 t)-"ll u IIH l n Lq.2, (ii) IlecB(Z-Po)uII C(1+ t ) - a - 1 ' 2 1 [ u I I H ~ n ~ q 1 2 , with a=(n/2)(l/q- 1/2).

+

Proof. (4.4.5)

Referring to (4.4.3) and by Parseval's equality,

\

llefBuII~z= (ItIkl)2111etB(x)li(k, t ) l l h R ; ) d k. Rn

Split the integral over S , [ K o ] and S2[liO], and denote the respective integrals as ZI and Z,. By virtue of Theorem 4.4.3 (ii), I z ~ C e - z u ~ t I l u, l l ~ , whereas, by (i) and Theorem 4.4.2 (i) for P,(k), I,I

CCY I ~+ e, - ~2 ~ o ~ ~,~ u ~ ~ ~ c ) 1-0

Solutions of the Boltzmann Equation

61

so by the aid of Holder's inequality and then, thanks to the well-known interpolation inequality for the Fourier transformation,

where

which is majorized by C(l+r)-(nt*)'2if m 2 0 . These estimates prove (i) of the theorem. T o prove (ii), it suffices to take account of Theorem 4.4.2 (i) and put m=q' in the above. Remark 4.4.5. mum at q= 1.

Thus, a > 1/2 if q E [l, (l/n+l/2)-l) and a=n/4 is the maxi-

The nonlinear operator the space HL,a defined by (4.4.7)

K , p 3 24

-

r of (4.3.4)is not well-defined

L,ARf ;H1(R:)) II~II1,~--suP<EYllu(.,

in H,, but it is, in

7

E

E)llHhR;)

<

*

This will be shown in the next section. Hence, the preceding decay estimate should be translated into that in this space, to solve the nonlinear problem. This can be done by Grad's nice idea [19]. Set, (4.4.8)

(1+t)"llu(t)ll1,,E.

lll~llla*l,p=~~

Theorem4.4.6. Letq€[1,2],Z€R,B20,m=O, 1,andput a=(n/2)(l/q-11/2). Then, there is a constant C 2 0 and for any u,

.

IlletB(~-~o)m~IIIat,,z*L*pl~II~llxz,B"Hz"L~'~

Proof. Define the operator A = -E-V,-v(E) Remark 2.2.2 and (2.4.2) that

x . We readily see from

S. UKAI

68 et.4u=

e-u(E)t

U(X-tE,

5) ,

and hence, by (4.3.7), that (4.4.9)

IletAll<e-"Ot,

in B(H,) and B(H,,,)

.

On the other hand, since B = A + K , etBsolves uniquely the equation (4.4.10)

e(t-a).4KeoBdS.

elB=etA+\* 0

For any Banach space X,we define the norm,

a special case of which is (4.4.8), with X=H,,,. (ii) and (4.4.9), we get readily from (4.4.10), (4.4.1 1)

llletB411a,~< c(llullX+ IlletBullla,u),

Using Proposition 4.3.1 (i), a20,

for the pairs X = H , , , , Y=H,, and X=H,,,+,, Y = H , , , , 1 2 0 . The iterative use of this with respect to then shows that this is also valid for the pair X= H,,,, Y=H,, for any 1 2 0 . Now, Theorem 4.4.4 giving a n estimate of IletBu]Ia,xz completes the proof of the theorem. In the space H,,,, the operator r happens to be unbounded, like Q in S 2.1, but can be handled similarly. For this, define the operator Y by (4.4.12)

, (?@f)(t)= e(t-s)B(Z-Po)Af(s)ds Jo

where A = v ( E ) x . By (4.3.7), v ( 5 ) may tend to 00 with [ E l , so A is a n unbounded operator. Nevertheless, Y is bounded because the integration in t plays a role of a smoothing operator, as in S 2.3. Theorem 4.4.7. LetO0andO O for n=2. Then,

IIIYflIla, t , ,

2 C(lI If1I12a'.r,a+

III Afl IL',H,"

L1.Z)

.

Proof. On account of nice behaviors of q5j of (4.3.10), we see that Pod is bounded. Define

Y,,, f = \ ' e(~--")A(Z-Po)mAf(s)ds,m=O, 1 . 0

Similarly as in the proof of Lemma 2.3.2, with E ( x , E ) replaced by can easily prove for any a 2 0 ,

v(E),

we

Solutions of the Bobzmann Equation

69

Combine this with Theorem 4.4.4 for q= 1, to verify the theorem.

4.5. The global existence Now we are ready to solve (4.3.3). Using V, we write the corresponding integral equation (4.1.3) as u(t)= etBuo+(VA-If [u, u])(t)= N [ u ] ( t ),

(4.5.1)

T o find a fixed point of N ,

where Lemma 4.3.3 (i) was taken into account. we need the

Lemma 4.5.1. Let a 2 0 , I>n/2 and ,8> (n+y)/2 where y is that of (4.3.7). Then, there is a constant C20 such that

II IA-lf[u,

vll I Iza, 1 , a+

IIIf[u, 4 I IIza, H p L 1 . 2 5 CIII4I l a ,

1,BI

I1vI lla.1.p

*

Proof. Notice the following three facts: H l , Bis a Banach algebra for I > 4 2 , HLis continuously imbedded in Hl,a if p>n/2, and if u, u € L z in x, then uv E L'. Consequently, the lemma can be substantiated as Lemma 4.3.3 (ii). Suppose q E [ l , min (n, 2)] and set a=a'=(n/2)(l/q-l/2) so that the condition of Theorem 4.4.7 is fulfilled. Combining Theorems 4.4.6 and 4.4.7 with Lemma 4.5.1, we see that N of (4.5.1) satisfies

I II"u1 II la, 1 , p I co IIuo IIH I , n L I , 2 + c,II IUIII", I"I] - "1 Ila, p CI(lII4I Iu. B+ II I4 I 1,

1,

1.

,b

Il,U,

7

@)II Iu-

vII lu.1.p

This gives (4.1.4) ( y = l ) , so N is contractive if uo is small. Thus, we have proved the

Theorem 4.5.2. Let qc [I, min (n, 2)], Z>n/2 and ,8>(n+y)/2. are positive constants a , and a , such that for any uo with I I ~ o l l i , p + IluollLq.2Ian 9

the equation (4.5.1) has a unique global solution uEL"([O,

H1.p)

9

satisfying, with a=(n/2)(l/q- 1/2),

-

III~lllu,l,p~alI1~ollH,,B"'~,~

Then, there

S. UKAI

70

It remains to discuss the regularity of u.

(4.5.2)

Hl,p={uOE

Hl.,9;

Define

IlxR(k, 8u0111,p40

(R-too)}

7

> R , and =O otherwise, k being the dual variable where xR(k,E)= 1 for Ikl+ of x . It can be seen that etAof (4.4.9), and hence elB, are C,-semigroups on Hl,,9,although not in H l , p ,with the domains

D ( A ) = D ( B ) ~ t j , b l , p +, r , r ' = m a x ( r , 1)

(4.5.3) for any I ,

(4.5.4)

/?. Also, A-lrl-,

-1

maps H t , p x H l , pinto H l , p

if

I>n/2, 8 2 0 .

For the proof, see [42]. Therefore, it follows that N[u] is in Co([O,a);H1,& if so is u and if u , € & [ , ~ . Using (4.5.3) with 1, /3 replaced by 1-1, /3-f, we then conclude the Theorem 4.5.3. then,

Let u , uo be as in Theorem 4.4.2. Zf, in addition, U ~ E H ~ , ~ ,

UE

co([o, m ) ; H1,,dncl([o, m); Hl-l,,9-r,),

and u is a classical solution to (4.3.3), and hence so is f=gO+g;% to (4.3.1). Since H l , , 9 c H l - t , p -for e any E > O , the above two theorems assert the

Theorem 4.5.4. Let u , uo be as in Theorem 4.5.2. Then, u is a classical solution to (4.3.3) such that

u€mto, for any

E

m);

K , ~n )co([o,a);L , ~ - A

n cl(to, m ) ; H ~ + ~ , ~ - ~,, - J

> 0.

Remark 4.5.5. ( i ) Thus f(t)+go ( t - a ) at the rate t-", provided f o is close to go, a physically natural conclusion. ( i i ) If n 2 2 , one can take q=2. Then, a=O, but one can show that u(t)-FO strongly in H l , p . Remark 4.5.6. It is in the form of Theorem 4.5.4 that the results of [37,38] were stated. Theorem 4.5.2 has been proved in [28, 32, 381 independently. The periodic case The previous arguments also provide the global existence for the Cauchy problem (4.3.1) associated with the periodic (in x) boundary condition. This is not a trivial result because the special initial boundary value problem for (4.3.1), where the domain Q is a parallelepiped with the specular reflection boundary condition (1.4.4) (ii), is reduced to the periodic case, [19]. 4.6.

71

Solutions of the Boltzmann Equation

In this case it is natural to use the Fourier series, instead of the Fourier transform. Thus, for example, (4.4.5) is replaced by

+

jlecRuII,,l=kzn(l Ikl)zLllecB(k)fi(k, -1 lliz

,

where ii is the Fourier coefficient and HL=L2(R;;H L ( T ; ) ) ,T" being the ndimensional torus. In applying Theorem 4.4.3 to this, we may assume O < K,< 1, from which readily follows the Theorem 4.6.1. There is a constanr oo>O, and for any l € R , ( i 1 IleWI 5 Cllull, ( i i ) ~lecB(Z-Po)ull n/2 and 9, > (n+r)/2. Then, there are positive numbers a,, a, such that for any uo with [luollr,a
eYu(t)llL,a
Remark 4.6.3. ( i ) Theorems 4.5.3-4 are also true, cf. [34]. ( i i ) If Pouo=O,f tends to go exponentially. (iii) The condition Pouo=O does not make a restriction. For, it can be always realized by choosing the Maxwellian go, i.e., its hydrodynamical quantities p, u, T o f (1.1.11) in such a manner that (4.6.1)

< j J ,gO>=
fO>

9

i=o, .

*

- 9

n+

9

<.,

hold, where h, are as in (1.1.8), f a the given initial in (4.3.1), and is the inner product of L z ( T ; x R ; ) . (4.6.1) is equivalent to P,u,=O, and owing to the conservation laws (1.3.1), P,u(t)=O. Notice that for the pure Cauchy problem such a go cannot be found because hJ, go are constant in x, so both sides of (4.6.1) diverge. Theorem 4.6.2 is due to [37], and is the first global existence theorem to (1.1.1). a>

Soft potentials If s < 4 in (1.1.5), the angular cutoff does not lead to (4.3.7), though (4.3.9) is still valid in case s>2. Instead, v ( E ) behaves like

4.7.

72

S. UKAI

(4.7.1) See [18]. For (1.1.5), we have 6=4/s-l. Whereas v ( E ) is bounded away from 0 in (4.3.7), it is not in the present case. A direct consequence is that the spectrum u(L) changes drastically; o,(L)=o,( - v ( E ) x ) = [ - v l , 01 (cf. Proposition 4.3.2). Hence, o e ( B ( k ) ) = { l € CI Re1€[-v1,0]},

(cf. Theorem 4.2.1), and Theorem 4.4.2 is no more valid. Nevertheless, it is possible to deduce decay estimates. First, note that is simply estimated as

where a>O is arbitrary and Cn=supt,o Two different methods which prove that etBtk)inherits this type of decay have been developed so far. In [9], the cutoff operator B,(k)= -ik-E+%,LX, has been introduced, where X R = 1 for /El < R and = O otherwise. Since X,v(E) behaves like (4.3.7), Theorem 4.4.3 holds, of course with modification. Another method, given in [42], is to extract pseudoeigenvalues of B ( k ) imbedded in u , ( B ( k ) ) which behave like p,(lkl) of Theorem 4.4.2. Again, a modified version of Theorem 4.4.3 1s obtained. The remaining step which leads to the global existence is the same. The one and two dimensional cases The estimates (4.3.7), (4.3.9) (and (4.7.1)), upon which the previous results all rely, are valid only for the case of the space dimension n 2 2 . On the other hand, the physically reasonable Boltzmann equation for n = 1 , 2 is not (1.1.1) itself, but is the one reduced from the case n=3 assumingf(t, x, E ) is constant in the variables x 2 , x3 or x , and keeping the variables E always 3-dimensional. Then, the Cauchy problem (4.3.1) is rewritten with the term E-V, replaced by C,"., E,a/dx,, where x € Rn(n= 1,2) and E E R3. The existence theorems 4.5.2-4 and 4.6.2 apply because the operator B ( k ) of (4.4.1) which is our starting point of the existence proof remains unaltered if restricted for k = ( k , , 0, 0) ( n = l ) and k = ( k , , k,, 0) (n=2).

4.8.

Global solutions near zero Theorem 4.5.2 which gives global solutions near go fails if go=O because the constant (1, given there becomes small with the magnitude p of go. Global solutions near 0 have been constructed in [20] and [23], by a method which also takes advantage of a decay of the linear part, although it is much simpler than, and seemingly quite different from, the preceding one. Here, we present a simplified proof of [201, [231. We consider the Cauchy problem (4.3.1) written in the integral equation (2.3.2), in which U ( t )is then (3.2.2); 4.9.

73

Solutions of the Boltzmann Equation

The key point is that, in This is the solution of (4.3.1) linearized at f = O . spite of its simple appearance, it has a decay if fo is nice. Indeed, define the norm

and write L">q=L"(R;; Lg(R;)). Then, if a, P > O , we get,

Illflll=lllflllu,g=~~P II~(-Mt)llu,p t t R

*

Evidently, we have

(4.9.2)

I f ( t , x,

<e-aI~-tE12-81€12

lllflll

f

from which follows the same decay as (4.9.1). Recall the bilinear operator No of (2.3.3). The global existence follows from the

Lemma 4.9.1. Suppose the cutoff (2.2.1) with the restriction O O . Then, IIINo[f,

sllllrclllflll lllslll *

Proof. Let Q,,j = 1 , 2, be given by (2.1.4) and N o , , defined by (2.3.3) with Q replaced by Q,. Noting the group property U(t+s)=U(t)U(s), and then using (4.9.2), we get IU(-t)NO,U,

s l k x, E l l =

11'

U(-s)QJf(s), g(s)ldsl

lJlllflll where

lllglll

Y

74

S . UKAI

(4.9.4)

q

e-alxta(€-€')12~S

-m

Since I X + ~ ( E - E ' ) ~ ~ ~ ~ ~ ( ~ + ( E - E ' ) . X v/ =D ~IE-E'I, ) ~ , we get W I C v - ' , and hence, J I C e x p (-a1x12-P1512) by means of (2.1.7). Now, the lemma follows for N,,,, and similarly for No,2more straightforward. Noting (2.1.5) then proves the lemma for No. Using this and noting the fact IllU(t)folll=Ilfollu,p by definition, we have, for the operator N of (2.3.4),

III"f1lll Ilfollu.p+clllfll12 III"f1 -"91 II I C(lIIf1I I + I I 191I I )I I If-91 I I * 7

As before, this implies that N is a contraction if f,,is small, and thereby, proves the Theorem 4.9.2. Let q(v, 0)be as in Lemma 4.9.1 and a, P > O . Then, there are positive constants a,, a, such that for each f, with IIfollu,8
lllflll ~a~llfolla,p Remark 4.9.3. ( i ) It is astonishing that, in this situation, the global solution exists backwards as well as forwards in t , which is never the case in the situation of S 4.5. ( i i ) The above condition for q(v, 0) is fulfilled by (1.1.4) and the cutoff of (1.1.5) for s>4/3, (n=3). 5. The Initial Boundary Value Problem The local and global existence for the inital boundary value problem to (1.1.1) can be established in the same manner as for the Cauchy problem: First, the linearized problem is solved, and then, the nonlinear term is added as a perturbation. The latter can be handled exactly in the same manner as before, but the former is much more involved. First of all, we must establish a trace theorem which makes sense of the boundary condition (S 5.1). The existence of solutions for the linearized problem which is necessary to solve the nonlinear problem locally is much more delicate to prove than for the Cauchy problem (S 5.2), and the decay estimate which promises the global existence for the nonlinear problem is even more (S 5.4). 5.1.

Trace theorem Many authors have discussed trace theorems associated with the operator

Solutions of the Boltzmann Equation

(5.1.1)

75

( f ,X ,E ) € R X R X R " ,

A = ~ , + ~ . V , + U ( XE).Ve, ,

and related ones, see e.g. [ 5 , 8 , 39,44,45]. Here we follow the line of [39] and show that trace operators (1.4.2) are bounded. In the sequel, 9 is a domain in R", bounded or not, and the boundary aR is piecewise C'

(5.1.2)

.

With S' defined by (1.4.1) and with a T>O,we set, V=QxR", D=(O, T ) x V , I*=(O, T)xS' , V'={T*}X V ,

(5.1.3)

aD+=z'U

v'

,

(same signs)

T + = T , T-=O,

.

The assumption on a ( x , 5 ) is always (2.2.5), but x is within

a;

4x9 E ) = -V,b(x) + a , ( x , 5) ,

(5.1.4)

( i ) ~ E C ~ ( Gb)( ,x ) > l , a, E CL(V ) , ( - a , = v e . a l = 0 . According to Theorem 2.2.1, therefore, (2.2.2) has a unique solution S,(x, E)= ( X , 5 ) for any initial ( x , E ) € V as long as X stays in R. Denote the forward ( t > O ) stay time by t + ( x ,E ) and backward one by t - ( x , E). By definition, S,€V, -t- O . In any case, S , € F at t = f t * . It is now convenient to write y = ( f , x , E ) . The characteristic (integral) curve of A passing through y € D u i?D+u 3D- is given by ( ii

YO, y ) = ( t + s , SJX, 0)

9

-l-(Y)<s
where Z'(y)=min (T*t+t,P ( x , E)), T-=0, T + = T . Obviously, Y E D (-Z-<s
y E F (s=kZ+)

and

Z*=O

(y€aD*).

We claim that if f € L'(D), then f( Y ( . , y ) )E L1(-Z-(y), Z'(y)) for almost all y €a D* and (5.1.5)

holds, where dy=dtdxdE and du*=ln(x)-51dfdo,dE on

I*,=dxdE on V* ,

n(x) being the outward normal to aR and du, the measure on aR.

Set,

S . UKAI

76

These will be used to denote Lp-Lq pairs. Our trace theorem will be established between the spaces, wp= tf€W D ) I LP(D)}7 L$**=Lp(dD'; @do*), B = W = m i n (1, Z + ( Y ) + ~ - ( Y ) )

,

where Af is defined in the following distribution sense. Let f € L:,,(D) and suppose there exists a g € L:,,(D) such that

(f,4) = -(a 9)

(5.1.7)

holds for every 9 € C,l(D) (C1 and support compact in D). Then we put g= Af. Note from (5.1.4) that A * = - A (formally). The trace operators 7; are defined primarily on CA(fi)by (5.1.8)

~ ; f = f ~ a D *

f c c ~ ( o- )

Theorem 5.1.1. Let p E [l, a]. 7; have extensions belonging to B( W,, LZ-') and denoted again by 75. Thus, it holds that (5.1.9)

Ilr$fll,gn*

Cllfllwp=C(IIUIILp(D~ f IIAfllLpCD,)

*

Proof. Let f € W p and write f(s, y ) = f ( Y(s, y ) ) . Since &=@/as holds for $ € C A ( D ) , we can deduce from (5.1.5) and (5.1.7) that for almost all y € aD*, f(s, y ) is absolutely continuous in s and ( 5.1.10)

f(s, Y)'

fb', Y ) +

1'

( i i S ) ( r y)dr ,

8'

holds for any s, s'€ [--2-(y), Z+(y)]. Now we define j $ f ~ f ( l * ( y )y,) , yEi3D' which coincide with (5.1.8) if f C €A(@ (note that CA(D)C~W,). It remains to prove (5.1.9). In (5.1.10), we put s=l'(y)=O but retain s' This gives (5.1.9) forp=oo, while forp
Integrate both sides first with respect to s' on ( - l - ( y ) , l + ( y ) )and then with respect to y over d o * . In view of (5.1.5), we are done. Observe that since l ( y ) = l + ( y ) + Z - ( y ) < Tby definition, 0 and Z are equivalent as weight functions so long as T < 03. In the above, we cannot remove the weight function B if p < 03. For this reason, some authors have obtained Lg:-traces only ([8,451). The present theorem is given in [39] for the case a ( x , E)=O. The space Wp is a nice space to solve the linear problem (S 5.2) but Lg,'traces which are natural traces in W , are not adequate for the boundary con-

Solutions of the Boltzmann Equation

ditions of

77

S 1.4; we need LP.*-traces where, Lp**=Lp(eD+;do+).

Note that Lp,*=LP,*'forp=a but Lp.+$ZLP,.*i f p < a .

w, ={f E wp I 7; f E LP'*} c w,

Define,

.

wp=

Theorem 5.1.2. ( i ) If p = a , then W,. I f 7; f E Lp,+, then 7; f € L p , - and vice versa. Let f E W,, p E [l, a ) . (ii) I n this case, it holds that for any 1€ R, (5.1.11)

JPllf IIPLPCD, +Ilr;f

ll:P.+

= Ilrif IIZP.--P

IflP-'sgn (f)(A+Rlfdy ,

Re SD

where sgn ( f ) = f / l f l gf(x)+O,= O i f f ( x ) = O . Proof. (i) is obvious. One can show that i f f € W,, p < a,then, IflpE W, In (5.1.10), put Ifl" in place o f f and set and Alflp=pRe Iflp-'sgn(f)Af. s=--I-(y), s'=l+(y), yEdD-. Then (5.1.11) follows by integration on dD- and by (5.1.5), proving (ii). The following Green's formula is essential in Theorem 5.1.3. (5.1.12)

Proof. 5.2.

Let f E

wp,g E

p-'+q-'=

1, and 1 € R. Then,

( ( A + 4 f , s)+(f, (A-J)9)=+--

-

Put fg in place off in (5.1.10) and proceed as before.

The linear initial boundary value problem Let A be as in (5.1.1) and M be as in S 1.4.

(5.2.1)

S 5.2.

We shall solve

in D , J E R , on I-, in V- .

(A+;l)f=O r-f=MrFf f(0)=fo

Evidently, 7'f and f(0) =f It=O should be understood as

r'f=r;firi

,

f(T')=rgflYi

.

If llM/l< 1, (5.2.1) can be simply solved by successive approximations

([8,39]), but the case llMll= 1 which involves the physically important examples (1.4.4) (ii)-(iv) is delicate. Three different methods have been developed so far, making use of Riesz' representation theorem [39], the limiting absorption principle [ 5 ] and the monotonicity [8],respectively. Here, following [39], we discuss weak solutions. Only L"-solutions are useful for the nonlinear prob-

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lem, a contrast to the transport equation which requires L'-solutions [8, 451. In the sequel, we assume (5.1.2) on 9, (5.1.4) on a ( x , E ) and the following on M . Set Yps+=Lp(S' I In(x).Elda,dE). (5.2.2)

with the norm

M E B( Ypp k, Yps-)

IlMll I 1

.

Denote the adjoint to M by M*.

Then, for p E [l, a), we have,

(5.2.3)

with

M*EB(Yqs-, Y " + )

llM*ll
For p = m , this is taken as a n additional assumption. See Remark 5.3.2. Note that since we are assuming that M does not act on t, Yp,' can be replaced by Lp,*l. = L * ( I ' ; In(x)-Eldtdu,d~). The weak solution is defined through Green's formula (5.1.12), with the space of test functions 2

W : = ( g E Wlq1 T + g = M * r - g , g ( T ) = O j ,

p-'+q-'=l

.

Suppose f E W p solves (5.2.1) (the strong solution). Recall (5.1.6). Then it follows from (5.1.12) that for any g E W';,

(f,( A-4g)

(5.2.4)

= - v -

.

Definition 5.2.1. Let f o E L p ( V ) . f € L p ( D ) is called a weak solution to (5.2.1) if (5.2.4) holds for every g E W z .

Theorem 5.2.2.

Suppose p E (1,

a],f,, € Lp(

Then a weak solution

V ) , R>O.

f~ Lp(D) exists. Proof. Apply (5.1.11) to g E W z replacing p , R by q, - A Note that q € [ 1 , m) if p E ( 1 , a]. Then by (5.2.3), we have,

(5.2.5) where 11

llgllq

llP is the

9

respectively.

~ ~ ~ " ' ~ - ' ' ~ l l l~l ( A~ -~~ )~ ! JI l ll q; ~

norm of Lq(D) and

9

I.11;

that of Lq( V ) . Define,

Z,={(A-R)g I g € W $ } C L S ( D ).

(5.2.5) shows that for each h c Z , , there exists a unique g € W$ such that h = ( A - R ) g . Therefore, F(h)= -,- is a well-defined functional on Z,. Further, F is bounded because by (5.2.5),

Consequently, F has a bounded extension P to Lq(D) (the Hahn-Banach theorem [15]) and P ( h ) = ( f , h) with some f ELp(D) for any h € L q ( D ) (Riesz' representation theorem [15]). Restricting h in 2, and putting h=(A-R)g, we see that this f is a desired weak solution.

Solutions of the Boltzmann Equation

79

R,emark 5.2.2. For the case p = 1, the above proof works only if llMll< 1, and gives a weak solution in Lm(D)*=ba(D) (the set of bounded additive set functions which vanish on sets of measure 0 [15])2L1(D). When l[M[l<1, the situation is fairly simple: Theorem 5.2.3. Suppose IIMII strong solution and has the estimate

for p = 00 and p

(1,

00)

< 1. Then f of Theorem 5.2.2 is a unique

respectively.

This follows readily from (5.1.11)once the following characterization of the weak solution is established. Theorem 5.2.4. Any weak solution f € L p ( D ) satisfies ( i 1 fe W p ,( A + J ) f = o , ( ii 1 f(0)=f,E Lp(V ) , , or weakly* (iii) r-fa-My+f6-0 ( € 4 0 ) weakly if p ~ ( 1 a) ), (l(y)<~). Lp,+(E+;du'), where f e = x e f and x E = l ( l ( y ) > ~ =O

if

p = a , in

Let us consider the case IIMII=l. Replacing M by KM with K E ( O , 1) in (5.2.1), and by Theorem 5.2.3, we have unique strong solutions f K satisfying (5.2.6) uniformly for K . Hence, passing to a subsequence,

p-.f

in

Lp(D) ,

p(T)+g

in

Lp(V)

as r-tl ,

both weakly (resp. weakly") for p € ( l , 00) (resp. p = 0 0 ) , with some limits f, 4. This f is a weak solution with f ( T ) = $ , and going to the limit in (5.2.6), we conclude the Theorem 5.2.5.

(5.2.7)

When IIM[I = 1, (5.2.1) has a weak soZutionfE Lp(D) such that Ilf(~)IILPCV,

IIlfoIlLPc",

*

The uniqueness is not known without additional conditions on M (see [5]). So far, we have assumed I>O. However, i f f is a solution to (5.2.1), so is eplfwith I replaced by A-p. In particular, (5.2.7) holds for 220. Since f ( T ) € L P ( V )by (5.2.7) and since T may be arbitrary (even negative), we can define the operator V ( t ) ,t = T € R, by

(5.2.8) Theorem 5.2.6.

~ ( t ) f o = f ( t* ) Z f p € (1,

a), V ( t )is

a C,-group on L p ( V ) .

This is not true when p = a , just a s in the case Q = R n

(S 2.2).

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5.3. The local existence We can now solve the initial boundary value problem to (1.1.1): (5.3.1) Let U ( t ) be as in (5.2.8) for 1=0. Then, (5.3.1) is reduced to the integral equation (2.3.2). With pa,Bof (2.2.10), we redefine the space X;,! of (2.2.11) on V . Clearly, Lemma 2.3.1 for Q is also valid in this space, so that if our U ( t )satisfies the estimate

(5.3.2)

I l ~ ( ~ ) f oI l l Ilfoll

in

x:,)4

7

for all a in some interval [a,, a,], a,>a,>O and for some p 2 0 , then we get (2.3.7) and conclude the local existence for (2.3.2). To prove (5.3.2), put f=p;,lPg in (5.2.1) and note that A P ~ , ~ = by O (5.1.4) (ii). We see that g also solves (5.2.1) with M replaced by a a ~ j = p a , f i M p i 9~ ~

and with the initial p;,lBfo=g,. Hence, if &fa,psatisfies (5.2.2-3) for p = w , then (5.2.7) applies to g; lIg(T)II I llgollin L”( V ) . This is nothing but (5.3.2). Thus, we have the same existence theorem as Theorem 2.3.1. Let X,,,,B([-T, T I ) be the space defined by (2.3.5) with our X:,B.

Theorem 5.3.1. Suppose (2.2.1) on q(v, B ) , (5.1.2) on aQ, (5.1.4) on a ( x , f) and (5.2.2-3) on & f a , B for p = w and f o r all a € [ a , , a,] with some aI>az>O and b 2 0 . Then, for any f o € X ; , B , a € [a,, a,], (5.3.1) (reduced lo (2.3.2)) has a unique solution f € [ - T , T I ) with some T , K > 0 satisfying

Ilf(0II-,a-z,t l ,,8 211 f o II-,a,B . Remark 5.3.2. If M is one of (1.4.4) (i)-(iii), then, a a , , = M and fulfills (5.2.2-3), for any a, B E R . If M is (1.4.4) (iv), G,,, also does, but only for a=(2TJ1, p=O. This does not suffice and we must assume that pw is a small parameter independent of T,. This is permitted in a convex linear combination of (1.4.4) (i)-(iv). Remark 5.3.3. If, in addition, M is nonnegativity preserving (i.e., like (2.4.5)), then Theorems 2.4.1 and 2.4.3 are also true for the present case. Thus we can get solutions belonging to the same function class as fa.

5.4. The global existence Up to the present, the only method available to establish the global existence for (5.3.1) is the one described in 5 4 . 1 which promises a solution One might expect that the method of 5 4 . 9 is also in the form f=g,+gh’2u.

Solutions of the Boltzmann Equation

81

useful. However, because of its special choice of norms, the bounded domain cannot be dealt with, and for the case of the unbounded domain with boundary, it seems difficult to deduce a nice estimate like (4.9.4) because 0 of (4.9.3), modified by the boundary condition, has n o longer any simple properties. To apply the method of 4.1, we shall first rewrite (5.3.1) in the form (4.1.1) putting f = g O + g i ’ Z u . Then, the Maxwellian go should be such that Ag,=O and

(5.4.1) Otherwise, inhomogeneous terms appear both in the equation and the boundary condition. The case where (5.4.1) is violated will be discussed in the next section. Now, with such a go, we have,

(5.4.2)

in

D

y-u=l\)r+u,

on on

I-,

u=u,,

where L,

(5.4.3)

,

u,=--E.V,u--a.VEu+Lu+r[u],

V-,

r are as in (4.3.4) and l\)=g;‘f2MgAf2

.

Therefore the linearized Boltzmann operator €3 is,

(5.4.4)

B = --5.Vz--a.V~+L

,

(x,6)€

v,

associated with the boundary condition r - u = f i r + u . The global solution to (5.4.2) can be found by the method of S 4.5, provided elB has a nice decay. In S 4 , we have observed two different types of decay of elB. One is Theorem 4.4.4 for the Cauchy problem, i.e., for the case SZ=Rn, which can be taken as a typical case of the unbounded domain. The other is Theorem 4.6.1 for the case SZ=Tn, to which the initial boundary value problem for a parallelepiped, a special example of the bounded domain, with the specular reflection can be reduced. For the general case, we may infer from this that (a) if 8 is bounded, e t B ( I - P o )decays exponentially, and (b) if Q is unbounded, erBdecays like I r a , a>O. Several works have been done to confirm this, under the assumptions of a ( x , E)=O and Grad’s cutoff hard potential described in S 4.3. Taking (4.3.2) as go, (a) has been shown for the diffuse reflection in [17], for the specular one in [35], both assuming that SZ is convex, and for the general 8 and M in [6] with rather restrictive conditions on M including (5.2.2) and (5.4.1). The proof can be carried out appealing to Theorem 4.2.3. As for (b), the case of the exterior domain to a bounded convex obstacle was studied in [6, 411. This is a special case (c=O) of the result described in the next section. See also [22, 271.

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6. The Stationary Flow The existence and stability of the stationary flow having a prescribed velocity c at infinity and passing arround a n obstacle is one of the classical problems in fluid and gas dynamics and has been discussed extensively. However, most works start from the fluid equations such as the Euler and NavierStokes equations, and few from the Boltzmann equation. Here, following [44], we will show that if c is small, the Boltzmann equation has a stationary solution which is asymptotically stable in t . The case where c is large, especially the case where c is close to the Mach number 1, is a physically more interesting problem in connection with the transonic flow in which the shock appears, but remains unsolved. However, we should mention the works [12, 291 on the one-dimensional shock profile described by the traveling wave solution to the Boltzmann equation, for c near the Mach number 1. Its stability is still a n open question. We assume always Grad's cutoff hard potential and a(x, E)=O. Nothing is known in other situations.

6.1.

The stationary problem Denote the obstacle by 0 ( c R n ) and its exterior by Q. Suppose that, at infinity, the gas is in equilibrium and moving with the velocity c € R". Then, our gas flow is described by,

(6.1.1)

ft=- E * V J + r-f =M r 'f

Q[f I

in

D ,

on C - ,

f-g,(E)=exp(-lE-c12/2)

(Ixl-ta)

f It=o=fo

(t,E ) € R x R " , in V ,

where 9,is the Maxwellian with p, T normalized appropriately. we shall solve the corresponding stationary problem, (6.1.2)

-E-F,f+Q[fl=O r -f = M r 'f

in V , on S - ,

f->9,

for all E E R " ,

(IXl -~~)

First of all,

where f=f(x, 6). Note that g, is not, in general, a solution to (6.1.2) if c Z 0 , because it violates the boundary condition on S - as is seen for M of (1.4.4) (ii) (iii). However, since these M satisfy (5.4.1) with g0=g,=,,, we may expect that if c is small, (6.1.2) has a solution which differs slightly from 9,. T o show this, put f=ge+g;% and reduce (6.1.2) to (6.1.3)

--E.V,u+L,u+r[u]=O @y+u f h , u+o (Ixl400) T-U=

in V , on S - , for ail E € R " ,

Solutions of the Boltzmann Equation

83

where r is as in (4.3.4), ?I? in (5.4.3) and (6.1.4) One might expect that it is more convenient to set f=g,+gE% because L , then becomes selfadjoint in L Z ( R ; )for all c, whereas our L , is not if c f O . However, ?I? then becomes unbounded, for exapmle, for M of (1.4.4) (ii) (iii), which makes (6.1.3) ill-posed. Let B, be the linearized Boltzmann operator,

B,=-E-V,+L,,

(6.1.5)

associated with the boundary condition r-u=?I?rtu, and suppose it have a n inverse B;l. Then, (6.1.2) can be reduced to (6.1.6) where

U+ $c

B,-'I'[u] -$c

is a solution to the linear stationary problem, -E.V,$+L.,$=O

(6.1.7)

=O ,

r-$=?I?r+$+h, $+O (1x1-00)

in V , on S - , for all e E R n .

Once the existence of B;' and $c is known, (6.1.6) can be solved by the implicit function theorem (S 6.3). A delicate problem is B;'. It will be seen in s6.2 that O € a ( B , ) and thus B;' does not exist, in L 2 ( V ) . However, the principle of limiting absorption which is familiar in the scattering theory and enables us to find the values of resolvents on the boundary of spectrum is applicable to our B,. Thus, B;' will be constructed as a limit of R ( I , B,) as 2-0. Denote the solution to (6.1.6) by u,. Then,

f , =9,+9:/2uc solves (6.1.2), and hence, is a stationary solution to (6.1.1). Now, set fc+g~'2v=g,+g~/2(u,+v) and rewrite (6.1.1) as

u,=-E.V,v+Lcv+2r[u,,

v]+I'[v]=O

in D , on 8 - , EER"

vl,=o=vo

f=

in

,

V.

By definition, the stationary solution f , is asymptotically stable if (6.1.8) has a global solution v which tends to 0 as f+w, whenever uo is small. This will

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84

be shown in

S 6.4 by

solving the integral equation

where we have put (6.1.10)

E,(t)=exp (tBJ

9

i.e., the semigroup generated by B,. As in S 4 , (6.1.9) will be solved by the help of a nice decay of E,(t). Note that the linear operator to (6.1.8) is not B, but Bc+2r[u,, -1 and that if the corresponding semigroup is used, the linear term of the right-hand side of (6.1.9) disappears. However, it seems difficult to deduce a decay for that semigroup. The extra linear term in (6.1.8) can be made small with u,, for small c. The limiting absorption principle To illustrate our method, we first discuss B, for the special case S2=Rn, i.e., the operator B," given by (6.1.5) but in V m = R ;x R ; . Then, as in S 4.4, it suffices to study B,m(k)=-ik.E+L,. Under Grad's cutoff hard potential, L, has the same properties as L = L , = , of (4.3.4) except that it is not selfadjoint unless c=O. In particular, 6.2.

(6.2.1)

Lc = -

Y A E ) + K

9

where v e ( E ) = v ( E - c ) and K , is a n integral operator to which Proposition 4.3.1 applies, continuously in c E R". Further, 0 € u,(L;) whose eigenspace is invariant in c, that is, if P, denotes the eigenprojection, then P,=P,, [44]. Using this and Theorem 4.4.2, we can prove the following theorem. Let L2, Lz are as in S 4.3, and lie, uo, p r ( r ) , S J r ] as in Theorem 4.4.2. Set, E(a, U ) = { R E C , ( - - U )

I -ReRla[ImA[2}.

) the maximal domain. Then, for Theorem 6.2.1. Define B; in L 2 ( V mwith any c o 2 0 , there is a positive number a , such that the followings hold for all C~ SJC O l.

( i ) P ( B 3 3 a a 0 , uo)\{O}, 0 € 0,"). ( i i ) R(R, B;)=C:f,2 V,(;C,c), for all ;Ce2'(ao,uo)\{O}, where, for O<j
(6.2.2)

V,(;C,c)=.Fz-lx(k)(R-;Cj(k,c))-'P,(k, C ) F Z , x(k)=l (kES,[rol) =o (keSl[rol) 9 4 =pJ( Ikl) i k . c Pj(k, C ) Bo(Si[rolx Si[col; W L 2 ,LF)) 1520 9 9

+

9

while for j=n+2,

85

Solutions of the Boltzmann Equation

(6.2.3)

U,+,(J,c) E Bo(,Wo,a,) x S1[cOl; B F 2 (v")).

Further, Uj'sare mutually orthogonal, and P,'s are mutually orthogonal projections 00 L2 with P,(k, O)=P,(k), C Pj(0, c)=P,=Po. According to (6.2.3), U,+,(O, c) is a bounded operator, whereas, since Aj(k, c)-l has a singularity at k=O as seen from the asymptotic expansion of ,u,(K) given in Theorem 4.4.2, U,(O, c), O l j l n f l , are unbounded, in L2(Vm). However, since this singularity is integrable, U,(17, c) can be made continuous at R=O, and hence, U,(O, c) become bounded, if the spaces of domain and range are chosen appropriately. This is the principle of limiting absorption. To state this more precisely, we set, (6.2.4)

L ~ , ~ = L $ , r ( V m ) = E{ )~I <E>~u€LI(RF; ~=~(~, Lp(R;))}

Theorem 6.2.2.

.

Let 1 < 4 < 2 < p < w , O € [ O , l), m=O, 1 with

(6.2.5)

4-1 -p-1>

(2-m)/(n+O).

Then, for O < j < n + l , lcleU,(17, c)(Z-P,(O, c))" € Bo(Z(a0,ao)xS,[c,];

B(L:B~,

.

Proof. It suffices to discuss the case m=O. By the interpolation for the Fourier transforms, and then, proceeding as in (4.4.6), IlUj(17, C)UIIL$'"ICII-iTZU~(R, C)UllL;;'.== Ic$llull'q.2

(p-'+p'-'=l)

Y

where, putting 7=4-l--p-l,

+=(

\slcco)

i l - q k , c)1-1/7dk>1.

After a lengthy calculation using the asymptotic expansion of ,u,(K), we see that (D
In order to link BY to B,, it is necessary to solve (6.2.6) for a given h € Yp,- (see (5.2.2)). Suppose, (6.2.7)

0 is a bounded convex domain and

W=aQ

Then, (6.2.6) can be easily (and explicitly!) solved.

is piecewise Cz.

Denote the solution by

86

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u=R,(l)h, R,(R) being the solution operator. Let e be the extension operator from V to V" by 0, and r the restriction operator from V - to I/. Further, set A?Z=r--&?y+ and (6.2.8)

T,(R)=A?ZrR(I, BT)eK,R,(R)

.

After some manupilations taking account of (6.2.6), we have an explicit formula of R(1, B J : (6.2.9)

,

R(2, B,) =rR(1, B:)e+S,(I)(Z- Tc(1))-lA?ZR(A, B,)e S,(R)=R,(l)+rR(R, B,")eK,R,(R)=(r-rR(I, B;)*e)*

.

Originally, this is derived as an equation in L2(V ) for 1 such that 1 € p(B:) n in other spaces as far as the right-hand side makes sense. A crucial point is the invertibility of Z-T,.(I).

p(B,) and l € p ( T c ( l ) )but , can be used to define

Proposition 6.2.3. Let n 2 3 , p~ [2, positive constants a,, c,, a1such that

001,

P>n(2-1-p-1).

Then, there are

(z-Tc(l))-l B o ( m l ,0 1 ) x S,[c,l; B( Y$*-Ni where Y;;,-={u 1 < t > @ uY~p 3 - J . We evaluate the right-hand side of (6.2.9) by the aid of Theorem 6.2.2 and this proposition. Besides, we need some estimates for & ( I ) and must appeal to Grad's argument used in the proofs of Theorems 4.4.6 and 4.4.7. Define L;S'=L;.~(V)by (6.2.4) with V" replaced by V . Set, (6.2.10)

X ; = L g y I p nL ; ' " ,

and set A , = Y , ( ~ ) x . Theorem 6.2.4.

(6.2.11)

zq=L2,2nL q J ,

Our result is, Let n 2 3 , 1 < 4 < 2 < p < m , ,B>n/2, O € [ O , l ) , m=O, 1 with

q-1-p-1>(2-mm)/(n+e)

,

p<1-2/(n+0)

Further, let a € [0, 11 andpur r=l+p-'--q-'. Also, with a,, c,, 6.2.3, set z = 2 ( a 1 , u , ) ~ S , [ c , ] . We have, ( i ) There is a constant C 2 0 andfor any (1, c) € 3,

. u,

of Proposition

+

IcPllR(1, B ~ ) ( Z - ~ ~ ) ~ A ~ u l C(ll~Ilx; l ~ $ ~ y / pl I ~~ 3 4 l z q ) ( i i ) Let e > O and 6>Or. U(C) E

then,

L "( s , [ ~ ,;l X $

If u=u(c) be such that

n B o ( S , [ ~ ,X$-J l; ,

A:,u(c)E Bo(S,[c,l;Zq),

Solutions of the Boltzmann Equation

87

IcldR(R, B,)(Z-P,)~A:U(c)€BO(Z';L;:;,p-E).

Compared with Theorem 6.2.2, the behavior of R ( I , B,) near c=O is worse than that of U,(;C,c). Put m=a=O and let u € X ; n Z q . Then, for c€S,[c,] fixed, R ( I , B,)u E Bo(Z(al,a,); L $ L ~ , ~ so - € that ) , B;'u= -R(O, B,)u E L;:;,p-a exists as a limit as k 0 . Using this inverse, we can solve (6.1.7) in the form, (6.2.12)

$c

Theorem 6.2.5. (6.2.13)

.

=R,(0)h,-B~lK,R,(O)h,

Let n 2 3 , p € [ 2 , 001, O € [ O , 1) with p-'< 1-2/(n+B)

.

Let /3 > n and suppose h, be such that

(6.2.14)

hcEBO(Si[cil; YF*-)t

llhcll=O(l~l) (c+O)

.

Then, $c solves (6.1.7) in Lp-sense and, with r=2-1/p, $,EBD(SJc,l;L;;*")

(6.2.15)

9

ll$cll=~(IcI1-o~).

So far, we have not mentioned the conditions to be imposed on M . Here, we only point out that all the arguments from Proposition 6.2.3 on are valid for M of (1.4.4) (i)-(iii), and for M of (iv) if IT,--T,I

(6.2.16)

Salcl

holds with some a 2 0 , where T,=l is T of our Maxwellian ge. The last condition comes from the second requirement in (6.2.14). The proofs of the statements in this section are all long, and we refer the interested readers to [44]. 6.3. Existence and stability From Theorem 6.2.4 and Lemma 4.3.3, we can see the Proposition 6.3.1. Let n 2 3 , [0,1) and ,!3>n/2+1. P € [2,41 n ((n+0)l(n+0-2), n+O)

(6.3.1) and put

(6.3.2)

r= 1+2/p.

Suppose,

,

There is a constant C20 such that, for c Sl[cI], IIB;'r[u, v]ll
in

X;

.

This and Theorem 6.2.5 enable us to apply the contraction mapping principle to solve the stationary problem (6.1.6). At the first glance, however, (6.3.2) does not seem nice because if we choose 0 f 0 , it diverges as c+O while

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88

in the physically important case n=3, the choice 8=0 is excluded by (6.3.1). It is the nice behavior of $, near c=O given in (6.2.15) that compensates for this defect. Let 8 E [0,2/7) and p 2 2 . Then, we can find (Y such that al = 8( 1+ p - ' ) < a < 1-8(2-p-')

(6.3.3)

= a,

.

Put u=Iclav and rewrite (6.1.6) as

By virtue of (6.2.15) and (6.3.2), it holds that

where II-II is the norm of X ; , C , and C, are positive constants independent of c, v, w, and u=a-a,, r=a,-a. Since u, r>O, G(., c ) becomes contractive for small c, which proves the Theorem 6.3.2. Let n 2 3 , 8 € [0,2/7), p>n/2+1, and suppose (6.3.1) and (6.3.3). Then, there is a positive number c, ( I c l ) such that f o r any c€S1[co], (6.1.6) has a unique solution u, in X ; satisfying

(6.3.4)

IIu,IIX;
a+r=a,= 1-8(2-

l/p) ;

Further, the continuity properties in c stated in Theorems 6.2.4-5 prove that u, € BO(S,[col;X;-J

,

E

>0 .

Also, it can be shown that u, E Wp(V ) and satisfies (6.1.3) in Lp-sense. With this u,, we now solve (6.1.9). Since the second term on its right-hand side is linear, Theorem 4.1.1 must be looked at with y=O, and hence E,(t) must decay faster than t - l . Taking the inverse Laplace transform of (6.2.9) gives a n explicit formula of E , ( t ) ; (6.3.5)

E,(t)=rE~(t)e+(y-rE,(t)*e)*

T D,(t) 9 n;lE:(t)e ,

where E,(t)=exp (tB,"),9 means the convolution in I and D,(t) is the inverse Laplace transform of (I-Tc(,2))-1,see (4.2.7). Knowing Theorem 6.2.1 and following the line of Theorem 4.4.6, we have, Theorem 6.3.2.

(6.3.6)

Let 1 < 9 < 2 < p < o o and m=O, 1. Then,

IIE,(t)(Z--P,)mull.$.-I

C(l+t)-~-~'~lluII,$~""Zs ,

with ~=(n/2)(1/q-l/p)and C 2 0 independent of c , t , u.

89

Solutions of the Boltzmann Equation

By this and Theorem 6.2.2, etc., we have,

Proposition 6.3.3. Let n 2 3 and l e t c , be that of Proposition6.2.3. each 8 € [0, l), there is a constant CTO such thar, ll(~c(t)-Z)ully;~- i Clcl -V +t)-rIIuII,;~-

(6.3.7)

holds for all c€S,[cJ, with r=(n-1+8)/2 is even.

Then, for

,

if n is odd and =(n-1)/2 is n

Substituting these into (6.3.5) yields a desired estimate. Write the righthand side of (6.1.9) as N [ v ] ( t ) . In order to evaluate the second (linear) term of "v], it is necessary that r > l in (6.3.6) ( m = l ) and (6.3.7), while for the third, it suffices that 7>1/2, according to Theorem 4.1.1. For the former, therefore, we should take 8> 0 in (6.3.7) when n=3. Otherwise, we can choose 8=0. If 8>0, a divergent factor IcI-8 appears, but this can be cancelled by (6.3.4). In any case, a careful choice of parameters is required. Write p , 8 of (6.3.1) as p o , 8, and impose the additional condition p ,
P [2,41 n ( ( 1 - 2 / n ~ , (112- U P ~ I - ,~ ) q € [ L 21n [ l , (l/p+l/n)-l) , Oc(0, a ) , p>n/2+1 , r=min ((n/2)(1/q-U~), (n/p,+l)/2, ( n / ~ + 1 ) / 2 ).

(6.3.8) Then,

r> 112.

Set

l l l ~ l l=y: l (1+ t)rllv ( t )11x5 . We have,

III"v1lll I I I"4

+

IC(Ilvollx$nzg ( I C I -@a+Ill~lll)llI~lll) 7 I I C(IC I +a+ II1411+ I IIWII l)llIy- WI II

- "wll

9

where a=lluell in X $ , p=p,. By (6.3.4), IcI-@a+O as c+O, so N is contractive if v, is small as well as c. Thus, we proved,

Theorem 6.3.4. Let n 2 3 and suppose (6.3.8). Then, there are positive numbers a,, a,, co such that f o r any c € S,[c,] and if IIvoll
v = v ( t ) e B o ( [ O00); , X;)

,

llv(t)llia,(l+t)-T .

Now, the stability off, has been established.

7. The Euler Limit and the Initial Layer The justification of the Hilbert expansion has been discussed in [31].

90

S. UKAI

Here, we follow [43] which simplified the argument of [31] on the one hand and is suitable for the study of the initial layer on the other hand. The problem of unboundedness arises in establishing uniform estimates of solutions as E -> 0, and will be resolved by introducing a Banach scale again. We always assume a ( x , 5)=0 and Grad’s cutoff hard potential, and deal with the Cauchy problem only. Otherwise, all are open. In particular, the case SfR” where the boundary layer appears as well as the initial layer is a physically important open problem. Our result is local in t . A long time behavior which may involve the shock layer is also a n open problem. In this respect, however, [12] is suggestive, in which the Chapmann-Enskog approximation of the one-dimensional shock is discussed. 7.1.

The uniform existence of solutions We consider the Cauchy problem to (1.6.2) for all E > O , with a fixed initial data f,, and seek a limit of the solutions f’ as E+O. If such a limit exists and coincides with the first term f of (1.6.1), then the Hilbert expansion will have been justified to the 0-th order. For such a limit to exist, it is primarily necessary that f E exists on the time interval [0, T ] independent of E . According to Theorem 2.3.1, f’exists on [0, ~ € 1 but , T~ is easily checked to tend to 0 with E . O n the other hand, f’ exists globally in t if f, is near go, as shown in Theorem 4.5.2, but f, should approach g, as E + O , i.e., a,+O. The desired solutions exist i f f , is near go and analytic in x . To prove E go this, we shall make use of the result from S 4.4. Put f ’ = g o + g ~ ’ 2 ~where is as in (4.3.2). Then, (1.6.2) is reduced to (7.1.1)

U;=B%c+-

1 r [ u e ],

UfI,=o=u,

E

,

where

B.=-E.V,+L L

.

E

As before, we shall investigate the integral equation, (7.1.2)

uE(r)=etBEuo+€

Roughly speaking, (7.1.3)

lt

.

e(c-s)Bar[~E(~)]d~

0

is uniformly bounded for E > O , while

elBE

1 E

e t B ( Z - P , ) = I V , l a + ~e-‘Ot’€b , E

with uniformly bounded operators a, b and a constant o,>O.

Thus, the un-

Solutions of the Boltzmann Equation

91

bounded factor a - l is replaced by the unbounded operator lVsl, a pseudodifferential operator with the symbol Ikl, The last term of (7.1.3) contributes to the initial layer. Now, our situation is much like that in S 3 in which Q is a pseudo-differential operator in E, and (7.1.2) can be solved by introducing a Banach scale to control the unboundedness of IV,1. Since its order is 1, the scale should be that of analytic functions in x. To be more precise, recall E ( k ) of (4.4.1) and set E e ( k ) = -ik-E+a-'L, the Fourier transform of EL. Since Ez(k)=a-lE(ak), Theorems 4.4.2-3 apply to Bc(k)with k , t replaced by ek, t / e respectively. We write the result as follows.

where L 2 = L 2 ( R ; ) . Now (7.1.3) is visible since (Z-P,)P:o)(i)=O by Theorem 4.4.2 (i). By Proposition 4.3.2 and proceeding as in the proof of Theorem 4.4.6, we can infer the

Y V )3 u=u(t)

-

III~III~=III~IIId,~*l,a,l=SUP llmIla-Tt,L,p< t€I

O0

9

where =(l+ly/2)1'2,y € R n , i=F& and Z c R is a n interval. sequel, we fix a, I , p such that, (7.1.5)

a>O,

l>n,

In the

/3>n/2+1.

Then, X is a Banach algebra, and u E X is analytic in x in the strip R"+ , P by i{lyl
Lemma 7.1.2. There is a constant C 2 0 depending only on a, I , and the following hold. ( i 1 IletBcuollO, t 2 0 .

B of

(7.1.5)

92

S. UKAI

( i i ) Let r > O andput r=a/y.

Then, writing ~ ~ ~ * ~ ~ ~ = ~ ~ ~ * ~ ~ ~ ~ , , ~ ,

l l l ~ ~ l l l ~ ~ ~ ~ ,+ r ~E >l O~.l l l ~ l l l Proof. By (7.1.4) and Lemma 7.1.1, (i) is immediate, and (ii) also, using thrice the inequality,

with 0, u ( E ) / c and u,/e as 5. Write the right hand side of (7.1.2) as N'[u'](t). Then, NE[u]=etBEuO+ P A - l r [ u ] . Since X is a Banach algebra, and by Lemma 4.4.3, IIIN"~1lIls c{IIklII

+( 1+ $)lIluIli~)

IIIN"ul-~"~IlllsC

9

~ l l l ~ l l l + l l l ~ l l l ~ l l l ~* - ~ l l l

This indicates that if uo is small, then N' is contractive, uniformly for E > O . Thus, we proved, Theorem 7.1.3. Suppose (7.1.5). Let r>O andputr=a/r. Then, there are positive constants a,, a, such that if IIuollO, (7.1.2) has a unique solution uEE Y ([0, T I ) with

lllUflll s a ~ l l ~ o* l l Consequently, u'(t) is analytic in x in the strip R"+i{jy]
X={ue

Let

I [ l x R ( k , E)u/140 (R400)} ~ = { u ~ I( e-Ttlfilue(t) t) E BO((O, ~ J )[O, ~ rIt; x X)}. 9

Then, we can infer that N e is a contraction in Z, and hence, [43], Theorem 7.1.4. Let ue, u, be those of Theorem 7.1.3. If, in addition, u, E 8, then u'EZand i s a classicalsolution to (7.1.1). 7.2.

Limit of solutions Define the space Y = Y ; ; : * l by P={u(t) I e - r t l k l u ( t ) ~ B ~ (T[I O ; 2)) , .

Evidently,

PC Y([O,r]). Going to the ,limit in (7.1.4), we can show,

Proposition 7.2.1. Let uo€ X and u(t) € P. Then, as e+O,

93

Solutions of the BoItzrnann Equation

etB'uo+E(t)uo,

W'u(t)-Fu(t) ,

strongly in Y ( [ 8 ,r ] )for any 6 > 0 with the limits, nfl

E(r)uo=.Fz-l

C #,(t, k)P:O)(R)ii,

3=0

n+lst

F u ( t ) = X z - l .C 3=0

where

# j ( t ,k)llclP:l)(O, L)(Z-P,)Aa(s)ds-L-l(Z-Po)Au(t),

o

#3(r, k)=exp (i2y)Iklr). Further, E(t)uo,Fu(t)E p .

In the above, 8, p cannot be replaced by X , Y([O,71). From this, it follows that if u, E 8, N g maps w={u*(t) I e-rtl*luE(t) E Bo([o,11x 10, 7]\{(0,0)}; 8 ) },

into itself. Since W may be regarded as a subset of Z , then u' of Theorem 7.1.4 is in W . Consequently, us(t)-uo((t) in Y ( [ & T I ) for any 8>0. The limit uo((t)satisfies (7.2.1)

u0(t ) =E( t ) u 0 + F k 1 f[uo](t ) ,

on (0, r ] , but since this can be solved in Y by the contraction mapping principle, we can say that u o ( t ) E I ' and satisfies (7.2.1) on [0, 71. Recall that LPo=O, by which (Z-PO)P:2)(O,k)(Z-P,)=O follows. Hence, (7.2.1) gives (7.2.2) (7.2.3)

(Z-P,)uo(t)=-L-lf[uo(t)],

Luo(r)+f[uo(t)]=O ,

or

P,u"O) =Pouo.

Set f f = g o + g ~ ' 2 u Lfor € 2 0 . (7.2.2) is equivalent to Q [ f o ] = O , so f o is a local Maxwellian. In view of Theorem 1.1.1, (1.6.6) holds also for f',E > O . Going to the limit, we have, E-c=

-

s:


which is nothing but (1.6.7). Also, putting (7.2.4)

E.V,f"S)>e

t=O

e= E

ds

9

here gives 7

which is just (7.2.3). Summarizing, we have,

Theorem 7.2.1. Let ue be that of Theorem 7.1.4. Then, uc(t)+uo(t) strongly in ~([6, 71) for any 6>0, with uo(t)EF. (i ( i i ) f O ( t ) = g o + g ~ / 2 u o ( tis) a local Maxwellian whose fluid dynamical quantities p(t, x ) , u(t, x), T(r, x ) solve the compressible Euler equation (1.6.7) with the initial condition (7.2.4).

94

S. UKAI

In (i), 6=0 is not permitted, i.e., the convergence is not uniform near t=O. In fact, f ' ( 0 ) = f o is not in general a local Maxwellian, but f o ( 0 ) is. Physically, this non-uniform convergence is called the initial layer. However, if the initial f, is itself a local Maxwellian then the convergence becomes uniform and the initial layer disappears;

Theorem 7.2.2. If,in addition, uo=Pouo,then (i) of Theorem 7.2.1 holds good with 6=0. The proof is simple but is referred to [43]. Note that we have, at the same time, constructed a solution to the compressible Euler equation, although within a class of analytic functions. An interesting converse is [lo]: Suppose (1.6.7) has a smooth (Sobolev) solution and construct the local Maxwellian f 0 ( t )by (1.1.11) from the solution. Then, we can construct a solution f S ( t ) to (1.6.2) which tends to f o ( t ) , on some time interval [0, r ] . In this case, initial layers are absent because f c ( 0 ) = f o ( O )is a Maxwellian. The method does not apply to non-Maxwellian initials. So far, we have justified the Hilbert expansion to the 0-th order. Justification to higher orders is not known, but a slightly different expansion is possible ([7]); f"(t)=f"(a,

t)+fO(E,

t/E)+Ef1,*(E,

t) ,

where ( i ) f , ( ~ t,) is sufficiently smooth in [0, 11x [0, T], with f o ( O , t ) = f o ( t )of Theorem 7.2.1 (ii), ( i i ) ~ O ( Ea), is sufficiently smooth in { ( E , a)/€€ [0, 11, EU E [0, r ] } ,and behaves like e-aa with a>O, (iii) f l r * is uniformly bounded and ~f'-* is sufficiently smooth, in [0, 1 ] x [O,r]\{(O,O)}. Further, f l , * has the form, f"*(E,

t)=f'(E,

f)+J'(E,

t/E)+ES2.*(E,

4,

and similarly for f 2 , * , fs2* and so on, each having a property like (i) (ii) (iii). As for the Chapman-Enskog expansion, we must mention [26] which shows that the solution of (1.6.2) with a fixed E > O approaches, as t + a , that of the compressible Navier-Stokes equation with the viscosity and heat diffusion coefficients proportional to E . This is not, however, sufficient for the justification of the expansion to the first order.

References [ 1 ] T. Arai, (in preparation). [ 2 ] L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal., 45 (1972), 1-34. [ 3 ] -, Intermolecular forces of infinite range and the Boltzmann equation, Arch.

Solutions of the Boltzmann Equation

95

Rational Mech. Anal., 77 (1981), 11-23. L. Arkeryd, Loeb solutions of the Boltzmann equation, Arch, Rational Mech. Anal., 86 (1984), 85-97. K. Asano, Local solutions to the initial and initial boundary value problems for the Boltzmann equation with an external force, I, J. Math. Kyoto Univ., 24 (1984), 225-238. -, On the initial boundary value problem of the nonlinear Boltzmann equation in an exterior domain, (in preparation). K. Asano and S. Ukai, On the fluid dynamical limit of the Boltzmann equation, Lecture Note in Numer. Appl. Anal., 6, North-Holland, 1983, 1-19. R. Beak and V. Protopopescu, Abstract time-dependent transport equations, (preprint). R. E. Caflisch, The Boltzmann equation with a soft potentials, Comm. Math. Phys., 74 (1980), 71-109. -, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 651-666. -, Fluid dynamics and the Boltzmann equation, Nonequilibrium phenomena I, The Boltzmann Equation, (Eds. J. L. Lebowitz and E. W. Montroll), NorthHolland, 1983. R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. T. Carleman, Probltme Mathtmatiques dans la Thtorie Cinttique des Gaz, Almquist et Wiksell, Uppsala, 1957. C. Cercignani, Theory and Application of the Boltzmann equation, Elsevier, Amsterdam, 1975. N. Dunford and J. Schwartz, Linear operators I, Interscience Publ., New York, 1957. R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl., 54 (1972), 1825-1856. J. P. Giraud, An H-theorem for a gas of rigid spheres in a bounded domain, Colloq. Intern. CNRS, 1975, N236, 29-58. H. Grad, Asymptotic theory of the Boltzmann equation, Rarefied Gas Dynamics I, (Ed. J. A. Laurmann), Academic Press, New York, 1963. -, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Symp. Appl. Math., (Ed. R. Finn), AMS, Providence, 1965. K. Hamdache, Existence in the large and asymptotic behavior for the Boltzmann equation, to appear in Japan J. Appl. Math. E. Hille and R. S. Phillips, Functional Analysis and Semigroups, AMS, Providence, 1957. A. G. Heintz, Solution of the boundary value problem for the nonlinear Boltzmann equation in a bounded domain (in Russian), Aerodyn. Rarefied Gases, 10 (1980), 16-24. [231 R. Illner and M. Shimbrot, The Boltzmann equation; Global existence for a rare gas in an infinite vacuum, (preprint). 1241 S. Kaniel and M. Shimbrot, The Boltzmann equation, Comm. Math. Phys., 58 (1978). 65-84. T. Kato, Perturbation Theory of Linear Operators, Springer, New York, 1966.

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[26] S. Kawashima, A. Matsumura and T. Nishida, On the fluid dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys., 70 (1979), 97-124. N. B. Maslova, Stationary boundary value problems for the nonlinear Boltzmann equation (in Russian), Aerodyn. Rarefied Gases, 10 (1980), 5-15. N. B. Maslova and A. N. Frisov, Solution of the Cauchy problem for the Boltzmann equation (in Russian), Vestnik Leningrad Univ., 19 (1975), 83-85. B. Nicolaenko, A general class of nonlinear bifurcation problems from a point in the essential spectrum, application to shock wave solutions of kinetic equations, in: Application of Bifurcation Theory, Academic Press, New York, 1977. T. Nishida, A note on a theorem of Nirenberg, J. Differential Geometry, 112 (1977), 629-633. -, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Commun. Math. Phys., 61 (1978), 119-148. T. Nishida and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation, Publ. R.I.M.S., Kyoto Univ., 12 (1976), 229-239. A. Parczewsky, Local existence theorem for the Boltzmann equation in L’, Arch. Mech., 33 (1981), 971-981. Y. Shizuta, On the classical solution of the Boltzmann equation, Comm. Pure Appl. Math., 36 (1983), 705-754. Y. Shizuta and K. Asano, Global solutions of the Boltzmann equation in a bounded convex domain, Proc. Japan Acad., 53A (1977), 3-5. C. Trusdell and R. G. Muncuster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monoatomic Gas, Pure Appl. Math., Vol. 83, Academic Press, New York, 1980. [37] S. Ukai, On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. Les solutions globales de 1’8quation de Boltzmann dans I’escape tout entier 1381 -, et dans le demi-espace, C. R. Acad. Sci., Paris, 282A (1976), 317-320. The Transport Equation, (in Japanese), Sangyo Tosho Publ., Tokyo, 1976. [391 -, Local solutions in Gevrey classes to the nonlinear Boltzmann equation with[401 -, out cutoff. Japan J. Appl. Math., 1 (1984), 141-156. S. Ukai and K. Asano, On the initial boundary value problem of the linearized Boltzmann equation in an exterior domain, Proc. Japan Acad., 56 (1980), 12-17. -, On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. R.I.M.S., Kyoto Univ., 18 (1982), 477-519. -, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 303-324. -, Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I, Existence, Arch. Rational Mech. Anal., 84 (1983),248-291,11, Stability (preprint). J. Voigt, Functional analytic treatment of the initial boundary value problem for collisionless Gases, Habilitationsschrift, Univ. Munchen, 1980. Department of Applied Physics Osaka City University Sugimoto 3, Sumiyoshi-ku Osaka 558, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 97-128 (19861

Equations of Motion of Compressible Viscous Fluids By T a k a a k i NISHIDA Abstract. We survey the global solutions of equations for one-dimensional motion of compressible, viscous and heat-conductive fluids. Initial value problems with fixed and free boundaries are treated about solutions global in time and about the asymptotic behaviors as time tends to infinity. Key words: equations of motion of compressible, viscous fluids, solutions global in time, asymptotic behaviors of solution, initial value problem with fixed boundary, free boundary problem

S 1.

Introduction

We consider the one-dimensional motion of viscous compressible and heat-conductive fluids:

in t 2 0 ,

-m<x<+w.

Here we take p : the density, u : the velocity, 8 : the absolute temperature as unknown variables, and e = e ( p , 8 ) : the internal energy, p = p ( p , 8): the pressure, p = p ( p , 8 ) : the viscosity coefficient and r = ~ ( p ,8): the coefficient of heat conduction are known functions of p and 8, and the subscript denotes the differentiation with respect to the variable t or x. The fluids satisfy in general (1.2)

~ ( p0),

, e(p, 8) >o ,

apiap

, a p m , aeiao >o

and also

(1.3)

p>O,

r>O,

in p > O ,

8>0.

The initial and initial-boundary value problems for (1.1) are solved in a general situation under the natural assumptions (1.2), (1.3) and p>O, 8 > 0 locally in time. [24], [5], [28], [29]. Received October 15, 1985.

T.NISHIDA

98

Thus the main concerns with (1.1) are the global solvability and behavior of the solutions of the initial and initial-boundary value problems. The present theory for global solutions is far from complete. But there are two important special cases of equations of state of fluids. [l]. ( i ) polytropic gas:

where R and y > 1 are positive constants. ( i i ) barotropic gas:

(1.5)

p=ApT,

A=constant

.

Theories assuming the condition (1.4) or (1.5) and also the constancy of the coefficients p=constant>O ,

(1.6)

r=constant>O

,

have been fairly developed recently. [7], [8], [13], [15], [14], 1271 and [26] etc. In this paper we survey the two basic initial boundary value problems for (1.1) with (1.4) or (1.5) and (1.6). ( I ) Fixed boundary problem in Q - = { ( t , x); t 2 0 , O<x<X}:

t

(1.7)

u(t, O)=u(t, X ) = O

,

O J t , 0)=O,(r, X ) = O

.

(11) Free boundary problem in Q = { ( t , x ) ; O < t , O < x < X ( t ) } , where X ( f ) is a free boundary.

(1.8)

‘i

B,(t, O)=O , u(r, O)=O , p(t, X(t))=O , B(r, X(t))=O and

dX/dt=u(t, X ( t ) ) .

Problem I is a standard one for the system (1) and Problem I1 is a model for interface of a compressible fluid with the vacuum. We use the following notations: Q,={(t, x) E R 2 ;O i t < T , xEZ}, where Z is [0, 11 or R . .gU(Z)is the Holder space of Holder continuous functions with exponent u on I . .glta(Z) is the subspace of ~ 3whose ’ ~ 1-th derivative is also Holder continuous with exponent u. G+’/2+’(Q,) is the Holder space of functions on QT which is Holder continuous with exponent 012 and u with respect u €U~ , ~, E~ ~~ ~~ ~~ ~W1+“(Q,)= ; * ~ } . totandxrespectively. . ~ 3 ’ ~ + ~ ( Q ~ ) =u,,{ u,, { u € & + ’ / ~ , ~ ut, ; U , E ~ ~ H ~ 1~denotes ~ ~ }the. Sobolev space of L2 functions together with their x derivatives up to and including the Z-th order. B(0, T ; H 1 ) denotes the bounded continuous functions of t E [0, TI with vector values in H L . Lz(O,T ; H L ) denotes the square summable functions of t € [0, TI with vector values in H 1 . We also use the space LP of functions whose p-th power are summable, 1 i p < 00.

Motion of Compressible Viscous Fluids

8 2.

99

Initial Value Problem with Fixed Boundary

It is convenient to transform the system (1.1) with (1.4) or (1.5) to that in the Lagrangian mass coordinate. After a suitable scaling of the variables we have the following system:

where v =l/p is the specific volume. The boundary condition (1.7) is given by (2.2)

u(t, O)=u(t, 1 ) = 0 ,

B,(t, O)=B,(t,

1)=0,

rTO,

and the initial conditions are supplied: (2.3)

( v , U, O)(O, x ) = ( v ~ ,~ o ,B,)(x) ,

0 1 ~.~ 1

Here we assume the positivity (2.4)

vo(x), O o ( x ) > O ,

01X<1,

and the compatibility condition (2.5)

i

U d O )= uo(U

=o ,

eo,,(o)

=Oo,x(l)

=o

9

~ a ~ o / v , ~ , - ~ ~ ~ o , , ./ ~ ~ ~ ~ l ~ = o , l = ~

And also we may assume while scaling without loss of generality the condition on the initial data (2.6)

~ ~ v , ( x ) d x =, l

The initial boundary value problem (2.1)-(2.3) has been solved globally in time. Theorem 2.1. If the initial data (2.3) satisfy (2.4), (2.5) and (2.6) and have the regularity v o E alto, uo, eo &PtU such that (2.7) then a unique solution of the initial boundary value problem (2.1)-(2.3) exists globally in time and satisfy for any T>O V G W1+"(QT) ,

u, OE.G'ztu(Q,),

T.NISHIDA

100

{ II ""8;";;

(2.8)

M(T)- -

Yllzl,

,

~ll$,'"'

I W T )< 0jM(T) in Q , . 1

This is the first and a nice global existence theorem for the polytropic viscous gas and given by Kazhikhov-Shelukhin (1977). It is proved by a standard argument of continuation of the local in time solution by the a priori estimates in the form (2.8). But the proof for the estimate is technically involved (See [15].). This solution decays to the constant state ( 1 , 0 , 1) as time tends to infinity.

Theorem 2.2. The solution of the initial boundary value problem (2.1)-(2.3) decays to the constant state (1, 0 , 1) in H'(0,l) as ?--too. For some T>O the decay rate is exponential for t 2 T . This is proved by using several excellent ideas by Kazhikhov [14]. Since the Journal is difficult to obtain, here we trace his proof of this Theorem for the solution obtained in Theorem 2.1.

Proposition 2.3. density satisfies (2.9)

There exist constants m,,M , such that for any t2O the

O < m , < v( t , x) < M, < m ,

where

(t,x)€Qm,

em=[O, a)x [O, 11.

Before the proof of Proposition 2.3 we notice three identities for the sohtions of (2.1)-(2.3).

(2.10) (2.11)

\:u(t,x)dx=

s:

v,(x)dx=l

,

\'0(t, x)+ue(t, x ) / 2 d x = ~ 1 0 , ( x ) + u , l ( x ) / 2 d x =5 l 0

0

s:

(2.12)

V(t)+ V(t)dt=E,<

where

(2.13)

V(t )=

s:

u2/2+a(Y - 1-log

Y)

,

+0-

1 -log 0 dx

(2.14) Lemma 2.4.

For each t > O there exists x ( t )E [0, 11 such that

(2.15)

a Iu(t, x ( t ) )I B,

(2.16)

a l 0 ( t ,X ( t ) ) l / 5 ,

Motion of Compressible Viscous Fluids

where

LY,j9

(2.17) Proof.

(2.18)

101

are two roots of the equation.

y-1-logy=

min (a, 1)

'

It follows from (2.12) that \~v-l--logv+0-l--log0dx~E1,

where E,=E,/min(a, 1). If v-l-logv+0-l-logO>E, for any x € [ O , 11, then it contradicts (2.18). Therefore at some point x(r) E [0, I], v-1-log v f 0-l-log0<E1, which proves (2.15) and (2.16). Lemma 2.5. For each t 2 0 there exists a t least one point xO=xO(t) € [0, 11 such that the following identity holds:

where

(2.20) Proof. Since by (2.1) we have ut=o,,

there exists a function +(t, x ) which satisfies u=(i),,

u=&.

Furthermore it follows from the definition (2.20) of u that this function the solution of the equation:

+ is

Multiplying this by v and using (2.1) this equation can be transformed to a divergence form:

( u + ) ~ - ( u + =p+,,--aO--U2 )~

.

Integrating it in Q, and using the boundary condition u=+,=O ary, we have the following equality:

on the bound-

102

T. NISHIDA

+

u2 a0 dxds = ?F( t ) ,

(2.21) where

Here we know that for each t > O there exists xo(t)€ [0, 11 such that

40, xo(t))=W

(2.22) In fact if not, there exists

*

such that

to>O

or

But multiplying this alternative by v(torx ) and integrating it on [0, 11 by using (2.10) we get an inequality for t = t o which contradicts the identity (2.21). Therefore since we can take

Lemma 2.5 follows from (2.21) and (2.22). Lemma 2.6.

The specific volume v(t,x ) has the representation:

v(t, x ) = D ( t , x ) exp

(2.23)

[

x 1 +-sf_

p

(-F1

so t

\ou2+aOdxds) 1

s‘ m, (+ \:1: 0

x)

exp

+

u2 a0 dydT) ds]

,

where

Proof. The first two equations of (2.1) give the following:

Integrating it with respect to t on [0, r] and integrating it on [ x o ( t ) ,x ] for t : fixed, we have

Motion of Compressible Viscous Fluids

103

= p l o g v , , ( ~ ) - 0\ ~0~ ~ u ~ + a O d x d s + - \ x o u o ( ~ ) d ~ + ]u-u0dy, x 20

where thelequality (2.19);is'used.

Therefore we have

and so we have

aB exp (\t*ds)=exp PV

PV

0

(Ljt j1 u2+aOdxds P

0

0

Thus integration gives e x p (pl \8\1uz+aOdxdr)ds 0 0

e x p ( \ t *0 dPVs ) = l + \ t * D - lO( s P, x )

and (2.23) follows from this and (2.24).

Q.E.D.

Let us define (2.25)

M,(t)=max ~ ( tx),

,

M,(t)=max O ( t , x) O i X 5 l

O L X i l

.

Now we show the uniform bound for v from above. Since we have (2.11), D is uniformly bounded: (2.26)

O
in

Qm.

If we use the identity u2+aO =2(O+ u2/2)

+ (a- 2)0

for a > 2

or

u2+a8 =a(O+ u2/2)+(2--a)u2/2

for 0 < a I 2,

and the conservation of energy (2.11), we can estimate for v in (2.23) as follows : (2.27)

M,(t) 5 C exp ( - a l t )

T. NISHIDA

104

where C=C(p) and a,=min (a,2)/p. On the other hand 19 can be estimated from the above by M , as follows: (2.28)

Me(t)
.

In fact it follows from Lemma 2.4 that for each t 2 0 there is a point x(t)€ [0, 11 such that aIB=O(r, x(r))ip. Then we have

Thus (2.28) follows from (2.11) and (2.14). Now if we use (2.28) in (2.27) and the basic equality (2.12), we can conclude (2.30)

l + e x p (-a,t)

This proves half of Proposition 2.3. equality: (2.31)

s:

V(s)exp (a,s)M,(s)ds

Similarly for (2.27) we can prove the in-

v(t,x)>Cexp(-a,r)

where a,=max (a,2)/p and C=C(p) is a positive constant. notation (2.32)

m,(t)= min v(t, x)

,

0<2<1

me(t)= min O ( t , x) 0<2_<1

Here we use the

.

Then (2.31) gives rn,(t) 2 C exp ( - a , t )

me(s)exp ( -aa,s)ds

and it follows from (2.29) and (2.30) that

where C, and C, are positive constants. These two inequalities give

Motion of Compressible Viscous Fluids

2C,/2>0

for

105

t2T,

where by applying (2.12) we use the following:

1:

lim exp ( - a d t-m

1:

exp (0,s)

.

%dxds=O

Q.E.D. of Proposition 2.3. Further we show the uniform boundedness of H' norm of

v, u, 0.

Lemma 2.7.

where C is a constant independent o f t . Proof. It satisfies

Let us consider the function w = 0 + u 2 / 2 of the solution of ( 2 . 1 ) .

(2.35)

w, = ( K

+),+(y ), -($) 2

Multiplying it by w and integrating it on [0, 11, we have the following after integration by parts and by using the Schwarz inequality:

On the other hand we multiply (2.1) by u3, integrate it on [0, 11 and we have (2.37)

(1'

1'

u'dx) + l o p

o

t

0

x d x i C 2 u

j:T

dx

Thus we have for a suitable constant c > 0

(1'

w2+cu4d

1:

+)

~

~

K$+C----

u2u,2 dx U

0

(2.38)

IC,j ' s d x 0 2 ,
max u2( t,x )

O$Z11

I'

0

w2dx

.

106

T. NISHIDA

By the way we know

So

< v(t)M,(r) edx
-

Thus we have for any t 2 0

S’

(2.39)

0

max u2(s,x)ds< C O i Z < l

s:

.

V(s)dsi C’

.

It follows from (2.38), (2.39) and Proposition 2.3 that by using the Gronwall inequality (2.40)

]]O(t)ll2+\‘ IIB,(s)llz

ds
for any r 2 . O

.

0

The first two equations of (2.1) give the following: (2.41) where ?=(log v),.

ae

a

V

V

Pvt+-v=Ut+-Ox,

Multiply (2.41) by 7 and integrate it on [0, 11 and we have

It follows from Proposition 2.3, (2.40) and (2.41) that

On the other hand multiplying the momentum equation by u, we have

These two inequalities (2.42), (2.43) give the following:

1:

v,zdx=(\lB+$dx) 0

\‘vXzdx 0

107

Motion of Compressible Viscous Fluids

Thus we have

1'

s: (1:

+\: uzdx) j' vzZdxds+j'

~ ~ ~ , ds< ( s ) ~ ~ OZ2dx ~

0

0

0

1'

OvZ2dxds
0

Q.E.D.

Lemma 2.8.

(2.46)

SUP t2O

Proof.

ll~,(~)llz+~m ll~,,(s)llz

ds
.

Multiply the second equation of (2.1) by u,, and integrateiit.

Then we have

and

Then (2.45) follows from (2.47), Proposition 2.3 and Lemma 2.7 by taking E small. The estimate (2.46) is proved similarly by multiplying the third equation of (2.1) by O,, and integrating it on [0, l ] and using Proposition 2.3 and Lemma 2.7. Q.E.D. Proposition 2.9.

(2.48) Proof.

l[v-1, u, O-111,1-0

as

r+oo

It follows from Lemmas 2.7 and 2.8 that

where we used the equality

.

T.NISHIDA

108

5: Thus as I+

00,

I[v,,

u,uZtdx= -

5'

u,,utdx

0

u,, Oz(t)ll- 0.

Therefore as t--r

llu-1, u, O - 1 1 1 p - t O

00,

,

and (2.48) follows.

Q.E.D.

Since we know that the solution becomes small in Hi norm for large t , we can conclude the solution decays to the constant state exponentially as t+w. See [13]. Q.E.D. of Theorem 2.2. Remark. There are other boundary conditions for system (1.1). ( i ) Dirichlet boundary condition for the temperature: O ( t , o)=Oi(r)

(2.49)

7

@(t,X ) = @ z ( t )9

where O,(t), O,(t) >O. ( ii ) Moving boundary condition (Piston problem): (2.50) where X , ( t ) and X z ( t ) are given functions such that

5:

vo(x)dx+

s:

{X,(s)-Xx,(s)}ds>constant>O

.

Two global existence theorems (similar to Theorem 2.1) of the initial boundary value problem for (1.1) and (2.50) with Neumann zero boundary condition or Dirichlet boundary condition (2.49) for 0 are given in [14] by introducing auxiliary functions which transform the boundary data to zero. Some asymptotic behaviors of solutions for ( l . l ) ,(1.5), (2.50) are considered by Kawashima 1101.

S 3. Cauchy Problem Here we notice the initial value problem (2.1), (2.3), (2.4), x € R , which is solved globally in time by Kazhikhov [14]. Theorem 3.1.

(3.1)

Zf the initial data satisfy

Motion of Compressible Viscous Fluids

109

then there exists a unique global solution which satisfies for any T > 0

Proof. In this case the equalities (2.10), (2.11) do not hold, but the equality (2.12) holds.

(3.3)

where U ( t )=

(3.4)

s'"_

+

u2/2+a( v - 1 -log v) 8- 1 -log 8 dx

Thus we have the following: Lemma 3.2. For each interval I , = [ n , n + l ) , there is a point xn(t)EZnsuch that

(3.5) (3.6)

a<j1,vdx,

1

Sdx
J ,

where a and /3 are two roots of the equation (2.17).

Proof is similar to that for Lemma 2.4, because on each interval Z, the inequality holds :

where U,,(t), V J t )are integrals of (3.4) on the interval I,. The inequality (3.6) is a consequence of (3.7) and Jensen's inequality for a convex function: y 1-log y . Using Lemma 3.2 the proof for bounds of v and 8 is similar to but a little simpler than that for Theorem 2.1. In fact the density has the representation

T.NISHIDA

110

(3.8) where

we know Thus for x , x,,(t)€Z%

Then there exist constants C, > 0 and K ( t ) (3.9)

c,-1< U(r, x)Ic,< 02 , C,-'
where C, and K ( t ) < K ( T ) for follows from (3.8) (3.9) that

re [0, T ] do

TIXI,,

not depend on n.

Therefore it

(3.10)

On the other hand similarly to (2.29) we have (3.11)

M,(t)IC(l+M,(t)W))

.

Then (3.10) and (3.11) give

(

M,(t)
s:

l+M,(s)V(s)ds

By the Gronwall inequality and (3.3) we have

(3.12)

M,(r)
9

t€

to, TI ,

and so (3.13)

('

MB(s)ds
Jo The remaining proof is the same as that of Theorem 2.1.

See [15].

111

Motion of Compressible Viscous Fluids

Remark. It is not known that the decay and asymptotic behavior of solution of the Cauchy problem in this section as t j m . Compare this with Theorem 4.1 for the solution of barotropic gas motion considered in S 4.

S 4.

Asymptotic Behaviors of Solutions of Barotropic Viscous Gas Burgers equation

(4.1) is the simplest model for the system of equations of viscous fluid motion. It was solved explicitly by Cole and Hopf. In particular by using the explicit solution Hopf investigated the asymptotic behaviors of solutions of the Burgers equation as time tends to infinity when M = $ymu,(x)dx is finite. By introducing a change of variables x = x / 2 / F t , f=log t , ii=z/(tj21-L)u, he obtained the asymptotic convergence -lim n ( f ,X) = - G’(X)/G(X)

(4.2)

,

t+m

where G ( x )=exp ( -M / 4 p )

1‘

jr exp (-y2/2) dy .

+

exp ( - y 2 / 2 ) dy exp ( M / 4 p )

-m

In the rescaled variables the Burgers equation can be written as follows: (4.3)

an ai

I

an -_L -+--* azu 1 a ( Z )

aX

2

a?

2

ax

Thus the limit function (4.2) is a stationary solution of (4.3) with the moment ii(i,X)dT=M/2p, which is a conserved quantity with respect to time. We do not know in general the asymptotic behavior of solutions for the system of equations of viscous compressible fluid motion as time tends to infinity. In [ 121 and [9] we treated asymptotic behaviors and equivalences for small solutions between the Boltzmann equation and the compressible NavierStokes equation as time tends to infinity. However in this case of more than one space-dimension the asymptotic behavior is described by linear partial differential equations. It is because the decay rate of solutions is so fast that the nonlinear part decays faster than the linear part as time tends to infinity. But in the one space-dimension this is not true and we have to consider the nonlinear part as well as the linear part for the asymptotic behaviors. In fact a reductive perturbation method [30] predicts that the Burgers equation describes the far field. i.e., the asymptotic behavior for the general system of viscous fluid dynamical equations.

-:5

T. NISHIDA

112

In this section we consider a system of equations of viscous barotropic gas motion (1.1) with (1.5) and show that the asymptotic behaviors are described by two Burgers equations with different propagation speeds. Similar asymptotic behaviors are investigated for the inviscid case, i.e., for the hyperbolic conservation laws in [2], [17] and in the references in them. Thus the asymptotic forms of the solutions as time tends to infinity are different from each other between the inviscid and viscous motions as noticed in [3] for the Burgers equation. We consider the viscous barotropic gas motion which is governed by the following nonlinear system of two equations:

where p is the density, u is the velocity, p=(a2/r)p7is the pressure for the barotropic gas, and a , y (ratio of specific heats) and ,u (viscosity coefficient) are assumed constatns. The initial data

(4.5)

~ ( 0x)=po(x) ,

9

~ ( 0 x, ) = ~ o ( xt)

x€R

are given and we want to investigate the asymptotic behavior of solutions for the Cauchy problem (4.4), (4.5)as time tends to infinity. System (4.4)has a hyperbolic-parabolic type, i.e., the first order part is hyperbolic but the velocity satisfies a parabolic equation. This is a main feature of viscous fluid dynamical equations. The Cauchy problem (4.4), (4.5)is solved globally in time by Kanel’ for rather genera1 HLinitial data by using the Lagrangian mass coordinate, which can be summarized as follows. Hereafter the equilibrium state is assumed p = 1 , u=O.

Theorem 4.1. Zf po(x)- 1, uo(x)€ H L ,1 2 2 , po(x)>0, then there exists a unique global solution p(t,x ) , u(t, x ) such that p(t, x)>O, p(r, x)-1, u(t, x ) e B ( 0 , 00; HL), p,(t, x ) , u,(t, x ) E L2(0,00; HL), and that ( p , u)+(l, 0 ) in L“(R) as t+m. Concerning the decay rate of solutions for Cauchy problem (4.4), (4.5)we have the following

Lemma 4.2. Zf po(x)- 1, u o ( x )E HLn L’, 12 12, and small in the norm, then IP(p(t, .)-l, u(t, . ) ) I ~ 2 ~ C / ( l + t ) ~ ~,’ 4 +a=o, ~ ~ ~ )1,2, 3

(4.6)

(4.7)

.

1, - - -8 ,. li34tu(p(t,- ) - l , u(t, . ) ) i . ~ ~ c / ( l + t ) ( ~, - ~ / ~a=O, )

The decay rate (4.6)is best possible i f

Motion of Compressible Viscous Fluids

113

This is a slight improvement of [12], [9] which is proved by the linear decay rate obtained by the Fourier transform and by the energy method using the convexity of the Sobolev norm. As in [12], [9] using this decay rate we make a comparison of solutions represented in the variation of constants formula in the Fourier transform between system (4.4) and the following uniformly parabolic system

with the same initial data ud0, x) = uo(x) .

Pl(0, x) = p o w ,

Lemma 4.3. The asymptotic equivalence between p, u and p,, marized as follows: (4.9)

~ a q p - ~u-ul)(t, ~,

ii,

can be sum-

,

X)I 1~/(1+t)(s/4-6+a/z)

a=O, 1 , 2 , f o r any 8>0 and f o r any t > O

.

Since the decay estimate (4.6) is optimal in general this estimate for the difference of solutions for systems (4.4) and (4.8) is meaningful and essential for further discussion on system (4.8). In order to make a further reduction of our system of equations we will use the Riemann invariants of the hyperbolic part of systems (4.4) and (4.8), namely the eigenvalues and the corresponding Riemann invariants given by the following:

(4.10)

t

I , = u-ap(r-1)/2 , 1,=u+ ap ( I - , ) ’2 ,

r =(2a/(y- l ) ) ( p ( r - l ) / *- 1)- u l))(p+l)’Z- l ) + u .

s=(2a/(y-

By using these Riemann invariants r=r(p,, u,), s=s(p,, u,) the parabolic system (4.8) can be written as follows:

where

+

f =(p/16ap(r-”)((7- r)rZ2 2(3- r)r,s,

-

(r+ 1)sZ2)

and

+

g =(p/16ap(r-l))((7-r)sZ2 2(3- r)r,s, -(r+ l ) r Z 2 ). Here we have the nonlinear terms rr,, ss, sr,, rs,, f and g which have the decay rate by Lemma 4.2 and the definition (4.10):

T. NISHIDA

114

lrr,, ss,IL1
(4.12)

Since there is a difference on the decay rate between these terms, we want to compare the solution of system (4.11) with that of system (4.11) without the terms f and g, i.e., system (4.14). But in so doing the cross terms sr, and rs, do not have a divergent form and prevent us from obtaining a n estimate for the difference directly. Thus to get around this difficulty we use the hyperbolicity of the first order part of system (4.11) following a n idea of Lax [16] after rewriting sr,=(sr),-rs,. Let us introduce the unknown functions

R=rA,

S=sB,

where A =(2a+ (y - l)r/2+ ( y + 1 ) ~ / 4 ) ( ~ - r ) ' ( r + ,')

+

+ + l)r/4) (3-r)'(r+1) .

B= (2a (y - l)s/2 (y

Then instead of (4.11) the functions R and S satisfy

(4.13)

t

R,- (aR+(y+ 1)Rz/8A),- (r- ~ ) ( S R / B ) , / ~ = , U R , , / ~ +7 F S,-(aS+ (y+ l)Sz/8B),-(y- 3)(SR/A),/4=pSZ,/2+G 9

where F = F ( r , s,f,g) and G=G(r, s,f,g) consist of those terms which decay as fast as f and g, i.e.,

IF, GILiiC/(l+r)"/" . Using this decay rate and the fact that the quadratic terms R2, S2 and R S have the divergent form in system (4.13) we can compare the solutions of (4.11) and of the following system: rz,t-(a+ (?'+ l)rz/4+ (y- ~ ~ Z / ~ ) ~ Z , Z = P ~ Z , Z Z / ~ 9

(4.14)

is,.,-(a+(l+

l)s2/4+(y- 3)r2/4)sz,,=r*s2,~~/2 9

for t 2 T , with the initial data

(4.15)

(rz, sz)(T,x) =(rr s ) V , x)

.

The corresponding system for R,= R(rz, sz) and S z = S ( r z rsz) is given by

(4.16)

+(y+ 1)RzZ/8Az),-(7 3)(SzRz/Bz),/4 =pRz, { Rz, Sz,,-(aSz+(y+1)Sz2/8Bz),-(y-33)(RzSz/Az),/4=~Sz,,,/2+Gz -

t

where

Fz=F(rzr sz, 090) Az=A(rz, sz) 9

9

Gz=G(rzr sz, 090) Bz=B(rz, sz) .

9

+Fz t

9

115

Motion of Compressible Viscous Fluids

Lemma 4.4. W e have the estimate for the difference of solutions ( R - R , , S - S , ) or equivalently for ( r - r z , s-sz) (4.17)

Iaa(r-re, s- sz)(r, x ) ILz IC/(1 + t - T )( 1 ’ 4 + n / 2 ) ( l + T ) ” z , a=O, 1 , 2 , for any t 2 T : fixed.

The estimate (4.17) is not as good as the estimate (4.9) which is valid for all t 2 0 , but this estimate in this form is expected to be best. In fact if we note the special case r=3, it has the nonlinear terms in the nondivergent form. Here we arrived at a n almost diagonal system (4.14) except for cross terms s2r2,zand rzsz,z. These cross terms have had the same decay rate estimate as (4.12). In order to distinguish these from the main quadratic terms r2r2,= and s,s,,, we use a property of the finite propagation speed of exponential decay with respect to x of parabolic system (4.14) which corresponds to the finite propagation speed of the hyperbolic system of the first order parts of (4.14). Lemma 4.5. If the initial data satisfy an additional exponential decay as x--tkoo, i . e . ,

lP(r(0,x ) , s(0, x))l
a=O, 1, 2,

3,

then

(4.18)

laa(r(t,x ) , s(t, x))l <min { C / ( l + t ) ( 1 / 2 + aKept/cosh )/2, x} a=O, 1,2, 3 , where j9 is a constant.

This is true for all systems (4.4), (4.8), (4.11), (4.13) and (4.14) because of the maximum principle. See for example [4] for the maximum principle. Thus the solution decays exponentially with respect to x for each t . In particular the initial data (4.15) have the estimate: (4.19)

laa(r,, s,)(T, x)l I m i n {C/(l+T)(1/2ta)/2, KeBT/coshx } ,

a=O, 1, 2, 3

.

Using the maximum principle for system (4.14) along each characteristic direction we can obtain the exponential decay estimate: Lemma 4.6. (4.20)

Under the condition (4.19) we have

[ P r , ( t ,x)l <min {C/(l+t)(l/z+a)/z, K,ea(c-T)/2/cosh (x-a(t-T))} ,

{laas ( t, x)l < m i n {C/(l+t)(1~2ta)/2 ,Kl e a ( t - T ) / 2 /cash ( x + a ( t - T ) ) } , a=O;l, 2, 3 , for any t r T , where K,=KepT .

Now we can distinguish the cross terms by a faster decay than (4.12) as (4.21)

Ida(rzsz)(t,.)IL1 2 min {C/(l + t ) ( l + a ) / 2Kle-at/2( z, Ir, sls)} CKe-a(t(ZB/a+l)T)/Z < min {C/( 1 + t ) (Itn)/Z , }

7

a=l,2,3.

T. NISHIDA

116

Using this decay rate we compare system (4.14) and the diagonal system (4.22) with the inital data

where T I=( 3+2P/a)T.

Lemma 4.7. We have the estimate for the diflerence of solutions between systems (4.14) and (4.22)for t 2 T , (4.23)

Iau(rz-r3, s,-s,)(t, .) lLz < C/{( 1+ t)(1/4ta12) eaT } ,

a=O,1,2.

This is proved by the representation of solutions in the variation of constant formula in the Fourier transform for systems (4.14) and (4.22) and by the decay estimate (4.21).

Theorem 4.8. If the initial data are close to a constant state (p,0 ) in the norm in Lemma 4.2 and decay exponentially as x++oo, then the solution of system (4.4) behaves asymptotically like that of two Burgers equations (4.22) as time tends to infinity. Remark. If the states of initial data at x = 00 are different from each other, then the solution is expected to behave like the solution of the Riemann problem as time tends to infinity. It is proved for the inviscid case by Liu [18]. In the viscous case if the perturbation has zero mean, a single shock wave solution is proved to be stable by Matsumura-Nishihara [22] for the barotropic gas and by Kawashima-Matsumura [ l l ] for the polytropic gas, namely, if the initial data is close to a single shock wave in H1norm a n d if the integral of the difference of the initial data and the shock wave can be taken to zero by a translation of shock wave with respect to x, then the solution converges to the traveling shock wave solution as t+m in L" norm. The case of two shock waves and the case of two rarefaction waves can be treated similarly. [23]. A more general stability theory for shock waves has been considered recently by Liu [19].

S 5. Free Boundary Problem We consider a free boundary problem where the density is continuous across the interface of the barotropic gas and the vacuum. First we notice the correspondence between the Lagrangian mass coordinate and the Eulerian coordinate. The barotropic gas motion is described by the following in the Eulerian coordinate:

Motion of Compressible Viscous Fluids

117

where x=O is the fixed boundary and X ( t ) is the interface of the gas and the vacuum : dX -- -u(t, W r ) ) 9 dr

(5.2)

p(t, X(r))=O

*

Here f is the external force. This problem can be reformulated conveniently in the Lagrangian mass coordinate by using the transformation. (5.3)

x’=S’

p(t, x)dx

,

t’=t

, i.e.,

0

System (5.1) is transformed by (5.3) to the following: (5.4)

pc.+p2u,,=0 , ucJ+pz.=(ppuu,!),,+f,

t’20 , X ’ l X ’ l O ,

where

is independent of t’, i.e., the free boundary is transformed to a fixed boundary. Then the boundary condition (5.2) becomes the following: p(t’, X’)=O

.

There are two cases even if X ’ > - m . ( i ) X(O)=--00. ( i i ) X(O)> - m . We will mainly consider the case (ii), i.e., a finite total mass on a finite interval, which is most interesting as a free boundary problem. Hereafter we drop the prime in the Lagrangian coordinate and consider the following problem after rescaling the variables: (5.5)

t

pt+p2uz=0

9

u,+p,=(ppu,),+g,

t20,

0 5 x 2 1.

Here p=apr, a=constant>O, r > l and p is assumed a positive constant, and concerning the external force we consider each of the cases ( A) 9=0, (B) g>O is the gravitation constant.

T. NISHIDA

118

The boundary condition is given by (5.6)

{

p(t, O)=p(t, O)u,(t, O)=O at the free end, u(t, 1 ) = 0 at the fixed end.

The initial data are supplied (5.7)

d o , x)=po(x) ,

4 0 , x)=uo(x) ,

O l X l l

,

where po(x) and uo(x) are Holder continuous on [0, 11, are smooth in (0, 11 and po(x)> 0 in (0, 11. Since the density becomes zero on the free surface (x=O), the known local existence theorem of solution for (5.5) by Tani [29], Kazhikhov-Shelukhin [15] does not apply directly. Thus we construct the solution by using the line method, i.e., we consider systems of 2 N ordinary differential equations when N goes to infinity:

(5.8)

Pzn,tfPzn

-0, -

2uzn+1-u21-1

Ax

n=1,2,

-..,N ,

n=1, 2,

..-,N ,

where Ax= 1/(N+ l/2). They are supplemented by the boundary conditions.

(5.10)

We explain how to obtain a global solution in time in the case (A). If the initial data are given as

(5.11)

pz,(0) = d o , nAx) > 0

,

...,N , the local existence of solutions (pz,(t), ~ ~ ~ - ~ (nr=)1 ), 2, , .- .,N , for ( 5 . 8 ) , (5.9), u,,-,(O)=u(O,

(n-l/2)Ax) ,

n=1, 2,

(5.10) and (5.11) is trivial for fixed N, and the solution satisfies the positivity of density inside: pz,(t)>O,

n=l,2,

..., N

.

Thus in order to obtain global solutions in time, we want to get the a priori estimates independent of N. If we multiply (5.9) by u,,-, and (5.8) by ~ p , , r - ~ and sum with respect to n = 1 , 2 , N, and use the boundary condition (5.10) and the summation by parts, we get

.- .,

119

Motion of Compressible Viscous Fluids

5 (ac+u) Ax+ ppzn( N

(5.12)

n=l

r-1

where we assumed

r> 1 .

2

t

n=l

Ax

Ax=O,

Thus we have a basic energy estimate

This energy estimate shows the global solvability of the system ( 5 . 8 ) , (5.9) for m, we fixed N. Next if we sum up (5.9) times Ax with respect to n= 1 , 2 , have

...,

where C,, C2 and C, are constants independent of N. The inequalities in (5.16) are easy to see, if we use (5.13) in the form:

we need Before we obtain the bound for luzn-l(r)l, Lemma 5.2. N

t N

(5.17)

)‘As
00

,

T. NISHIDA

120

l p zn

(5.18)

u,,+1- U,,-I Ax

IC
where C is a constant independent of N . Proof. Differentiate (5.9) with respect to t , multiply it by u ~ ~ -sum ~ . ~up, with respect to n= 1,2, N, and use (5.8) a n d the boundary condition (5.10). We have

- - .,

-

PPen -

Ax

arp2,7 u,, t 1 - k n - 1 Ax

Ax l,t-%-l,t

u2.t

Ax

Thus by using (5.14) and (5.16) we obtain

N

N

1+C U , , - ~ , , ~ A X

Combining (5.12) and this, and using (5.13) we conclude (5.17). The inequality (5.18) is a consequence of (5.17), (5.14) and (5.16). Q.E.D. Lemma 5.3.

(5.19)

The density has the Holder continuity:

+

Ip d t ) - p,,(t) I IC, I ( n- m)AxI , ( 1 C z t ) ,

where a is the minimum of 112 and the Holder exponent of the initial density. If the initial density satisJies f o r some a, 0 < a < 312 and f o r a constant C, (5.20)

p(0, x ) 2 coxu

I

then it holds f o r t > 0 (5.21)

where C, and C, are constants independent of N

Motion of Compressible Viscous Fluids

121

Both estimates (5.19) and (5.21) come from the representation for pzn(r) in (5.15) and from (5.13). Remark. Since we are considering the case (ii), the condition (5.20) is not restrictive. In fact, if p(0, x)=C,,xU, l g a , then it corresponds to the case (i). The velocity has the bound and Holder continuity:

Lemma 5.4.

lu2n-l(t)l
(5.22) (5.23)

I uZm-1( t )- uzn-,( t )I IC(1+ C,t 1 9 (( m-n)

.

Proof. It follows from (5.14) and (5.21) that N

luzl-l(t)lI

F luzmtl(t)-uzm-,(t)l

ICz(1 + Cltl/r) ,

provided a < 3/2

The Holder continuity (5.23) can be proved similarly.

Q.E.D.

Lemma 5.5. For any 6 > 0 the density and the velocity are boundedly diflerentiable in x E [a, 11, sf the initial density is so in x € [a, 11. If n A x 2 6 , (5.24) (5.25) where C and C, are constants independent of N . If one uses the representation (5.15) for density, the estimate (5.24) comes from the differentiability of the initial density and the boundedness (5.22) of the velocity. The estimate (5.25) is a consequence of (5.18) and (5.21). Now let us define functions (PA, ud)(t,x) extended by (pn, u J ( t ) ,n= 1,2, ., N.

.-

pd(t, x ) = { ( x - ( n - l ) A x ) p , , ( t ) + ( n A x - x ) p , , - , ( t ) } / A x

+

,

(n-l)Ax<x
uA(t,x )= { ( x - ( n - l/2)Ax)uZ,,,(t) ((n+ 1/2)Ax--x)uZn-,(t)}, (n- 1/2)Ax<x<(n+1/2)Ax ,

T. NISHIDA

122

p d ug , , =p 2 n (t )(u 2 n + , ( r ) - - z , - l ( t ) ) i A x

,

(n- 1/2)Ax<x< ( n + 1 / 2 M x

.

Then by using the above lemmas we can prove that a subsequence of ( p d , u,,) ( t , x ) converges boundedly and almost everywhere in [0, T ]x [ 0 , I] for any T>O to functions (p, u ) ( t , x), when pdud,,(t,x) converges boundedly and weakly to pu,(t, x) along the subsequence, and that the limit (p, u)(t, x) is a generalized solution of system ( 5 . 9 , (5.6) and (5.7). The solution satisfies the corresponding limit versions of the estimates (5.13), (5.16), (5.17), (5.18), (5.19), (5.21), (5.22), (5.23), (5.24) and (5.25), and also has the equalities corresponding to (5.14) and (5.15). More regularity of solution can be proved by considering the energy estimates for higher time derivatives of p and u. Theorem 5.6. There exists a generalized solution globally in time for the free boundary problems ( 5 . 5 ) with g=O, ( 5 . 6 ) , (5.7) under the condition (5.20).

System ( 5 . 5 ) in which the gravitation is acting, i.e., g=constant>O, can be treated similarly. In fact we have the conservation laws corresponding to (5.12)

and the representation for the density corresponding to (5.15),

c

(5.27)

pZm7(o)exp - (u2,-,(t)-u2,-,(O))Ax+gtmAx P ' pZm7(d= n 1+Tapz,7(0) exp - C (uZ,-,(s)-uz,-,(O))Ax+g~m~x ds ' P

5: t$1 "

m=l,2,...,N

1.

Here the third summation on the left-hand side of (5.26) gives a finite sum because of the condition (5.20). Then we can follow the same arguments as the proof of case (A) g=O except that the various constants now depend on t (but not on N). Remark. ( i ) Detailed arguments for the proof, discussions when the viscosity coefficient p depends on p, and asymptotic behaviors of solution will be given in [25]. We have not yet proved the uniqueness of the generalized solution. ( i i ) (a) An existence theorem of global in time solution for a free boundary problem of viscous polytropic gas was obtained in [15]. But there the density is strictly positive up to the free boundary. (b) Padula [27] obtained a global in time existence theorem of solution for viscous polytropic gas in the case that the density goes to zero continuously to the free boundary and the temperature has Neumann zero condition on the

Motion of Compressible Viscous Fluids

123

free boundary. In this case the temperature is strictly positive up to the boundary. (c) Dirichlet zero boundary condition (1.8) for the density and temperature seems a natural extension of our free boundary problem (5.5) and (5.6) because the entropy remains bounded in this case by virtue of the equation of state for polytropic gas ( 1 . 4 ) . See also (2.08) in [ l ] . An existence theorem of global in time solution is a n open problem in this case. (iii) In the case of inviscid gas in R 3 a n interesting local existence theorem is given by Makino [20] in this book, and open problems are raised. (iv) Our method (5.8) and (5.9) can be discretized with respect to time to give a finite difference scheme for ptn=p(kAt, nAx), uE,_l=u(kAt, ( n - l / 2 ) A x ) :

We do not know a convergence proof for this scheme, but it conserves the positivity of density and works very well for the numerical computation. Here we have computation results. E X . 1. y=2.0, a=1.0, g=0.5, p=0.015625, p2,(0)=sin RnAx , u,,_,(O)=sin ( n - 1/2)nAx , n= 1 , 2 ,

-,64

.

The velocity a n d density are plotted on the Eulerian coordinate (the orientation of x axis is reversed) in the figure.

Fig. 1

...

1 Fig. 2

T. NISHIDA

124

Fig. 3

Fig. 4

I=

1.43

Fig. 5

I

Fig. 6 I=

4.69

IP

u Fig. 7

Fig. 8

Motion of Compressible Viscous Fluids

f=5.86

IP

I Fig. 9 I= 6.64

I

If

Fig. 10

I=

8.01

Fig. 11

I=

I=

9.96

IP

92.98

I' Fig. 13

f=

96.89

I'

t-Fig. 14

125

T. NISHIDA

126

The trace of the free boundary is plotted in the t-x plane in the following figure.

I

1.0

19.5

39

1

Fig. 15 5

1 .o

39.1
78.2 0

-

78.2

. . 7

I

1

97.7

Fig. 17

t

117

Motion of Compressible Viscous Fluids

127

These numerical results r emi n d us of water wa ve s on the beach. I n fact the ratio of specific heats r = 2 corresponds t o the shallow water wave equation. Related t o this w e refer t o an interesting paper by R. E. Me ye r “On the shore singularity of water waves, P ar t I. T h e local model”, Technical Summary Report # 2872, Mathematics Research Center, University of Wisconsin-Madison, 1985. ( v ) W e refer t o a s u r v ey of Matsumura-Nishida [21] a n d the references in it for a theory on the viscous compressible fluid equation i n the three space dimension. References

t51

[61

[81 191

151

R. Courant and K. 0. Friedrichs, Supersonic Flow and Shock Waves, SpringerVerlag, New York, 1948, 1976. R. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J., 24 (1975), 1047-1071. E. Hopf, The partial differential equation ur+uu2=pusZ,Comm. Pure Appl. Math., 3 (1950), 201-230. A. M. Win and 0. A. Oleinik, Asymptotic behavior of the solutions of the Cauchy problem for certain quasilinear equations for large time, Mat. Sb. (in Russian), 51 (1960), 191-216. N. Itaya, On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluid, KGdai Math. Sem. Rep., 23 (1971), 60-120. -, Some results on the piston problem related with fluid mechanics, J. Math. Kyoto Univ., 23 (1983), 631-641. J. I. Kanel’, On a model system of equations for one-dimensional gas motion, Differencial’nye Uravnenija (in Russian), 4 (1968), 721-734. -, Cauchy problem for the dynamic equations for a viscous gas, Sibirsk. Mat. Zh., 20 (1979), 293-306. S. Kawashima, The asymptotic equivalence of the Broadwell model equation and its Navier-Stokes model equation, Japan. J. Math., 7 (1981), 1-43. -, Asymptotic behavior of solutions to the equations of a viscous gas, preprint. S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, to appear in Comm. Math. Phys., 1985. S. Kawashima, A. Matsumura and T. Nishida, On the fluid dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys., 70 (1979), 97-124. S. Kawashima and M. Okada, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ., 23 (1983), 55-71. A. V. Kazhikhov, To a theory of boundary value problems for equation of onedimensional nonstationary motion of viscous heat-conduction gases, Boundary Value Problems for Hydrodynamical Equations (in Russian), No. 50, 1981, 3762, Inst. Hydrodynamics, Siberian Branch Akad., USSR. A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time

128

T. NISHIDA of the initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282. P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. T. Liu, Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 585-610. -, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796. -, Nonlinear stability of shock waves for viscous conservation laws, Dept. Math. Univ. Maryland Tech. Note BN-1034, 1985. T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in this book, 459-479. A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous fluids, Contemporary Mathemmatics, vol. 17, Nonlinear Partial Differential Equations, ed. by J. A. Smoller, Amer. Math. SOC.,1983. A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, to appear in Japan J. Appl. Math., 1985. -, Asymptotics towards the rarefaction waves of the solutions of a one-dimen sional model system for compressible viscous gas, preprint. J. Nash, Le probltme de Cauchy pour les tquations difftrentielles d'un fluide gtntral, Bull. SOC.Math. France, 90 (1962), 487-497. T. Nishida and M. Okada, Free boundary problems for the equation of onedimensional motion of viscous gas, in preparation. M. Okada, The free boundary value problem for the equations of one-dimensional motion of compressible viscous fluids, preprint. M. Padula, Existence and continuous dependence for solutions to the equations of a one-dimensional model in gas dynamics, Meccanica, 1981, 128-135. A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kyoto Univ., 13 (1977), 193-253. -, Two-phase free boundary problem for compressible viscous fluid motion, J. Math. Kyoto Univ., 24 (1984), 243-267. T. Taniuchi, Reductive perturbation method for nonlinear wave propagation, part 1, general theory, Progr. Theoret. Phys. Suppl., 55 (1974), 1-35. A. I. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb., 16 (1972), 517-544.

Department of Mathematics Kyoto University Kyoto, 606, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 129-155 (1986)

Predation-Mediated Coexistence and Segregation Structures By Masayasu MIMURAand Y u k i o KAN-ON Abstract. From an experimental point of view, it is reported that predator may have a tendency to increase species diversity in competitive communities, which is the so called predation-mediated coexistence. One predator-two competing prey diffusion models of Lotka-Volterra type are considered. It is shown that even if one predator and two competing prey can not coexist in the absence of diffusion, the three species are able to coexist by exploiting the differences in the diffusion rates of the prey and the predator. This points to the possibility that coexistence of competing species is enhanced by the interaction of predation pressure and by the diffusion effect. Key words: predation-mediated coexistence, segregation structures, singular perturbation method, ecological reaction-diffusion model

1. Introduction

It has been suggested that in some circumstances, predation may have a tendency to increase species diversity in competitive communities, which is the so called predation-mediated coexistence. From both experimental and observational points of view, there are a number of studies by, for instance, Paine [20], Harper [S] and Conell [2]. On the other hand, from theoretical points of view, Parrish and Saila [21], May [16], Vance [23], F u j i [6], Hsu [lo], Huston and Vickers [ l l ] and Takeuchi and Adachi [22] studied the possibility of predator-mediated coexistence by using predator-prey models of LotkaVolterra type. Intuitively, the mechanism of such coexistence can be interpreted as follows: Consider two-prey species where one is extinct due to competition from the other. If one-predator species is introduced into the two competing species community and exerts higher predation pressure on a competitively dominant species, then the competitive pressure is relaxed and hence coexistence of the two prey species is possible. In this paper, we assume that all migration occurs solely by diffusion and study segregation structures of two competing prey mediated by one predator. The model treated here is of the form Received March 10, 1986.

130

M. MIMURA and Y. KAN-ON

-d,Aul+u,(a,-bb,u,-clu,-k,v) (7-

1Lau= d 2 A u 2 +u2(a2-b,u,I

I?= DAv+ v( -r+ 1

c2u2-k,v) ,

at

a,k,u,+ a2k2u2)

at

where A is the Laplace operator in R", u,(t,x), u,(t,x) and v(t, x ) are the population densities of two prey and one predator species at time r € (0, m) and x E Q c R n . a,, a, and r are the intrinsic growth and death rates, respectively. b,, c2 and c,, b, are measures of the intraspecific and interspecific competition, respectively. k,, k , are the predation rates of the prey. a,, a2 are the transformation rates of the predator. All of the parameters are positive constants. Q is a bounded domain in R" with smooth boundary as. To express (1.1) in nondimensional variables, we let

t =a,t ,

ii2=c,u,/a, , ii, =b,u,/a, , f=r/a,, ii!,=a,k2/b, , ii!,=azkz/cz, a=a,/a, , b=b,/b, , c=c,/c, , d;=d,/a, , d,=d,/a, , D=D/a, .

i j =k,v/a,

,

k=k, / k, ,

Then (1.1) becomes

[dl-d,Au, +u,( 1 (1.2)

'

-

- u, -

cu, - kv)

-d,Au,+u,(a-bu,-u,-v)

,

r>O

,

X

~

where we drop the overbars of all variables and parameters. and initial conditions are taken to be

,Q

The boundary

and (1.4)

u,(O, x)=uto(x) ( i = l , 2)

,

v(0, x)=vo(x),

x€S,

respectively, where a p n denotes the outerward normal derivative on First we state the global existence theorem on (1.2-4).

fi=QuaQ.

Theorem 1. Suppose that there is K , > 0 such that

aQ

and

Predator-Mediated Coexistence o s ~ , o ( x ,) uzo(x)

&(X)

9

131

SKl

, that for any x €a. Then (1.2-4) has a unique solution (ul(t,x ) , uz(t,x ) , ~ ( tx)) exists for all time. Moreover, there is K z > O such that

OSu,(t, X) , u ~ ( TX ,) , d t , x ) S K z for any

~ €and 0t>O.

The proof will be stated in Appendix. When the predator is absent (v=O), (1.2) is simplified to be - d,Au,

(1.5) -= at

+u,( 1 -u, -cu2) +

,

r>O,

x€Q.

dZAu2 u , ( u - ~ u ~ - u ~ )

For (1.5) with zero flux boundary conditions (1.3), de Mottoni [19] and HSU [9]proved the following: ( I ) If a
(11) If b
(a-b)/(l-bc))

.

(111) If l/c
Case (111) is more precisely studied. Kishimoto and Weinberger [14] showed that when Q is convex, any non-constant stationary solutions are unstable. That is, (1, 0) and (0, a ) are only stable stationary solutions of (1.5), (1.3). On the other hand, Matano and Mimura [15] have proved that there exist stable non-constant stationary solutions for suitable non-convex domains Q. Coming back to the complete system (1.2-4) in the presence of a predator, we know the following result:

Theorem 2 (Conway, Hoff and Smoller [3]). Consider (1.2-4) under the assumption of Theorem 1. Let G be di- m = o > 0

where d=min (dl, d,, D), I be the smallest positive eigenvalue of - A on Q with the zero flux boundary condition and rn be the maximum norm of F'(u) in S= {u=(u,, u2, v ) € R:: OSu,, u2, V ~ K , } .Then there are K 3 ,K,>O such that

M. MIMURA and Y.KAN-ON

132

and

where G(t)=-

la

u ( t , x)dx

n

.

Theorem 2 indicates that for appropriately large d,, d2 and D , solutions of (1.2-4) become spatially homogeneous and the asymptotic behavior is determined by that of solutions of the following ordinary differential equation:

(1.6)

In this paper, assuming that at least one of the diffusion rates is not large, we intend to discuss the possibility of predator-mediated coexistence which exhibits a segregated pattern between two competing species. We consider Case (I) only, that is, when a predator is absent, one of the species (u,) always survives but the other (I(,) is led to extinction in competition even if all of the species can migrate by diffusion. Other cases will be treated similarly. uz

Figure 1.1. Dynarnical orbit of (1.6) which approaches the interior equilibrium point u* in R3, where a=0.992, b=1.5, c=1.0, al= az=0.5, k=10.0 and r=2.0.

Predator-Mediated Coexistence

133

As a preliminary stage, Section 2 is devoted to the qualitative study of solutions of (1.6). Since (1.6) has 7 parameters, a complete classification of asymptotic states of solutions will be extremely complicated. Therefore we use k and r as free parameters, leaving the remaining parameters fixed. It can be shown that for particular choices of k and r , the presence of a predator increases the possibility of coexistence of two competing prey. Temporal patterns of coexistence are classified into three types: (1) equilibrium point, (2) periodic solutions and (3) chaotic behavior by cascades of period doublings. Figure 1.2-4 shows temporal segregations of two competing species. In Sections 3 and 4, we consider the complete problem (1.2-4) under the

Figure 1.2. Periodic orbit of (1.6) where a, b, c, al,a2 and k are the same values as in Figure 1.1 except for r=1.2.

Figure 1.3. Biperiodic orbit of (1.6) where a, b, c, al,a2 and k are the same values as in Figure 1.1 except for r=1.0.

M. MIMURA and Y. KAN-ON

134

Figure 1.4. Chaotic orbit of (1.6) where a, b, c , al, a2 and k are the same values as in Figure 1.1 except for r=0.9. condition that the diffusion rates of the prey species are sufficiently small. It is shown that there appear spatial segregations of two competing prey species mediated by the addition of one predator, by using a singular perturbation technique introduced by Mimura and Fife [17] (Figure 2). It is shown that even if one predator species and two competing prey species cannot coexist in the absence of diffusion, it is possible for the three species to coexist by exploiting the differences in the diffusion rates of the prey and the predator. This points to the possibility that coexistence of competing species is enhanced by the interaction of predation pressure and by the diffusion effect. Finally in Section 5 , we give a few comments on spatio-temporal segregations of competing species (Figure 3).

‘(1

11

I’

Figure 2. Stationary pattern exhibiting segregating coexistence where a=0.95, b=1.5, c=l.O, al=a2=0.5, k=10.0, r=2.8, dl=d2= 0.005 and D=5.0.

135

Predator-Mediated Coexistence

Figure 3. Chaotic pattern exhibiting segregating coexistence where b, c, ( Y ~ , ( Y Z , k and D are the same values as in Figure 2 except for a=0.992, r=0.9 and dl=dz=0.028.

2. A Predator-Prey Model without Diffusion In this section, assuming that the diffusion rates d,, d , and D are all large, we concentrate on the study of the qualitative behavior of solutions of (1.6) as dependent on two free parameters r and k . The initial value problem is

=u,( 1- u, -cu,-kv) ( 7 dul

(2.1)

and

(2.2)

ui(0)=ui,>O ( i = 1 , 2 ) ,

Define A by

b1

A=[

C 1

v(O)=v,>O

.

:].

-a,k Theorem 3. Suppose IAIfO. If there is not an interior equilibriutn point in R:, then there are equilibrium points on the boundary of R3,, some of which are locally stable. A solution of (2.1), (2.2) approaches one of thein as t tends to infinity. Theorem 4. Suppose 1AI=0. matrix of A ( i , j = l , 2, 3).

Let A t j be the (i,j)-element of the cofactor

M. MIMURA and Y.KAN-ON

136

( i ) If A,,,+aA,,,,-rA,,,#0 Theorem 3 is still valid; ( i i ) IfA,,+aA,,-rA,,=Ofor bY

for some io {1,2, 3}, then the assertion of any i , then the solution of (2.1) is represented

U l ( t ) A " u Z ( t ) A " 2 v ( t ) A ~ ~ = u l ( o ) A ~ ~ u z ( o )(i'l, A ~ " ( o2,) A3)~ .~

The proofs are stated in Appendix. Denote by E,++=u*=(uT, u$, v*) a n interior equilibrium point of (2.1). Then the linearized matrix of (2.1) around u=u* is represented by

(2.3) so that the eigenpolynomial of M is A3+(~@+~$)AZ

+{( 1 -~c)u?u:+ ( a l k 2 ~ ~ + a z ~ ~ ) ~ * } AI =O+ ~ ~ ~ ~ ~ * l A i

It is obvious to see that when 1AI <0, u=u* is unstable, if it exists. In other words, coexistence of the two competing species is not realizable in a sense of predator-mediation. Therefore, in what follows, we assume 1Al >O.

Theorem 5 (Takeuchi-Adachi [22]). Suppose

.

(2.4)

(CU',+~L~Z)~<~CX~U'~

Then ( i ) IAI>O; ( i i ) If E,,, exists, then it is globally stable: (iii) If E,,, does not exist, then one of the equilibrium points Eooo=(O, 090) Etoo=(l, 090) E o t o = ( O , a, 0) E , =( ( 1- a c ) / ( l - bc), ( a - bM1- bc),0) 9

9

Eto,=(r/alk, 0 , (ka,-r)/a,k2) ,

.

7

Eo,,=(O, ria,, a-rYIaA

is globally stable. We fix a, b, c, a1and aZ appropriately and assume k and r to be free parameters. Then we have the following lemma on the existence of E , , , :

Lemma 1. Let / A \> O and a < b, l/c point E,,, in R3, if ak> 1 and

- - - (I).

(2.1) has an interior equilibrium

Predator-Mediated Coexistence

137

k(alc-1) ak- 1 =r(k) (Figure 4). 0 S r ( k )=az-
Theorem 6. Consider (2.1) under the assumption of Lemma 1. Let r be a bifurcation parameter. Then either ( i ) E,,+-branch is stable or ( i i ) E,+,-branch is unstable for r c (r*, r*) and Hopf bifurcations from u=u* occur at r=r* and r*, where r* and r* are some constants satisfying r
r

a2a

Figure;4. Shaded region means the existence of the interior equilibrium point E+ttin ( k ,r)-space. k* is a positive solution of IAl =O.

M

E I

E,,+

:

Eot t

Ell t o

I

r

:

I !I

i-

7,

r

Figure 5. Schematic global bifurcation diagram of equilibrium points with respect to r E R + . means stationary bifurcation point.

M. MIMURA and Y.KAN-ON

138

The proof of Theorem 6 is stated in Appendix.

Remark. If b, c, a1 and a, satisfy (2.4), (i) is realized. The occurrence of Hopf bifurcation in (2.1) was already suggested by F u j i [6], Vance [23], Kawasaki and Teramoto [13] and Takeuchi and Adachi [22]. (ii) in Theorem 6 can be considered more precisely. For some parameter choices, it is numerically shown that there is a cascade of period doublings for some parameter choices. Figure 6 shows the transition to chaotic orbits by cascades of period doubling bifurcations. As r is increasing, there occurs a Hopf bifurcation from E , , , to a stable periodic solution a n d as r continues to increase, the periodic solution undergoes cascades of period doublings and is eventually involved i n apparently chaotic behavior. Finally it condenses back into E , , , . Such a transition also occurs when k is varied. The following theorem suggests the possibility of the above phenomena.

Theorem 7. Suppose that there is a new periodic solution primarily bifurcating from a periodic solution with the period 0. Then the period of the new one is 0 or 20. This will be proved in Appendix. Calculating Lyapunov characteristic exponents of (2.1) for the values of Figure 6, we find that the dynamical behavior is asymptotically reduced to almost two dimensional behavior, and hence the solution of (2.1) is approximated by solutions of a one dimensional difference equation of the form X,,,=F(X,) (see Figure 7). We see from this observation that the values of the parameters at which chaotic behaviors occur can be determined by evalu-

7

Figure 6 . Global picture of equilibrium points and periodic solutions with respect to r E ( 0 , +m) where n=0.992, b=1.5, c=l.O, al=az= 0.5 and k=10.0. -: stable equilibrium branch. _ _.-unstable . equilibrium branch, a : stable periodic branch. o 0 0 : unstable periodic branch.

--

Predator-Mediated Coexistence

139

Figure 7. One-dimensional map obtained by plotting the uz-component at v=v* plane. ating period doubling Feigenbaum sequence [4]. It is not our purpose to discuss the above in detail, so we omit this discussion.

3. A Predator-Prey Model with Diffusion

As we stated in Section 1 , when the diffusion rates of all species are large, solutions of (1.2-4) become spatially homogeneous. I n other words, there occurs n o spatial pattern formation. In this section, assuming that the diffusion rates of the two prey species are considerably smaller than that of the predator, we consider pattern formation of the three species in one dimensional habitat I=(O, 1 ) . The equations treated here are

with zero flux boundary conditions of the type (1.3). Let us first assume

(H.1)

a
and

aI=a2=a.

The former implies that when the predator is absent,

We next assume

03.2)

O<E<<~

and

d,D=0(1)

M. MIMURA and Y . KAN-ON

140

which mean that the diffusion rates of the prey are sufficiently small rather than that of the predator. Under the hypotheses ( H . l ) and (H.2), we use a singular perturbation technique by Mimura and Fife [17] to construct nonconstant stationary solutions of (3.1). The stationary version of (3.1) is represented by

d2u 0 =?----1+ u,( 1 - u,-cu2-kv) dx2 d2u O = E 2 dx2 d ~ + U 2 ( a - b u l - u z - ~ ), 0=D

d2v -+ dx2

+

~

€

1

+

v( -r aku, au2)

with

(3.3)

du, du -(O)=--(l)=O dx dx

( i = l , 2) ,

dv &(O)=-(l)=O dx dx

We first consider the reduced problem ( E = O ) of (3.2), (3.3) O=u,( 1 - u1-cu2-kv)

0 = u,(a- bu, - u2- v)

(3.4)

d2u O=D-+v(-r+aku,+au,), dx2

XEZ

with

q o ) = - (dlv) = o . dx dx

(3.5)

The first and the second equations of (3.4) lead to

(3.61,

ul=uz=o,

(3.6)i

u,=l-kv,

(3.61,

ul=O,

(3.6)s

u, =

u~=O,

uz=a-v,

1-kv-c(a1-bc

u)

,

u2=

a - v -b( 1 -kv) 1-bc

Substituting (3.6) into the third equation of (3.4), we have

(3.7)

d2v O=D-+g,(v) dx2

x€Z ,

,

Predator-Mediated Coexistence

where s=O, 1 , 2 , 3. forms

141

Here the functional forms of g,(v) and g,(v) are of the g,(v)= { ( a k - r ) -ak2v}v

and g2(v)= {(aa-r )- av}v ,

respectively.

Assume

(H.3)

k>r/a>a,

which implies that when u, is absent, (1.1) has a coexisting equilibrium point of (u,, v) which is globally stable. On the other hand when u, is absent, there is n o such coexisting point of (u2,v) while ( u 2 ,v ) = ( a , 0) is globally stable. Define g(v; 5) by

which has the discontinuity of the first kind at v=e by (H.3), and then consider the following scalar equation with respect to v:

(3.8)

d2v O=D-+g(v;E) dx2 q O ) = - (dlv) = o . dx dx

y

X€Z

Using phase plane methods, Mimura et al. [18] proved that (3.8) has non(&/2, with constant positive solutions v(x; e C1(& for arbitrarily fixed &=(ak-r)/ak2 (& is a positive solution of g,(&)=O), satisfying

e)

e), we have a solution (uy, u!jyvo) of

By this function v ( x ; the form

e€

e*)

(3.4) which has

(3.9) indicates that only u, and v can exist in one region where t < v while uz and v exist in the other region where v<E, that is, uy and u! exhibit completely segregating coexistence with intermediate boundary points where

M. M ~ M U Rand A Y. KAN-ON

142

0

1

x*

Figure 8. Shapes of (u:(x;E ) ,

u!(x; E ) , vo(x;5 ) ) with

one point x=x*.

v(x*; E)=E. This is somehow interesting because in the absence of a predator, there is no coexisting stationary solution of u, (Figure 8). By singular perturbation methods, it is expected that the function (u:, K:, vo) becomes a candidate for the lowest order approximation to a solution (ul(x; E ) , u,(x; E ) , v(x;E ) ) of the problem (3.2) with a sufficiently small but nonzero E . In fact, the function (u!, ui, vo) is a n (outer) approximation outside a neighborhood of x=x*. We must seek another approximation in the neighborhood of x=x*. Suppose that there is only one point x* where v(x*; E)=E for simplicity only. Rewriting (3.2) by making use of the stretched variable z = ( x - x * ) / e , we have

(0

(3.10)

{

d2u =L+ u,( 1- u, - cu, -kv)

dz2

O = d ~ + u , ( o - b n , - u,- v)

The third equation of (3.10), when E=O, leads to system for u, and u,, which is represented by

v=t, so

that we have a

(3.11)

where hl(E)=l-k[ and h,(E)=a-E. with the boundary conditions

Let us consider (3.11) in - w < z < + m

(3.12) To solve (3.11), (3.12), it is convenient to rewrite (3.11) as

Predator-Mediated Coexistence

143

(3.13)

where * = d / d z . We denote solutions of (3.13) by X ( Z ) = ~ ( Up ~t ,,uzrp z ) ( z ) . Clearly Q,=c(h,(E),0, 0,O) and Qz=c(O, 0, h,(E), 0) are equilibrium points of (3.13). It is our task to find E such that there exist trajectories connecting Ql and Q2. The linearized system of ( 3 . 1 3 ) about Q, ( i = l , 2 ) are respectively x=MiX

(i=l, 2 ) ,

where

ro

1

r

o

0

0

and 1

0

01

Here we assume

(H.4)

mk> 1

Define E, and E2 by

and

(el,

Ez), M , ( I =1,2) have respectively. Then it turns out that for any fixed 5 E respectively two positive eigenvalues a n d two negative ones. That is, (3.13) has a two-dimensional unstable manifold and a two-dimensional stable manifold a t each Q, ( i = l , 2 ) . Thus our problem falls within the framework of suddle connection problems in R4.

c2)such that

Theorem 8 (Mimura and Fife [17]). There i s some e*E (El, there is a trajectory X ( ~ ) = ~ ( u : (pz;)(,z ) , u:(z), p : ( z ) ) connecting Q , and

Q2.

144

M. MIMURA and Y.KAN-ON

We have thus obtained two approximate functions. One is (uy(x;E*), u t ( x ; E*), u o ( x ;f * ) ) which is valid outside a neighborhood of x = x * and the other is ( U ~ ( ( X - X * ) / E ; f * ) , u $ ( ( x - x * ) / E ;c*), f * ) which is valid in the neighborhood of x = x * . Matching suitably these two functions, we may construct the uzo,u,) to a solution ( u l ( x ;E ) , u , ( x ; E ) , v ( x ; E ) ) lowest order approximation (ul0, of ( 3 . 2 ) in the whole interval I . These are of the form u,,(x; E ) = U : ( X ; E*)-r(x-x*)

(-;

{u: x - x *

E*) - H , ( x - x * )

(3.14)

where y ( x ) is a C" cut-off function satisfying

and

0

1

0

I

Figure 9. Shapes of the lowest approximation (3.14) to a singularly perturbed solution of (3.2).

Numerical experiments confirm that (3.14) is a nice approximation to a non-constant solution of ( 3 . 2 ) when E is sufficiently small, though we have not yet been able to prove its validity. We call such non-constant solutions exhibiting coexistence with sharp internal layers in u, and uz singularly perturbed solutions. We should remark here that the approach used here can be performed independently of the question not only of whether the interior coexisting equi-

Predator-Mediated Coexistence

145

XI

Figure 10. Shapes of singularly perturbed solutions for several values of r. a=0.95, b=1.5, c=l.O, al=ay~=O.5,k=10.0, d = l , D=5.0 and E =0.0001.

librium point E + + , exists or not, but also whether E+,+ is stable or not if it exists. Figure 10 numerically shows singularly perturbed solutions ( u l , u2, u) of (3.2), (3.3) for several values of r . The case when r = r 6 @ ( _ vP), is quite interesting, for which E,,, does not exist in the absence of diffusion. In other words, in the absence of diffusion, there never occurs coexistence mediated by a predator. However a suitable difference between the diffusion rates of the prey and the predator leads to coexistence of the two competing prey. We now address the question of why such a singularly perturbed solution exists for some r @ (c, P) and whether it is stable or not. This will be discussed in the next section. 4.

Discussion about Singularly Perturbed Solutions

T o make clear the reason why singularly perturbed solutions exist for some r @ (r-, P), we try to draw a picture of a global parametric dependency of E and r on non-constant solutions of system (3.2), leaving a, b, c, k and a fixed to satisfy (H.l-4) and D fixed to be large. The largeness of D is assumed to avoid mathematical technicalities. The analytic tool used to study such solutions is the local bifurcation analysis. We first consider the case when

M. MIMURA and Y. KAN-ON

146

( r , 4 E (r, P ) x R+ 9

under which there exists a coexisting equilibrium point E,,,. T o consider the stability of E , , , , we study the linear eigenvalue problem associated with (3.2)

J

A(€, r ) @ = -9(E)-+M(r)@ dz@ dx2

where @=“$I,

92,

-Q(E)=

0 -0

$8)

?

dE2 O

0

D

and M ( r ) is the Jacobian matrix (2.3) at E , , , , associated with the dynamics of (3.1). The set of ( r , E ) E (c, ?) x R, where (4.1) has zero eigenvalues corresponds to primary (stationary) bifurcation points where I?, ,, loses its stability property. One finds that this set consists of a n infinite number of curves {rn}Es1 by using the eigenfunctions {cos(n~x)};=,for r E ( r , P ) . Here we note that when D is large, {rn};., has the following properties: (i) {rn};=l never intersects with each other (there is n o double criticality), (ii) there is a n order with respect to n and (iii) r, is the primary bifurcation curve between {rn}zz1 first appearing when E is decreasing (Figure 11). It is to be noted that for D not so large, {rn};=l does not satisfy the above properties and the situation may be extremely complicated (see Fujii et al. [5] for one prey-one predator models). Fix arbitrarily re € (L, p) and take E as a bifurcation parameter. Then, with the result of local bifurcation analysis, we know that for any ( E ~ ,r,) E r,, there exists u,>O such that (3.2) has a unique one-parameter family of solutions ( ~ ( u )u(u)) , E R, x X bifurcating from ( e C , u*) for Iu]
147

Predator-Mediated Coexistence

E

Figure 11. Primary bifurcation curves sufficiently large.

{r,& in

( r , €)-space when D is

U

UU

EC

&

Figure 12. Shapes of ul, uz and v on D1-branch for chosen r E ( r 0 , f o ) . a=0.95, b=1.5, c=l.O, ( Y I = ~ z = O . ~ ,k=10.0, d = l , D=5.0 and r=2.0.

branch (Of course D is taken to be large). Let us consider the stability of D,-branch. First of all, we should recall Theorem 6 which indicates that in the sense of a n ordinary differential equation, either E,,, is stable for any rc(L,f ) , or it loses the stability for re (r*, r * ) ( c ( c , P)) where a (spatially uniform) periodic solution or chaotic behavior appears. Here we fix parameters except for r and E such that the former case occurs. (The latter case will be treated in the next section.) Then Figure 12 indicates that D,-branch is stable and never changes the stability for _yo< r < Po (super-critical case). This implies the well-known difusion-induced instability. On the other hand, if r is increasing ( P , < r < F ) , the branch is deformed to a sub-critical state, proceeds to the right and turns back to the limit E\O, that is, D,-branch is originally unstable but recovers its stability. If r is continued to increase to f , then E,,, is absorbed by E,,,

148

M. MIMURA and Y.KAN-ON

&

Figure 13. Shapes of uI. u2 and v on D1-branch for chosen r>F. The solid curve is a stable branch and the broken curve is a n unstable branch. a=0.95, b=1.5, c=1.0, a1=a2=O.5, k=10.0, d = l , D=5.0 and r=3.1.

Figure 14. Schematic DI-branch sheet in ( r , €)-space. The upper sheet is stable and the lower one is unstable.

and there occurs no bifurcation from E , , , . Let r be further increasing beyond F ( r > F ) . In this situation, a coexisting equilibrium point E,,, no longer exists and either E,,, or E,,, is the stable equilibrium point. However D,-

Predator-Mediated Coexistence

149

branch still exists with a limit point S,. It is noted that there are two different types of solution branches to the limit €LO. When E is sufficiently small, the upper branch corresponds to the singularly perturbed solutions constructed in the preceding section, while the lower branch to that of new singularly perturbed solutions which we have never seen. Numerical results suggest that the upper branch is stable while the lower one is unstable (Figure 13). Moreover, we find that the limit point s, as a function of r is decreasing with r and the D,-branch disappears for large Y (Figure 14). In summary, we conjecture that there exist stable singularly perturbed solutions even for r @ (!, P), which exhibit sharp spatial segregation between u, and u2. 5.

Concluding Remarks

In the previous sections, we have shown mainly two different types of asymptotic states: one is the spatially homogeneous periodic or aperiodic solutions (Figures 1.2-4) and the other is the spatially inhomogeneous stationary solutions when E is sufficiently small (Figure 2). Both solutions exhibit coexistence mediated by a predator. We should remark that the argument in constructing the latter solutions is valid independently of the stability of E,,,. In this section, we are concerned with spatio-temporal segregated patterns as shown in Figure 3. Fix appropriately all of the parameters except for a parameter E to realize the situation where E , , , is unstable and there coexists a stable spatially homogeneous periodic solution branch when E is large (Figure 1.2) and D,-branch bifurcating from E,,,-branch which enters into the region where our singular perturbation approach is applicable when E is sufficiently small (Figure 12). It is noted that D,-branch in a neighborhood of ( E ~ u*) , is unstable, if it is super-critical, because E , , , is unstable. In this situation, it is observed by numerical simulations that when E is decreasing, D,-branch is still unstable but the spatially uniform periodic solution becomes unstable and a spatio-temporal periodic solution appears. This solution exhibits

u2

111

V

Figure 15.1. Spatially homogeneous periodic solutions where a =0.95, b=1.5, c=l.O, a1=a2=0.5, k=10.0, r=1.0, d = l , 0 = 5 . 0

and ~=0.04.

M. MIMURA and Y.KAN-ON

150

111

U

U?

Figure 15.2. Spatially inhomogeneous periodic solutions where a, b, c, al, at, k , r , d, D are the same as in Figure 15.1 except for ~=0.03.

Figure 15.3. Spatially inhomogeneous stationary solutions where a, b, c, a ~az, , k , r , d, D are the same as in Figure 15.1 except for ~=0.0025. S

spatially liorno~cneoos periodic solution

....-

3000000000000000000000~*********

.**

***

--. ..

spatially inliornogeneous periodic solution

'\,D,-branch \

,_ _ - -- - - a ___________ E+++-branch

\

-_--- ----

&

spatio-temporal segregation between 2-competing species. Furthermore, when E continues to decrease, the spatio-temporal periodic solution disappears and D,-branch becomes stable. That is, singularly perturbed solutions are stable (Figure 15). This behavior can be understood as a result of the inter-

Predator-Mediated Coexistence

151

action of Hopf and stationary bifurcations (see, for instance, Guckenheimer [7]). Thus, we confirm the existence of a secondary branch connecting the spatially uniform periodic solution branch with D,-branch (Figure 16). This suggests recovery of the stability of D,-branch when E is sufficiently small. That is, singularly perturbed solutions are stable. Moreover, a situation can be considered where there coexist spatially uniform periodic and aperiodic solutions and the globally existing D,-branch. For this situation, it is numerically confirmed that when E becomes sufficiently small, D,-branch recovers the stability after fairly complicated interactions of periodic and aperiodic solutions and D,-branch. Appendix

Proof of Theorem 1. Since a standard theory of semilinear parabolic equations yields the local existence and uniqueness of solutions of the problem (1.2-4), we may only show a n a priori estimate of L"-uniform boundedness on solutions. By using the assumption Osul0(x), u,,(x), vo(x)5K1,it is easy to see that

OSu,(t,x ) S M a x (1, K , ) and 0 5 u,( t, x) 5 Max (a,K,)

Let us show L"-uniform boundedness of v(t, x).

. From (1.2), we have

Applying the inequalities u ~ ( tx)dx ,

(i= 1,2)

to (A.l), we find that there is some K,>O such that (a,u,+azuz+v)(t, x)dxSK,

for any t > O . Thus, by using L"-uniform boundedness of u, and uz and L1uniform boundedness of v, Alikakos' theorem ([l, Theorem 3.11) shows L"uniform boundedness of v. Thus, we can secure the global existence of solutions of (1.2-4). Proof of Theorem 3. Since E , , , does not exist, there are equilibrium points on the boundary

152

M. MIMURA and Y.KAN-ON

of R:. By a simple calculation, one finds that some of them are locally asymptotically stable. Therefore we may prove that the solution of (2.1) approaches one of them as t tends to infinity. We may assume 1Al >O. The case of IA I < 0 can be treated similarly. Rewrite (2.1) as

(A.2)

d -log dt

u=e-Au

where e = t ( l , a, - r ) and log u=f(log u l , log uz, log v). Multiplying (A.2) by the cofactor matrix (Atj) of A , we have

('4.3)

Let u=zi be the solution of Au-e=O. Since E , , , does not exist, at least one of zi,, ii2 and V is negative. Suppose El be negative. Note that the first equation of (A.3) becomes

Then the right hand side of the above is negative, so that uf11u$2vA13 must be strictly monotone decreasing. By the boundedness and positivity of solutions, uf%f12vA13 tends to zero as t tends to infinity. Thus it turns out that the w-limit set of the orbit u belongs to the boundary of R3,. O n the other hand, if the initial values (ulo, uzo,u,) are on the boundary of R:, the solutions (ul, uz, v) are always on the boundary of R3, a n d approach one of the stable equilibrium points in the sense of a suitable two component system of (2.1). Moreover, by using phase space analysis, it is found that these equilibrium points are stable in a sense of the complete system (2.1). Thus , w e conclude that the w-limit set consists of a stable equilibrium point o n the boundary of R3,. Proof of Theorem 4. From IAI=0, (2.1) becomes

(A.4)

In Case (i), by assumption, one of the right hand side is not zero.

Therefore

Predator-Mediated Coexistence

153

the proof of this case can be reduced to that of Theorem 3. In Case (ii), the right hand side of (A.4) are zero. So the proof is obvious.

Proof of Theorem 6 . The eigenpolynomial of A4 is

By IAI>O, n o eigenvalues of (A.5) can be zero. When r is a bifurcation parameter, leaving the remaining parameters fixed, the critical value of r is obtained by R e 1(r)=O. The necessary and sufficient condition that Re 1(r)=O is f(r) =(uF+u$){(1 -b c ) u ~ u $ + ( a , k z u ~ + a z u f )v *ufu$v*lA } I=O

.

Sicne u* is the solution of Au+e=O, u:, uf and v* are all linear with respect to 1. Then f ( r ) is cubic with respect to r. By simple calculations, we know that limf(r) =c~,u$~v* r\c

and

limf(r)= a,k2ufZv* . r/?

Therefore we have two cases: (i) if f(r)+O on (c, P ) , E,,,-branch is stable and (ii) iff@) has two real roots r,, r*, E,,,-branch is unstable on (r*, r*) and Hopf bifurcations occur at r=r, and r*. Proof of Theorem 7. Let Ue='(Uel, uez,ve) be a periodic solution with period 0 of (2.1). stituting the transformation

By sub-

(A.6) into (2.1), we have the system for U = c ( U , ,U,, V ) of the form (A.7)

d U=A(t)U+ O(I UIZ) dt

where

i

-uei

-cue, A ( t ) = -hue, -ugt alkuo,

azuez

0

Let Y ( t )be the fundamental matrix of the linear part of (A.7). The stability of the periodic solution ue is determined by the eigenvalues of T(0).Denote the eigenvalues of V(0) by A,, A, and 1, which consist of lAllS lApl and A3= 1.

154

M. MIMURA and Y.KAN-ON

It follows from Abel's theorem that I,R,=det V(O)=exp {$," tr A(t)dt}
121 [31 [41 [5

1

[6I [71

81 91 [lo] [I11

[12] [13] [14]

[15] [16] [17] [18]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 32 (2) (1979), 201-225. J. H. Connel, A predator-prey system in the marine intertidal region. I. Balanus glandula and several predator species of Thais, Ecol. Monogr., 40 (1970), 497s. E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, HAM J. Appl. Math., 35 (1978), 1-16. M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformation, J. Statist. Phys., 19 (1978), 25-52. H. Fuji, M. Mimura and Y . Nishiura, A picture of global diagram in ecological interacting and diffusing systems, Physica, 5D (1982), 1-42. K. Fujii, Complexity-stability relationship of two-prey-one-predator species system model: Local and global stability, J. Theoret. Biol., 69 (1977), 613-623. J. Guckenheimer, On a codimension two bifurcation, Lecture Notes in Math., 898 (1981), 99-142. J. L. Harper, The role of predation in vegetational diversity, in Diversity and Stability in Ecological Systems (Eds. G. M. Woodwell and H. H. Smith), Brookhaven National Laboratory, Upton, N.Y., 1969, 48-62. S. B. Hsu, On general two-species competition model with diffusion, Preprint. -, Predator-mediated coexistence and extinction, Math. Biosci., 54 (1980), 231248. V. Hutson and G. T. Vickers, A criterion for permanent coexistence of species with an application to a two-prey one-predator system, Math. Biosci., 63 (1983), 252-269. G . Iooss, Bifurcation of Maps and Application, North-Holland, 1979. K. Kawasaki and E. Teramoto, Relaxation of Competition Due to Predation; in Mechanism of Foraging Behavior, Sangyo Tosho, 1982,217-235 (in Japanese). K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibrium of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21. H. Matano and M. Mimura, Pattern formation in competitive-diffusion systems in nonconvex domains, Publ. RIMS Kyoto Univ., 19 (1983), 1049-1080. R. M. May, Stability in multispecies community models, Math. Biosci., 12 (1971), 59-79. M. Mimura and P. C. Fife, A 3-component system of competition and diffusion, Hiroshima Math. J., 16 (1986), 189-207. M. Mimura, Y.Hosono and M. Tabata, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math.

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Anal, 11 (1980), 33-46. [19] P. de Mottoni, Qualitative analysis for some quasi-linear parabolic systems, Inst. Math. Polish Acad. Sci. Zam 11/70., 190 (1979). [20] R. T. Paine, Food web complexity and species diversity, Amer. Natur., 100 (1966), 65-75. [21] J. D. Parrish and S . B. Saila, Interspecific competition, predation, and species diversity, J. Theoret. Biol., 27 (1970), 207-220. [22] Y. Takeuchi and N. Adachi, Existence and bifurcation of stable equilibrium in two-prey, one-predator communities, Bull. Math. Biol., 45 (1983), 877-900. 1231 R. R. Vance, Predator and resource partitioning in one predator-two prey model communities, Amer. Natur., 112 (1978), 797-813. Masayasu MIMURA Department of Mathematics Hiroshima University Hiroshima 730, Japan Yukio KAN-ON Information Processing Center Hiroshima University Hiroshima 730, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPp. 157-219 (1986)

On the Structure of Multiple Existence of Stable Stationary Solutions in Systems of React ion-Diff usion Equations By Hiroshi FUJII,Yasumasa NISHIURA and Yuzo HOSONO Contents Abstract. This article is intended to survey the results about pattern formation in a class of reaction-diffusion systems. The focus is on the phenomenon of multiple existence of stable stationary solutions, which has biologically or physically significant consequences. The mathematical structure and stability of stationary solutions is investigated in a certain parameter space. Especially, a-local stability and instability theorems for D1-sheet are given, and stabilization of D2-sheet is proved via two approaches: the spectral method and the singular perturbation-theoretic one. Key words: reaction-diffusion, bifurcation, singular perturbation, stability, pattern formation

1. 2. 3. 4.

Introduction Local Bifurcation Structure Shadow System and Global Problem for Large dz Stability and Instability of u) for Small u 2 0 , and Instability of D"(E,0) ( n 2 2 ) on the Shadow Ceiling 5. Structure of Solutions on and near the Singular Wall 6. Singular-Shadow Edge as the Organizing Center of the Ultimate Structure 7. Concluding Remarks # References

157 169 185 191 199 208 216 217

1. Introduction

Reaction-diffusion systems have for a long time attracted the considerable attention of researchers in various fields such as mathematical biology, biochemistry, physics and so on, as model systems describing the phenomenon of Received October 28, 1985.

158

H. FUJII,Y. NISHIURA and Y. HOSONO

pattern formation. In fact, in contrast with the simpleness of the form of equations, they are known by the variety of their qualitatively different solutions, which include not only stationary patterns,. but also spatio-temporal patterns like travelling waves, spiral patterns, etc. This article concerns the formation of stationary patterns, observed in a class of reaction-diffusion systems. The origin of our problem may be traced back to the work of A . M . Turing [40] written in 1952, with the basic aim of finding the mathematical mechanism of pattern formation in morphogenesis. This article is intended to suinmarize the results to date concerning understanding of this phenomenon of pattern formation in a class of reaction-diffusion systems. It is not, however, intended to give a thorough survey on this subject. The focus will be on the phenomenon of multiple existence of stable stationary solutions-a phenomenon which seems to have biologically or physically significant consequences. We shall investigate the mathematical structure and the stability of the totality of stationary solutions in a certain parameter space, which may lead to such a complex structure of solutions. The main part of this survey consists of results obtained by the authors. It is noted with sincere appreciation that contributions, directly or indirectly, of Prof. M. Mimura, Hiroshima University, have always been indispensable to the above works. The paper of Prof. M . Yamaguti et al. [26] motivated the authors to the study of these topics. They wish to express their sincere thanks to Prof. Yamaguti on the occasion of his 60th birthday. Let us introduce the system with which we are concerned, and then, refer briefly to its origin. The model system is the reaction-diffusion equations of activator-inhibitor type, which takes the form:

where the no-flux conditions are imposed on the boundary: --I)=au -

ax

ax ’

t>O,

x€{O,l}=L3Z.

Throughout this paper, the space dimension is assumed to be one. The functions f = f ( u , v) and g=g(u, v) represent nonlinear interaction effects. The crucial feature, which we assume here, is that the nullcline o f f takes a sigmoidal form in the (u, v)-plane. This assumption comes from biological or physical considerations, and hence, will be discussed later.

On Multiple Existence of Stable Solutions

159

On the origin of

our system: There are two origins, both from mathematical biology. In the field of morphogenesis, A. M. Turing considered reaction-diffusion systems in 1952 to discover how self-organization of structures could happen in biological systems. His idea may be summarized as “diffusion-induced instability’’, or, in modern mathematical terminology, “a pattern formation due to symmetry-breaking bifurcations”. His idea has been followed by many people in various fields, for instance, the Bruxelles group in chemical reactions [19, 291, the Tubingen group in morphogenesis [16], and so on. Another origin lies in mathematical ecology, as a model describing population dynamics with migrational effects of several ( 22) species, for instance, prey-predator interactions with diffusion. The basic aim is to discover the mechanism of appearance of spatial heterogeneity-known as patchiness-in population densities, even under a homogeneous environment. For more discussions, we refer to [2, 21, 28, 35, 401. In order to motivate our basic assumptions to be imposed on the nonlinearities f and g, let us suppose, for a moment, that (P) is a n ecological prey-predator system. Then, u and v denote the population densities of the prey and predator, respectively. (Hence, we are interested only in non-negative u, v20.) The nonlinear interaction terms f and g are, in general, expressed in the form:

f = H ( u , v)u

and

G=K(u, v)v ,

where H and K describe the effective growth rates of u and v, respectively. That the prey and predator are in a relation of activator and inhibitor implies : In ( u , v)€ D (an appropriate d o m a i n ) c R:, ( i ) the growth rate of u decreases as v increases, i.e., He< 0, and ( i i ) the growth rate of v increases as u increases, i.e., K , > 0. Also, in many ecological models, it is assumed that K,< 0. Concerning the conditions on f, we have already assumed that the nullcline of f is sigmoidal. In ecological terminology this means that in the absence of the predator, the prey has a n optimal population density in its growth rate. In other words, there is a n autocatalytic effect in H , and S O , H(u, v)=O has a “hump” as in Fig. 1.1. The necessity of such a “hump” effect in the formation of patterns is discussed in Mimura-Yamaguti [28]. From ecological considerations, f > O in the lower part of the curve H(u, v)=O. See, [2] and [28] for more discussions. Keeping in mind the preceding considerations, the main assumptions in this article may be classified into two groups. The first group of assumptions, which we call “Turing Assumptions” (Assump. ) is concerned with a small neighborhood D, of a rest (critical) point U = Ue (=(uC,vc)€ R2,). See, Fig. 1.2.

H. FUJII,Y. NISHIURA and Y. HOSONO

160

Fig. 1.1.

0

.

I

( 1 1 1

U

Fig. 1.2. Turing Assumptions.

Fig. 1.3. Local Singular-Shadow Assumptions.

Fig. 1.4. Global Shadow Assumptions.

The second group, called “Local Singular-Shadow Assumptions” (Assump. ), is imposed on f and g on a band-shaped region D , c R ? , i.e., in a neighborhood of a line U = Y * (defined later). See, Fig. 1.3. The “global Shadow Assumptions” () on Da. Turing Assumptions (Assump.
Then, it holds that

161

On Multiple Existence of Stable Solutions a,,>O ,

al2
and

a,,>O

a,,
,

and that Tr (Ac)
and

det ( A c ) > O .

Local Singular-Shadow Assumptions (Assump. g > _ O and da>bs>_O,with 6s-bs=26>0*)) that the following properties hold: ( 1 ) Let Z be the zero level set o f f : Z E { ( U , O ) E R ZIf(u,u)=O}. ,

Then, the intersection respectively by

nZ

consists of three curves R - , R , and R , , defined

u = h - ( u ) , h,(v) and h + ( v ) , u € F , with and

h t ( v ) ~ C w ( Y ) , i=-

,0

and

+.

The curves R - , R , and R , divide Ds into four open subregions D g ) , i = l , .*., 4 (from the left to the right). Then, f < 0 in DF) U OF), and f > 0 in D:) U DP’. Let

and (1.2)

( 2 ) There is a unique u*€ Y , such that J,,(v)=O if and only if u = u * , and

*)

Without loss of generality, we may take: ba=v*+6,

b_a=v*-6.

162

H. FUJI],Y.NISHIURA and Y.HOSONO

( 3 ) f u < O on ( u , v ) € R , U R - . On the nonlinearity g, it is imposed that the curve g=O lies between R , and R-. Namely, (4,

IR-
lR+.

Fig. 1.5.

(V6>0, small).

Remark 1.1. Assump. (3) defines a bistable kinetic system for u, when v € 7 is fixed. In other words, the ODE uC=f(u,v) has two stable critical points u, =h,(v) for each v E Y. Global Shadow Assumptions (Assump. ): There is a n open rectangle D,=(g, Cf)x(b, 6)= %X F in R:., (O<_a
u = h - ( v ) , h,(v) and h + ( v ) , v c y ; where the equalities h-(v)=h,(v)

holds if and only if

v=b,

and h,(v)=h+(v) holds if and only if

v=6

It holds moreover that Cf=max h+(v) and g = m i n h-(v) . v E P-

V € 3

Also,f>O in the lower part of the curvef=O.

.

On Multiple Existence of Stable Solutions (2)

163

There is a unique v*= Y, such that

J,(v)=O if and only if

U=Y*

,

and

( 3 ) f, 0 > S I X -

*

(b) The curve g=O traverses the curve f = O once and only once at U = u" ( = ( u C , uc))E R,; moreover, U c is a Turing point satisfying Assump. . Global Singular assumptions (Assump.
,

vc F ,

where the equalities h-(u)=h,(u) holds if and only if u = b , and

h,(v)=h+(u) holds if and only if u = 6 . Also, f > O in the lower part of the curve f = O . ( 2 ) There is a unique V* E "t, such that

Jo(u)=O if and only if

U=V*

,

and

on ( u , u ) € R , U R-. ( 4 ) (a) The curve g=O lies between R , and R-, and hence, ( 3 ) f,
gl R+ >

(b) It holds that

> gl R-

*

H. FUJII,Y.NISHIURA and Y.HOSONO

164

where RZ are

Remark 1.2. (1.3)

Let:

G , ( v ) g g ( h + ( v )v) , ,

V€

Then, it holds that

Hence, in view of assumption (3) above, (4)-(b) is equivalent to assuming: G: ( v ) / ~<;0 . Local Stability Assumptions (Assump. ): Let RT-(h+(v*), Y*). We say that the local stability assumptions are satisfied if one of the following two sets of assumptions (I) and (11) is satisfied. (1) ( 1 ) J Z Y * ) < O , ( 2 ) f,gu
Remark 1.3. ( 1 ) A number of known activator-inhibitor models satisfy both (I) and (11) of . ( 2 ) In prey-predator models, galR: may become positive. However, they satisfy the assumptions (I), if K,O on RT or RX, but still g,,O on RT U RX. Hence, the setting (11) is satisfied. Although these Assumptions are derived from properties common to activator-inhibitor interactions, it is worth noting that so long as our Assumptions, especially local ones, are satisfied, corresponding conclusions hold equally even for reaction-diffusion systems with interactions of the non-activator-inhibitor type.

On Multiple Existence of Stable Solutions

165

In this survey, the discussion in Sec. 2 proceeds under Assump. , and in all other Secs. Assump. are assumed. I n Sec. 3, the structure of solutions is discussed under , and as well. Sec. 4 discusses stability under and , Sec. 5 under , and also with . The main conclusion of the present survey, “the multiple coexistence result”, holds for small u only under Assumps. and . We show several examples. They satisfy all of the preceding Assumptions. Example 1 (Diffusive prey-predator model [26]). This is a model system of a prey-predator interaction with diffusion:

where k is a positive constant; fo(u) is a smooth function such that ( i ) fo(u)20,

>o,

o
for some positive constant c ; go(u)=co+clum (co, cl, m>O). The zero level curves of the nonlinear terms are shown in Fig. 1.6. Note that ( El) satisfies (I) of Assump. .

Fig. 1.6. Functional forms of prey-predator model.

Example 1’. The following system will be used as the model of numerical computations: (See, [9], [ l l ] and [12].)

fO(u) =(35+ 16u-u2)/9 2 g,(u)=l+--v. 5

,

Numerical data which will be introduced in this paper are always for this system. Note, however, that the interval Z is taken as Z=(O, 5).

166

H. FLJJII, Y . NISHIURA and Y . HOSONO

Example 2 (Gierer-Meinhardt model with saturation [ 161). The following model was proposed by Gierer and Meinhardt [16] for the study of morphogenesis:

h, =dhh,,

+c’P’~

-I

,

J ~

where O.

Fig. 1.7. Functional forms of GiererMeinhardt model with saturation.

See,

Fig. 1.8. Functional forms of Seelig’s model.

Example 3 (Seelig’s model with diffusion [25]). The final one is a model of a substrate-inhibition reaction-diffusion system:

where r(u, u)=uv/(l+u+u+Ku2), and j l , j z ,p, r and K are all positive constants. This model without diffusion was originally proposed by Seelig [39]. See, Fig. 1.8. Note that (E3) satisfies (11) of Assump. . Let us turn to our main topic, “stable solutions”. It is only recently that the existence of such a phenomenon has been found in the real physical world-though from a purely theoretical standpoint, it is not such a n unusual phenomenon. In a survey of results of a series of physical experiments concerning the Rayleigh-BBnard convection written in 1980, J. P. Gollub pointed out that “One important discovery is the observation that there are a number of stable convection patterns in cells with small aspect ratio, depending on the past history or initial conditions of the system” [17]. Concerning our reaction-diffusion systems, a similar phenomenon has been discovered numerically by M . Mimura(*), during his stay a t Oxford in (*)

Personal communication.

On Multiple Existence of Stable Solutions

167

1976-77. His system is a diffusive prey-predator one, Eq. (El)’. We can find also several papers which report, directly or indirectly, numerical evidences of such a coexistence phenomenon. See, [19] for Brusselator model, and [4],[24]. The biological or physical consequence of this fact seems important, since this means that several biological or physical states are admissible in the nature. Therefore, at this stage, we would like to begin by admitting this multiple coexistence phenomenon as a physical/mathematical reality. The goal is to answer the question, “how and why can multiple, stable solutions appear?’ Our standpoint is to understand “what kind of nonlinearity of the system may lead to such a complex structure of solutions”. There are three approaches. The first one is the Turing setting (as stated before), where the basic mechanism is that “diffusion destabilizes the uniform equilibrium state U = U c - a stable critical point (in the sense of ODE) of the corresponding kinetic system, yielding new spatially nonuniform states. However, there are several criticism to this Turing setting. This idea is exactly the setting of the local bifurcation theory-or, symmetry-breaking bifurcations from a trivial state. Hence, it shares all the weak points of the theory, as well as advantages (such as the linearized stability can be easily investigated as a by-product, it works equally well for higher dimensional problems, and so on). This may give a n explanation of the onset of formation of patterns, but they are small amplitude ones. Another criticism is that the Turing assumption is not necessary for pattern formations! See, E. Conway [ 2 ] . I.e., even though there are no local bifurcation points, there can exist large amplitude patterns. (See, discussions in Sec. 5 ) . The third opposition is that if we regard the problem as a bifurcation problem with respect to the diffusion parameters ( d , , d,) E R:, it is only the neighborhood of a countable number of one-dimensional lines in the (dl, d2)space, i.e., near the bifurcation lines (see, Sec. 2 ) that patterned solutions exist. Moreover, it is only in the vicinity of the most outward bifurcation line where solutions may be stable. See Sec. 2 . One might say that the situation may well be compared with that for a n explorer of a new continent, whose “map” is almost white except in the neighborhood of the coast line. However, in reality, formation of patterns, especially of “big patterns”, is observed for a (dl, &)-region far from those primary bifurcation lines. However, despite the local nature of the bifurcation theory, “evidences” and explanations of multiple existence of stable solutions may be possible to some extent as consequences of bifurcation structures near multiple singularities. See, Sec. 2. Perhaps, the most crucial limitation of the local bifurcation theory is that

H. FUJII, Y . NISHIURA and Y . HOSONO

168

it gives almost n o information on the relation of the structure of solutions with the global structure of nonlinearities of the system. The second approach is the shadow method developed by the second author in 1301 and [311. This method has been introduced to investigate the global behavior of solutions of the limit system of d2 T 00 - the shadow system. It is based on tha fact that the second equation (concerning v) of the stationary problem of (P) degenerates to a n integration condition. The results in [31] show that under Assump. (defined later). An interesting conclusion is that even under Assump. (with a situation possibly violating Assump. (within each symmetry D;, n= 1,2, holds near the singular wall d,=O. Destinations of the Turing branches, or global behavior of Dn-large amplitude solutions in such a case can be discussed also using the shadow method. These results are summarized in Sec. 3. Now, the third approach is the singular perturbation method, which gives the existence of large amplitude stationary patterns. This gives a constructive proof when one of the diffusion parameters, say Eis small enough. P. C. Fife [ 5 ] has given a basic framework for Dirichlet boundary value problems, and M. Mimura et al. 1271 have extended it to our Neumann boundary value problems. We shall refer this solutions, and use the symbols D:(E,u), where E z l / d l and a d ; ' (n=1, 2 , Here, the integer n > l corresponds to the fact that D:(E,u)=(uD;(x; E , u), vD:(x; E , u ) ) are periodic functions of mode n with n interior transition layers. This class I large amplitude family D ; ( E ,a) exists locally in u, i.e., in Qn= {(c, a) I O < E < E , , O. It is known also that under {global Sg), it exists a-globally until u u:. See. [20]. Concerning the stability of the class I family, it has been a n open question until very recently, "though numerical evidence leaves little doubt that the pattern is quite robust" (according to E. Conway [2]). The paper of the authors [34] shows the stability of D1(c,u ) in Q1,ul: small, under Assumps. +. It proves stability of large amplitude D1(c,a) in terms of the nonlinearities of the system.(*) A n outline of these results will be introduced in Sec. 4. An instability result is also discussed under a situation violating {local st>. It is to be emphasized here that the stability property of Dn's ( n 2 2 ) is not the same as D 1 ( ~ , u ) The . reason is that there exists a mechanism of symmetry-breaking destabilization a n d of recovery of stability in a neighborhood of . m e )

v'z,

a ) .

(*)

Recently, a 0-global stability result has been obtained by the authors. See, [*I].

On Multiple Existence of Stable Solutions

169

o)=(O, 0). I n fact, on the “shadow ceiling u=O”, D ” ( E0) , ( n 2 2 ) are all unstable; they have exactly (n- 1) unstable (=positive) eigenvalues, corresponding to D n symmetry-breaking eigenfunctions. See, Sec. 4. The central concept of the papers ([15], [33]) appears here. I.e., the ultimate symmetry-breaking stabilization of D % ( Eu)’s. , This means roughly that all members of the Dn( 2 2 ) family recover their stability as E 1 0 for u > 0 small. There are two aspects in the proof of this proposition, i.e., the spectral aspect and the structural one. The first aspect is the study of asymptotic behaviors of spectra of Dn’s. The second one is the structural porblem of studying the mechanism of symmetry-breaking stabilization mentioned above (if it exists). Sec. 5 and the first half of Sec. 6 are devoted to such structural studies. The results may be summarized as follows. ( 1 ) The stabilization mechanism of the class I family can not be understood within the class I family. More precisely, the class I are not the all possible large amplitude solutions. Sec. 5 introduces such a result, with discovery of a second class of large amplitude solutions, possessing interior transition layers and new type of transition layers called Neumann layers. (See also [14].) This Neumann layered family will be called the class ZZ family. Interrelations of these two classes of large amplitude solutions on the singular wall E = O are discussed. ( 2 ) In Sec. 6, it will become clear that the class I1 solutions are the global E 1 0 destinations of the secondary branches from the class I-Dn family (so long as D ~ + (uE) are , concerned). ( 3 ) The origin of this stabilization (or,destabilization) associated with the D2 symmetry-breaking bifurcations lies in (or, frozen at) the structure of “solutions” at the singular-shadow edge ( E , u)=(O, 0 ) . Note that a t the singularshadow edge, our problem (P)is not any more a system of differential equations (Sec. 6). ( 4 ) The spectral study for the D2(e,u ) sheet (Sec. 6) shows that the class I1 solutions are responsible to the recovery of stability of the class I solutions. We have some asymptotic estimates of the spectrum of D2 and so on. As consequences of these mathematical structures of large amplitude solutions, we prove the multiple existence of stable stationary solutions in (P), although our analysis is restricted to the simplest case, i.e., D ~ ( Ea), and D ~ , ( uE),sheets of solutions. (E,

2. Local Bifurcation Structure -Results from the Turing assumptions This section describes consequences of the “Turing setting”, and of some symmetry considerations of the system. Recall that Assump. is local and concerns only a neighborhood D , of a critical point Uc=(uc,vc)€ R2,.

170

H.

FUJII,

Y . NISHIURA and Y . HOSONO

We let

The main assumptions are that

and Tr ( A c ) < O ,

det (Ac)>O

Let us begin by considering biological aspects of Assump. . The conditions a,,,O come from f , < O and g,>O, a natural global assumption to activator-inhibitor systems; < O is a consequence of: gvl,,o,O=(K(u, v)v)ulK=o=K"vIK=o
i

The condition U , , ~ > O claims that u" must lie on R,. Since det ( A c ) f O , two curves f = O and g=O intersect transversally at Uc, and both slopes there are positive by virtue of the assumption for sign ( A c ) . Moreover, det ( A c )>O implies that the slope of g=O at UE is larger than that of f=O. Also from sign (Ac)),the positive valued regions o f f and g are on the right-hand side of each curve. Thus, qualitatively, the arrangement of the nullclines f = O and g=O is uniquely determined as in Fig. 2.1.

0'

Fig. 2.1.

1

Functional forms of f = O and g=O near U=Uc.

Through the change of variable U = Uc+ V, the stationary problem of (P), denoted by (P)omay be written in the form: D-V+A"+N(V)=O, dz

X€Z,

xar,

On Multiple Existence of Stable Solutions

171

where we put:

and

N( V )=F( V + V )-A c V ; in which

Note that N ( V ) = 0 ( I V l 2 ) ,in view of F ( V ) = O . (P)o has a trivial solution V=O, corresponding to the constant solution U = Uc, for all (d,, d,) € R:. Let us study the stability of the constant state U = Uc. We introduce the linearized eigenvalue problem:

By the Fourier cosine expansion Y= C;=o On cos (nnx), (E) is reduced to

(EL

(A,-AI,)O,=O

,

n=O, 1,2,

--- ,

where

The characteristic equation of (E), is, for each n 2 0 ,

First of all, one can confirm from the assumptions Tr(Ac) O that Re 2; < 0, which means the stability of U = u" for space-clamped perturbations, i.e., the stability in ODE sense. Secondly, ford,, d, large enough, ReR,O, for all n 2 0 , A,, is never purely imaginary, ruling out the possibility of Hopf bifurcations. Hence, it is enough to consider the case R,=O for some (dl, d,) E R2,. Solving Eq. (2.1), one finds that R,=O on (d,, d,) € f,,where

--

H. FUJII,Y . NISHIURA and Y.HOSONO

172

( d l ,d2)~dz=rn(dl)"'--[ 1 ~1,zaZ.l + a z v 2 ] } n2n2 n2n2d,-a,, I

(n=1,2,

rn's are hyperbolic curves, and they are similar in the sense that

See, Fig. 2.2.

V

J>O

\

C

Fig. 2.3. Turing instability does not occur in this case. Note, however, that (assuming aZ,,O. Namely, whenf,l,c
r,

(2.3)

rn,,%nrm

( n , m 2 1 , m>n) ,

where two primary bifurcation lines intersect, are double critical points. define rl,( n r l ) by

(2.4)

We

rprnimut n r,,, .

Then, on each point rl,,the linearized operator of (P),,a t V=O has a onedimensional kernel N , = { @ , cos ( n n x ) } , and on r,,, it has a two-dimensional kernel N , , , = N , u N,.

On Multiple Existence of Stable Solutions

173

Let us introduce the concept of the symmetry group of the system, namely, the maximal covariance of (P),, which will play a basic role throughout this paper. In appearance, the symmetry of the equation (P), seems to be only the reflection with respect to the midpoint x = 1 / 2 . However, as is shown in [12], (P), has a much larger symmetry group D" in a generalized sense-the Lie group of plane rotations and a reflection. (Such a symmetry may be called "a hidden symmetry".) In fact, we identify the Neumann boundary value problem (P), with a section of the periodic boundary value problem (F), defined on the circle f= 1-1, I]:

B(D, V ) E D -d2 B + A " B + N ( V ) = o ,

X€f.(*)

dx2

x:, s

Then, the set S of the solutions of (P), is identified with the section Sn being the set of solutions of (F),, 3: the closed subspace of the solution space 2,such that

X:={B€XI

V(-x)=V(x),x€f}

i

s (e),,

Thus, we take the strategy to look for all (bifurcating) solutions of and then take the section S of .!?.(**I Now, we define the symmetry covariance of (P), in terms of the extended operator IT. Let G be a compact continuous group.

Definition 2.1. We say that (P), is covariant under G in a generalized sense, if @ is covariant under G in the sense that T ( g ) B ( D ,V ) = @ ( D ,T ( g ) V ) ,

(2.5)

VgEG, Vt€Dom(H),

where G-GL( P) is a linear continuous representation of G on Y.(***) Let us define the "symmetry of a function V in X". (*I

2 is an appropriate solution space, e.g.,

2=(Hi(I",)Z =(closure of {exp(innx));:,

in

HZ(T))Z ,

and F=(L2(f))2(a complex L2 space): i?: R:xDom(G)(c&?

(O€Dom(I?)).

A solution V of (P), is identified with P, a solution of ($0, by taking the even extension of P to 7. (***) Assuming the inner product of d is stable under T, T is a representation of G on d, also. (**I

174

H. FUJI[,Y . NISHIURA and Y . HOSONO

Definition 2.2. A function V in X is invariant under G in a generalized sense, if

T ( g ) 8 =8 ,

(2.6)

for all g E G

where is the even extension of V to f = [ - 1 , It is easy to see the following

,

11.

Lemma 2.1. LP)ois covarianr under D", where T : D"-GL( representation on Y ;

(2.7)

p) is a

unitary

(T(r0)8)(x)= 8(x+B/n) and ( T ( s ) ~ ) ( x8)(=- x )

.-

Let Dn, n= 1 , 2 , -,denote the dihedral group of order 2n which sends a regular n-polygon onto itself: Dns{r:, srk}:zt

,

where r, is a rotation of 2n/n, and s a reflection at x=O. A representation T of Dn is given by the one naturally induced from T defined in Lemma 2.1. The following "Fold-up Principle" has a close relation to the D--covariance of (P)o. Proposition 2.1 (Fold-up Principle). Suppose W = W ( X ;_d) is a solution of (P)o,at_d=(d,,d,). Then, R n ( W ) ( x )isasoZutionof(P),at&ln2,f o r n = 1 , 2 , . - - . Here, (2.8)

W(n(x-i/n; 4 ) , W(n(l/n-(x-i)/n);_d) ,

for i / n < x < ( i + l ) / n (i=O, 1,

- - -,n-1).

i =even, i=odd,

Note that R"(W ) is D"'4nvariant if W is Dk-invariant. Now, coming back to the simple bifurcation point at r',=(d,O, d;), we can immediately see the Proposition 2.2. ( 1 ) The kernel N , at rl, is spanned by a function I , invariant under Dn (n= 1,2, .) (In=@n cos (nnx)). Note. The kernel corresponding to is two dimensional, spanned by @,exp(+innx). ( 2 ) The bifurcation equation at rl,itself preserves the D"-covariance in a generalized sense ( S e e , [ 381 and [ 121). ConsequentZy,

--

(2.9)

n(E;ul,a 2 ~ = E [ - ( a l u , - a , u 2 ~ + ~ E 2 + ( h . o,. t . ~ (l E E R ) ,

where (h.0.t.) denotes higher order terms 0([fl4, 1uI2),and

On Multiple Existence of Stable Solutions ~i

( i = l , 2) ,

=dl -df

at=7

m2 L

175

q5;q5?*

>0

(i= 1,2)

(Here, @ , , = t ( ~ l ,q52); @$=t(q5F, $$): the eigenvector of the adjoint matric A:.) Hence, all bifurcating branches from rl,are one-sided. ( 3 ) The primary bifurcating branch from rl,consists of Dn-invariant functions, n= 1 , 2 , .

---

This DGnvariance continues to hold, even at limit points, or other simple bifurcation points (where secondary branches are invariant under a subgroup D k c D n ) . See, [12]. Thus, one can label each primary bifurcating branch by its symmetry group D". A remark on the stability of bifurcating branches. At a cross section: d,=fixed (large enough), the bifurcation diagram is either Fig. 2.5 (a) or (b), depending on the sign of ,B in Eq. (2.9); the numbers of eigenvalues with positive real parts are indicated along each branch. We can see from these dia-

LT=d;'

Fig. 2.4. Each primary bifurcating sheet is labelled by its symmetry group D" ( n = 1 , 2 , a * . ) .

Fig. 2.5. (a)

Fig. 2.5. (b)

H. FUJI],Y . NISHIURA and Y . HOSONO

176

grams that their possible stable branch is only D', and all the other branches Dn's ( n 2 2 ) are unstable at least in a neighborhood of the bifurcation points

r;.

Another remark on the diagram (b). Depending on parameters of (P)o,/3 may take both positive and negative signs. However, the argument in Sec. 3 shows that under
Proposition 2.3 (Classification theorem). ( 1 ) If n/m=integer=r(z2) E[A13(u)+ A ! ~ ) ( U ) ~ + A ~ ~ ) ( U ) ~ ~(h.0. + At.)l ~ ~, ) ((rU=2) ) ~ ,~ +(12)

II,

+ +

= E[A',$(u) A~~'(u)Ez++AI.T;'(o)EgfA1~';1(u)7jr2+ (h.o.t.11

1'

+

5[A',:)(u) A$;'(o)E2+A',T;'(u)v2 (h.0. t.11

,

, (r= 3) , (13) (t-241,

(2.10) n , = A ~ l " l ' ( u ) ~ + A : ~ d ( u ) 5 ~ + ~ ~ ~ ~ ( u ) 5 ' ~ + A ' , ~ ~ ( u ) 7 j.r s + ( h . o . t . ) (2)

(2.11)

If n/m=non-integer, then

t

I T , = E [ A ~ ~ ~ ( u ) + A ' , T b ' ( u ) 5 z + A ~ ~ ~ ( u ) ~ ,z + ( h . o . t . ) ] (NU ITn =v[A',l;'(u) + A l ~ ~ ( u ) +A&)(u)q2+ E2 (h.o.t.)] .

( 3 ) It is if and only if n/m=2 that second order terms (as 57, 5,) remain in IT, or in IT,. Also, when nlm=r=2, IT, is odd in 5 and II, is even in 5.

Preimage of zeros of the system (ITm, IT,) can be further classified according to their coefficients. See, [12]. For instance, let us consider the case 12, namely the case r=2. Then, the bifurcation equations become (2.1211

IT,=5{ -( a , ~-, a d + ~ ~ ~ v + ~ ~(h.o.t.11 ~ 5 =O ~ +, ~ ~ ~ v ~ +

(2.m

IT,= -(biui-bz~z)v+ q z o E 2 +qz,E2v+ 403v3+ (h.0.t .) =O ,

where

On Multiple Existence of Stable Solutions

177

Here K , and K , (resp. K: and Fz) are the normalized eigenfunctions corresponding to the zero eigenvalue of the linearized operator 9 ( T m , ,(resp. J ptj and ql, are the corresponding coefficients of the its adjoint P*(f,,J), Taylor expansions with respect to and p of

e

and

where is the orthogonal complement to the kernel space in the LyapounovSchmidt decomposition; ( , ) denotes the inner product in (Lz(Z))z. We note that by a direct claculation (2.13)

azlal> bz/bl .

We assume, for a moment that, pllq20f0, to which case we refer "nonwe obtain by the implicit function degenerate 12". Dividing (2.12), by theorem

e,

H. FUJII, Y. NISHIURA and Y. HOSONO

178

(2.14) represents a solution branch projected on (f, v)-space for fixed u9. Since (u(u,) is positive for small u2, we can see that there are two different types according to sign(p,,q,,) ( =sign(~,,qzo)). (Z2-h)-type (pllqzo>O,Fig. 2.6 (a)): in this case (2.14) is a hyperbolic curve. The solution branch where pl1>0, q2,,>0 is drawn in Fig. 2.6 (a). (12-e)-type ( p l l q z O < O Fig. , 2.6 (b)): (2.14) is a n elliptic curve for this case. The typical figure for pI1>O, q z O < O is shown in Fig. 2.6 (b). Fig. 2.7 shows the local bifurcation diagram (I2-h), near r1,2. It is observed that on D2-branches there appear secondary bifurcation points Rkf!, or Lkf’. Despite the local nature of the classification theorem, its consequences are themselves interesting, and diagrams near double singularities are by far more complicated than those near “simple” bifurcation points. For instance, let us show a bifurcation diagram on a tubular domain Q near r,. rlitself is a simple bifurcation line, and moreover, on rl there lie a countably many double bifurcation points rl,k (k=2, 3, .). As a n illustra( k 2 2 ) as: tive example, let us assume the types of double singularities rl,k

-.

r1,*: 12-11 , rl,3: 13-s, rl,4: IE-e’ , rl,5: IOD-s , rl,8: IE-e’ , and so on. (See, [12] for the classification.)

u,>o

u,=o

Fig. 2.6.

a,<0

(a)

On Multiple Existence of Stable Solutions

'I

'I

Fig. 2.6.

Fig. 2.7.

179

(b)

Local bifurcation diagram (12-h) near rl,2:r,,2=(1.600, 36.75). R$t?+is the first recovery point of 0;;L$t' is the first losing point of D,.

In a particular system, it is the types of singularities, and not concrete values of coefficients of the bifurcation equations, which are important to grasp qualitative situations. I n Fig. 2.8, we see a number of secondary bifurcating branches exist.

H. FUJII,Y. NISHIURA and Y. HOSONO

180

rl,

6

Fig. 2.8. Bifurcation diagram on a tubular domain near TI.

A numerical result of tracing branches of solutions from D2 and D8 is shown in Fig. 2.9. This picture is obtained through a numerical method making use of group theoretical “standard decomposition” of the numerical solution space. See, [9] and [ l l ] for details. The model used here is the diffusive prey-predator system Eq. (El)’. The theoretical diagram Fig. 2.8 represents correctly the secondary branch connecting the primary D 2and D 8 branches.

0.7280 0.8672

I

Fig. 2.9. A numerical result of tracing branches of solutions [12].

The following diagram Fig. 2.10 (a), numerically obtained, shows the z , (with d,=fixed, d, > d i * z ) ,for the behavior of branches near r 1 , 2 = ( d : ~dis2), diffusive prey-predator model (El)’. Fig. 2.10 (b) shows the functional forms of bifurcating solutions on each point located on the branch D’.

On Multiple Existence of Stable Solutions

181

( a ) at A (d,z2.800) ( b ) at SI(d1z1.144) ( c ) at Ri,- (d,~1.325) ( d ) at B (d,~0.100) ( b)

Fig. 2.10.

Numerical result at d2=50; here, r1=3.108, fz=1.730, R$!'= 1.325, S1=1.144=S:.

When we compare this with the theoretical diagram near f , , 2 , Fig. 2.1, we find that Fig. 2.1 does not explain the following two points. ( i ) In Fig. 2.1, the primary D2-branch has only one secondary bifurcation point R:!: (and so, only one side of the D2 recovers stability), while the numerical result shows that the both sides of the D2-branch recovers stability. ( i i ) Moreover, the secondary bifurcating branch from R$!F of D2 is the primary D1-branch itself, showing the destination of the D1. The D1-branch (=the mode-1 branch) does not exist at the left of the limit point S. A reconsideration on rls2 Firstly, if the nonlinearities f and g give an additional odd symmetry at U = V , i.e., if F ( U c + V ) = - F ( V - V ) , then N ( - V ) = - N ( V ) . The maximal symmetry group is now D m x Z z . Consequently, the bifurcation equations are all odd in E and 7. f , , z thus reduces to (N1)-type, since pl1=qZ0=0. Even if F is not odd at U = Uc,the key parameter p l l q z 0may traverse zero in some cases. In the former analysis, they are assumed fixed and non-zero. See. p. 20 of [12] for such a n example. Also, one finds that the diffusive preypredator model has a small pll. These considerations may justify regarding the 12-singularity as a n unfolding of the NI-singularity, by some small unfolding parameters pIs1and q2,0, where p3,,,, p l , z , q2,1and are assumed non-zero and fixed.

182

H. FUJI],Y . NISHIURA and Y . HOSONO

Let us consider 12, again, under this program. We unfold I7, and I7, near the degenerate parameter values p l , l q 2 , 0 = 0 . As may be the case for the , let P ~ , ~ =beE diffusive prey-predator model, we suppose that q 2 , 0 = q ~ , 0 # 0and the unfolding parameter as IEI
e=O

and blu1--b2uz=qo37jr2+(h.0.t.)

near the origin of ( 5 , ~ul, ; u2). The Jacobian S o n D2 is given by:

where u l = u l ( ~a,,) is solved (by the implicit function theorem) as

Thus, the D2-sheethas a secondary bifurcation line in a neighborhood of the origin as:

EBET-AA7j12+(h.0.t.) , where (h.0.t.) denotes O(117I3)).See, Fig. 2.11. In the ( T , u2)-plane, the peak Z*=(q*, u?) is given by T * = z

E

and

B2

4A

For I E ~ small enough, this peak comes into a n arbitrary small neighborhood of the origin, and consequently, so does the parabolic lines near the peak. Let us see next how the secondary bifurcations on the D2-sheet are. 2 )in Solving (by the implicit function theorem) uI=O (EfO), u l = u l ( ~ , ~ ; uas Eq. (2.12). Then, we have

On Multiple Existence of Stable Solutions

183

tbZ

-7

Fig. 2.11. (a)

Fig. 2.11. (b)

v

i -

Lt

=- +

Ic=small

-~

=

I<-0

IE

Small

Fig. 2.12.

where

and Co=albz-azb, .

To see the qualitative picture of zeros of 3 on ( E , v ; uZ)-plane,it is enough to consider the two cases, after a suitable change of variables (assuming that A,, B , and C,fO): 3'=- 9 ' k

E$+

+

uzp+ E2 (h.0.t .) =0

,

(E

> 0) ,

where the signs t- correspond to sign(q,O,pll). The zeros of 3*, when a2=0, are drawn in Fig. 2.12. Fig. 2.13 shows that there are two qualitatively different pictures according to sign(q&q,,). The rectangular regions surrounded by dotted lines are ex-

H. FUJII.Y.NISHIURA and Y.HOSONO

184

rl,,-double hyperbolic

RG Z’-elliptic

Fig. 2.13.

.--_____... ._.______ -.-..-..___ ‘.+. ...c, r, 2

z:

...I

...._____ .* i....... .__.,: : .’..::..*.. .____ ...___ .._. i..

(4 Fig. 2.14.

,

(b) The “DI pot” near f l , z ; 2:’ is a simple degenerate singular point of elliptic type ((a), pllq20>O); of hyperbolic type ((b), p11q20 < O ) .

actly the regions which are covered by Fig. 2.6(a) a n d (b), where p l l was assumed to be a fixed non-zero constant. In fact, when €=pll is a fixed constant, the parts of the pictures outside the rectangles go outside a n e-neighborhood of the origin. This is the reason why we have missed the existence

185

On Multiple Existence of Stable Solutions

of a secondary bifurcation a t one side of the branch D’. When p L 1 = € ,there appears the point Z* in a n &-neighborhoodof the origin, which we have seen in Fig. 2.11. Schematic bifurcation diagrams near rl,2 are now shown in Fig. 2.14 on (u,, 02,(E, ?))-space. Note that the diagrams in Fig. 2.14 completely realize the numerical diagram in Fig. 2.10. To summarize, our former local bifurcation analysis “explains” in a sense the coexistence phenomenon! See, Fig. 2.14, which shows the possibility of occurrence of such a phenomenon. A numerical diagram, obtained in [ l l ] for the diffusive prey-predator system (El)’, clearly shows that it really happens. (Fig. 2.14.) One may point out that “a local structure can be changed by a local change of nonlinearity”. But, we have strong evidence that this coexistence phenomenon is a structural one (namely, not a happening), and not affected seriously by some local changes of nonlinearities. This means that our local diagrams must be a reflection of some deeper structure of the system, which we can not see through the use of local bifurcation theory. 3*’. Shadow System and Global Problem for Large d ,

The concept of the “shadow system” provides us with a useful tool to investigate the global problem of the Turing bifurcated branches as d , 10, for large enough d, ([30], [31]). Here, “shadow” means the limit system of d, T 00 in Eq. (P). The original stationary problem can be regarded as a “regular” perturbation from the shadow system with respect to u=d,-’=O, the global result for the D“’s equally holds for small enough o>O. (Remark. This statement is true for global “existence” of Dn-branches, but not for their stability. See, Sec. 4.) The results in [31] may be summarized as follows. Proposition 3.1. The primary D”-branches appearing under the Turing assumption , can be extended to the limit d , 10, f o r large enough d,, under and without the Turing assumptions. The conclusion is that there are D:-branches uniquely, for each n r l , for dl>O small enough and a=O. They coincide with the singularly perturbed solutions of class I for small u . *) For simplicity, we use in this Section a stronger hypothesis in place of (2). I.e.,

dv


,

where E:(v)=J(ho(Y), h,(v);

Y)

.

186

H. FUJII,Y.NISHIURA and Y.HOSONO

Let us begin by introducing the outline of the method under . The shadow system is the limit system of u 1 0 of (P)? (where we put e=dz and a=d;l): x€Z,

d P=O,

x€dZ,

where vE=vcis a constant function. A shadow solution is the pair (UE,v ' ) C,Zx R , satisfying (P);. A shadow solution is of D : ( E ) if , U E is strictly increasing in Z. We shall restrict our discussion to 0:. (Other D;'s can be constructed by the fold up principle. See, Sec. 2 . ) The strategy is to reduce (P); to find zeros of a scalar equation r ( E , v ) = O , where r :T c R 2 - R is a smooth mapping, and E the energy level of the system. ( T is defined below.) We first construct u=u6(x; v ) , regarding v as a parameter, by the energy method. In fact, if we define

(3.1)

F(u, . ) E l u

ve F,

f(s, v ) d s ,

hg(w)

v,

then for each ve F(u, v ) is monotone decreasing in ue (h-(v), ho(v)),and , and so on. increasing in u € ( h 0 ( v )h, + ( v ) ) ,sincef
(3.2)

€2

-uu",+F(u, v ) = E , 2

where E is the energy level of the system. regarded as functions of E and v :

(3.3)

u = u ( x ; E, v )

Then, as in [31], u and

E

can be

and E=E(E, v)

There is a solution u if and only if E>F(u, v), and the monotonicity of u requires that the equality holds only at x=O and 1. Let the definition domain of E=E(E, v ) be denoted by T . Then,

where Eb =ma x {E;(hs), E?(bh,)}and E"(y)=min {E:((y), EZ((y)}i

On Multiple Existence of Stable Solutions

187

Then,

(3.4) and

(3.5) where ii(E, Y) and _u(E,Y ) are two consecutive zeros of

F(u, Y ) = E with h-(v)
r ( E , Y)-'!

(3.6)

g(u(x;E , Y), v)dx=O

.

I

Lemma 3.1 [31]. (1) r(E,v)€CO(T)nC"(T) (2)

ar

z ( E , Y ) E Co(T/6,),where 6, is the union ofn-neighborhoods ofthe two corners of T , ( E , Y)=(O,

6)and ( 0 ,b).

Lemma 3.2 [31]. lim r(E,v)=g(h-(D),C)
( S e e , Fig. 3.2.)

( E . u ) - ( E - ( C ) ,i7)

V

t

u

Fig. 3.1.

Fig. 3.2. The definition domain T.

H. FUJII,Y.NISHIURA and Y. HOSONO

188

Note that Jo(v)=E~(v)-E:(v)by definition, and Jo(v*)=O means that the two curves E=Ey(u) and E=E?(v) take the same value at the peak v=v*. lirn r ( E , v)=g(h+(;),D ) > O , u*
ar

( 3 ) ->0 dv

in a neighborhood of (E”(v*),v * ) .

Note. r ( E , Y) can be continuously extended to T\{E“(v*),

Y*}.

Lemma 3.3 [31]. lirn

t Ern(,)

E

for any ve 7

e(E,v)=O,

Remark 3.2. The above lemma shows that any “solutions u ( x ; E , v)” on the boundary Em(u)correspond to E = O . Moreover, it is shown in [32] that Together with they are boundary layered (“boundary slit”) solutions if u f u * . Lemma 3.2, (1) and (2), this means that they do not satisfy the integral constraint Eq. (3.6), in general. (I.e., unless u ( x ; E , v)-U a.e., as E+E“(v), and ( U , v ) is a zero of g ; g( U , v ) = O . ) Thus, from Lemma 3.2, r ( E , v ) is monotone increasing as a function of E , in a neighborhood, say T*, of (E”(v*),v*). Starting the lower boundary of T with the value g(h-(v),Y ) < 0 , increases until the upper Hence follows boundary with the value g(h+(v), v ) >O. v , for each fixed

r

Theorem 3.1. There is a unique branch S F c T * : v=v?(E), such that r ( E , $(E))=O, and vF(E)+v* as E->E“(v*). Moreover, as E-E“(u*), the limit function of u = u ( x ; E , vF(E)) converges to the step function:

uniforrnZy in [0,x * - K ] and [ x * + r , I] determined by the relation lim

c t I:“(”*)

(VK>O).

r ( E , v?(E))=

L

The point of discontinuity x* is

g(u*(x), v*)dx=O

.

Hence, x* is given by (3.7) Under Assump.
189

On Multiple Existence of Stable Solutions T=

U

(0, E ” ( v ) ) x{ v }

.

”PY

By a direct calculation, we have the following Lemma 3.4. Under
in

3,, where 3,=

E 10 (_bfK,6-K)

(K>O).

( 3 ) limu(x;E,v)=ho(v), uniformly in ~ € and 1 ~ € 3( v,l i > O ) . El0

(4)

= det

drj E=O,v=uC

(AC)
f & C

For the proof, see [31]. Thus, together with the fact: r(0,u”=g(h,(uC),

uC)=O,

we have, by the implicit function theorem, the unique branch of solutions S , : v = v l ( E ) of r ( E ,v,(E))=O, which crosses ( E , v ) = ( O , vc). (Note. One can extend T to E
.

(The
7

for O<E<Em(v*). Hence, follows the global existence of S,. By the uniqueness of ST near (E”(v*),v * ) , S , = S ? where both branches are defined. Remark 3.3. The above global existence of S , does not rule out the existence of possible secondary bifurcations, or limit points between O < E <

H. FUJII,Y. NISHIURA and Y. HOSONO

190

V

1

Fig. 3.3. V

1 V

I\

/ S I ("Turing branch!')

I

I

0

E"(v*)

0

Fig. 3.4. (a)

Fig. 3.4.

'E

(b)

However, thanks to the Sard lemma, S , is a onedimensional smooth curve in T i n a generic sense. See, [31] for details.

Em(u*). See, Fig. 3.3.

Remark 3.4. It is a n interesting problem to see what happens when Assump. is violated, but still under . This problem has been studied in [32]. For example, suppose that g=O behaves as in Fig. 3.4. (a), where g=O intersects three times withf=O. Also, the critical point on h&v) violates the Turing condition in the sense that it is already unstable in ODEsense. One can see that (1) the large amplitude branch Sr exists uniquely in a neighborhood of (Em(v*),v*), but it does not connect to the "Turing branch S,". Sf has a limit point, and connects to a boundary layered large amplitude branch, mentioned in Remark 3.2. (2) The "Turing branch S," continues to exist to E 10 (but, not to E E"(v*)), and finally it connects to another boundary layered solution in the limit. (See Fig. 3.4. (b).) We remark finally that the crucial point in our future discussions is the v*), under the assumpunique existence of ST in a neighborhood of (Em(u*), tions , whether or not it is a continuation of Turing branches.

On Multiple Existence of Stable Solutions

191

4. Stability and Instability of D'(E,a) for Small a20, and Instability of Dn(e,0) (n22) on the Shadow Ceiling

So far, we discussed mainly construction of solutions near the shadow ceiling, i.e., u=d;'>O small enough. In the next section, we discuss also construction of large amplitude solutions using the singular perturbation technique. An important problem which is not included in these arguments is the stability of D"(E,a) solutions for n > l . In this section, we first show the stability of D ~ ( E a), for small enough E > O and a20, under Assumps. and . As a remark, a violation of leads to a n instability theorem. We next show that for n22, D*(E,0) (large amplitude solutions with mode n 2 2 ) on the shadow ceiling u=O, are all unstable. We note here that the "recovery of stability" of Dn's when o > O is the main subject in our future discussions. The results in this section are due to [34] and we ask the readers to refer to it for details. (See also [33].) To show the essence of our argument, we first consider the case o=O. So, we denote Sf as D:(E,0) in this section. The linearized stability problem of D:(E,0) is reduced to the study of the eigenvalue problem : iL'W'+

,

f:7j+=REWL

x EI , x€dZ,

J%=O

where 77' is a constant function, and

(4.1) Here,f,"=f,(uE,

vf),

and so on.

{c,

Lemma 4.1. Let Sturm-Liouville problem :

fn}

(n=O, 1,

-

- ) be the orthonormal system of the

(4.2)

( 1 ) Then, the principal pair of eigenvalue and eigenfunction (C;, &)

H. FUJI].Y.NISHIURA and Y.HOSONO

192

satisfies:(*)

< C , exp ( - r / E )

0<

(4.3)

(E

>0

small)

and

oi>o

(4.4)

( X E U

&dx = O(dT),

7

where r and C,are positive constants independent of E > 0. ( 2 ) Moreover, there is a positive constant p > 0 , such that for n= 1 2 , (4.5)

<:L~-,u
VE>O

*

-

a ,

small.

Let, for Re R > - p ,

namely, (Ls-R)t is a compact operator of Lz(Z)into L Z ( I ) n{&}', for &>O. The next lemma We let Q; be the projection of Lz(Z)onto Lz(Z)n {#,}l. is the first key to our arguments.

Lemma 4.2. More precisely,

(P-R)t becomes a nzultiplication operator in the limit

for any bounded u E L 2 ( Z ) ,and Re R >

-,u,

where fZ=f,(u*(x),

E

LO.

y*);

The following prepares the second key.

Lemma 4.3. 1 ( 1 ) lim -
dT

lim

1 -
610

where K*

@,>=,r;k(g:-g?)>O.

f,", &>=

> 0 is a positive constant.

Now, before stating our main Theorem, we need to divide the eigenvalues into two classes. Let A , ( i = O , 1) be defined as:

Non-critical eigenvalues (4.8) (*)

A,"'{;jEl

IR'126>0, & > O } ;

To extend the argument to ~ 2 0 we , need to show: OO).

On Multiple Existence of Stable Solutions

193

Critical eigenvalues

A o g { P l ( l E I + O (E'O)}

(4.9)

.

We note first that any eigenvalue 1' with ~ ' = 0has a negative real part and is uniformly bounded away from zero for all small E > 0, and hence, is a non-critical one. To see this, it is enough to observe that such 2''s satisfy the Sturm-Liouville problem, and hence is equal to some 6 . Thus, if k 2 1 , then, Re R"=P=G< - p , by Lemma 4.1. Next, we point out that (C:, (&, 0)) can not be a n eigenpair to (E)f, since from (1) of Lemma 4.3, the left side of the second equation of (E)€can not be zero. Accordingly, it suffices to consider only those eigenfunctions which satisfy V'fO,

E>O.

Using the Sturm-Liouville eigenfunctions, I is now written as:

(assuming that Re I > - p without loss of generality!). From Lemma 4.2, it follows that

where f z = f u ( u * ( x ) ,Y*)

;:, I:zf'"'

, etc.,

and A*=-?-

-

Lemma 4.4 (Non-critical eigenvalues). Let I' be a non-critical eigenvalue which stays in A , for small E > O . Then, there exist positive constants 6 , and €dl independent of E such that (4.12)

ReP<-6,<0

,

for

O<E<E~~.

Proof. We only show that the limiting value of a non-critical eigenvalue as E 10 is strictly negative. Let R*=lim,,, 2. Then, from Equations (4.10) and (4.11),

(4.13)

I*=! I

-stf,xdx+ E-1"

g$dx=(Z),+(l),,

in view of the fact that the second term of Eq. (4.10) is of O(le1). Hereafter, let us assume that R" is real. First, we show that under
H. FUJII,Y . NISHIURA and Y . HOSONO

194

S> and the option (I of) , ( I ) , is strictly negative.

(Z),=g,(h+(v*),v*)lZTl +g,(h-(v*), v*)IZ?l

I

where x*) ,

Z?=(O,

ZT=(x*,

1)

,

x*=gT/(gf-g%)

.

Hence,

For a real Re, if 1 x 2 0 , the right side of Eq. (4.13) is strictly negative, since the first term is obviously nonpositive, leading to a contradiction. For the option (11) of , the same conclusion follows easily from the third expression of Eq. (4.11). For complex 2 , Re A*< -6, < 0 (6, >0) can be similarly shown. Concerning the critical eigenvalues A,,, we have the following Proposition 4.1 (Critical eigenvalues). A , includes onZy one eigenvalue A:, which is real and simple. When E 1. 0 , 2: behaves a s : (4.1411

2;--Aloa

,

where

See, Fig. 4.1 (a).

Corollary 4.1.

Under , it holds that

Remark 4.1. Suppose that J J ( v * ) > O bility occurs, i.e., &>O.

1, < 0 .

(see fig. 4.1 (b)), the crinical insfa-

Proof. First, we rule out the two cases: ;i&><;>O, P E A , , . Next, from Eq. (4.10), R=R' is a solution of (4.15)

F(I, &)=R2--R

<-f:,

and l;;>RE>O

&>+C

\,

for any

[*Idx] = O ,

where the integral $ I [*]dx is the first term of the right side of Eq. (4.10). In view of exponential decay of C: (Lemma 4.1), Lemmas 4.2 and 4.3, and using the implicit function argument, we can show our proposition.

On Multiple Existence of Stable Solutions

Fig. 4.1. (a) Spectra of D:(E,0).

195

Fig. 4.1. (b) Functional form of f ( u , v) when .Ti("*) >O.

Stability o f D:(E, a) for small a>O. An extension of the preceding arguments to D:(E, a), ( E , a) E Ql, namely, D'-singularly perturbed solutions of class I (see, Sec. 5 ) is given in [34]. Here, we show the outline of the discussion. Proposition 4.2.

There is a small rectangle Q , E { ( E , ~ ) I O < EO
in which the D ~ ( Ea)-sheet , of solutions has a unique, real simple eigenvalue ;I=;(:*", which converges to zero from below as E 10; all the other eigenvalues have strictly negative real parts. ;I:,"behaves as ;j;.ae

- CaE

i

where Ca is a positive constant for 020, and converges to

1, in Prop.

The eigenvalues problem on D ~ ( Ea), can be written as: (E)E,O

{

Le,aws.a+ f;L,aZe,a=RE.aWe,a g~a~e.a+ME,OZE,o=;(L,az~,a

'

with no flux boundary conditions on ill, where

X € Z ,

4.1.

H.Fum, Y. N~SHIURA and Y.HOSONO

196

and f:.a=fw(uE,a,

and so on.

w ~denotes ) the singularly perturbed solutions of class I of mode 1 Here, type. See, Sec. 5. The proof of Prop. 4.2 proceeds as follows. We decompose z as ( u L z o ,

Then, (Eye" may be rewritten as

I

LE'W +f,' *"( 1+( T i ) =Iw ,

x€Z,

[g',."w+g:*"(1+ a i ) ] d x = I ,

(-&-d)i= R-gYw - g:'"( 1 + 0 2 ) ,

x€z

X€dZ.

Let:

Next, we show that, for a given such that

i'c B,,

there is a unique A=&(E, u$')

I ( ) ( €0, ; i')=C(u, i')€ ,

satisfying

I= (Cf. Eq. (4.13)).)

Sz

[g2aWE,a(I, i ' ) + g : , a ( l + a i ' ) ] d ~.

On Multiple Existence of Stable Solutions

197

In the final step, we define a mapping fP,a: ?'-ti of B , into itself, by (-&--020)5=10-g~%yP,

d dx

5')-g;ql+ff5')

,

X€

I ,

x € X ,

-z=o,

where R o = R 0 ( ~ , a ; 5'). Note that the integration of the right hand side vanishes due to the construction of l o ,and hence, i is uniformly solvable in Hio(Z)up to a=O. Now, making use of the contraction mapping argument, we can show the unique existence of a fixed point i E B M of sz",", for small enough E > O and 020, which leads to Prop. 4.2.

Instability of D n ( q0 ) ( n 2 2 ) on the shadow ceiling Let D;(E,O) ( n 2 2 ) denote the shadow solutions ( U ; , + ( X ) , Y ~ ) which are invariant under the dihedral group D" (in a generalized sense), and constructed from D:(E, 0) by a reflection and the fold-up principle. See, Sec. 2. (See, Fig. 4.2, for D t . )

0

Fig. 4.2. u i , + ( x )is shown in the above; u ( 3 . J ~ )is the reflection at x = & of u!,+.

The next proposition shows that all these shadow solutions are unstable. They have exactly (n- 1) symmetry-breaking unstable eigenfunctions.

Proposition 4.3 (Instability of Dn ( n 2 2 ) shadow solutions). Assume . On each D:(E,0 ) branch of shadow solutions with n 2 2 , the critical set of eigenvalues consists of n eigenvalues 2; ( i = O , 1, * ., n- 1). 2; (i=1, 2, -,n- 1) are real, simple, and positive eigenvalues, with D,-symmetry breaking eigenfunctions, while 28 is a real simple eigenvalue with Dn-symmetry preserving eigenfunction. I f , in addition, is assumed, then, 2; is negative. They are order-preserving:

-

--

(4.16)

and behave as

A;>;(;>

-.->2:-,>0

(E>O),

H. FUJII,Y. NISHIURA and Y. HOSONO

198

(4.17)

%=exp(-T/&),

E>O,

and, therefore, all of 2; ( i = 1,2,

- - -,n- 1) tend to zero exponentially as

(4.18)

Jc-In 0-

o&

E

-1 0 , while

(E>O),

where 2; =do, which is negative under . S e e , Eq. (4.14)2. All the other eigenvalues, belonging to the non-critical set A1, have negative real parts bounded away from zero uniformly in E > 0. S e e , Fig. 4.3.

Fig. 4.3. Spectra of D:(E,0) ( n 2 2 )under .

Outline of the Proof. This Proposition is closely related to the notion of symmetry covariance of the system. Coming back to Sec. 2, the function spaces P and are reduced to p o = L 2 ( P ) x C ,and so on. The unitary representation TO: Dn+ GL( Y o ) ,naturally induced from T defined by Eq. (2.7), is now T;=(T;'), TF)), g € D", with TL2)=1 for all gc D". (We simply write T, instead of T(g).) Defining the orthogonal projections;

x

yf"( F!") is called "the Dn symmetry preserving (breaking) space", in view of the following characterization: v(%+,I?+)€ Ff" , Tg(')Gt=%+,

VgE D"

,

and V(%-,

I?-)€ F!" , q-=o .

Let pEaL be the linearized operator on ( u E , v L )the , D* invariant shadow solution,

On Multiple Existence of Stable Solutions

199

gC be the one corresponding to ( 2 ,P), the even extension of

( u c , vs) to 1. Then, s@ commutes with T,O, for all g E D n , and consequently with P:", thanks to the D" (cD") covariance of the nonlinear operator in the generalized sense. Accordingly, we have the direct sum decomposition: @€=*:@@:, where @; x:"- yf". Thus, follows

and

Lemma 4.4. An eigenfunction ( w , v ) of -Pe belongs to either of the following two cases. ( 1) (w,7) is a D n symmetry preserving eigenfunction; and is identified with the eigenfunction of Dl(nE,O) shadow branch by using Fold-up Principle (see Prop. 2.1). ( 2 ) ( w , 7) is a D* symmetry breaking eigenfunction with v = O ; w=wC is a non-principal eigenfunction of the Sturm-Liouville problem : LEwE=PwE,x Z, with the Neumann conditions on the boundary x=O, 1. Proof. Even extension (a,?j)belongs to either 2" : or ,80" thanks to the above characterization. Hence follows the conclusion.

For the D n symmetry preserving eigenvalue A;, (4.18)is a direct consequence of Lemma 4.4 (i). For the symmetry breaking ones, note the following two facts; Firstly, the x-derivative of D : ( E ,0) is the n-th eigenfunction with n nodal points of LE associated with 0 eigenvalue under homogeneous Dirichlet boundary conditions. Secondly, the principal eigenfunction of Lf under Neumann boundary conditions is obtained by applying Fold-up Principle to 4; in Lemma 4.1, and the associated eigenvalue 2; satisfies (4.17). Then, using the nodal property of the eigenfunctions of the Sturm-Liouville operator and comparison arguments, the D" symmetry breaking eigenvalues {,?:};:p-l satisfy (4.16)and (4.17). 5.

Structure of Solutions on and near the Singular Wall -Two classes of large amplitude singularly perturbed solutions

The third approach for getting "patterns" is to construct solutions for small enough E >0, where the basic technique is the singular perturbation method. Works along this line appear to have begun with P. C. Fife for Dirichlet boundary value problems [ 5 ] , and Mimura et al. [27] extended his framework to Neumann problems, including activator-inhibitor models. See, also M . Ito 1201. In this paper, we refer this . We shall give a precise characterization of large amplitude solutions later.

200

H. FUJII, Y . NISHIURA and Y . HOSONO

Stability of this solutions has been discussed in the preceding section, for small enough u 2 0 . However, the stability result in Sec. 4 concerns only D’ (=mode 1) solutions, the other Dn ( n 2 2 ) solutions are shown to be unstable on the shadow ceiling u=O. If one wants to give a reasonable explanation for, e.g., the recovery of stability of D”for u>O, destinations of secondary branches (observed so many times in Sec. 2) and so on, the question of whether the , and consequently, responsible to the recovery of stability of primary Dn-branches. Only this conjecture can explain the structural and stability questions mentioned above. Existence of patterns has been announced in [13], and a proof was given in [14]. The structural relationship between these two classes of patterns will be the subject of the next section. As may be well-known, the singularly perturbed solutions of have, in addition to interior jumps, Neumann slit(s) at one or both of the boundaries and/or at points of symmetry. The depths of these slits are determined by the second diffusion coefficient u-l. See, Fig. 5.2. Why do we call them O , and the reason for this layer phenomenon is the presence of boundary conditions of Dirichlet type. See, e.g. [5]. In this context, we may refer to such as Dirichlet layers. O n the contrary, as may be clear from our construction, the layers which we consider here appear either at Neumann boundaries or at points of group symmetry of spatial patterns of solutions.(*) This means that the appearance of Neumann layers depends essentially on “boundary” conditions of Neumann type.

Remark 5.1. Neumann layers may appear even in Dirichlet problemsat the mid-point of the interval, since they can appear a t folding points of symmetry. (*) More precisely, for a D”-class I solution, there are ( n + l ) possible places of Neumann layers, i.e., x j = i / n , i=O, 1, . . ., n.

20 1

On Multiple Existence of Stable Solutions r

J

I

€

0

I

I

E

E - 0

10

€20

Fig. 5.1. Profiles of solutions of
r

Fig. 5.2.

I

Profiles of solutions of . (upper: @'#,with an interior transition layer and two Neumann layers at both ends) P

(lower: D",, with a Neumann layer at the center and two interior transition layers)

The characterization of singularly perturbed solutions of two defferent classes-class I and class I1 may be clearly given in terms of "reduced set" corresponding to each class. Let us define the reduced set R ( o ) of class I. We begin by the definition of the reduced problem. The reduced problem is to find solutions of (P), with E = O ; namely,

( zdu = o ,

.y

E

a

H. FUJI],Y.NISHIURA and Y.HOSONO

202

Solutions (u, v) of (P):," are called reduced solutions, which are the first candidates to singularly perturbed solutions for E >O. Note that not all reduced solutions can be extended to singularly perturbed solutions for E > 0; the following two classes are "well-behaved'' (=extendable) reduced solutions. Reduced solutions of class I are constructed in the following way. Let:

Then if one puts U'(x)=h,( V'(x)), for any V', then, f ( U', V')=O for all V'(x) € fuIu*).

So, for a given a 2 0 , let us define G*(Vd"Fg(MV), V ) t

and consider the boundary value problem : Find Y o = V " ( x )belonging to C1(r),satisfying

We consider only a monotone increasing V a for definiteness, which corhas a disresponds to a 0: solution. It follows under that G*(v) continuity of the first kind at u=u*, and G,(v)>O for v>u*, and vice versa. Under (global Sg), G*(v) is monotone decreasing in each of the intervals v < v < u * and u*
.

Num (v(x))={min v ( x ) ,max v ( x ) } z e l

z e i

Proposition 5.1. Under Assump. , ( 1 ) there is a o-family of monotone increasing C'-solutions V'(x) of (R)"; for O
lim Num (v"(x))={u*}, 010

(3)

moreover, under Assump. , V" exists for O
On Multiple Existence of Stable Solutions

203

Fig. 5.3.

Proposition 5.2.

(

1 ) There is a unique "matching point"

x*(a) € I , such that

vq x*(a))=Y* , due to the monotonicity of V " . In the limit o 10 , x*(O)=lim,, x*(a) is determined bY

where U*(x) is

U*(x)glim h,( Vg(x)) 010

0 1x < X*(O) Hence, it follows that

x*(0) =gf/(g* -g?)

.

(Cf.Eq. (3.7), Sec. 3.) (2)

Let uf(u)=max,,,-V"(x) (= Vg(l)), and u*((a)=min,,~V'(x) ( = V"(0)).

204

H. FUJII,Y.NISHIURA and Y.HOSONO

Then, vT(o)and -v?(o) are monotone increasing functions of a 2 0 . ( 3 ) By construction, the function U‘(x)=h*( V‘(x)) hasa “jump” a t x = x * ( a ) . See, Fig. 5.5.

Whenever the reduced solutions ( U “ , V‘) exist, let us define the closed sets R,(o), R* and R-(o). Let: Z-(u)=(O, x*(u)) and Z+(a)=(x*(a), 1). R-(o)gCI {(Uo(x),V‘(x))I x € Z-(u)} in R2 ,

R*={(s, Y*) I h - ( v * ) I S < h + ( v * ) } , Rt(u)=C1{(Uu(x),V g ( x ) ) I x € Z t ( a )in } R2. Then, the reduced set R ( a ) of class I is defined by:

R ( u ) g R , ( o ) U R * UR-(a) . The corresponding functions ( U‘, V‘) are plotted in Fig. 5.4. The “jump” corresponding to the set R, becomes a n interior transition layer when E>O. Note also that this is a jump along R, connnecting the two bi-stable states (v*, h-(v*)) and ( v * , h + ( v ) ) . The “depth” of the jump is determined by the condition: (5.1)

J(h-(v*), h,(v*); v*)=

s

h+(u’)

f(s, v“)ds=O

.

k-(v*)

The reduced sets of Neumann layered class, i.e., class 11, R(o)’s are obtained by attaching Neumann slits to one and/or both ends of the interval Z. Let us define two functions k + = k * ( v ) ,defined on Y*, respectively, by the relations : J(k,(v), h , ( v ) ; v ) = O ,

and

U E Yi.

On Multiple Existence of Stable Solutions

205

The reduced solutions (U#n(x),V;(x)), corresponding to R,(u) are plotted in Fig. 5.5. II=

h,

(c'

( c))

I

U 1

!

"t

I 1

We note that

and

By definition, the "heights" af(a) (See, Fig. 5 . 5 above) of Neumann slits are determined by

af(d=k*(v34)

,

and hence, satisfy the generalized Fife condition: (5.2)

J(k*(vZ(o)),M v 3 u ) ) ; vZ(a))=O

.

0 '

Fig. 5.6. Reduced set R#(o)of (the bold solid line).

H. FUJII,Y. NISHIURA and Y. HOSONO

206

Proposition 5.3 ([27]). Assume . Then,for each n 2 1, it holds that There exists a rectangle 0 ) [ O < E < E , , O
en={(€.

S(E,o ) " " { ( u ~ ~E(, xu;) , v ~ ~ E ( ,xu );) I X € I } c R 2 .

Then, the limit set limtloS(E,a) in R2 coincides with R ( u ) .

Remark 5.3. Under Assump. , the above proposition holds u-globally, i.e., 0 1a < ut. See, [20]. Remark 5.4. The construction of Mimura et al. [27] breaks down in the limit u 1. 0. However, it is possible to repair this breakdown, and to construct solutions uniformly until u 10. The basic idea is, first, to decompose the function v as V=Y+UW,

(YER).

Then we claim the continuity of (l/u)v,, namely, of w,, at the interior transition point, in place of u, itself. See, [34] for details.

Proposition 5.4 [14]. Assume . Then, there exist three ( E , a)a),, # D ~ ( uE) ,and # D ~ ( uE), in Q1. families of class I1 solutions D ~ ( E Their ( u , +plots S&E,u ) , $ ( E , u ) and # S # ( Ea) , tend as E 1 0 to R,(a), # R ( u )and #R,(a),respectively. Numerical solutions With a DF singularly perturbed solution of , there are three associated brothers of Neumann layered solutions, depending on where the Neumann layer exists: at the left end # D 1 ,a t the right and D; or at the both ends #Dt. In Fig. 5.7, we show the profiles of those brothers, i.e., D1, #D1, Dt and as well as their (u, 0)-plots, at e2=1/10, and 0=1/200, for the diffusive prey-predator model Eq. (El)'.

Remark 5.5. The functions in Fig. 5.7 are obtained by the numerical method making use of the group representation theory. See, for details [9] and [lo]. Note that every evidence shows that Neumann layered solutions are unstable. One cannot obtain patterns as shown in these Figs, by a conventional numerical method, e.g., solving (P) as a n initial value problem, and letting t T +m. View on the singular wall On the basis of the previous discussions, let us consider the structure of singular solutions of and on the singular wall. Here, by

On Multiple Existence of Stable Solutions

Profile of D,.

207

Profile of @

the singular wall we mean the limit space R , x X o ( 3(0, ( u , v))) of E 1 0, where Xo=L2(1) x Ifj(1). We assume ) is plotted with a double solid line. At u=O, it starts from the singular-shadow solution (U*, Y*), and hence, connects to the shadow branch D:(E,0) at the edge ( E , u)=(O, 0). As u tends to OF, D:(O, a) ceases its existence, since R+(o*) covers the whole R + . (Similarly, we can easily see that #R&uT)=#R(oT).) Thus, the critical point a = u T may be called the singular limit point. At the singular limit point, as we noted before, Di(0, uT)=D1(O, uT). There appear the Neumann layered branch D i ( 0 , u ) upwards to u 10. For more discussion, see [13]. Note. These singular branches should be measured by a n “asymptotic norm”, since they have the same X,-norm for each U . We omit the details. See, [13].

H. FUJII, Y. NISHIURA and Y . HOSONO

208

u = h , (o:(U))

p=J

pj

a=o

O
a= a;

Fig. 5.8.

Remark 5.6. The picture in Fig. 5.9 is valid under .

U

Fig. 5.9.

6. Singular-Shadow Edge as the Organizing Center of the Ultimate Structure Let us consider the structural problem between the two classes of amplitude solutions. As a consequence, together with some spectral siderations, we shall obtain a multiple existence of stable solutions. Section is based on the authors' paper [15]. To make the situation clear, we shall first focus on the D$-sheet of tions of . Definition 6.1.

The singular-shadow edge (SSE) is the space

where X o = L 2 ( Z ) x H . j ( Z ) .

large conThis soh-

On Multiple Existence of Stable Solutions

209

Let (U*, V*) be the limit E , u 1 0 of the D ~ , ( a)-sheet E, of solutions (u(x;E , a), v ( x ;E , 0 ) ) . Then, V*-Y*, and U* is a step function

I

h-(lJ*),

U*(x)= h+(u”),

(6.1)

h-(v*),

where (6.2)

OlX
,

( = ~ * / 2 Cf. . Eq. (3.7).)

i*=gT/2(g?-gJC)

U*(x) is symmetric with respect to a reflection a t x=1/2, and has two interior jumps. The pair (U*, Y*) satisfies, of course, the limit relation:

Needless to say, (P):., possesses many (even continuously many) solutions belonging to X , . However, what we are interested in here is the following fact. (P):,” has a one parameter continuum of “solutions”, say, U q = U(x;4); V=IJ*, - l < q < l , such that Uo=U*, the limit of 02, solutions of class I, and that Up ( - 1 q< 1 ) are obtained by “shifting” the block of U*,as Fig. 6.1.

q= -1

q=

q=o

fl

Fig. 6.1.

In fact, let Uq’s be step functions: h-(lJ*), W x ; 4)= h+(u*) ,

i

O<x
h-(v*) ,

l-r:(q)<x
t:(q)=i*(l+q) ,

tf(q)=i*(l-q)

where

(See, Fig. 6.2.) Then such Up’s satisfy the integral relation:

.

,

H. FUJII,Y . NISHIURA and Y . HOSONO

210

where we put

rF( q)=I* = 1- @(q)- r:(

q)

.

= -g”2(g?-gg”)

Note that q=O corresponds to the

symmetric solution, and other Ug’s

(qf 0) do not keep any symmetry.

Fig. 6.2.

At q= 1 and - 1, the “block” collides with the right/left boundaries. They are the terminal states. It is to be noted that these terminal states are a special o=O case of the reduced solutions of class I1 associated with D: and DL, respectively. Thus, this q-continuum of solutions connects two class I1 Neumann layered solutions ,D? and D:#, and a class I solution 0:. Symbolically, we have a diagram:

pD1 - .D: q=-1

q=o

-

D:t

.

q=+l

(Similarly, for D?, the diagram is: *D:-DD?-DDi,.)(*) However, this q-continuum of solutions can not be extended to E >0, on the shadow ceiling a=O. (Due to the uniqueness of SF. See, Sec. 3.) On the other hand, there exists the shadow branch D:(E, 0), for E >0, which connects to the limit 0:. Thus, we have a “bifurcation diagram” on the shadow ceiling, with respect to E > O (small enough). See, Fig. 6.3(a). One may interpret the origin (u, q)=(O, 0) as a “Dz-symmetry breaking bifurcation point”, and the continuum as a global bifurcating branch. Such a bifurcation may be called “the ultimate symmtery breaking bifurcation”, (*) lime,,,,o (D!(E, o)=(U?, v * ) ; Uf(x) is the function defined by Eqs. (6.1), (6.2), exchanging h t with h-, and g? with gf.

On Multiple Existence of Stable Solutions

211

O
U=O

Fig. 6.3. (a) “Frozen” bifurcation diagram showing an ultimate symmetry breaking bifurcation.

Fig. 6.3. (b) Unfolded bifurcation diagram.

although at the edge ( E , a)=(O, 0), our system of differential equations degenerates to (P):ro, which are no longer differential equations. solutions exist in a rectangle region Q2= Together with the fact that { ( E , U ) I O < B < E ~ , OO (small enough), one may have a bifurcation diagram as shown in Fig. 6.3 (b). In other words, the singular-shadow continuum at the edge is the a J O limit (“frozen branch”) of the global branch bifurcated from the D:(E, 0 ) branch. The main problem is, therefore, to show the validity of this conjecture. This is now a conceptually simple, but technically complicated work, to unfold the q-continuum to a global q-parametrized branch with respect to u > 0. The main part of [15] is devoted to this task.

Proposition 6.1 (The ultimate D2,-symmetry breaking bifurcation). (a) For any q E (- 1, l), and E > O ( s m d enough), there are ( E , q)-family of triplets : u=u(E,

u=@;

q)>O €7

,

d E , 4),4)

c,z

9

u=u(x; € 7 d E , d , q ) E c2

t

such that they satisfy ( P ) ; ~ ‘ ( e ~ q ~ . (b) (Convergence to the q-continuum) For each q € (-1, l), U ( E , 4)-0 as E 1 0. Moreover, lim

IIU(.,

€9

d E ,

q), 4)--Q(.)IIL2u)=O

€9

o(E9

417 q)-Y*II+(I)=O

9

610

Iim IIu(., €10

.

H. FUJII,Y.NISHIURA and Y . HOSONO

212

( c ) (Intersections with D ~ + (a)) E, When q=O, (us v)=(u(x;E , a(€,0 ) ,0 ) , v ( x ; ~ , a ( ~ , 0 ) , 0are ) ) D2-invariantfor all ( E , a ) = ( € ,a(€,0 ) ) ,E > O (small). More precisely, they are a part of the D:(E,a)sheet, namely, D:(E,a(€,0 ) ) . (d) (Neumann layered solutions) For each U=U(E, q) fixed (small), if we let q++l, then. E 10. Moreover, the corresponding solutions are class I1 Neumann layered ones (see, See. 5 ) near €20. (e) (Asymptoticform of the unfolded sheet a(€,q ) ) The unfolded sheet U = U ( E , q) has the asymptotic form as: a(€,4)"

2Ey(v*) exp [-c*f*(l-q)/~]-exp [-c*f*(l+q)/e] 9 J;(Y*)T*J*g* 4 ( E > 0 , 4 € ( - 1, 1)/{01)

where Y*)~ , c5=2/231f,(h-(~*), i*=gt/2(g"-g5) ,

and _t*=

-g*/2(gT-g'")

.

Fig. 6.4 shows the form of the sheet a(€,q ) : (a) shows the curves U = U ( E , q ) , with - 1 < q < 1. (b) is the graph showing ( E , q)-curves at the cross section o(e, q)= o,,, a. varying in ( 0 , aJ. Fig. 6.5 gives the conceptual picture showing the structure of various sheets of large amplitude solutions, i.e., the D;-unfolded sheet a = ( € ,q ) , the D;-contirnuum at the singular-shadow edge, the D;(e, o)-sheet, and

Fig. 6.4. (a)

U=U(E,

q), with -1 < q < l .

Fig. 6.4. (b) ( E , q)-curves at with uo E (0, UI).

U(E,

q)=uo,

On Multiple Existence of Stable Solutions

213

the two sheets of D : ( E ,u ) and D ~ ( E u ) ,. We observe that the unfolded sheet U ( E , q) and the D : ( E ,a)-sheet intersect transversally, at q=O. This line is the secondary bifurcation line of the large amplitude solutions (of class I) D : ( E ,u ) . This line, say r+,is obtained by taking the limit q 10,of U ( E , q ) . In fact, we have the following

(f) (The secondary bifurcation line of the D : ( E o)-sheet) , Let y + g i ( E , o)lo=o.,(E)}

,

where U S ( € ) behaves as:

Then, r+ is the line showing the ultimate symmetry breaking bifurcations of the large amplitude solutions 0 9 ( E , a). We confirm now that under Assump. , ( 1 ) The D?(E, u)-sheet (and similarly, E ( E u)-sheet) , has D2-symmetry breaking bifurcations of the line y + (7-). ( 2 ) These bifurcations originate in the singular-shadow edge ( E , u)= (0,0), where the bifurcated branch degenerates to a q-continuum of discontinuous “solutions”. In this sense, we may call these the ultimate symmetry breaking bifurcations. ( 3 ) The bifurcated branches from D ~ + (uE+, ( E ) )exist globally with respect

Fig. 6.5.

214

H. FUJII.Y . NISHIURA and Y . HOSONO

&

the eigenfunctions at (a)

Fig. 6.6. Numerical solutions of and . Model: Diffusive prey-predator system (El)'. a=&' = l / l O O ; the ( u , v)-plot (Zefr), and the functions ( u ( x ;E , a), v(x; E , a)) (right) are shown for each point from (a) to (g).

On Multiple Existence of Srable Solutions

215

to E 10, where the limits are the Neuniann layered large amplitude solutions, i.e., of solutions. We may thus have a catchword, “The class I1 solutions stabilize (or, destabilize) the class I solutions”. Moreover, if is assumed, then, ( 4 ) The unfolded sheet U=U(E, q) and the D : ( E u)-sheets , (class I) connect at the singular limit point ( E , a)=(O, ul). See, Fig. 6.6. In Fig. 6.6, we reveal a numerical result showing various solutions of and . Stabilization or destabilization ? The final stage is to investigate the asymptotic behavior of the spectrumeigenvalues of the D ~ + (uE) ,sheet of solutions. As its consequence, we shall show that the bifurcation line 7 + is actually the recovery line of stability. Prop. 4.3 shows that there are two critical eigenvalues A:.o and on the shadow branch D ~ ( E 0);, is negative for E > O under +, while 2;jo is positive decaying exponentially as E tends to zero. on D ~ ( E a)-sheet , is “similar” to the critical For a>O, the behavior of eigenvalue A;,“ on D:(E,u ) , thanks to the fold-up principle. In fact, we have that 2 ; z 0

& ( D ? ( Eu))=;1,(D1(2~, , u/2*))

for any

(E,

a) € Qz .

Thus, it is enough to consider the asymptotic behavior of I;r“=A,(D2L(~, a)) as E

1 0.

Proposition 6.2. Assume . Then, there exist positive constants r and A , independent of ( E , a ) € Qz, (Q2E{(c, a) 1 O < E < E ~ , O
; ~ , ( D ; (a))eAe-”E-BEu E, ,

(E,

u)€ Qz ,

in which B= where

K*

2(r”)2J,’(v”)(g”Z

st-s?

,

(>O) is a normalized constant.

Moreover, the associated eigenfunction (w,z ) ; , ~is the principal D2-symmetry breaking eigenfunction. Thus, the criticalline u=u;(E), on which I;.“=O, is what we have obtained in Prop. 6.1 (f). Corollary 6.3.

Assume +. Then, y + c Q2 is the recovery line of stability f o r the 0 9 ( E , a)-sheet, in the sense that

H. FUJII.Y . NISHIURA and Y . HOSONO

216

J

Fig. 6.7.

Fig. 6.8. The behavior of ~ , ( D : ( E u ),) , under Assumps. . 1 , ( 0 2 , ( a))
J1(D2_(&, a))>O in Q; , where

and Q:dPf{(c, a ) € Q21u>a1(~)} . Moreover, in Q;,both &(D2,(&, u ) ) and R,(D?(c, a)) are negative. See, Figs. 6.1 and 6.8, showing the stable and unstable regions of D ~ + (a), E, and the asymptotic form of the critical symmetry breaking eigenvalue 4(D?(E, 0)). In conclusion, if both and hold, then, both the P ( Ea), and D:(E,a) are stable in the region Q = Q , n Q;. (And, the same is true for the D?(E,a)-sheet.) We can say that solutions stabi2ize solutions.

7. Concluding Remarks The intention of this survey, or that of the series of our joint papers on which this survey is based, has been not only to establish stability theorems for large amplitude patterned solutions, but also to provide insight into the mechanism which allows the multiple coexistence of stable patterns.

On Multiple Existence of Stable Solutions

217

In summary, this survey has introduced the results as: [ 1 ] a-local stability a n d instability theorems. O u r principal stability results (including the multiple existence theorem of stable patterns) a r e given i n the region Q : = Ql n Q;. So, they a r e a-local results. [ 21 In particular, stability a n d instability theorems a r e given for D ~ ( E a)-, under Assumpts. . sheet of solutions in the region ( E , a) € Q1, E, a r e proved via two approaches: the study Stabilization of D ~ ( a)-sheets a),, a n d the investigation of singular perturbationof spectral behavior of D ~ ( E theoretic mechanism of their stabilization. Thus, one can see that the following problems a r e left unresolved. [0.1] a-global stability theorem of D1(&, a), so long a s it exists. a),via spectral study. [0.2] a-global stability theorem of D ( E a). , [0.3] Stabilization mechanism of P ( E Concerning problem [0.1], the authors h a v e obtained very recently a aglobal stability theorem, in terms of global assumptions o n the nonlinearities f a n d g. See, [*1] a n d [*2]. By using the same idea, the problem [0.2] is also resolved.(*) Question [0.3] seems to require further study. References D. G . Aronson, A. Tesei and H. Wienberger, On a simple density-dependent diffusion system, preprint. E. Conway, Diffusion and predator-prey interaction: pattern in closed systems, Research Notes in Math. 101, Pitman, 1984, 85-133. E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-states solutions for predator-prey equations, Adv. in Appl. Math., 3 (1982), 288-334. T. Erneux and J. Hiernaux, Transition from polar to duplicate patterns, J. Math. Biol., 9 (1980), 193. P. C. Fife, Boundary and interior transition layer phenomena for pairs of secondorder differential equations, J. Math. Anal. Appl., 54 (1976), 497-521. -, Pattern formation in reacting and diffusing systems, J. Chem. Phys., 64 (1976), 554-564. -, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. SOC.,84 (1978), 693-726. -, Mathematical Aspects of Reacting and Diffusing Systems (Lecture Notes in Biomath, Vol. 28), Springer-Verlag, Berlin, 1979. H. Fujii, Numerical pattern formation and group theory, Computing Methods in Applied Sciences and Engineering (Eds. R. Glowinski and J. L. Lions), NorthHolland, Amsterdam, 1980, 63-81. H. Fujii and M. Yamaguti, Structure of singularities and its numerical realization in nonlinear elasticity, J. Math. kyoto Univ., 20 (1980), 489-590. H. Fujii, Looking for the picture of global bifurcation diagram-its numerical realization (in Japanese), Numerical Analysis and Nonlinear Phenomena (Ed. (*I

In preparation.

218

H. FUJII,Y. NISHIURA and Y. HOSONO

M . Yamaguti), Nihon-Hyoronsha, Tokyo, 1981, 45-122. [12] H . Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Physica D., 5 (1982), 1-42. [131 H. F u j i and Y. Nishiura, Global bifurcation diagram in nonlinear diffusion systems, Nonlinear Partial Differential Equations in Applied Science; U.S.-Japan Seminar Tokyo '82, Math. Studies 81, North-Holland, Amsterdam, 1984, 17-36. [I41 H. Fujii and Y. Hosono, Neumann layer phenomena in nonlinear diffusion systems, Recent Topics in Nonlinear PDE: Hiroshima '83 (Eds. M. Mimura and T. Nishida), Math. Studies 98, North-Holland, Amsterdam, 1984, 21-38. [15] H. Fujii and Y. Nishiura, Ultimate symmetry-breaking stabilization of large amplitude singularly perturbed solutions in systems of reaction-diffusion equations, manuscript. [16] A. Gierer and H . Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30. [17] J. P. Gollub, Recent experiments on the transition to turbulent convection, preprint. [18] M. Golubitsky and D . Shaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Math. Phys., 67 (1979), 205. [19] M. Herschkowitz-Kaufman and T . Erneux, T h e bifurcation diagram of model chemical reactions, Ann. New York Acad. Sci., 316 (1979). [20] M. Ito, A remark on singular perturbation methods, Hiroshima Math. J., 14 (1985), 619-629. [21] J. P. Keener, Activators and inhibitors in pattern formation, Studies in Appl. Math., 55 (1976). 187. [22] P,. M. May, Stability and Complexity in Model Ecosystems, Monogr. Popul. Biol. 6, Princeton Univ. Press., Princeton, New Jersey, 1973. [23] -, Models for two interacting populations, Theoretical Ecology-Principles and Applications, (Ed. R. M. May) Blackwell Scientific Publications, Oxford, England, 1976, 49-70. [24] M. Marek and M. Kubicek, Morphogen patterns formation and development in groth, preprint. [25] M. Mimura and J. D. Murray, Spatial structures in a model substrate-inhibition reaction diffusion system, Z. Naturforsch., 33c (1978), 580. [26] M. Mimura, Y. Nishiura and M. Yamaguti, Some diffusive prey and predator systems and their bifurcation problems, Ann. New York Acad. Sci., 316 (1979), 490. [27] M. Mimura, M. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980). 613. I281 M . Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Adv. Biophys., 15 (1982), 19-65. [29] G. Nicolis and I. Prigogine, Self-organization in Non-Equilibrium Systems, J. Wiley & Sons, 1973. [30] Y. Nishiura, Global branching theorem for spatial patterns of reaction-diffusion systems, Proc. Japan Acad., 55 (1979), 201-204. [31] -, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.

On Multiple Existence of Stable Solutions

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[321 Y.Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems and their stability problems, Computing Methods in Applied Science and Engineering (Eds. R. Glowinski and J. L. Lions), North-Holland, Amsterdam, 1982, 185-204. Every multi-mode singularly perturbed solution recovers its stability-from a 1331 -, global bifurcation viewpoint, Lecture Notes in Biomath. 5 5 , Springer, 1984, 292-301. [341 Y. Nishiura and H. Fujii, An approach to the stability of singularly perturbed solutions in reaction-diffusion systems, manuscript. [35] A. Okubo, Diffusion and Ecological Problems: Mathematical Models, (Biomathematics Vol. lo), springer-Verlag, Berlin, 1973. [36] F. Rothe and P. de Mottoni, A simple system of reaction-diffusion equations describing morphogenesis I: Asymptotic behavior, Ann. Mat. Pure Appl. c.IV, 122 (1979), 141. [37] J.-P. Serre, Reprtsentations Lintaires des Groupes Finis, Hermann, Paris, 1971. [38] D. H. Sattinger, Group Theoretic Method in Bifurcation Theory, Lecture Notes in Math. 762, Springer-Verlag, Berlin, 1979. [39] F. F. Seelig, Chemical oscillations by substrate inhibition. A parametrically universal oscillator type in homogeneous catalysis by complex formation, Z. Naturforsch, 31a (1976), 731. 1401 J. M. Smith, Models in Ecology, Cambridge Univ. Press, Cambridge, London, 1974. 1411 A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. SOC.,B237 (1952), 37. [*1] Y. Nishiura and H. Fujii, Stability theorem for sungularly perturbed solutions to systems of reaction-diffusion equations, Proc. Japan Acad., 61 (1985), 329-332. Stability of singularly perturbed solutions to systems of reaction-diffusion [*2] -, equations, manuscript. Institute of Computer Sciences Kyoto Sangyo University Kyoto 603, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 221-258 (1986)

Chaotic Phenomena and Fractal Objects in Numerical Analysis By Shigehiro USHIKI Abstract. Studies on chaotic and irregular behavior in dynamical systems derived from difference schemes and population models are surveyed. Analytic formula of invariant manifolds associated to singular points of dynamical systems and normal forms of singularities of vector fields are discussed. Ghost solutions, strange attractors and fractal objects are treated. Key words: bifurcation, chaos, difference scheme, dynamical system, fractal, homoclinic point, invariant manifold, Julia set, Mandelbrot set, normal form, population model, singularity, strange attractor, vector field, versal family

Contents Part I. Chaos in numerical analysis § 0. Numerical methods and dynamical systems § 1. Euler’s finite difference scheme and “chaos” in the sense of Li-Yorke § 2. Mixed difference scheme and HBnon’s mapping § 3. Central difference scheme and chaos § 4. Saddle connection curves for 2-dimensional analytic dynamical systems § 5 . Analytic formula for invariant manifolds § 6. Ghost dynamics in numerical studies near the Hopf bifurcation point § 7. Population models and chaos Part 11. Normal forms theory and strange attractors § 8. Formal normal forms for degenerate singular points of vector fields § 9. Truncated versal family $10. Application of versal family to “renormalization” of strange attractors Part 111. Fractal objects in holomorphic dynamical systems $11. Chaotic dynamics on a complex domain s12. Julia sets for quadratic maps and rational maps §13. Mandelbrot sets for rational functions Received March 10, 1986.

222 222 225 221 230 233 231 239

24 1 245 246 241 250 251

S. USHIKI

222

Part I.

S 0.

Chaos in Numerical Analysis

Numerical Methods and Dynamical Systems

Strange phenomena have often been observed in the integration of nonlinear differential equations. Even the simplest scheme as the Euler’s finite difference or the central difference scheme can produce a very complicated behavior of numerical solutions. M. Yamaguti and his colleagues, inspired by the work of Li and Yorke [14], R. May [18], and M. H h o n 1111, studied numerical solutions from a dynamical-systems viewpoint. They insisted on the importance of studying the asymptotic behavior of numerical solutions for nonlinear equations for the following reasons. Much study has been done for the justification of numerical methods for the integration of ordinary differential equations. Most of such justifications are satisfied if one takes sufficiently small At and/or Ax. And the validity holds for some finite interval of time O < n A t < T . However, when the numerical method is applied in practical problems, it is very rare that one verifies the relevance of his time step size At. Often numerical integrations are executed far beyond the justifiable range of time interval. For example, beautiful pictures of strange attractors seldom have such a justification. The behavior of solutions of nonlinear differential equations is, in most cases, very difficult to predict a p r i o r i . If one sees some strange phenomenon displayed by numerical integrations, one cannot say whether it is properly generated by the differential equation or is due to the inconvenience of the numerical method. Another motivation of the dynamical-systems-viewpoint study of numerical methods is to examine the relationship between the bifurcation phenomena of nonlinear differential equations and numerical solutions of the discretized version.

S 1.

Euler’s Finite Difference Scheme and “Chaos” in the Sense of Li-Yorke

Let us begin with a very simple example of a nonlinear ordinary differential equation. Consider a scalar differential equation (1.1)

d x / d t =x ( 1-x)

.

The exact solution of this equation for initial condition x ( O ) = x , is given by x (t )=

x,et 1+x,(et-

1)

’

The equation has two equilibrium points, x=O and x = 1. Equilibrium point x=O is asymptotically unstable and the other equilibrium point x = l is asymptotically stable. If initial value is taken as O < x < 03, then the solution

Chaotic Phenomena and Fractal Objects x(t)

223

is asymptotic to 1, i.e.,

(1.3)

lim x(t) = 1 . t-m

Now consider the Euler’s finite difference scheme:

withf(x)=x(l-x) and the time step h>O. For any initial condition x(0)=xo, time interval 0 t i T and E > 0, there exists a sufficiently small time step h > 0, such that the numerical solution obtained by (1.4) starting from u,=xo gives an approximation for the solution x(t), i.e., (1.5)

Ix(nh)-uu,l<~

for O i n h i T .

Even if h is taken very small, however, the discrete dynamical system (1.6)

FtL:R--R

defined by (1.7) has some different features from the original ordinary differential equation. In our case, Fhis a quadratic mapping and is not one to one. Hence Fhis not very faithful to the original equation, at least outside some bounded region. The asymptotically stable equilibrium point x = 1 of the ordinary differential equation has a n unbounded open interval (0, m) as its basin of attraction (consisting of the points whose orbits tend to the attractor). On the other hand, the basin of attraction of the asymptotically stable fixed point of the discrete version is a bounded set. Moreover, the fixed point loses its asymptotical stability if h is irrelevantly large. The eigenvalue of Fh at u= 1 is given by (1.8)

FL(l)=l-h

.

Hence, if h > 2 then u = l is not asymptotically stable any more. One must note that one doesn’t know, a priori, the eigenvalue of a n equilibrium point when one is given a n ordinary differential equation. Moreover, if the stability of the equilibrium point is very strong so that the real part of the eigenvalue of the linear part of the equation is negative and large in norm, then the corresponding fixed point of the dicretized version is apt to turn out to be unstable. As is well known, a discrete dynamical system can have a quite complicated dynamical behavior, which is now called “chaos”. Motivated by the work of R. May [18] on the dynamical behavior of quadratic maps, Li and Yorke [14] gave a definition of “chaos” for one-dimensional mappings as follows.

S. USHIKI

224

Theorem 1.1 (Li and Yorke). L e t J be an interval and let F : J-J be continuous. Assume there is a point a € J f o r which the points b=F(a), c=F2(a) and d= F3(a),satisfy

d
(or d > a > b > c ) .

Then T1: for every k = 1,2, -, there is a periodic point in J having period k . Furthermore, T2: there is an uncountable set ScJ (containing no periodic points), which satisfies the following conditions: (A) For every p . qc S with p f q ,

--

(1.9)

lim sup IFn(p)-Fn(q)I > O n-co

and (1.10)

lim inf IF"(p)-F"(q)l = O

.

n-ca

(B) For every p € S and periodic point q € J ,

(1.11)

lim sup JFn(p)-Fn(q)l> O n-m

.

The set S in this theorem is called a scrambled set. We say a continuous map F of a n interval into itself (or a continuous map of the real line into itself) is chaotic in the sense of Li and Yorke if it satisfies the condition of this theorem. Their theory can be applied to quadratic maps. As the discrete dynamical system (1.4) is nothing but a family of quadratic mappings, all the complicated behavior and the beautiful series of bifurcations detected in the full family of quadratic mappings can be observed in this simple scheme, too. M. Yamaguti and H . Matano [34] proved that in a certain class of nonlinear ordinary differential equations including the logistic equation (1.l), the chaotic phenomena in the sense of Li and Yorke occurs for some values of time step. Consider a nonlinear scalar ordinary differential equation of the form (1.12)

duldt =f(u ) ,

where f ( u ) is continuous in R. We assume that (1.12) has at least two equilibrium points one of which is asymptotically stable. After a linear change of coordinates if necessary, we can suppose the following conditions (1.13) (1.14)

f(O)=f(l)=O

f(u)>O

1

for O < u < l ,

Chaotic Phenomena and Fractal Objects (1.15)

f(u)
for

225

l
+

Here the constant K is possibly 00. Equilibrium point u= 1 is asymptotically stable. Euler’s difference sche-me for (1.12) takes the form (1.16)

xnti=x,+ A f f ( x n )

We denote the right hand side of this formula as FAt(x)=x+Atf(x). The theorem of Yamaguti and Matano can be stated as follows. Theorem 1.2 (Yamaguti and Matano). ( i ) Let (1.13), (1.14) and (1.15) hold. Then there exists a positive constant c, such that for any At > cl the diference equation (1.16) is chaotic in the sense of Li-Yorke . ( i i ) Suppose in addition that K = 00; then there exists another constant c2, O < c, < c2, such that for any time step At with O < A 1. Moreover, when c, < At < c2, the above mentioned chaotic phenomena occurs in this confinement interval.

+

S 2.

Mixed Difference Scheme and HCnon’s Mapping

M. Yamaguti and S. Ushiki [36] studied some numerical methods which might produce “ghost” solutions. The apparition of “ghost” solutions in the numerical integration of nonlinear ordinary differential equations by the central difference scheme (midpoint method) had been detected more than a hundred years ago. Detailed analysis for the central difference scheme was done by S. Ushiki [29],which will be reviewed in the next section. M. Yamaguti and S. Ushiki [36] examined a mixed difference scheme, which was designed as a remedy to remove the numerical instability produced by the central difference scheme at asymptotically stable equilibrium points. Let us consider again the logistic equation

(2.1)

dx/dt=x(l-x) ,

and consider the following mixed difference scheme parametrized by p with O<,Lf
This scheme is an interpolation of the central difference scheme (2.3)

Untl-Un-l=f(Un)

2h

S. USHIKI

226

and the Euler’s finite difference scheme (2.4) If p is set to 0 then (2.2) is reduced to the central difference scheme (2.3) and if p is set to 1 then it becomes the Euler’s difference scheme (2.4). By setting U , + ~ = U , , the scheme (2.2) can be rewritten in the form

(2.5) which we regard as a dynamical system of the plane defined by mapping Tp,h: R2+R2, i s . , T , t , h ( ~vn~, ) = ( u , , +v R~+,A . This system has at least two fixed points (0,O)and (1, l ) , which we denote 0 and P respectively. These points correspond to the unstable equilibrium point x=O and the asymptotically stable equilibrium point x = 1, respectively. For positive time step h, the fixed point 0 is a saddle point if 1 >p>O. And the other fixed point P is a n asymptotically stable fixed point if the mixed “Euler component” is sufficiently important according to the time step h. More precisely, if l > h > O and p > h / 2 then the fixed point P has two real eigenvalues with norms smaller than 1, so that P is a sink. If we take a sufficiently small time step h and select parameter p satisfying p>h/2, then the numerical integration can be executed for O < t < 00 without producing ghost solutions for appropriately chosen initial values. If we fix h and vary p from a value p0>h/2 to a value p1
i

1-ax:

Yn+l=bXn,

where a and b are parameters. transformation

To see this fact, just apply an affine coordinate

Chaotic Phenomena and Fractal Objects

227

A strange attractor was discovered by M. HCnon [ l l ] for a=1.4 and b= 0.3. Much study has been done on the chaotic behavior of such plane mappings and on the bifurcation structure as the Feigenbaum’s series [ 5 ] of period doubling bifurcations, Mira’s “box in the box” structure [6], [ 7 ] , etc. All such strange and complicated phenomena should be expected naturally also for our mixed difference scheme (2.5). M. Yamaguti and S. Ushiki [36] reported the existence of Smale’s “horseshoe” dynamical system [25] as a subsystem of (2.5) for some values of parameters. If h is very near 3 and if p is sufficiently near 1, then there exists a rectangular region in the neighborhood of the u-axis, which is mapped by ( T , , J 2 into a horse-shoe shaped region and the image intersects the original region in two pieces. The rectangular region is stretched in the u-direction and shrinks in the v-direction. Then it is bent to form a horse-shoe like shape. Finally the horse-shoe like shape is put on the original rectangular region so that both of the two ends are mapped to the left of the rectangle and the turning part of the horse-shoe is mapped to the right of the rectangle. As a n example, p=0.99, h=2.995 satisfies the condition for the existence of Smale’s horse-shoe for (2.5). The existence of Smale’s horse-shoe does not necessarily imply that the chaotic phenomena should be observed. But it at least implies complexity of the dynamical system. In fact, there exists a n uncountable Cantor set 9 in the uv-plane, which is invariant under T,t,,, and the mapping T,,,, restricted to SZ is conjugate to the so-called Baker’s transformation. The nature of “chaos” observed in the HCnon’s mapping seems to be more complicated than Smale’s horse-shoe dynamical system. In spite of many efforts from both a numerical and theoretical approach, a complete analysis of such “strange” behavior has yet to be obtained.

S 3.

Central Difference Scheme and Chaos Central difference scheme

(3.1)

unt1-

u”-l=f(u,)

2h

for the integration of ordinary differential equation

(3.2)

dxldt =f(x )

is known to have the numerical instability.

In order to simplify the argument, we consider only scalar equations for (3.2). As in the preceding section, we ~, regard (3.1) as a dynamical system F,: R2->R2 defined by Fh(un,~ , , ) = ( u , +v,,,) with

(3.3)

S . USHIKI

228

Suppose (3.2) has a n asymptotically stable equilibrium point, say at x = p , which satisfiesf(p)=O and f ( p ) < O . The Jacobian matrix of Fh is given by

(3.4) Note that the determinant of DF, is always equal to - 1. Hence the iterated map Fi=F, 0 F,: R2+R2 is a n area preserving map. Fixed point, (u, v)= ( p . p ) , of Fh corresponds to the asymptotically stable equilibrium point x=p. The eigenvalues of (3.4) at ( p , p ) are given by

Since f ' ( p ) < O and h > O , one of the eigenvalues, R-, is smaller than -1 and the other, A,, is between 0 and 1. The eigenvector of linear map DF, are given respectively by

(3.6)

tE,

and

tE-,

where Et=(l+, 1) and (-=(A-,

1)

for r € R . Let E s = { r f + l t € R } and P = { t f - I t € R } denote the eigenspaces. Note that if w e Es, then (DF,)"w tends to the origin as n tends to infinity, and that if w e E" then (DF,&)-"wconverges to the origin when n tends to infinity since R- < - 1 and 0 < A, < 1. Linear space Es is called the stable eigenspace and Eu is called the unstable eigenspace at ( p , p ) . Linear space E" is called the unstable eigenspace. If h is very small, then A, is near 1 and I - is near -1. Eigenspace Es is near the "diagonal line" A = { ( u , v)€R21u=u} and E" is almost orthogonal to A . Now, let us look at the mapping Fh near the fixed point ( p , p ) . By the theorem of Hartman-Grobman [8], we know that F, is topologically conjugate to the linearized map DF,. More precisely, there exist a neighborhood V of ( p , p ) , a neighborhood U of the origin ( 0 , 0 ) € R 2 , and a homeomorphism q5: U- V such that for any ( u , v) € U ,

holds.

If w € $ - ' ( E S ) ) ,then F,l(w) converges to the fixed point ( p , p ) . If converges to the fixed point as n tends to the infinity. The set fioc=q5-1(E*)is called the local stable manifold and W;.,,=q5-'(Eu) is called the local unstable manifold. The stable manifold w" and the unstable manifold Wu of the fixed point are defined as

w € q5-'(Eu) then F;"(w)

They are smoothly immersed curves i f f is smooth. In our case, the fixed point ( p , p ) is said to be a saddle point since both w" and W' are non-trivial.

Chaotic Phenomena and Fractal Objects

229

Next, suppose that the ordinary differential equation has a n unstable equilibrium point, say x=q, withf(q)=O andf’(q) >O. By a similar argument, fixed point ( q . q) of Fh has its own stable manifold and unstable manifold, becuase, in this case, the eigenvalues satisfy - 1 < L ( q )< 0 and R+(q)> 1. The stable eigenspace Es is defined a s the linear subspace spanned by eigenvectors belonging to the eigenvalue which is smaller than 1 in norm. The unstable eigenspace Eu is spanned by eigenvectors with eigenvalues greater than the unity in norm. Fixed point (4, q) is a saddle point, too. Consider the following situation. A point, say w E R 2 , is said to be doubly asymptotic if F;(w) tends to a saddle point, say P, as n tends to infinity and if F i n ( u )also tends to a saddle point. If there exists a doubly asymptotic point w such that

(3.9)

lim Ph(w)=lim F;”(w)=P n-oa

n-sm

then o is called a homoclinic point. The existence of homoclinic points was first discovered by H . PoincarC [22]. A homoclinic point belongs to the stable manifold W s and the unstable manifold Wu of the saddle point at the same time. S. Smale [25] proved that if these two curves, w” and W u , intersect at w and are not tangent to each other at 0, then there exists a n uncountable closed set S, which is invariant by the mapping Fhr such that the dynamical system Fh restricted to S (or its some iterated composition) is equivalent to “coin tossing”. The existence of a homoclinic point with transversal intersection of invariant manifolds implies the existence of infinitely many homoclinic points and periodic points (and a n uncountable number of aperiodic points). By numerical experiments, M . Yamaguti and S. Ushiki [35] observed the existence of homoclinic points in several cases. They considered it as a n origin of the strange phenomena of numerical solutions of (3.1). S. Ushiki [29] gave a rigorous proof for the existence of Smale’s horse-shoe in Fh for the case of f(x) =x( 1 -x). According to this proof, Fh has horse-shoes for any non-zero time step h. The proof is based on a transcendental property of complex analytic functions. This method can also be applied to some holomorphic dynamical systems. The outline of the proof is as follows. Suppose f(x)=x(l-x). Then ordinary differential equation (3.2) has two equilibrium points. The point x=O is a n asymptotically unstable equilibrium point and x = 1 is a n asymptotically stable equilibrium point. Corresponding to these two equilibrium points, there are two fixed points, P=(O, 0) and Q= (1, 1). Both of these are saddle points. Stable nanifolds and unstable manifolds associated with these saddle points are denoted as W ( P ) ,Wu(P),W ( Q ) ,and W’(Q). They are depicted in Figure 3.1. Since we consider only the case f ( x ) = x ( l - x ) , so that f(1-x)=f(x), we see that Ws(P)and W u ( Q )are symmetric with respect to the line L = { ( u , v ) l u + v = l } c R 2 . Similarly, Wu(P)and

230

S. USHIKI

S

P

Fig. 3.1.

W”(Q) are symmetric with respect to L . By an elementary argument, we obtain that W ( P ) n L f Q ) and W u ( P ) nL # @ . Using the symmetry of invariant manifolds, these imply that W ( P )n W s ( Q ) #0 and W’(Q)n W ( P ) f 0 . As non-empty intersection does not imply a transversal intersection, we must study more precisely how they intersect each other. In general, unstable manifolds and stable manifolds can intersect with tangency. Especially, if we treat the system as a family of dynamical systems parametrized by h, then such “degenerate” situation is inevitable. In fact, non-transversal intersention of invariant manifolds can occur and it can be observed numerically. Even though the transversal intersection is not always expected, “transversal intersection” in the topological sense can be proven. And “topologically traversing” intersection of unstable and stable manifolds of a saddle point can give rise to a “horse-shoe’’ dynamical system in the topological sense, which was studied by K . Yano [37]. Finally, to prove the topological transversality of the intersection of invariant manifolds, we must appeal to the analytic property of invariant curves. In order to prove this fact, a detailed analysis of invariant manifolds is necessary. It is found in S. Ushiki [29], which will be reviewed in the next section.

S 4.

Saddle Connection Curves for 2-Dimensional Analytic Dynamical Systems

As we have considered dynamical systems on the plane, the dimension of the unstable manifold and the stable manifold associated with a saddle point is always one. These invariant curves are always smoothly embedded curves if the dynamical system is smooth. They are analytic if the system is analytic. Note that F is not supposed to be diffeomorphic. In this case the invariant “curves” W s and Wu are not necessarily manifolds any more. Let F : R2-+R2be a real analytic mapping. Assume that the origin is a fixed point, i.e., F ( O ) = O . Let a and p be the two eigenvalues of a differential map of F at the origin. Assume that 0 is a saddle point, i.e.,

Chaotic Phenomena and Fractal Objects

23 1

The saddle point is possibly critical when P=O. In this section, we treat only the analyticity of unstable “manifold”. When F is a diffeomorphism, then by considering the inverse map F-I: R2+R2 in place of F , a n analytic formula for stable “manifold” can be obtained. If F is not invertible, then unstable “manifold” will be the image of a n analytic map and the stable “manifold” (stable set) will be a complicated entangled set with many branching points. An analytic parametrization of unstable “manifold” is given as follows. By applying a linear change of coordinates if necessary, we can assume that the Jacobian matrix DF at the origin is diagonal. The mapping F : R2+R2 can be expanded into Taylor series around the origin as,

(4.2) A mapping $: R+R2 which parametrizes the unstable “manifold” W” of the origin is constructed by setting its Taylor coefficients as described in the following. Supposing $(E)=( f ( E ) , g(E))to be analytic in a neighborhood of the origin, expand the components f ( E ) and g(E) into the power series:

Observe that if $: R-R2 parametrizes the unstable “manifold”, it could satisfy the following function equation

(4.4)

F $(El =$((YE) , 0

which was called the fundamental equation by H . PoincarC [21]. Moreover, the linear part of $ should be tangent to the unstable local manifold at the saddle point. Hence, we can assume f l = l and gl=O. Now compute both sides of (4.4) as a formal power series and compare the coefficients with respect to p. The formal power series to be computed are given by

and

All the coefficients f n and g, can be determined inductively, starting from

S . USHIKI

232

f,=1 and gl=O.

For non-negative integers p and q with p f q 2 2 , let

where the summation is done over all combinations of positive integers i,, ..., i,,jl, . . - , j , with i,+ +i,+j,+ ..- +jq=n. Since p + q > 2 , all the indexes i,, . ., i,, j , , . , j q are smaller than n. gn-lare known, then @:.q can be determined Hence, if f,,..-,f n - , , g,, by (4.8). Coefficients f , and g , are given by

..-

-

--

and (4.10)

for n 2 2 . The formal power series computed by this procedure converges in a neighborhood of the origin. Theorem 4.1. If 101 > 1> then formal power- series f(E) and g(E) defined above converges in a neighborhood of the origin. The proof needs a delicate estimation for the coefficients using a majorant series. See S. Ushiki [28] for details. The theorem assures the analyticity of the mapping q5: R-R2 near the origin. We constructed a n analytic mapping 0: R-R2 in the neighborhood of the origin of R . It satisfies the fundamental equation (4.4). Theorem 4.2. If F : R2-R2 is analytic on the entire plane, R 2 , and if lal>l>[email protected] h e n $ : R - R 2 d e f i n e d a b o v e i s a n a l y t i c o n R . Proof. An analytic continuation of Q can be dfined by using the fundamental equation (4.4). Suppose that power series Q(E) is convergent for ]El < r . For any E € R , find a positive integer k such that < r . Define $ ( E ) by (4.11)

$(E) =Fh(Q(a-XE)) .

The value $(E) does not depend on the choice of Ic, since (4.4) holds in the neighborhood of the origin. In such a way, analytic function Q is defined on the entire space R .

Chaotic Phenomena and Fractal Objects

233

Now, let us consider holomorphic dynamical systems. Suppose F : C2+C2 is a complex analytic mapping defined on C2. Assume the origin, O=(O, 0), is a fixed point a n d let a and /3 be the eigenvalues of a Jacobian matrix at the fixed point. By the same argument, we have the following theorem. Theorem 4.3. Zf la1 > 1> 1/31 then there exists a holomorphic parametrization $: C-C2 of unstable “manifold”, satisfying the following conditions: ( i ) q5(0)=0, ( i i ) Image ( d $ , ) = P , (iii) Image (+)= W , and (iv) F o $(O=+(at). The Taylor coefficients of q5 are computed as in the case of the real analytic case. This theorem, for the case of a rational map F : C2->C2,was proven by H . PoincarC [21]. Note that the obtained mapping F : C-,C2 is defined on the entire complex plane. It is a n entire mapping. See S. Ushiki [27] for the proof. Let us now make use of this fact to proved the non-existence of “saddleconnection curves”. In the preceding section, we proved the existence of intersection points of unstable manifold Wu(P)and stable manifold W*(Q). These invariant curves are analytic. If they intersect, the intersection takes place at discrete points locally, or two curves coincide entirely. Let h : Z-R2 be a n embedding (one-to-one non-singular differentiable mapping) of the unit interval into R2. Suppose F : R2->R2 has two saddle points, P and Q (P and Q may be identical). We call h a saddle-connection curve if ( i ) h(0)= P , h( 1)= Q, ( i i ) h(10, 1[) does not contain fixed point of F, (iii) h(Z)is invariant under F. Theorem 4.4. Zf real analytic diffeomorphism F : R2-bR2 can be extended to a complex analytic autornorphism F : C2-C2 of two dimensional complex Euclidean space, then there exists no saddle-connection curve. This theorem shows that in our dynamical system, W u ( P )and W s ( Q ) cannot contain a common arc in their intersection. For the proof of this theorem, see S. Ushiki [27].

S 5.

Analytic Formula for Invariant Manifolds

When we look at global bifurcation phenomena as the apparition of homoclinic points, saddle connection curves, or degenerate tangency of invariant manifolds, we need some numerical approach to study the behavior of invariant manifolds near the bifurcation point. In the case of analytic

S. USHIKI

234

dynamical systems on the plane, invariant manifolds associated to saddle points are calculated by the formula given in the preceding section. For higher dimensional dynamical systems, there exists analytic formula for unstable manifolds and stable manifolds associated to saddle points. Let f:Rn-R" be a real analytic map defined in a neighborhood of the origin. Suppose the origin, O=(O, - ,0 ) E R", is a fixed point of f, i.e., f ( O ) = O . We assume that the Jacobian matrix D F , at 0 is diagonal. Let al, -,a , be the eigenvalues of DF,. We assume

--

--

(5.1)

so that 0 is a saddle point. Let azL=(al, ax)and a , = ( a k C I , an). Let 6=(6,, -,6,) denote a multi-index with k components. All components of 6 are non-negative integers. We denote by 161 the length 6,+ +6,. If x = ( x I , * . -,x,) is a vector with m components and if p=(pl, p,) is a multiindex with the same number of components, we adopt the abbreviation - . a ,

--

...,

...

xP=~fl

...

X Pmm .

Using this convention, we assume

(5.2)

asfat

for any multi-index 6 with 161 2 2 and i = l , k. We call a point P € R* a n unstable point of 0 if there is a sequence of points P , € R " , i = O , - 1 , -2, - . - ,such that P,=f(P,-,) for i=O, - 1 , -2, ..-, P=P,, and that P, tends to the origin as i tends to -m. We denote the set of unstable points of 0 by W". Wu is called the unstable set of 0. I f f is a diffeomorphism, then W uis nothing but the unstable manifold of 0. Local unstable manifold W: is defined for small E > O as follows. Let BE denote the ball of radius E centered at 0. A point P E B , belongs to W: if there is a sequence of points P, Be, i=O, - 1 , -2, -,such that P,=f(P,-,) for i=O, - 1 , -2, P=P,, and P, tends to the origin as i tends to -a. Note that m a - ,

-

..-,

m

(5.3)

W"=

u f"W:)

k=n

.

Let E" denote the linear subspace spanned by the eigenvectors of DF, associated with eigenvectors a,, -,a,. Let 8 : E'-Eu be the linear map DF, restricted to the invariant subspace E".

--

Theorem 5.1. There exists a real analytic mapping q5: U-R" defined in a neighborhood of the origin of E". such that ( i 1 q5(0)=0, ( i i ) D@ is non-singular,

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235

(iii) $( U ) = WE, (iv) f o q5=# 0. 0

Taylor coefficients of $ are given below. Note that iff: R"-Rn is defined globally on R", so that the iterated compositions f k = f o f are always defined, then using the fundamental equation (iv) in the theorem, q5 is defined on the entire space Eu and q5(Eu)= W" holds. Moreover, i f f is a n analytic diffeomorphism, then the same argument holds for W . When f is a n analytic mapping of a real analytic manifold into itself, a similar theorem holds. So it is true in the case of complex analytic mapping. In order to give the explicit formula for Taylor coefficients of q5, we x , ) € R n , and d=(dl, d,) be a introduce several notations. Let x = ( x l , multi-index with n components. Let f ( x ) = ( f l ( x ) , . , f , , ( x ) )and the Taylor expansion as 0

.-.

. ..,

. . a ,

where the summation is done over all multi-indexes d with length [dl 2 2 . For the sake of consistency of notation, let d(j)=(O,

. . a ,

0, 1,0,

. . a ,

0)

3

denote the multi-index with n components of length 1, which has only one positive component. Similarly, let a(j)=(O,

. - - , O ,l,O, ..-,O) 3

denote the multi-index with k components of length 1. Let f i , d ( j ) = aifY ti=j and f,,,cn =O if ifj. We have D F o = ( f i , d ( j ) ) .Using this notation, (5.4) can be rewritten as fi(x)=

(5.5)

C

ft,dxd

for

i = l , ..-,n.

Id121

..

Let (=(El, -,5,) denote a vector in E u = R k . Suppose q5: E'-Rn be a real analytic map with q5(0)=0. Let $(f)=(Q,(E), -..,q5,(()) and their Taylor expansion be

q5Yt(E)=

(5.6)

c $,.SP,

i=l, ...,n.

161>1

We define the Taylor coefficients $ i , S as in the following. For indexes k , of length 1, and for i= 1, -,n, set

S ( j ) , j = 1,

(5.7)

--

a ,

-

S . USHIKI

236

so that the Jacobian matrix, D$o, at the origin is the inclusion map E u c R " . Hence it is of maximal rank. For the sake of consistency, we set $ L , c o , . . . , o , =O. Next, let us formally calculate the coefficients of products of (5.6). Let d= ( d l , * -,d,) be a multi-index. Consider d,-th power of $((E). Just set ( e = d , for simplicity of printing)

-

c $t,sS")"= c $:,sEs

(5.8)

16121

161>e

Then the Taylor coefficients $:,s are given by

$h=c$d* *

(5.9)

$z.P

Y

where the summation is done over all the combinations of multi-indexes r', - . . , y e such that yl+ +ye==6. For e=O, we If e=l then $:,a=$t,s. set

--.

(5.10)

$P,co,

. o ) = l,

if

$P,s=O

161 21

.

Observe that if e 2 2 , then all f , - - . ) r ewhich appear in (5.9) have lengths strictly smaller than 161, since they must have positive lengths. Next, to compute the product

we set the Taylor coefficients (5.11) as

(5.12)

(Q(E)Id=

c $;Es

.

Then, for Id1 2 1, we have

(5.13)

$:=

c

* *

-

$:yrn

,

where the summation is done over all the combinations of multi-indexes - ,y" with k components, such that r'+ + y n = 6 . When Idl=l then

f ,

---

-.

(5.14)

'$~'"'$z.d

The Taylor coefficients of f,($(E))

.

are computed by setting

as (5.16)

ft,s=cft,d$:

9

where the summation is done over all multi-index d satisfying 1612ldl. (5.16) can be rewritten as

Chaotic Phenomena and Fractal Objects

237

(5.17)

Now put all these into the fundamental equation (5.18)

fo$=$oB.

Then we obtain the equation of formal power series (5.19)

That is to say (5.20)

for 161 2 2 . For 161 =1, the equalities of coefficients hold automatically. As we noted before, (5.20) gives all values of $i,8 since the right hand side contains only $ t , r ’ ~ with lyI < 161, and the divisors do not vanish by our assumptions (5.1) and (5.2). Theorem 5.2. (5.20).

The function $: E”-R” in Theorem 5.1 isgiven by ( 5 . 6 ) , (5.7),

The proof of the convergence of this formal power series is found in S. Ushiki [28]. 6.

Ghost Dynamics in Numerical Studies near the Hopf Bifurcation Point

F. Brezzi, S. Ushiki and H . Fujii [3] studied the relevance of numerical methods in view of the bifurcation theory. Much work had been done for numerical methods used in the numerical study of stationary states. In their study, they regarded the relevance of Euler’s scheme for a numerical method employed in the Hopf bifurcation problem. When we treat a n ordinary differential equation or a partial differential equation which has a Hopf bifurcation, some “dynamic” aspects intervenes in the bifurcation, since a generation of oscillatory solutions is concerned. In their work, they concluded that Euler’s finite difference scheme can reproduce sufficiently well the phenomenon of Hopf bifurcation in a neighborhood of the bifurcation point. At the same time, they showed that the relevance of the numerical approximation holds within a certain limited extent. Global bifurcation behavior can be quite different from the original differential equation even though the time step is taken very small. And if the time step is not “sufficiently” small, then the numerical dynamics produce a strange behavior similar to the so-called “chaos”. We consider a family of ordinary differential equations

238

(6.1)

S . USHIKI

du/dt=f ( u , s ) ,

u € R"

, s€ R .

Suppose (6.1) contains a Hopf singularity, say at (uo,so)=(O, 0). Let us consider Euler's finite difference scheme

As a classical result, it is well known that if we fix the time step r > O , then (6.5) possesses a n s-family of "invariant circles" whenever F, satisfies the Hopf condition in the mapping sense. It seems quite natural to expect that, as r tends to zero, there exist invariant circles, uniformly in r , and this r-family of circles converges to the limit cycle of the original system. The uniform convergence of this family of invariant circles to a family of limit cycles is not guaranteed by the classical Hopf bifurcation theory for discrete mappings. It is because the eigenvalue of discrete dynamical system (6.3) approaches 1 as r tends to zero. The Hopf bifurcation point, say (u,,s,) depends on the time step r. Hence a delicate analysis and a more detailed error estimate are necessary to bring a uniform convergence of invariant circles. Proposition 6.1. There exist positive constants ro, so and c,, and a smooth function 6=6(r), r € N,'=]O, r o ] ,such that for each r E N : , F, has a Hops bifurcation point ( 0 , 6 ( r ) )€ R" x R in the sense of mappings. Here, 6 ( r ) satisfies that 16(r)l
f o r s E N , +a n d r c N : . For a detailed analysis, see F. Brezzi, S. Ushiki and H . Fujii [3]. Now, let us just look at the global dynamics of (6.3). Consider, as the simplest example, a n ordinary differential equation

(6.4)

dzldt = z( i+s- I z /z,

Chaotic Phenomena and Fractal Objects

239

defined on the complex plane C z R 2 . It is rotationary symmetric and has a Hopf point (zo,so)=(O, 0). For sO, (6.4) has a globally attracting limit cycle with radius &. The basin of attraction of the limit cycle is R2-{O}. The Euler’s finite difference scheme for (6.4) is given by:

(6.5)

zn+l=zn+rzn(i+s- Iz,12)

.

By virtue of the rotational symmetry, we can analyse the global dynamics of the discrete version (6.5) by examining a one dimensional mapping

(6.6) where qa=Izn12 (hence q n 2 0 ) . It has three fixed points, namely, qo=O which corresponds to the equilibrium point z=O, q-=s+(l-(1-r2)1/2)/rwhich corresponds to the attracting limit cycle, and q+= ~ + ( l + ( l - r ~ ) ~which / ~ ) /does ~ not have its counterpart in the continuous version (6.4). exist if T < 1. If T is very The invariant circles with radii V‘Zand small, then the “ghost” invariant circle is located far away from the origin. It defines the boundary of the attractive basin of either the asymptotically < 1-( l - ~ ~ ) l / ~ ) / r ) , stable equilibrium point z=O (when -(1+( 1- r 2 ) 1 / 2 ) / r < s -( the asymptotically stable limit invariant circle lzI2=q- (when - ( l - ( l - ~ ~ ) ~ / ~ ) / r < s < ( 4 - 2 r 2 ) / ( 2 r ( l - ~ 2 ) 1 ~ 2 ) -and l/r s < ( l + ( l + ~ ~ ) ~ / ~ or) some / r ) , other type of attractor near the invariant circle. In the last case, a variety of phenomena can occur. The attractor can be periodic invariant circles, or a band of circles on which the orbit wanders etc. All the complicated dynamics observed in one dimensional mappings (period doubling, chaos in the sense of Li and Yorke, etc.) can occur. The phenomenon of “chaos” can be observed even when we take r very small, if we look at (6.5) as a n s-family of mappings. Moreover, if we consider a similar equation and its discretized version without the radial symmetry, even more complicated bifurcation and chaotic strange behavior can be observed. No effective theory of understanding for the global bifurcation scheme of two dimensional mappings is available as of yet.

z/z

S 7.

Population Models and Chaos

By looking at plane mappings derived from the Euler’s finite difference scheme for systems of ordinary differential equations such as population models, chaotic phenomena were detected. S . Hayama [lo] reported the existence of “chaotic” behavior in a difference scheme

240

S . USHIKI

for prey-predator system (7.2)

dX/dt=(e, --A,,x+ -A,,y)x dy/dt= -(~z--R,,x)y .

S. Ushiki, M. Yamaguti and H. Matano [33] reported that similar strange behavior can be observed also in competition systems (7.3)

du/dt= ( a ,- b,,u--b,,v)u

i

du/dt= ( a ,-b,,u- b,,v)v

.

In these cases, Marotto’s theorem can be applied to prove the existence of “snap-back repellers” and “chaotic” orbits. We call a fixed point z of a smooth mapping F:Rn+Rn a snap-back repeller if there is a sequence of compact sets {B,}, --oo
xn+1

is “chaotic”. That is, there exists: ( i ) a positive integer, N , such that f o r each integer p 2 N , F has a periodic point of period p ; ( i i ) a “scrambled set” of F , i.e., an uncountable set, S , containing no periodic points of F such that: ( a ) W)IS, (b) for every x , y € S with x f y , lim sup llFk(x)-Fk(y)l\> O , k-oo

( c ) for every x E S and any periodic point y of F , lim sup IIFX(x)-Fk(y)II> O , k-m

(iii) an uncountable subset Soof S such that for every x , y € So, lim inf IIFL(x)-Fk(y)jl=O . k-m

The proof is found in S. Marotto [17]. A generalization of this theorem in the case of a “hyperbolic snap-back point” was obtained by K . Shiraiwa and M. Kurata [23]. M. Hata [9] studied the existence of a scrambled set in expansive maps on Rn.

24 1

Chaotic Phenomena and Fractal Objects

Part 11. Normal Forms Theory and Strange Attractors

S 8. Formal Normal Forms for Degenerate Singular Points of Vector Fields Motivated by the existence of “strange” attractors in nonlinear systems of ordinary differential equations, the author a n d his coworkers were interested in the “normal forms” of degenerate singular points of systems of ordinary differential equations. Usually, the interest of mathematicians is directed toward the normal forms of simplest singularities (least degenerate singularities). If we want to understand global bifurcation of strange attractors, however, we must look at a very degenerate system, since a degenerate system has quite a complicated structure and can give rise to strange attractors in its unfoldings. Let us begin with the definition of jets of vector fields. Let &?denote the Lie algebra of the (germs of) smooth vector fields defined in a neighborhood of the origin, 0, of R*. For a positive integer, k, let -.’/x denote the subset of Z consisting of those (germs of) vector fields whose Taylor coefficients at 0 of orders strictly smaller than k, vanish. Subspaces -Hk, k = l , 2 , are Lie ideals of &Y.We have the following sequence of linear subspaces: . . a ,

Define quotient Lie albebras, H A ,for k = l , 2 , ..., by H k = = / - / . f l k + l . If we fix a system of coordinates around the origin, then there is a one to one correspondence between H k and the polynomial vector fields of degree k which vanish at 0. The correspondence is given by the Taylor expansion and the truncation at degree k. Two elements, say X and Y , of 2’are said to have the same k-jet at 0 if their Taylor coefficients at the origin of orders smaller than or equal to k coincide. The Lie algebra H k can be regarded as the space of k-jets at the origin. Define linear spaces, H k , for k = 1, 2 , . -,by Hk=A‘,/J//X+,. Note that the linear space, H,, is isomorphic to the linear space of homogeneous vector fields of degree k. This isomorphism depends on the choice of the system of coordinates. We use the standard coordinates of R n in the following. We denote the canonical projections as j k : P + H k and j k , p : Hp+Hk, for p > k . Since we fix the standard coordinates and identify the elements of H k with polynomial vector fields of degree k, we have natural inclusions H-Hk and H,+Hk for 1I i l k. Lie algebra Hk has a decomposition as a linear space:

Decomposition H k = H k - ’ @ H, is often used. For X E 2 , X k will denote the k-jet in Hk represented by X , i.e., X k = j k ( X ) . We denote the Hk-component of X k by X,. X , is the degree k homogeneous part of the vector field, X . Similarly as decomposition (8.2), X k can be decomposed as

S. USHIKI

242

(8.3)

Xk=X,+...+X,

and

Xk=Xk-l+Xk.

In order to compute the normal forms of singular points of a system of ordinary differential equations, one must look at how a coordinate change works on the vector fields. When we compute the effect of infinitesimal changes of coordinates, we must use the Lie algebra calculus. Let us introduce some notations for "graded Lie algebra". For X and Y in Z, [ X . Y j = X Y - YX denotes the Lie bracket and [ X , Y ] , denotes the k-jet of [ X , Y ] . After our systematic use of superfix and suffix, [ X , Y ] , denotes the H,component of [ X , Y ] , . If X , € H , and X , € H , then [ X , , X,] E H k + , - , . Lie bracket [ X , , X,] defines a bilinear map H k xH,+Hk+,-,. If Xx€ H k and Yk€ H k , then [ X k , YkIkis a well defined element in FP. It does not depend on the choice of the representatives for X k and Y k . The adjoint operator, adk(Xk):Hk-.Hk, of Xk€ H k is defined by adk(XA)(Y k )= [ X k , YkIk

(8.4)

for

Y kE H k ,

Its i-th component with respect to the decomposition (8.2) is denoted as ad:( X k ): Hk+H,

(8.5)

(1Ii I k )

and is defined by

(8.6)

adl(Xk)(Y k ) = [ X k Y , k ] t = [ X k Y, k ] % for

- + X , and Y * = Y , + - - - + Y,, then adk(Xk)(Y k ) = [ X , + .- - + X k , Y,+ - - - + Y,jk

In other words, if X x = X , + . (8.7)

Y k EH x .

a

=[XI, Y , l + [ X , , Y , I + . * * + [ X , ,Ykl + [ X , , Y , l + . . . + [ X , , Yk-I1

+...

+IX,, Yll and (8.8)

a d : ( X k ) ( Y k ) = [ XY, ,J + [ X 2 ,Y , - , l + - - . + [ X , , Yll

.

Now let us look at how a coordinate change transforms a vector field. Let X 6 2" be a vector field defined on a neighborhood of the origin, O€R", and let q5 be a local diffeomorphism around 0. The ordinary differential equation defined by X : (8.9)

dx/dt= X ( X ),

x

R"

is transformed by the coordinate change, y = $ ( x ) into (8.10)

d y / d t = ( d ~ / d x ) . X ( ~ - ' .( v ) )

Chaotic Phenomena and Fractal Objects

243

Hence the coordinate change y = $ ( x ) transforms X into (d$/dx).Xo $-I, where Jacobian matrix d$/dx is evaluated at $ - ' ( y ) . The transformation formula for jets of a vector field and diffeomorphism is given as follows. Let Sk denote the group of k-jets of local diffeomorphisms a t 0 € R". Dkis a Lie group. Let Xk€ H k and $k € 9fh. Although k-jet does not define a vector field nor diffeomorphism, the k-jet of transformed vector field (8.11)

-

(d$'/dx)

Xk

0

(p) -1

does not depend on the choice of representatives for X k and p . We denote the k-jet of (8.11) by Adk(qV)(Xk). We say two k-jets Xk,Y k€ Hkare k-equivalent as truncated vector fields if there exists a k-jet of diffeomorphism, @k, such that (8.12)

Yh=Adk(qP)(Xk).

In other words, two vector fields X and Yare k-equivalent as truncated vector fields if there exists a coordinate change such that the transformed vector field (8.10) and Y has the same k-th order Taylor expansion at 0. The truncated k-equivalence is a n equivalence relation. Let - T k ( X k denote ) the set of all kjets YkEHkwhich are k-equivalent to X k as truncated vector fields. Since -Qk is a Lie group and its operation (8.11) on H kis a Lie group action, the set Sk( X k ) is a manifold.

Definition. The k-th order normal form of vector field singularity on R n is the set of representatives of the orbit space Hk/GJh. The choice of representatives for each equivalence class is not unique. The choice must be strategically done in view of the use of normal forms in various problems. In order to select the representatives, we must compute explicitly the orbits S h ( X k ) . This computation can be done using the calculus of the Lie algebra Hkof the Lie group S k . Infinitesimal transformation of coordinate change and the infinitesimal deformation of vector fields can be calculated by looking at the tangent spaces of the orbits 9 k ( X k )which is obtained by the action of Lie algebra. Detailed computation is found in S. Ushiki [30]. The normal forms of first order are given by the theory of Jordan's normal forms of matrices, since H 1 is isomorphic to the linear space of n x n matrices, S ' = G L ( n , R), and Adl($l)(X1)=$l.X1.($l)-l. Hence higher order normal forms, which we are going to compute, can be considered as the generalization of Jordan's normal forms. If none of the eigenvalues of the linear part of a vector field has a zero real part, then the vector field in a neighborhood of the origin is topologically conjugate to its linearized mapping. Hence, we consider only the cases where the linear part has multiple zero eigenvalues or purely imaginary eigenvalues.

S . USHIKI

244

In the following statements of normal forms, we adopt notations for vector fields used in geometric theories. For the sake of simplicity, we employ notations as a, in place of atax, 8, for a/ay, etc. By identifying these differential operators with a basis of the tangent space, the system of ordinary differential equations, dx,/dt=f,(x,, * * * , x J , i=17 . - . , n ,

(8.13)

can be rewritten as a vector field in the form: (8.14)

X=f,(x,,

*

-

*

,X n ) a s l + * * - +fn(xl,* * * x,1azn 7

The k-jet, Xk,of (8.14) is represented by a polynomial vector field of degree k obtained by truncating its Taylor expansion at the origin. Theorem 8.1. Assume vector field, X , on R2 (or C z )has ya, as 1-jet at the origin. Then X can be transformed by a change of coordinates into a vector field whose 4-jet is one of the following form: (a) ya, + ( i - x 2 f u x y wx3+qx3~)a,, x y v,x3 v,x2y+ qx3y+sx4)a,, (b) ya, (i or (c) ya, ( w,x3 w,x2y+ qx3y+sx4)a,, with w: w;= 1. Parameters u, v,, v,, w, q, s, w,, w, are uniquely determined fr om the 4-jet at the origin of the given vector field, X .

+ +

+ +

+

+

+

If we use the terminologies and notations defined above, this theorem can be re-stated as follows. If X,=yd, then X4 is 4-equivalent to (a), (b), or (c) above in the truncated sense. In other words, (a), (b), and (c) are normal forms of fourth order for vector fields with X,=ya,. In the above theorem, cases (b) and (c) are special cases. Almost all vector fields have their normal forms belonging to (a). I n the following theorems, we use the term “generically” to mean that the transformation can be executed for all vector fields satisfying the assumption in the theorem, except those vector fields which satisfy a n additional algebraic condition on Taylor coefficients. Let X be a vector field around the origin, O E R 3 (or C3). Theorem 8.2.

+

If X I = -ya,+xa,,

+

then X 6 is generically 5-equivalent to

+

+

+

+

( 1 bz+ ez2+gr4)i3, (azr dzZr frK)i3, ( ir 2 k zz czs)i3, , where a,= -yd,+xa,, ra,=xa,+ya,, determined f r o m the 5-jet X5.

andparameters a , 6 , c , d , e ,f , g are uniquely

Chaotic Phenomena and Fractal Objects

If X,=yd, then its third order normal f o r m is generically

Theorem 8.3. yd,

245

+( t x 2+axy-t z2+cyz+ e x s+hxyz+ iyz2)d,+(bxz+dz2+fx3+gz3)d, ,

where parameters a , 6 , c , d , e , f,g , h , i are uniquely determined f r o m X s .

Theorem 8.4. yd,

If X,=yd,+zd,

then the third order normal f o r m isgenerically

+zag+ ( -tx2+axy+ bxz+cx2y+dxz2 +ex3)&,

where paramters a , b, c, d , e are uniquely determined f r o m X 3 .

We refer to S. Ushiki [30] for the proof.

S 9. Truncated Versal Family Suppose X k € Hk is a n element in normal form of degree k. A family of k-jets of singular vector fields, A : Rm-Hk, with A (0) = X k , is called a truncated deformation of X k . Definition. A truncated deformation A : RT'-Hk, A ( 0 ) = X k , is a k-th order versal f a m i l y if, for any deformation A : Rm-Hk, there exists a continuous mapping I : Rm-RT with 1(0)=0 such that A ( a ) and A(1(a))are k-equivalent as truncated vector fields for all a in a neighborhood of the origin 0 c Rm. Since k-equivalence relation in H X is defined by a Lie group action Adk: Hk-Hk, versal families can be obtained as transversal sections to the orbit d k ( X k ) c H k . Especially, families of normal forms given in the preceding section give transversal sections of the orbit 9 * ( X k )restricted to affine sub@ H k . Hence, by adding a versal family of its linear part space X,+Hz @ X I , in the linear space H,, a versal family of k-th order is obtained. Versal family for linear part X , is given by V. I. Arnold [l]. As a n example, let us consider a 4-jet X4, at OER2, in normal form: 9

k

x

--.

(9.1)

X4=Ydz+(X2+U0xY

+w,x3+qox~Y)d,.

The versal deformation of the linear part X,=yd, is a two parameter family (9.2)

X1+(P,x+PzY)d,

*

Hence five parameter family (9.3)

+

+ +UXY +wx3+9X3Y)a,

X4 ( P1x PZY

with parameter ( p l ,p z , u, w,q) E R6 is a fourth order truncated versal family. Versal families for other normal forms can be obtained similarly. Versal families of linear parts are given as follows (see V. I. Arnold [l]). (9.4)

X,+plrd,+p2as+p3zd,

for X I = &

at O c R 3 ,

S. USHIKI

246

(9.6)

X,+(P,X+P,Y+P,Z)~,

for X,=yi?,+za,

a t O E R8 .

When we consider vector fields which have some symmetry, similar arguments can be made and analogous computations respecting the symmetry give normal forms under the presence of symmetry. S. Ushiki, H . Oka and H . Kokubu [31] calculated normal forms of vector fields around 0 E RS with X,=yd, under the symmetry ( x , y , z)+(-x, - y , z). Its third order normal form is generically given by

(9.7)

X 3 = X 1 + ( ~ x z + a y z + c x ~ + d x ~ y + e y z ~ ) t ?i ~x 2++( b z z + f z 3 ) d z.

H . Oka and H. Kokubu [I91 gave the third order normal form for XI= yl,+za, with symmetry ( x , y , z)->(-x, - y , - z ) as (9.8)

X 3= X I +(_ ~ x ~ + ~ x ~ ~ + / ~ x ~ z + .~ x ~ z + ~ x z ~ ) ~ ~

Versal families for symmetry cases can also be given. families for the linear parts are given by

(9.9) (9.10)

~ 1 + ( P , x + P z ~ ) ~ , + P 3 z & for (9.7)

Symmetric versal

7

X l + ( ~ l x + ~ z ~ + ~ 3 z for ) & (9.8) .

Normal form theory for parametrized families of vector fields was developed by H . Kokubu [13]. An application to the bifurcation problem of some reaction-diffusion equations is studied there. Normal form theory for constrained differential equations are developed by H . Oka and H . Kokubu. See their article in this volume for details.

S 10.

Application of Versal Family to “Renormalization” of Strange Attractors

The study of normal forms and versal families of degenerate singularities was motivated by the existence of “strange attractors” such as the Lorenz attractor and Rossler’s attractors. The existence of strange attractors and their structures have global features. When we want to study them in a n analytic manner, we need to localize them in some sense. We supposed that the global aspect of strange attractors and their bifurcations can be treated by looking at a very degenerate singularity if it contains such strange behavior in its unfoldings or versal families. In fact, S. Ushiki, H. Oka and H. Kokubu [31] found that the total family of the Lorenz system [I51

(10.1)

+

a(y- x)a, ( - X Z Y X -y)a,

+( x y -bz)&

is equivalent to a subfamily of the versal family of (9.7). Just set X = d T x ,

Chaotic Phenomena and Fractal Objects

241

Y = d T ( y - z ) , and Z=(1--b/20)(20z-x2), then (10.1) becomes (10.2)

Ya,+ ( AX+B Y f 0 X . Z - X 3 ) a ,+ ( C X + X z ) a , ,

where A = u ( r - 1 ) , B=-0-1, C=--6, a=2o--6. (10.2) is a subfamily of (9.7)+(9.9). Family (10.2) includes four parameters whereas (10.1) contains only three parameters. In other words, Lorenz family (10.1) is embedded into (10.2). In this large family, we can construct a “homotopy” which connects the Lorenz system to a n integrable system, by smoothly changing the coordinates and the time scale. The global property is not affected except the limit system. Change the time variable t into t’/p and set the “renormalization” change of coordinates as x = p X , y = p 2 Y , z=pZ. Then the “renormalized system” is

Similar results for Rossler’s family is found in S. Ushiki, H . Oka and H. Kokubu [31].

Part 111. Fractal Objects in Holomorphic Dynamical Systems

S 11.

Chaotic Dynamics on a Complex Domain

Modern research in holomorphic dynamical systems began with Mandelbrot’s experiments on Julia sets and the so-called Mandelbrot sets (see B. Mandelbrot [16]. A . Douady and J . H. Hubbard [ 4 ] , D . Sullivan [29] and M. Herman [12] revealed the strikingly rich and beautiful dynamical behavior of holomorphic dynamical systems. Since real analytic dynamical systems include holomorphic dynamics, a study of holoinorphic dynamical systems is necessary to understand, at least, the apparently mysterious property of real analytic dynamical systems. Let us begin with one dimensional holomorphic dynamical systems. Let c = C U {a} denote the Riemann sphere, and let f: be a complex analytic map. Holomorphic mapping is a rational map and can be written as a quotient of mutually prime polynomials, say f ( z ) = P ( z ) / Q ( z ) . We call d=max (deg (P), deg (Q)) the degree off. Holomorphic dynamical systems are found in numerical analysis, where iterations of complex analytic functions are concerned. As an example, consider a polynomial equation

c->C

(11.1) in complex variable z.

(11.2)

P(z)=O The Newton’s method applied to this equation:

S . USHIKI

248

C-tc

gives a holomorphic dynamical system f: with f(z)=z-P(z)/P’(z) a rational function. In terminologies defined below, a solution of (11.1) corresponds to a super attractive fixed point of (11.2). The problem is how the basin of attraction of these fixed points are composed, what their boundary is, etc. A point z G C is said to be normal if there exists a neighborhood, U , of z such that the sequence of iterated composition { f k l u } k = l , . .restricted ., to U is equicontinuous. The set of normal points (11.3)

F ( f ) = { zC ~ I z is normal for

is a n open set and is called the Fatou set o f f . (11.4)

f)

Its complement set

J ( F )=C\F(f)

is called the Julia set o f f . The Julia set is a closed set. The Fatou and Julia sets are both totally invariant under f, i.e.,

Here are some fundamental properties of the Julia set. Theorem 11.1. Z f d 2 2 then J ( f ) # @ . Theorem 11.2. J ( f ) is perfect and uncountable.

In most cases, the Julia set is a “fractal” set. Although J ( f ) is never empty, the Fatou set can be empty. A point p € c is said to be a fixed point if f ( p ) = p . The value I=f’(p) does not depend on the choice of coordinates and is called the eigenvalue of the fixed point. Fixed point p is classified by its eigenvalue as follows. It is super attractive if I=O, attractive if I R I < l , repulsive if 111 >1, and neutral if 1R1=1. When p is neutral, let R=ezrrie. Neutral fixed point p is rationally neutral if 0 is rational, irrationally neutral if 0 is irrational. A n irrationally neutral fixed point, p , is called diophantine if there exist positive constants C and E , such that (11.6)

IB-qlrl > C / r Z t f

holds for all integers q and r with r>O. For fixed point p , let A ( p ) denote the set of points z E C such that fn(z)--tp as n + a . A ( p ) is called the basin of attraction of p . If p is a n attractive fixed point, then A ( p ) is a n open set and p E A @ ) . I n this case A ( p ) is called a n attractive basin. If d 2 2 and p is rationally neutral, then p € a A ( p ) . In fact the “Flower Theorem” by Fatou and Julia shows that at least some “petal” region is included in A ( p ) .

Chaotic Phenomena and Fractal Objects

249

Theorem 11.3. I f d 2 2 , P = l , and I m f l f o r l < m < n , then there are an integer k and nk real analytic curves which are pairwise tangent at p and which bound petals. The union of the petals is forward invariant, and any orbit in a petal is asymptotic to p . In this case, A ( p ) is called a parabolic basin. When f is topologically conjugate to a n irrational rotation in a neighborhood of the fixed point p , then there exists a n invariant disk around p . Such a disk is called the Siegel disk. If p is a n irrationally neutral fixed point and its rotation number 8 is diophantine, then p has a Siegel disk. An attractive basin, the interior of a parabolic basin and the interior of a Siegel disk are included in the Fatou set F ( f ) . If f “p)=p and f “ p ) f p for 1 Ii< k , then p is said to be a periodic point of period k . The eigenvalue of a periodic point is given by ;i=(fk)’(p). According to its eigenvalue, periodic points are classified similarly as the fixed points. Classification of the basin of attraction etc. are given similarly. Besides these components of F ( f ) , the Fatou set can have a (periodic) invariant annulus called the Herman ring, on which f works as a n irrational rotation. D. Sullivan gave a classification for connected components of F ( f ) . Each connected component of F(f) is called a stable region.

Theorem 11.4 (Sullivan). All stable regions are eventually periodic. Periodic stable regions are either attractive basins, parabolic basins, Siegel disks, or Herman rings. For the proof, see D. Sullivan [29]. For a n introductory survey of holomorphic dynamical systems, we refer to P. Blanchard [ 2 ] . M . Shishikura [24] developed a surgery theory of complex analytic dynamica1 systems and perturbation of systems. He proved the following theorems.

Theorem 11.5. For all integers p 2 1 , there exists a rational function of degree 3, which has Herman rings of order p . Theorem 11.6. For an irrational number 8 , the existence of a rational function with a Siegel disk of rotation number 8 implies the existence of a rational function with a Herman ring of the same rotation number, and vise versa. Theorem 11.7. A rational function of degree d has at most 2(d-1) cycles of stable regions, with cycles of Herman rings counted twice. Moreover, there exist at most ( d - 2 ) cycles of Herman rings. Conversely, f o r any prescribed combination of numbers of cycles for each type of stable region, satisfying the restrictions above, one can find a rational function of degree d , which just has the prescribed number of cycles. S. Ushiki, H . - 0 . Peitgen and F. v. Haeseler [32] studied the structurally

S. USHIKI

250

stable regions in the parameter space parametrizing a family of rational functions of degree 2.

s 12.

Julia Sets for Quadratic Maps and Rational Maps

Numerical experiments with computer graphic techniques are said to have inspired mathematicians working in holomorphic dynamical systems. Here, we show some pictures of Julia sets. As the simplest case, consider a family of quadratic maps

(12.1)

f(z)=z2+c.

The infinity z=oo is always a super attractive fixed point. We have the following cases. (a) J ( f ) is toally disconnected, (b) there is a (cycle of) super attractive basin, (c) there is a (cycle of) attractive basin, (d) there is a (cycle of) parabolic basin, (e) there is a (cycle of) Siegel disk, (f) none of the above. The coloration principle of the pictures is as follows. Take a small disk around the (super) attractive (periodic) point. For each point z € C , compute f " ( z ) , n = 1, 2, Select a color for the pixel corresponding to z as a function of the smallest n such that f " ( z ) falls in the small disk. Pictures 1 and 2 represent case (a). If c=O, then (12.1) has a super attractive fixed point at z=O. Pictures 3 and 4 represent cycles of super attractive basins of period 2 and 3, respectively. Picture 5 is for case (c). Pictures 6, 7 and 8 are for case (d) with A = l , - 1 , and i respectively. A quadratic map with a Siegel disk is represented by picture 9. Values of parameter c are as follows.

-

- a .

picture

C

1

1

2

-0.75+0.23

3

-1

4

- 1.22561 +0.7448621'

5

-0.15

6

0.25

7

-0.75

8

0.25+0.5i

9

0.37418+0.1934111'

Chaotic Phenomena and Fractal Objects

25 1

As the second example, consider the two parameter family of rational functions of degree two: f(z) =z(z+l)/(l+ p z )

(12.2)

.

Picture 10 represents the case with two attractive fixed points. Picture 11 represents the case with one attractive basin and a Siegel disk. In picture 12, f(z) has a Siegel disk and a parabolic basin. Picture 13 represents the case where there are two distinct parabolic basins. In picture 14, there is only one cycle of attractive basin. Both of the critical points are attracted to a n attractive cycle of period three. Parameters for these pictures are as follows. picture 10

R

P

-0.5+0.71'

,u=R

11 12

eo.sr

13

ezni/s

i

14

-0.692683+ 1.1997621'

p=R

el

O.5etIb errt/z

-

Rational mappings of a degree smaller than 3 do not have Herman rings. Picture 15 is a n example of a Herman ring discovered by M. Herman [12] for (12.3)

f(z)=(et/z)(z-a)z/(l-2)2

with a=0.25, t = 1.9. The green annulus represents the Herman ring and the other annuli are its preimages. Picture 16 is a cycle of Herman rings of period 2 found by M. Shishikura [24]. His mapping is given by (12.4)

f(Z ) = z'( Z- b)/(z- C) +a

for a =0.864375+0.2103381', b= 0.076868 -0.2503721' and c= - 0.080948 0.249204i. The brown annulus appears to be a cycle of Herman rings. Other rings are their preimages.

S 13.

Mandelbrot Sets for Rational Functions

B. Mandelbrot [ 161studied the bifurcation diagram of holomorphic dynamical systems. He executed numerical experiments for the family of quadratic maps (12.1), and found surprisingly beautiful and complicated objects. The Mandelbrot set M c C is defined by

M = {c f?C I {f"(O)},=,,,, ... is bounded}

.

S. USHIKI

252

Pictures of the Mandelbrot set are found in B. Mandelbrot [16]. Many beautiful full color pictures are found in H. 0. Peitgen and P. Richter [20]. Here are just a few of pictures of the Mandelbrot set. Picture 17 is the global shape of the Mandelbrot set. Pictures 18 to 20 are its enlargements. The regions of parameter c represented in these pictures are as follows. picture 17

region -2.11Re (c)<0.5,

-1.21Im (c)<1.2

18

0.2
0.471Im (c)10.66

19

0.288cRe (c)
0.48365gIm (c)<0.48385

20

-0.751Re (c)< -0.749,

-0.04sIm (c)<0.041

Similar numerical experiments for families with multiple parameters are also possible. In this case, coloration is done by examining the type of cycles of stable regions. Pictures 21 to 23 show sections of the “Mandelbrot set” for two parameter families of rational functions

+p .

(13.1)

f ( z ) =A(z+ 1/21

These pictures represent the A-plane. Values of ,9 are fixed to p=2erLr4,p= 2e3rI/S, ,8=2e5ri’s respectively. Another choice of a family of rational functions of degree 2 is given by

In this family of mappings, A and p are exactly the eigenvalues at 0 and 03 respectively. Pictures 24 to 26 represent the “Mandelbrot set” in the A-plane witj p=0.8eZrifixed. The I-regions are as follows. picture

region

24

-31Re (413,

25

- 2 < R e ( A ) < -1.3,

26

-1.891Re (A)<-1.82,

-31Im ( 4 1 3 -0.711m (A)< -0.05 -0.471Im (A)<-0.4

Strange “fractal” objects are found also in cubic functions. are sections of the “Mandelbrot set” for

Pictures 27 to 29

These pictures represent the a-plane with p fixed to 0.2, 0.1+0.4i, and O . l + O . l i respectively. Pictures 30 to 34 are some of their enlargements.

253

Chaotic Phenomena and Fractal Objects

picture

region

30

,8=0.2+0.li

- 0 . 7 1 R e ( a ) <-0.3,

-0.21Im (a)10.2

31

B=O.l+0.3i,

- 0 . 7 1 R e ( a ) <-0.0,

-0.9
32

a=0.1+0.3i

-0.3
33

a=0.1+0.3i,

34

a=0.1+0.3i,

(p)< -0.24,

-0.55
-0.49, )I: -0.2251Im -0.531Re @ -0.2895gRe

(p)< -0.282,

( a ) <-0.2 (/?)I

-0.495

(b)< -0.165

-0.367c Im (b)< -0.3605

References [ 11 V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equa-

tions, Springer, 1983. [ 2 1 P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. SOC.,11 (1984), 85-141.

[ 3 1 F. Brezzi, S. Ushiki and H. Fuji, “Real” and “Ghost” bifurcation dynamics in difference schemes for ODE’S, in Numerical Methods for Bifurcation Problems, eds. T. Kupper, H. D. Mittelmann and H. Weber, ISNM 70, Birkhauser, 1984, 79-104. 4 1 A. Douady and J. H. Hubbard, ItCration des polynbmes quadratiques complexes, C. R. Acad. Sci. Paris, SCr I, 294 (1982), 123-124. [ 5 ] M. J. Feigenbaum, Quantitative universality for a class on nonlinear transformations, J. Statist. Phys., 19 (1978), 25-52. 161 I. Gumovski and C. Mira, Accumulation de bifurcation dans une rtcurrence, C. R. Acad. Sci. Paris, Str A, 281 (1975), 45-48. [ 7 1 H. E. Hamouly and C. Mira, Lien entre les proprietCs d’un endomorphisme de dimension un et celle d’un diffeomorphisme de dimension deux, C. R. Acad. Sci. Paris, SBr. I, 293 (1981), 525-528. [ 8 ] P. Hartman, Ordinary Differential Equations, John Wiley and Sons Inc., New York-London-Sydney, 1964. [ 9 ] M. Hata, Euler’s finite difference scheme and chaos in R”,Proc. Japan Acad., Ser A, 58 (1982), 178-181. 1101 S. Hayama, Dynamics of a discrete prey-predator model and chaos, (in Japanese). 1111 M. HBnon, A two dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976),60-77. 1121 M. Herman, Exemples de fraction rationnelles ayant une orbite dense sur la sphere de Riemann, Bull. SOC.Math. France, 112 (1984), 93-142. [13] H. Kokubu, Normal forms for parametrized vector field and its application to bifurcations of some reaction-diffusion equations, Japan J. Appl. Math., 1 (1984), 273-297. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 [141 T.-Y. (1975), 985-992. [15] E. N. Lorenz, Dterministic nonperiodic flows, J. Atmospheric Sci., 20 (1963), 130-141. [161 B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freedman and Co., 1977.

254

S. USHIKI

[171 S. Marotto, Snap-back repellers imply chaos in R", J. Math. Anal. Appl., 63 (1978), 199-223. R. M. May, Biological population with non overlapping generation: stable points, stable cycles, and chaos, Science, 186 (1974), 645-647. 1191 H. Oka and H. Kokubu, Constrained Lorenz-like attractors, Japn J. Appl. Math., 2 (1985), 495-500. [201 H.-0. Peitgen and P. Richter, Beauty of Chaos, Springer, 1986. [211 H. Poincart, Sur une classe nouvelle des transcendantes uniformes, J. Math., 4" strie, 6 (1890). -, Les MCthodes Nouvelles de la Mtcanique CCleste, Gauthier-Villars, Paris, 1899. K. Shiraiwa and M. Kurata, A generalization of a theorem of Marotto, Nagoya Math. J. 82 (1981), 83-97. M. Shishikura, Surgery of complex analytic dynamical systems, (preprint). S. Smale, Diffeomorphism with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, 1964, 63-80. D. Sullivan, Quasiconformal homeomorphisms and dynamics, I, 111. (I.H.E.S. preprint). S. Ushiki, Sur les liaisons-cols des systemes dynamiques analytiques, C. R. Acad. Sci. Paris, Str. A, 291 (1980), 447-449. -, Unstable manifolds of analytic dynamical systems, J. Math. Kyoto Univ., 21 (1981), 763-785. Central difference scheme and chaos, Physica, 4D (1982), 407-424. [291 -, Normal forms for singularities of vector fields, Japan J. Appl. Math., 1 I301 -, (1984), 1-37. S. Ushiki, H. Oka and W. Kokubu, Existence d'attracteurs Ctranges dans le dtploiement d'une singularitt dtgCnCrte d'un champ de vecteurs invariants par translation, C. R. Acad. Sci. Paris, 298, SCr. I, (1984), 39-42. S. Ushiki, H.-0. Peitgen, F. v. Haeseler, Hyperbolic components of rational fractions l ( z + l / z ) , The Theory of Dynamical Systems and Its Applications to Non-linear Problems, World Sci. Publ., 1984, 61-70. S. Ushiki, M. Yamaguti and H. Matano, Discrete population models and chaos, Lecture Notes in Numerical and Applied Analysis, 2, Kinokuniya, 1980. M. Yamaguti and H. Matano, Euler's finite difference scheme and chaos, Proc. Japan Acad., 55 (1979), 78-80. 1351 M. Yamaguti and S. Ushiki, Discrttisation et chaos, C. R. Acad. Sci. Paris, 290 (1980), 407-424. M. Yamaguti and S. Ushiki, Chaos in numerical analysis of ordinary differential equations, Physica, 3D (1981), 618-626. K. Yano, A remark on the topological entropy of homeomorphisms, Invent. Math., 59 (1980), 215-220.

Institute of Mathematics Yoshida College Kyoto University Kyoto 606, Japan

Chaotic Phenomena and Fractal Objects

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256

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Chaotic Phenomena and Fractal Obiects

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258

Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equationspp. 259-278 (1986)

Fractals in Mathematics By Masayoshi HATA Abstract. In this paper we shall survey two topics concerning the Fractals: nowhere differentiable functions and self-similar sets in Euclidean space such as Cantor set, Koch curve, and Peano curves. Key words: fractals, nowhere differentiable functions, difference equations, self-similarity.

What is a “fractal”?

S 0.

The terminology “fractal” was created by Mandelbrot in the description of Nature. T o quote from his book [30]; “A fractal is by definition a set for which the Hausdorff dimension strictly exceeds the topological dimension.” For example, Cantor’s ternary set X, and von Koch’s curve X , are typical fractal sets, since it is known that dim,(X,)=(log 2)/(log 3)>0 and dim,(X,)= (log 4)/(log 3) > 1 where dim,(X) denotes the Hausdorff dimension of a set X . The notion of “fractal” is surely based on the classical mathematical works done by Cantor, Weierstrass, Peano, Lebesgue, Hausdorff and so on. It is quite surprising that such pathological counter-examples have something to do with certain fields of Natural Science. Thus, it is desirable to clarify the structure of such “singularities”. The measure theory is one of the most powerful mathematical tools to handle fractal sets. See e.g. Rogers [37] and Falconer [S]. However in this paper we shall survey some topics concerning nowhere differentiable functions and fractal sets in Euclidean space. We hope that this approach will make a contribution toward shedding light on the structure of “strange attractors” in dynamical systems.

S 1.

Nowhere Differentiable Functions

The question of the existence of a continuous nowhere differentiable function was settled affirmatively by Weierstrass. Namely he showed that (1.1)

Received April 2, 1985.

c

W(x)= an cos (b”7rx) , nZ1

260

M. HATA

where b is a n odd integer and O 1+(3/2)r, at n o point has a differential coefficient, either finite, or infinite with a fixed sign. After slight generalizations by a number of writers, Hardy [lo]proved that W ( x )does not possess a finite differential coefficient at any point in any case in which O < a 1 and ab2l. It is now common sense that most continuous functions are nowhere differentiable in the sense of the Baire category (see e.g. Jarnik [18]). However, a concrete example of such functions sometimes leads to interesting speculations.

(C)

Figure 1. (a) Weierstrass function (1- W( x ) ) / 2( b = a - l = 2 ) . (b) Takagi function. ( c ) Riemann function.

Fractals in Mathematics

26 1

In 1903, Takagi [40] discovered a quite simple example of a nowhere differentiable function (1.2)

T(x)=

c 2-n$(2n-'x) ,

7l21

where Q ( x ) = 2 1 x - [ x + 1 / 2 ] 1 . This example is highly instructive; that is, T ( x ) is a typical example of "Condensation of Singularities" (see e.g. Hobson [16, p. 4011, since it is a superposition of so-called sawfunctians. The Weierstrass function ( 1 . 1 ) and the Takagi function ( 1 . 2 ) have cusps at countably many points (see Figure l(a) and (b)). Hobson [16, p. 4101 also studied the series (1.3)

C anQ(bnx),

O
7l21

and showed that the conditions a b > 4 when b is a n even integer or ab> 1 when b is a n odd integer forbid the existence of a differential coefficient finite or infinite, applying a simple method of Knopp. For b=a-l=lO this was the example given by van der Waerden [42] in 1930. Also de Rham [35] pointed out that if we take b = a - l , b being a n even integer, then it has no finite differential coefficient. Consider now the following functional equation: (1.4)

f(x)-af(bx)=g(x)

*

It was de Rham who remarked that the Weierstrass function (1.1) and the series (1.3) satisfy ( 1 . 4 ) for g ( x ) = a cos ( b a x ) and g ( x ) = a $ ( b x ) respectively. Kuczma [24, p. 821 noted that if we take g ( x ) = a cos ( b a x ) , the equation (1.4) has a C" solution in ( - 0 0 , 00) depending on a n arbitrary function, although the unique bounded (continuous) solution is the Weierstrass function. On other non-differentiable functions, Faber [ 6 ] considered the function

c 10-"$(2"'x) ;

n2l

he showed that this function does not satisfy a Lipschitz condition of any order. Recently Cater [ 4 ] studied the function 2-"' cos (2'2n)'x) ; ntl

he showed that this function has n o cusps and satisfies some extreme properties. It was supposed by Riemann that the function (1.5)

R(x)=

2 x 2sin (n2ax) at1

is nowhere differentiable (see Figure l(c)). Weierstrass had attempted to prove

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262

Riemann's statement, did not succeed, and was led to the series (1.1). Hardy [ l o ] proved that R(x) has n o finite derivative at irrational points nor at rational points of the form 2p/(4q+l) or (2p+1)/(4q+2). Gerver [9] proved that R(x) has a derivative -x/2 at points of the form (2p+ 1)/(2q+ 1) and that no finite derivative at points of the form (2p+1)/2", n 2 l . Finally Smith [39] gave a complete answer to the problem, showing that R(x) has no finite derivative in the remaining cases. Consider now the following functional equation

This equation was studied by Artin [ l ] in characterizing Euler's Gamma function as a unique smooth solution of certain functional equations. The author [ l l ] regarded (1.6) as a n eigenvalue problem for some Perron-Frobenius operator and investigated various solutions of (1.6) according to the eigenvalue R. He also remarked that if 6 2 2 is a n integer, then the Weierstrass function W(2x)+cos (2rrx) satisfies (1.6) for p = b , p = l and R=a; the Takagi function T ( x ) - l / 2 also satisfies (1.6) for p=2, p = l and 2=1/2; and the Riemann function R(2x) satisfies (1.6) for p = 2 , p = 2 and R=1/4. The Weierstrass function ( 1 . 1 ) is a typical example of a lacunary series, that is a series where the terms different from zero are very sparse. More generally, Kaplan, Mallet-Paret and Yorke [20] studied the series

(1.7)

f(x)=

2 a"r(b*x) ,

O
n21

where ab> 1 and r(x) is a n almost periodic function. They showed that under certain smoothness conditions on r the series (1.7) is either continuously differentiable or nowhere differentiable; moreover, in the latter case the metric (capacity) dimension of I', is equal to 2+(log a)/(log b), where r, is the graph of the function (1.7). It is possible for rgto have the Hausdorff dimension greater than 1 if g is sufficiently singular. Besicovitch and Ursell [2] have shown that if g(x)belongs to the calss Lip(@, 0<0"<1, rg has a finite k-dimensional measure for k = 2-6, and they have constructed g for which the k-dimensional measure is actually positive for 1 1 k 1 2 - 6 . More generally, Love and Young 1291 have shown that if x(t) belongs to the class Lip(@ and y ( t ) to the class Lip(#) where 6+6'> 1, 0 < 6 ' < 6 < 1 , the curve (x(r), y ( t ) ) has a finite k-dimensional measure for k=2-(8+6'l)/S. Kline 1211 constructed a curve (x(t),y ( r ) ) for which the l ) / 8 is actually attained. Falconer [8] showed that if dimension k=2-(6+6'-

where O < s < 1 and {An} is a sequence of positive numbers satisfying

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263

then dim, (fg)=2--s. However, it will be difficult to determine the exact value of dim, for the Weierstrass function (1.1) and the series (1.3). We conjecture that in both cases log a dim, (r) =2 log b

(r)

+-

This value seems to be quite reasonable since Hardy has shown that i f f = -(log a)/(logb) < 1,

W(x+h)-W(x)=O(lhlE)

and

W(x+h)- W(x)#o(lhlE)

for any value of x.

S 2.

Chaotic Mappings

Consider a one-dimensional dynamical system Q(x)=4x( 1- x ) on the unit interval I . It is well known that the n-fold iteration $" can be expressed by $n(x)=sinz (2" arcsin

dX).

It was Prof. Yamaguti who had the inkling to combine I $ ~ with the Weierstrass function (1.1). Indeed, we obtain the fine relation

therefore the generating function F(a, x) is nowhere differentiable for 1/2
therefore F(1/2, x) is nowhere differentiable. See Yamaguti and Hata [44]. Note that +(x) is chaotic in the sense of Li-Yorke [27]; moreover there exists a probabilistic invariant measure d x / ( n z / x (1-x)), absolutely continuous with respect to the Lebesgue measure. It is also known that $(x) is topologically conjugate to the piecewise-linear function #(x); that is, $(XI=#

O

WX)

9

where H(x)=sin%x is a homeomorphism of Z. The above examples (2.1) and (2.2) raise the following problem: What kind of function w : I-I causes the non-differentiability of its generating function

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264

F ( a , x )=

(2.3)

C anwn(x) ?L>O

with respect to x ? Intuitively, the cause will be the sensitive dependence of initial value for the dynamical system a. For example, consider a family of quadratic functions $,(x)=Jx(l-x) with 0<1<4. Then it will be quite interesting to consider the smoothness of its generating function according to the parameter A. Suppose that there exists a bounded domain D, containing a n open segment (0,1) such that $,(D,)cD,. Then it is easily seen that F(a, z ) is analytic in D , for any la1 < 1. For 0 < 1 < 3 we may take D, to be the interior of the Julia set of the rational function $,; in particular, for 1=2 we can take D, to be a n open disk of radius 112 centered at z= 112. On the other hand, for A=4 the Julia set of $, is a segment [0, 11 and F(a, x ) is nowhere differentiable for 1/2
C anRn(z),

la1 < 1 ,

?l>O

where R ( z ) is a rational function, and obtained some conditions under which J(eto) has n o finite derivative with respect to 19; in other words, J ( z ) has a natural boundary IzI = 1. For example, we get the Weierstrass function if we take R(z)=zb. We now give some criteria for the smoothness of the generating function (2.3).

Theorem 2.1 (Differentiability). Suppose that o:I-Z is continuously diflerentiable and possesses stable periodic points; that is, there exists a point p such that wq((p)=p and I(oq)'(p)I < 1. Then the generating function F ( a , x ) is continuously diferentiable in the attractive region W corresponding to { p , w ( p ) , * -, wq--'(p)}f o r any fixed la1 < 1.

-

Proof.

Let K be any compact subset of W a n d put p,,=sup{Iw'(x)I; x € w " - ' ( K ) }

for n 2 l

.

- ..,

Since o " ( K ) converges to the set { p , ~ ( p ) , wq--'(p)} as n - a , we have lim sup,,, pntl * /I.+~ < 1. This implies the boundedness of {pl pn} and therefore

--

..-

Thus the series differentiated term by term converges uniformly and this completes the proof.

Theorem 2.2 (Non-differentiability). Suppose that

w : I-I

is continuously

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265

differentiable and possesses a repulsive fixed point p o such that I = --o'(po) > 1. Suppose further that there exists a homoclinic orbit { p l < p - $ < .-< p o < < ~ - ~ < p such - ~ }that ~ ( p - ~ ) = pf -o r~ n+>~l and p-*+po as n-m. Then, f o r any fixed

.

p-sup nP1

IP-.-l -Pol Ilal
the generating function F(a, x) has no finite derivative at any point x f o r which w"(x)=p, and ( w n ) ' ( x ) f Of o r some n>O. Proof.

It suffices to show the non-differentiability at x=po.

We note that

I p 2 1, since P-n-1-Po

P-n-Po

P-n-1-Po

-

~

W(P-,-l)--PO

1

as n - w .

W'(P0)

Suppose, on the contrary, that there exists a finite derivative A=(i?/i?x)F(a,p o ) . Then, the equality F(a, p-,-,)-aF(a, p-,)=P-,-, implies A=(l+aA)-I. On the other hand, we have A =F(a,p-zM-l)-FF(a,po) M-

=l+a

P - 2 M - I -PO P-2M-pO

+

...

+a2X

P-ZM-I-PO

First consider the case a
P-l-PO P-ZM-I-PO

Then we have A,>1

since

>O .

~ - 2 M t 2 , - 2 - ~ O + ~ ~ ~ - 2 M + 2 j - 1 - ~ o ~

Hence A 2 1, contrary to the assumption A-'=l-Alal< l-!rllal 10. Next, consider the case a>O. Then we also have A M > l since a2p2P-2Mt2j-2-P0

PO - P - Z M Hence A 2 1 , contrary to A - ' = l + a I > l .

t 2 j-

1

This completes the proof.

0

In general, it is a difficult problem to study the differentiability of (2.3) at repulsive periodic points. It is a n open problem whether there exists a nowhere differentiable generating function for which the dynamical system w is not onto.

266

M. HATA

S 3.

Substitution Operator S, The Weierstrass function (1.1) for b=2 can also be represented in the form

c an cos

7l2O

(27TX) =

c a" cos (qP(x)).

7L20

Thus the Weierstrass function and the series (1.3) corresponding to b=2 are particular cases of the following series: (3.1) where F(0, x)=g(x) is a smooth function on Z. It is easily seen that the series (3.1) is a unique continuous solution of the functional equation

To deal with the series (3.1), it will be convenient to introduce a substitution operator. Let E be a complex Banach space of all complex-valued continuous functions on Z with uniform norm. For a given continuous dynamical system o:ZkZ, we will define the substitution operator S, by

(3.3)

S,(f)(x)=f(w(x))

for x € Z .

As is easily shown, S, is a bounded linear operator of E and its spectrum o(S,) is contained in the unit disk. It is known that the substitution operator (3.3) is one of the Bourlet operators satisfying a multiplication formula (Targonski [411). Moreover it is a linear ring endomorphism of our Banach algebra. Note that the eigenvalue ) problem for the substitution operator leads to the Schroder equation f ( w ( x ) = Af(x). Using the operator S,, the series (3.1) can be written as

where the operator (Id-uS,)-' is known as the resolvent operator of S,. Therefore (Id-aS&' maps g,(x)=cos zx to the Weierstrass function and gl(x)=x to the series (1.3) for b=2; that is, it maps some snooth functions to nowhere differentiable functions. If the operator S, is completely continuous, then a family on functions { w , 02, ...} must be a compact subset of E . In this respect, we have the following :

Theorem 3.1 ([44]). Suppose that there exists a sequence { p - , } , , , such that w(p,,)=p,fp-, and o(p-,,)=p-,+,for n > l . Then we have u(S,)={z; lzl
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267

to possess the unit disk as a spectrum of S,. On the other hand, Bonsall [3] gave an interesting example of S, which is completely continuous in a cone C and not in any subspace of E containing C. Let C be a complete positive cone in E consisting of all increasing and convex functions f with f ( O ) = O and let LY be a n element of C satisfying a( 1)< 1 and d ( 0 )> 0. Then the cone map S, has the desired properties. Note that its partial spectral radius is given by C such that Sau=nl(0)u. d ( 0 )and there exists a n eigenvector More generally, we will consider the operator

(3.4)

T,(g)(x)=

c amSWn(g)= c a,g(w"(x))

9

b>O

Tl>O

where C a,, is a n absolutely convergent series. Plainly T , is a bounded linear operator with llToll < Cnzo lanl. By the well known representation theorem, there exists a function T ( X , y ) , defined on ZxZ, satisfying T,(g)(y)=

1:

g(x)dT(x,Y )

where ~ ( xy ,) is of bounded variation with respect to x for each y and is continuous with respect to y as x = l . Actually, we can obtain the concrete expression for ~ ( xy ), as follows:

On the operator (3.4), we have the following:

Theorem 3.2. Suppose that a power series C n t O a,,zn has a radius of convergence > 1 and has no roots in the unit disk. Then the operator T , is a homeomorphism of E. Proof. It is clear that the series C,,,b,z"=(C,,,~,z")-~has a radius of convergence > 1 and therefore C b, is absolutely convergent. Then, for any f E E , define

Hence,

This implies T,(E) =E.

Next we assume Cnro a,S;(g)=O.

Then

This implies that T, is one to one. Thus, T , is a homeomorphism of E.

0

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268

c,"==o

Corollary 3.3. Suppose that a polynomial c,zn has no roots in the unit c,S; is a homeomorphism of E. disk. Then the operator C,"=o Note that the conclusion of the above corollary is equivalent to the fact that the linear functional equation

+ - +C,S(W(X))+Codx)= f ( x )

CNS(W"(X))

* *

has a unique solution g € E for any f € E. It is also interesting to consider the higher dimensional substitution operator in the form

swl,...,"n(f)(xl,*

e . 7

Xn)=f(W1(x1)+

---

+W,(X,))

,

which maps E into the space E , of all continuous functions defined on the n-dimensional unit cube. In this respect, there is a remarkable result:

Theorem 3.4 (Kolmogorov [22]). There exists a family of continuous monotone increasing functions w P g ,defined on I , l < p < n , l < q < 2 n + l , such that the substitution opeartor S* on E Z n + l defined by

S*(fl,

-

2nt 1 *,f2,+1)=

c s,l,.....on, (f,,

q= 1

is onto; that is, S*(EZnf1) =En. This is known as the representation theorem of continuous functions of n variables by superposition of continuous functions of one variable and addition.

S 4.

Difference equations

In this section, we will discuss various properties of the function in the form

Obviously the Takagi function (1.2) and the series (2.2) are particular cases of (4.1). First of all, Hata and Yamaguti [14] proved the following theorem using particular orbits of the dynamical system $.

Theorem 4.1. Suppose that the series (4.1)converges everywhere. Then the series C c, is absolutely convergent. Moreover, they showed that the operator L defined by

Fractals in Mathematics

269

is a linear homeomorphism from the space of absolutely convergent series onto its image. They also generalized this result to the series (3.4) for o=$. Faber [7]showed that the series (4.1) has no finite derivative at any point if lim sup,,, 2"lc,l >O. This result was accomplished by K6no as follows: Theorem 4.2 (KGno [23]). The series (4.1)has nofinite derivative at any point Moreover, if lim sup,,, 2nc,=0, it is difer-

if and only i f lim sup,,, 2"lc,l > O . entiable on a set of continuum.

He also studied further properties on the series (4.1). In particular, he showed that the family { p ( x ) - 1/2}n20is a concrete example of a multiplicative system but not strongly multiplicative. In [14],we showed that a continuously twice-differentiable function in the form (4.1) must be a quadratic function. This result was also strengthened by KBno so that it holds true even in the class of smooth functions in the sense of Zygmund. Although there are n o simple functional equations the series (4.1) must fulfill in general, we can obtain a family of difference equations whose unique continuous solution is the series (4.1). It is convenient to denote the set of lattice points {(n, rn); O l } by Q. Then, the desired equations are (4.2)

f( 2n+F 1 )f ( $$ y ) +f(% { ) }n- 'f" l

-

for all (n, rn) E Q

with boundary conditions f(O)=O and f ( l ) = c o . Note that the left hand side of the above equation is essentially the so-called central difference scheme for f. Indeed, if we take c,=4-", m > l , then the equations (4.2) will shift to the differential equation f " = - 2 , so that f ( x ) = c , x + x ( l - x ) . Modifying the equations (4.2), consider

with boundary conditions f(O)=O and f ( l ) = l where O < a < l is a constant. In [14], we showed that a unique continuous solution of (4.3) satisfies the following functional equation:

(4.4)

f ( x )=

i

af(2x)

for O ~ X C 1- , 2

(l-aa)f(2x-l)+a

for -1< x < l . 2

This is a particular case of de Rham's functional equations: actually he proved the following

M. HATA

270

Theorem 4.3 (de Rham [36]). Suppose that F, and F , are contractions in

R". Then the functional equation

(4.5)

Fo(f (2x))

for

O < x < - ,1 2

Fl(f(2x-1))

for

-1< x < l 2

f(4=

possesses a unique continuous solution i f and only i f F,(p,)=F,(p,), where p o and p1 are unique fixed points of F , and F , respectively.

Moreover, de Rham showed that the solution L ( a , x) of (4.4) is strictly monotone increasing and its derivative vanishes almost everywhere if a# 1/2. Such functions are known as Lebesgue's singular functions. The solution L ( a , x) was also studied by Lomnicki and Ulam [28] and Salem [38]. It is known that L ( a , x) is the distribution function for the Bernoulli trials of unfair coin tossings. In [14], we obtained the following expression

where m ( p ) = p - z C , , , [p/2"] and

which is known as the Schauder base of E . From this formula, we can obtain a fine relation between the Takagi function (1.2) and the solution of (4.4) --L

(4.7)

a:

(:'- x )=2T(x)

The expression (4.6) is also valid for complex parameter a € { z ; Izi
(b)

(a)

Figure 2. (a) LBvy curve. (b) von Koch curve.

Fractals in Mathematics

27 1

is the curve studied by LBvy [26] (Figure 2(a)). Note that the n-th partial sum of (4.8) gives a n approximation broken-line curve. It is also interesting to consider the following equation instead of (4.4):

(4.9)

f(x)=

I

1 2

afrn

for O I X I - ,

(l-a)f(2x-l)+a

for -1< x < l . 2

De Rham pointed out that the solution of (4.9) becomes the von Koch curve for a = 1 / 2 + ( 1 / 7 / 6 ) i (Figure 2(b)) and Pdlya's space-filling curve for a= 1/2+ 272. The corresponding difference equations to (4.9) are particular cases of the following equations:

for all (n, m ) € Q with conditions R(O)=O, R ( I ) = l and R(1/2)=a, where O < A m J p m < l , m 2 1 are constants. Indeed, if we take R,=IaI2 and p m = l ll-alz, then the continuous solution of (4.10) also satisfies (4.9). It is easily seen that the equations (4.10) possess a unique continuous solution if

O < inf;i,< s u p p ( , < I . n>l

n>l

The curve R(Z) is clearly contained in the triangle with vertices 0, 1 and a. R(Z) becomes a Jordan curve if R, l and becomes a Peano curve if R,=pn, n 2 l . Also it is easily verified that 2-dimensional Lebesgue measure of the curve R(Z) is given by

Thus, for a suitable choice of {;in} and {p,,},we can get a Jordan curve of positive area as a unique continuous solution of (4.10).

S 5.

Self-similar Sets

In this section, we will discuss self-similar sets in Euclidean space R P . The self-similarity is an important notion in Mandelbrot's book. The LBvy and von Koch curves illustrated in Figure 2 are typical examples of such self-

M. HATA

272

similar sets. It is known that the Levy curve has a positive 2-dimensional Lebesgue measure and that the Hausdorff dimension of the von Koch curve is given by (log 4)/(log 3 ) ; therefore both curves are fractal. T o deal with self-similar fractal sets in RP, there are at least two methods as far as the author knows. One is accredited to Dekking [5] who used endomorphisms of words in free groups and the other is a method of Hutchinson [17] using a set of contractions; the latter is used in this section. A mapping F : R p - + R pis said to be a contraction provided that there exists a constant I € (0,1) for which 1IF(x)-F(y)ll
Definition 5.1 (Hutchinson). A non-void subset X of RP is said to be invariant with respect to a set of m contractions F,, F 2 , -,F , provided that X satisfies the equality

--

(5.1)

X=F,(X) UF J X )U

- *

U F,(X)

.

This method describing the self-similarity was refound by the author Although Hutchinson’s motivation probably has its origin in geometric measure theory, the author studied invariant sets from a general topological point of view. Some results of Hutchinson were strengthened by Mattila [ 3 2 ] . For a set of contractions F , , -,F,, we can define the mapping [ 121 recently.

--

(5.2)

@ ( X ) = F i ( X )U F z ( X )U

* * *

UF , ( X )

for a n arbitrary subset X of RP. Obviously the invariant set (5.1) becomes a fixed point of @. First of all, we have

Theorem 5.2 (Williams [43], Hutchinson [17]). For a set of contractions F,, , F,, there exists a unique non-void compact invariant set K. Further, f o r an arbitrary non-void compact subset X of Rp, @ ( X ) converges to K in the Hausd o r - metric as n+m.

-.-

The existence and uniqueness of invariant sets were essentially proven by Williams in 1971 toward a study of generic properties of the action of free (non-abelian) groups on manifolds. The author extended this result for weak contractions. For example, the Cantor set is a unique compact set invariant under two contractions of R , F , ( x ) = x / 3 and F 2 ( x ) = ( x + 2 ) / 3 . The LBvy curve is a unique compact set invariant under two affine contractions of R 2 , F l ( z ) = a z and F2(z)= (1-a)z+afor a=1/2+i/2. Also the von Koch curve is invariant under F l ( z ) = aZ and F2(z)=(l-cr)5+a for a=1/2+(2/3/6)i. We will illustrate in Figure 3 some other examples of invariant sets for two affine contractions in R 2 . In

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273

(d)

(C)

Figure 3. (a) (0.4614+0.4614i, 0, 0.622-0.1961', 0).

(d) (0.4614+0.4614i, 0, 0, 0.2896-0.5851').

all cases, we define F,(z)=a z f j 3

+

F,(z) =y ( z - 1) 6(5 - 1) + 1

and

The corresponding parameters ( a , p, y, 6) are given in the captions of Figure 3 respectively. Modifying the equation (5.1), the author [13] studied the following inhomogeneous equation

(5.3)

X=@(X)UV=F,(X)U

*-.

UF,(X)U V ,

where V is a given compact subset of R P . He proved that there exists a unique non-void compact solution X satisfying ( 5 . 3 ) . Moreover, he showed the following analogy to Alternative of Fredholm:

.-

Theorem 5.3. Suppose that F,, -,F, are continuous mappings such that @ " ( X )is pre-compact for any compact X . Then the following statethe set Un20 ments (a) and (b) are equivalent; (a) there exists a unique solution of ( 5 . 3 ) for every compact V; (b) @ has a unique fixed point.

M. HATA

214

On the Hausdorff dimension of invariant sets we have Theorem 5.4 (Marion [31], Hutchinson [17]). Suppose that each contraction Fj, l < j < m , is a composition of a dilation, a rotation, a translation and a rejection. Suppose further that there exists a bounded open set U satisfying @ ( U ) c U and F,(U) n F,(U)= 0 for i+j (The open set condition). Then the s-dimensional Hausdorff measure of the invariant set K is finite and positive; that is, dim, (K)=s, where s is defined by Lip(F$+ +Lip(FJS=l.

---

We now turn to the connectedness of invariant sets. First we have

...

Theorem 5.5 (Williams [43]). Suppose that Lip(F,)+ +Lip(F,,) that each Fj is injective. Then K is totally disconnected and perfect.

< 1 and

To study the connectedness of invariant sets, the author [ 121 introduced the structure matrix M,=(mcj) of K as follows: if F r ( K ) n F J ( K ) = 8 otherwise. Then we have Theorem 5.6. The invariant set K is connected if and only if its structure matrix MK is irreducible. Moreover, if K is connected, it is also a locally connected continuum and arcwise connected. If two contractions F, and F, satisfy F,(Fix(F,))= F,(Fix(F,)), then we can get a parameterization of the invariant set K applying de Rham's Theorem 4.3. In fact, let f ( x ) be a continuous solution of (4.5). Then,

Therefore f ( Z ) is a compact invariant set under F, and F2,so that K= f ( Z ) as required. In this respect we have Theorem 5.7. Let f (x) be a continuous solution of (4.5). Then (a) ifLip(F,)-Lip(F,)< 114, then the Fre'chet derivative off vanishes almost everywhere; (b) if each F, is a homeomorphism and Lip(F,-').Lip(F,-')<4, then f is not Frtchet diferentiable almost everywhere; moreover, if Lip(F,-') < 2 f o r j = 1, 2, then f is nowhere diferentiable. Note that the above result gives a generalization of Lax's [25] theorem. With these kinds of parameterizations, we can easily get the well known classical Peano curves given by Peano [33], Hilbert [151, and P6lya [34] using certain affine contractions of R2.

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References [ 11 E. Artin, The Gamma Function, Holt, Reinehart and Winston, 1964. [ 2 1 A. S. Besicovitch and H. D. Ursell, Sets of fractional dimension, V: On dimensional numbers of some continuous curves, J. London Math. SOC., 12 (1937), 18-25. [ 3 1 F. F. Bonsall, Lectures on Some Fixed Point Theorems of Functional Analysis, Tata Inst., Bombay, 1962. 14 1 F. S. Cater, A typical nowhere differentiable function, Canad. Math. Bull., 26 (1983), 149-151. [ 5 1 F. M. Dekking, Recurrent sets, Adv. in Math., 44 (1982), 78-104. [ 6 1 G. Faber, Einfaches Beispiel einer stetigen nirgends differentiierbaren Funktion, Jahresber. Deutsch. Math. Verein., 16 (1907), 538-540. [ 7 1 -, u b e r stetige Funktionen. Math. Ann., 69 (1910), 372-443. [ 8 1 K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985. I 9 I J. Gerver, The differentiability of the Riemann function at certain rational multiples of x , Amer. J. Math., 92 (1970), 33-55. [lo] G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. SOC., 17 (1916), 301-325. 1111 M. Hata, On the functional equation (l/p)[f(x/p)+ . . . +f((x+p-l)/p)}=2f(px), J. Math. Kyoto Univ., 25 (1985), 357-364. On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381-414. [12] -, On some properties of set-dynamical systems, Proc. Japan Acad., Ser. A, [13] -, 61 (1985), 99-102. [14] M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan J. Appl. Math., 1 (1984), 183-199. 1151 D. Hilbert, Uber die stetige Abbildung einer Linie auf ein Flachenstuck, Math. Ann., 38 (1981), 459-460. I161 E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Cambridge, 1926. [17] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. [18] V. Jarnik, Uber die Differenzierbarkeit stetigen Funktionen, Fund. Math., 21 (1933), 48-58. [19] G. Julia, Fonctions continues sans dtrivtes formtes avec les ittrtes d’une fraction rationnelle, Ann. Sci. Ecole Norm. Sup., 48 (1931), 1-14. [20] J. L. Kaplan, J. Mallet-Paret and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergotdic Theory Dynamical Systems, 4 (1984), 261-281. [21] S. A. Kline, On curves of fractional dimensions, J. London Math. SOC., 20 (1945), 79-86. [22] A. N. Kolmogorov, On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR, 114 (1957), 953-956. [23] N. K h o , On generalized Takagi functions, to appear in Acta Math. Acad. Sci. Hungar. [24] M. Kuczma, Functional Equations in a Single Variable, PWN-Polish Sci. Publ.,

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Warszawa, 1968. [25] P. D. Lax, The differentiability of Pblya’s function, Adv. in Math., 10 (1973), 456464. 1261 P. Ltvy, Les courbes planes ou gauches et les surfaces composkes de parties semblables au tout, J. Ecole Poly., Serie 111, 7-8 (1938), 227-292. [27] T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. [28] Z. Lomnicki et S. Ulam, Sur la thkorie de la mesure dans les espaces combinatoires et son application au calcul des probabilitts I: Variables indkpendantes, Fund. Math., 23 (1934), 237-278. [29] E. R. Love et L. C. Young, Sur une classe de fonctionnelles IinCaires, Fund. Math., 28 (1937), 243-257. [30] B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [31] J. Marion, Le calcul de la mesure Hausdorff des sous-ensembles parfaits isotypiques de R”, C. R. Acad. Sci. Paris, 289 (1979), Strie A, 65-68. [32] P. Mattila, On the structure of self-similar fractals, Ann. Acad. Sci. Fenn. Ser. A, 7 (1982), 189-195. [33] G. Peano, Sur une courbe qui remplit toute une aire plane, Math. Ann., 36 (1890), 157-160. [34] G. Pblya, u b e r eine Peanosche Kurve, Bull. Acad. Sci. Cracovie, A (1913), 305313. [35] G. de Rham, Sur un exemple de fonction continue sans dkrivte, Enseign. Math., 3 (1957), 71-72. [36] -, Sur quelques courbes definites par des Cquations fonctionnelles, Rend. Sem. Mat. Torino, 16 (1957), 101-113. [37] C. A. Rogers, Hausdorff Measures, Cambridge, 1970. 1381 R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. SOC.,53 (1943), 427-439. [39] A. Smith, The differentiability of Reimann’s function, Proc. Amer. Math. SOC., 34 (1972), 463-468. [40] T. Takagi, A simple example of the continuous function without derivative, Proc. Phys. Math. SOC.Japan, 1 (1903), 176-177; The Collected Papers of Teiji Takagi, Iwanami Shoten Publ., Tokyo, 1973, 5-6. [41] G. I. Targonski, Seminar on Functional Operators and Equations, Lecture Notes in Math., 33, Springer-Verlag, 1967. [42] B. L. va3 der Waerden, Ein einfaches Beispieleiner nicht-differenzierbaren stetigen Funktion, Math. Z., 32 (1930), 474-475. [43] R. F. Williams, Composition of contractions, Bol. SOC.Brasil. Mat., 2 (1971), 5559. [44] M. Yamaguti and M. Hata, Weierstrass’s function and chaos, Hokkaido Math. J., 12 (1983), 333-342. Department of Mathematics Faculty of Science Kyoto University Kyoto 606, Japan

Fractals in Mathematics

211

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278

LCvy curve. This is a unique continuous solution of (4.4) for a= (1+i)/2 (S 4). Von Koch curve. This is a unique continuous solution of (4.9) for a = 1 / 2 + J T i / 6 ( S 4). Unique invariant set for FL(z)=(0.4614+0.4614i)z and Fz(z)=(0.6220.196i)(z-l)+1 ( S 5 ) . Unique invariant set for Fl(z)=(0.3+0.3i)Z and Fz(z)=0.82(Z-1)+l

( S 5). Unique invariant set for F I ( Z ) = ( O . ~ + O . ~ ~a n) Zd Fz(z)=(-0.5+0.5i) ( 2 - l ) + l (S 5 ) . Unique invariant set for Fl(z)=(0.4614+0.4614i)z and Fz(z)=(0.2896 ( S 5). -0.585i)(Z-l)+1 This is a unique solution of (5.3) for Fl(z)=(0.5+0.6i)z, F2(z)=(0.50.6i)(z-l)+l and V is the closed triangle with vertices Po, F1(po)and Fz(p0) where pO=(l-i)/2. This is a unique solution of (5.3) for Fl(z)=(0.5+0.2i)z, Fz(Z)=(O.50.2i)(z- 1) 1, Fs(z)=(0.6+O.li)(z-0.5-22i)+0.5+2i and V is the union of three segments connecting p o with Fi(po)for i = 1,2,3, where p0=0.5-0.2i.

+

Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equationspp. 279-318 (19%)

Parallel Computation By Tatsuo Nocr Abstract. This paper aims to change the world’s tendency inclined to vector pipeline processors towards parallel processors of new types. It first reviews standard schemes, algorithms and data structures for simulations based upon PDE models from the point of view of natural parallel processing, and gives some comments about adaptability of vector pipeline computers or parallel computers of the ILLIAC-IV type. Consideration on multi-dimensional simulations leads us naturally to one-dimensionalization methods and ADE (Alternating Direction Edition) concepts which are essential for effective parallel processing. Those processors mentioned above, however, have some defects for standard ADE’s due to memory conflicts or poor capability of data exchange among slave processors. The ADENA machine is introduced as only an alternative which allows universal ADE’s and promises to open highly parallel processing. Some examples of application illustrate its usage and its effectiveness. Key words: parallel computation, simulation, vector pipeline, ADENA machine, AD1 method

Newtonisch Weiss den Kindern vorzuzeigen, Die padagogischem Ernst sogleich sich neigen, Trat einst ein Lehrer auf, mit Schwungrads Possen, Auf selbem war ein Farbenkreis geschlossen. Das dorlte nun. ((Betracht es nur genau! Was siecht du, Knabe?)) Nun, was seh ich? Grau! ((Du siehst nicht recht! Glaubst du, dass ich das leide? Weiss, dummer Junge, Weiss! so sagt’s Mollweide!)) (DEM WEISSMACHER, J. W. Goethe)

1. Introduction Some surveys W e already hav e s o me s u r v ey papers about parallel numerical computation. W e will here take up representative ones, which serve to find more references. 1.1.

Received September 17, 1985.

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Miranker 119711 gave his survey of a broad aspect of numerical analysis: optimization, root finding, differential equations and solutions of linear systems. But many parallel algorithms which he was able to survey are purely conceptual in their origin. In a subsection, he said that “when one deals with finite difference approximations to partial differential equations, the opportunities for parallel computation are ubiquitous. Thus the work in parallel modes of solving partial differential equations has taken the nature of organizing the computation for parallel execution. There has been little motivation for devising new algorithms and indeed none have appeared.” He continued, “Analysis of parallel computational phenomena will probably be motivated when some body of experiments with parallel processing of partial differential equations has been built up. In addition to a large amount of work of this sort being performed at the ILLIAC IV project at the University of Illinois.” In fact, such features in dealing with partial differential equations are generally seen even today, and parallel computation is considered by combining simple explicit schemes with machines of ILLIAC IV type. DAP of ICL (Flanders et al. [1977]) and PAX of Tsukuba University (Hoshino et al. [1983]) also stand on the same line. Unless such adhesion is abandoned, we can get neither excellent algorithms nor parallel machines. Heller [1978] gave a comprehensive survey of parallel techniques for problems in linear algebra. His specific topics include: relevant computer models and their consequences for programs, evaluation of arithmetic expressions, solution of general and special linear systems of equations, and computation of eigenvalues. Here, both parallel and pipeline computers are considered, and many results of accounting arithmetic operations are quoted, covering almost all well-known algorithms of linear algebra. Touching upon systems arising from differential equations, he says “The area of differential equations has exerted a great influence on parallel computation, for it provides a range of difficult and important problems. Because of these special applications, parallel computers have been designed to support in hardware some of the operations naturally occurring in the solution of diKerential equations. As examples, the interprocessor connections on the ILLIAC IV are precisely those of the five point finite difference molecule used for two dimensional elliptic equations, and the CDC STAR provides vector instructions for differencing and averaging.” Since his consideration is mainly restricted to the Single Instruction Stream-Multiple Data Stream model (SIMD), it is only said for example about the nested dissection method (George [ 19731) that the vectorization is complicated enough so that the standard band methods are expected to be more efficient both in terms of runtime and programming costs. As mentioned in 6.1 of our survey, it is a promising method in the Multiple Instruction StreamMultiple Data Stream (MIMD) environment. This example also calls upon us to abandon adhnesion to existing models of parallel or pipeline machines.

Parallel Computation

28 1

Sameh [1977] presented a summary of direct linear system solvers suitable for parallel computers, not for vector computers. He surveyed some results concerned with the speedup and redundancy parameter of parallel algorithms. Zakharov [ 19841 has given a most comprehensive survey, especially from his point of view of machine architecture together with historical perspective. He classified parallel computers into four kinds of architecture: pipelining, using many functional units within a uniprocessor, having many cooperating processors and special-purpose computing systems. He looked at some limitations to parallelism from the point of view of machines, language and software. Schendel [1984] presents a concise review of the current status of the design of parallel algorithms, and includes both the numerical procedures of well-tried classical methods and new parallel algorithms specifically selected for the problems under consideration. Finally, We will add some references, which are not of pure survey, but of fairly exhaustive study around some selected machines: Hold [1982] about ILLIAC-IV, Wallach [1982] mainly about SMS, Hockney and Jesshope [ 19811, Paddon [1984] about DAP and Kowalik [1985] about HEP. 1.2. Our survey In this paper, we will give a tutorial introduction of parallel computations for simulation, instead of a n exhaustive survey. The reason, while owing to the inability of the author on one hand, lies on our observation that parallel computations have been considered only on very simple schemes and algorithms avoiding sophisticated structures, and on our judgement that it is necessary for finding good schemes, algorithms, languages and machines to study again some fundamental schemes and algorithms, or to compare the effectiveness of these on several machines. Section 2 first considers characteristics of simulation data, and explicit schemes allowing natural parallel processing. Further, two kinds of machines for highly parallel processing are introduced for preciseness of discussion: processor array and vector pipeline. Section 3 stresses implicit schemes and follows some realization algorithms. Section 4 deals with a typical problem, Poisson problem, examines closely some classical algorithms from the point of view of parallel processing, arrives at a keystone, AD1 method, from which ADE concept is led and introduces data expression useful for parallel processing. Finally, the problem of memory conflict is introduced as a fundamental bottleneck for parallel processing. Section 5 is for our machine, ADENA (Nogi [1982, 1982]), which carries out ADE successfully with no memory conflict, and is to show its applicability for wide problems. Section 6 presents some sophisticated algorithms, especially aiming to solve equations from finite element methods.

282

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Parallel Simulation

Simulation data The starting point of simulation of physical phenomena is on a mathematical model which represents some physical laws by mathematical language. It is usually a system of partial differential equations, which is replaced by a computational model with proper discretization. Such a model has some characteristics : 1) it deals with natural array data; according to the space dimension, 1, 2 and 3, they are, for example, as follows; 2.1.

var u: array [ 1. . N ] of real ; var u : array [ l . . M , 1 . . N ] of real; var w : array [ 1.. L , 1. .M, 1. . N ] of real ;

2) computational models generally give each component of the array some relation to neighbouring components, and such relations are of homogeneous type all over components of the array. These characteristics lead us to natural parallel processing. Let us first comment how such array data are stored on usual uni-processor computers or multi-processor systems sharing a common memory. Even for multidimensional cases, they are stored in the one-dimensional form after all. For example, a two-dimensional array u[i,j] is replaced by V[i*M+j] with some stride number M . This fact itself brings some trouble for parallel computers with share memory to access, as mentioned later. 2.2.

Explicit and implicit scheme Let us first talk about the concept of parallel processing for typical onedimensional cases. Usually, a n unknown array variable u is found by such a command as (1)

for

i : = l to L do u[i]:=

;

For exapmle, when using an explicit scheme for a heat equation on a n interval region, we shall introduce variables var u: array [O..N+l] of real ; var v : array [ l . .N] of real ; and have

(2)

for i : = l to N do u[i]:=f(u[i-11, u[il, u [ i + l l ) ;

for i : = l to N do u [i] := u [ i];

for a time step cycle, where f is a function defined beforehand, and both u[O] and u [ N + 11 are given by boundary conditions.

Parallel Computation

283

With a n implicit scheme, the unknown variable is again found by a procedure like ( l ) ,which takes the form

(3)

for

i : = l to N d o u[i]:=g(u[i-11) ;

where g also is a function defined beforehand. Now, we must point out a difference between (2) and (3). The former allows parallel processing all over the index i, while the latter does so to determine successively with increasing i.

2.3. Natural parallel processing Natural parallel processing here means parallel computations of the same formula all over index i. Its realization necessarily leads to a scheme of a processor array. Its main part is, as seen in Fig. 1 , composed of many arithmetic units (slave processors) and attached memory blocks with the network allowing each processor to access not only its proper memory block but also its neighbouring blocks. For (2), all slaves read data u[i-11 from respective neighbours, then data u[i+ 11 from opposite neighbours, and finally take u[i]and compute to find v [ i ] . Let us write such realization anew in the following form: var u : array [(O. . N + l ) ] of real ; forall i : = l to N pdo u [ ( i ) ] : = f ( u [ ( i - l ) ]u,[ ( i ) ]u, [ ( i + l ) ] ); where u [ ( i ) ]is, we assume, determined by the (i)th processor and is stored in the (i)th memory block. Each parenthesized number (i) is a corresponding number modulo P, P being the real number of processors. The value u[(O)] and u [ ( N + l ) ] are here boundary values, and they need not be assigned to special processors. The command “forall” and “pdo” mean “for all” and “parallel do” respectively, and they denote parallel processing for all i. When

H S B

... Host processor . . . Slave processor . . . Memory bank Fig. 1.

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284

P < N , a whole processing is, indeed, divided into some steps, in each of which parallel computation is done. 2.4.

SIMD and MIMD To realize the above pure parallel processing, we can consider two kinds of machine architecture. One is the SIMD: the host processor broadcasts the same sequence of machine codes to all slave processors, which run altogether with the same machine clock. This is a scheme accepting just the characteristics of simulation mentioned in 2.1. Well-known ILLIAC IV and DAP are included in such category. The other is the MIMD: all slave processors have respective suquence of commands in their private memories, and run simultaneously, only with synchronization when transfering data among those processors. In this case, processors may have different programs, and hence the MIMD is generally more flexible than the SIMD. HEP and PAX are classified in the MIMD. ADENA developed by the author is also in this class. 2.5.

Vector pipeline We shall next consider, say, quasi-natural parallel processing. It is given as its concurrency being relaxed and added by some serial processing. While the aspect of processing same type operations is retained, those operations are restricted to some fundamental operations as adding and multiplying, etc. This is just the scheme of vector pipeline, which forms the main route of development of modern super-computers. It usually consists of a single processor and some memory blocks accessed by the so-called interleave way as seen in Cray-1 (see, for example, Hockney and Jesshope [1981]). Let us give a short comment about the way to get high speed by the concerned scheme. Return to realization of (2), but suppose here the function f being given simply by the formula

f ( ~ [ i11, - ~ [ i ]~, [ i + l ] ) = ~ * ~ [l]+b*~[i]+c*u[i+l] iwhere a , b and c are constant.

,

The procedure is to read three vectors

( u ~ ' - ~ l ) = ( ~ [ o 1 , ~ 4 [ 1 1 , u [ 2* 1- . ,, 4 N - 1 1 ) (u[*l)=(u[lI, @I, u[31, -,u"I) , (4' 11)=(u[21, 431, u[41, * -,u"+ 11)

--

+

7

-

from memory blocks and place them on vector registers, and perform the vector operation

+

(v[*])=a*(u[. - 11) b*(u[. 1) -I- c*(u[ by the pipeline scheme. There are three ways to get high speed:

+ 11)

Parallel Computation

285

( i ) To read vector data from memory blocks (as well as to write into), high speed access is realized by the interleave scheme, which is as follows: with some number of memory banks, in practical, 16 to 32, components of vector data are allotted to corresponding banks successively, and they are transfered to registers, group by group, as one group with such number of components being done during one memory cycle. ( i i ) Vector pipeline processing; it is to repeat a kind of operation (add, multiply and so on), where each operation is divided into a sequence of subprocesses (as for a n exponent, mantissa and normalization, etc.), and some subprocesses for different data are duplicated on each process pitch (pipeline pitch). After the first process for the first component, two processes, the second for the first component and the first for the second component are duplicated and so on. This scheme allows for a long vector to be operated as producing one result component by a pipeline pitch, except during rising steps. Such pipeline pitch is usually so small as to be a machine clock cycle (more or less 10 ns). (iii) Some kinds of pipelines with different functions, or some number of same pipelines; those pipelines are operated concurrently, and are connceted as a result sequence of a pipeline is immediately a n input for another pipeline. The last scheme is called ‘chaining’. We dare to express pipeline processing as follows;

var u : array [O. . N + l ] of real ; for i : = l to N pipedo (u[i]):=u*(u[i--l])+b*(u[i])+c*(u[i+1]) ; where pipedo means pipeline processing. The parentheses ( ) including a n array variable indicate the vector type. We must here notice that the above procedure is mainly serial even though subprocesses are done concurrently with small pitch.

3. Approach to Implicit Schemes Introduction of iteration With a n implicit scheme, it is necessary to solve a corresponding linear system of equations. So, we face the difficulty of parallel processing. One way to overcome such difficulty is to introduce a n iterative process, that is, a n explicit process. According to the degree of explicitness introduced, we have several kinds of algorithms. We shall take a n example, which is near the original Gauss algorithm. This and the next section are due to Traub [1973]. Suppose to solve the equation

3.1.

Au= f

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286

of the matrix form, where

,

u=

Put A into the sum of the lower triangular A,, the unit diagonal Z and the upper triangular A , :

(2)

A=A,+Z+Au

9

0

A L = i

g:

0

0

0

0

Sl

O

.. . . - .

0

tN

2 ..

sg

.O j . .

0

To solve (1) by Gauss elimination is only to use the LU decomposition of A : (3)

A=LU.

Due to the speciality of A , the lower triangular L and the upper triangular U can be found in the following form:

(4)

L=A,+D,

U=I+J,

where

dl D=[

dZ

d,

. 0 9

0

J=

0

d*

From (2)-(4), we get the equation ( 5 )

(I-ALJ)J=Au

to find J and the formulae (6)

L=Z+AL(Z-J),

U=Z+J.

.. .. 0

287

Parallel Computation

The problem (1) is reduced to a pair of triangular systems

(7)

uv=w.

Lw=f,

The former and the latter correspond to the forward and the backward substitution respectively. The procedure of LU decomposition can be expressed by components of the matrices and the vectors as follows:

Two procedures of substitution are

(9)

wI=fi

(10)

u N = f N ,

wi=

9

f,-

tiwi-1 l-tjei-l

u,. = w . - e . u

- *, N ) ,

( i = 2 , 3,

.-.,1)

(i=N-1, N-2,

I

There are typical serial processes. To avoid such pure serial processing, Traub has introduced the following iteration procedure:

(11)

(Z--A,J‘k-”)J‘X’=A,

(12)

( k = l , 2,

(Z--AA,J‘~’)W‘p’=f-AA,w(P-~’

V(’) =w ( 4 )

(13)

-,1) ,

( p = l , 2,

( s = l , 2,

-J(C)~‘S-l’

* *

* * *,

* *,

t)

q)

,

,

where I , q and t are iteration numbers of corresponding subprocesses. It is easily seen from the above equations that those processes can be done in parallel. For example, we have from ( 1 1 )

Its one time iteration is naturally realized by a n array of processors as follows: ; forall i : = 2 to N - 1 pdo e[(i)]:=s[(i)]/(l.O-t[(i)]*e[(i-l)])

It is also possible on a vector processor: for

i : = 2 to N - 1 pipedo

(e[i]):=(s[i])/(l.O-(t[i])*(e[i-11)) ;

But it is not always true that the above method with iteration is better than proper iteration methods such as the Jacobi method, and such method produces only several times speed-up. (See Traub [1973].)

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288

Cyclic reduction method Another parallel algorithm for problems like (1) of Section 2 is given as one repeating local eliminations on the formulae themselves. It is originally used for serial processing with success. We frequently call it a fast Poisson solver. According to Bunemann [1968] and Hockney [1965], we shall mention some characteristics of their odd-even reduction method. It is based on the assumption of constant coefficients: t's=s's=constant. For simplicity, we put

3.2.

I.

We now have for an even number i u,-2--au,-,+

ut

=ft-l

u,-,--aut+ 4 t 4-Wt

=ft

1 14-

ut tz = f t

9

9

t1

From these equations, we have the following equation by elimination: (16)

4 - 2

f (2-Qz)ut f u, t 2 =fi'"

f;"

=ft-i

+aft

+ f t tI *

The obtained set of equations (16) contains only unknowns with even numbers, and the number of such unknowns is N/2 in total. Such set itself constitutes again a linear system with a tridiagonal coefficient. Repeat such elimination procedure. We then have at lth step (17) (18)

~ ~ - ~ ~ + a ( ~ ) u , + u , ~ ~ (i=2l, ~ = f , '22l, L ) 32l,

a(i)=2-(a(l-1))2

,

fit)

= f y H -a '"-"ft

- ., N + 1 - 2 9

L;+f1:

The final step is attained with I=L=log(N+l)-l.

(a( 0 ) =a)

.

Its equation gives a relaWith known bound-

tion among variables indexed i=O, (N+1)/2 and N + 1 . ary values, the center variable is found as (. N'+ l ) / 2 k N t l )/ 2 - (f"

- uo - u,tl)/a(L'

.

Using the obtained value, we can find the center value between the number 0 and (N+1)/2 and that between (N+1)/2 and N + 1 . By repeating such substitution, we can get all values successively: (19)

~t

=

(fit) -u ~ --~Uz~ ~ ~ ~ ) / Q ( ' ) (Z=L-1, L-2,

- - -,1; i=2l, 2Z+21t1,2l+22lt+', - - -,N+1-29

.

Parallel Computation

289

In conclusion, the total process of the odd-even reduction method is composed of finding ( u ( ~ and ) } {f:[)} by (18), and { u ~by } (19). Consider parallel processing for the above algorithm. At the first step of elimination, allf,’L)with even number i can be found in parallel by using ( N + 1)/2 processors, at the second stepf,’,) by ( N + 1)/4, as well as fit) be found by ( N + l)/ZG processors. Only one processor is sufficient for the final fiI?,,),,. Also for the substitution steps, one, two, four, and ( N + l)/2 processors can take part in respective parallel computaion. The above method needs only log, ( N + 1)- 1 steps, while the ordinary Gauss elimination does N steps. It is certainly a considerable improvement, but (N+1)/2 processors do not work to the full so that the imrpovement is up to O(log, N ) times speed up. Such a situation can always be seen in algorithms with a binary tree structure, and we call such parallel procedure treeheight reduction (Kuck [ 19761). With vector pipeline, lengths of considered vectors become shorter step by step, so that the efficiency again decreases.

---

Scheme and algorithm An important property to be mentioned about computation schemes is the relation between its implicitness and convergence or stability. It can be said generally that better convergence/stability is obtained by schemes with stronger implicitness. The stronger the implicitness, however, the lower the simplicity/parallelism and the more is the dependency upon serial computation. Hence, it might be taken that only explicit schemes allow parallel processing with a very high efficiency. This is not true, and efficient implicit schemes are found especially for multidimensional problems, which we desire to solve by super computers. Such schemes have both better convergence/ stability and high simplicity/parallelism, and well claim the name of economical schemes. They also are classified into semi-implicit schemes, and are usually based upon some decomposition of original problems/operator into a number of one-dimensional subproblems/operators. (See, for example, Yanenko 119711, Mitchell [1969].) The whole set of subproblems are dealt with in parallel, each being processed in serial We call such parallelization that by

3.3.

T. NOGI

290

dimension.

4. Two Dimensional Simulation 4.1. Poisson problem For simplicity, we consider only a five point difference problem to solve the homogeneous Dirichlet problem for the Poisson equation in a rectangular region. Suppose that a net of square meshes covers the region, and u [ i ,j ] denotes a n unknown function defined on nodes, i and j being coordinate numbers. The problem is to solve the equation 4u[i, j ] - u[i- 1, j ]- u[i+ 1 , j ]- u[i,j - 11 - u[i,j + 11 =f [ i , j ] ,

(1)

u[O,j ] = u [ M + l , j ] = u [ i ,O]=u[i, N+1]=0 (i=l,2, - . . , M ; j=1,2,

- a * ,

N)

.

This is a simultaneous linear equation to find M N unknowns u[i,j ] ( i = l , 2, -,M ; j = 1 , 2, -,N ) . Its matrix form is

.-

--

AU=F,

(2)

where u=('17

u29

* * ' 9

x -I

F2,

F=(F19

9

. ' * 9

FN)

7

-1

x -I

-I

x

-I

O -I

I

-'Ix

and further

(4)

and I is the M x M unit matrix. X is a n M X M tridiagonal matrix and A is an N X N block tridiagonal matrix.

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291

4.2.

Point Jacobi method It is the iterative point Jacobi method that allows the most natural parallel processing all over the components of unknown vector U . This method is, in fact, to introduce pure explicitness. The iteration is carried out on the equation u[i,jl=O.25(u[i+ 1 , jl +u[i+ 1 , jl+u[i, j - ll+u[i,j+ l l + f [ i , j l ) instead of ( l ) , a new value u [ i , j ] being computed from old values on the right hand side. In the case of parallel processing by a n array of processors, the sentence repeated is forall i : = l to M , j : = 1 to N pdo

+

+

u [ (i, j ) ]:=0.25*(u[(i- 1 , j ) ] u[(i+ 1 , j ) ] u [ ( i ,j - l ) ]

+ u [ ( i , j + l ) I + f [ ( i ,i ) l ) ;

where the array of processors are assumed to be two dimensional and the (i, j)-th processor plays a role to renew u [ ( i , j ) ] , and it is also supposed that each processor can access to corresponding private memories of neighbouring four processors, as seen on ILLIAC-IV. In the case of a vector processor, the two-dimensional array ( u [ i , j ] )is considered as a one-dimensional array as really stored in the data storage: (u[O,o],U[l,O],* * - , u [ M + 1 , O ]u[o, , 11, U[l,11, - - -u[M+1, , 11, - * . , ~ [ O , N + l ] , u [ l , N + l ]* ,. . , ~ [ M + l , N + 1 1 ) = ( U [ l ] ,U [ 2 ] U[31, ,

m e - ,

U[(M+2)(N+2)])=U.

( f [ i ,j ] ) is also supposed to have the corresponding one-dimensional array F . Such arrangement, if necessary, can be realized as follows:

for j : = 1 to N do begin for i:=l to M do J [ i ]:= i+ 1 +j*(M+ 2) ; for i : = l to M pipedo ( U [ J [ i ] l ) : = ( u [j i],)

end ; ( J [i]) is usually called a list vector. One-time iteration of the point Jacobi method for such a vector is written as for k : = M + 4 to (M+2)(N+1)-1 pipedo ( U [ k ] )=0.25*((u[k: l ] ) + ( U [ k +lI)+(U[k-(M+2)1)

+( U [ k + ( M + 2 ) l ) + ( F [ k l );.)

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4.3.

SOR method The point SOR method is widely used since it converges at roughly twice the speed of the point Jacobi method. With ordinary numbering of unknowns, the sweep computation repeats to renew unknowns successively. So, the procedure is purely sequential and difficult to d o in parallel. But, for many cases such as the example considered, parallel processing can be introduced with the so-called red-black ordering. For simplicity of illustration, we consider only the degenerated SOR method, say, the GaussZeidel method. Apply the iteration scheme used by the point Jacobi method on all points (i, j ) with even sums i+j (we call Red points): (5)

u

+ 1)

[ i, j ] =0.25*( u (n' [ i - 1, j ]

+u

[i

(n)

+ 1, j]

+u("'[i,j- l]+u("'[i,j+l]+f[i,

j])

.

The right hand side contains then only unknowns with odd sums (Balck points). On the other hand, new values on Black points can be computed only from renewed values on Red points:

(6)

uCnt1)[i-l, j]=O.25*(~'"~''[i-2, j]+u("+')[i,j ] + ~ ( ~ + l )1 [, ji- - l ] + ~ ( ~ + l ) [ i -j l+, l ] + f [ i - 1, j ] ) ,

The schemes (5) and (6) constitute one step of iteration together. For parallel processing by a processor array, it is convenient to deal with a pair of unknowns on two neighbouring Red and Black points by a processor:

4.4.

One-dimensionalization The number of processors is usually far less than that of computation mesh points. For such cases, it is very natural that each processor takes over a portion of computation to find some row vectors (u[i,( j ) ] ,i= 1, 2,

* *

.,M )

or column vectors

( u [ ( i )jI,.i=l, , 2,

- * a ,

N).

For the set of row vectors, some processes run in serial along increasing or

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293

decreasing i and in parallel over ( j ) . The processor dealing with the j-th row is, of course, the (j)-th one modulo P. It is parallel processing by dimension mentioned in the last section. With a one-dimensional array of processors, the point Jacobi method is realized by the following program : forall j : = 1 to N pdo

i : = l to M d o

for

v[i, ( j ) ]:=0.254 u [i- 1 , ( j ) ]+ u [if 1 , ( j ) ]+u[i, ( j - 111 + 4 i , ( i + l ) l + f [ i ,(All ; forall j : = 1 to N pdo

i : = l to M do u[i, ( j ) ] : = v [ i (, j ) ];.

for

With a vector pipeline computer, it takes the following form: for j : = 1 to N do begin for k : = 2 to M + l do begin

JO[k]:=k +j*( M + 2 ) ; J l [ k ]:=k- l+j*(M+2) ; J2[k]:=k+ 1 +j*(M+ 2 ) ; J3 [k ]:=k (j - 1)*(M+ 2 ) ; J4[k ]:=k+ ( j + 2)*(M f 2 ) end. for k : = 2 to M+1 pipedo

+

+

+

(U[JO[kll):=0.25*(( U [ J 1[ k l l ) ( U [ J 2 [ k l l ) ( W J X k I l ) ( U[J3[kll) (U[J4[klI) ( F [ k l ) );

+

+

+

end;. It looks awkward to make some list vectors in the last program. -DDDDDD+ -DDD

D D D e

-DDDDDD+ -DDDDDD+ -D

13- D D D D +

Processor array

Usual computer

Vector pipeline

Fig. 3. Processing of array

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In comparison between both programs for a processor array and a vector pipeline processor, we can see a n obvious difference. While the processor array takes parallel processing over the index j’s, the vector pipeline does parallel processing to the i-direction, in order to find the set at vectors (u[i,(A]).

Block Jacobi method With a serial computation to a one dimensional direction, as mentioned in the last subsection, it may, of course, contain usual recursive algorithms. It is, for example, a solution of a tri-diagonal problem. Then, we can consider parallel processing even for algorithms with fairly high implicitness. We will take a block Jacobi method as a n exapmle. The procedure of a one-time iteration is as follows: 4.5.

where ‘procedure to solve’ is really replaced by Gauss elimination method or cyclic reduction method, etc. The above algorithm also can be realized on a vector pipeline computer. However, we can not write it down unless part of ‘procedure to solve’ is explicitly defined. In fact, the algorithm is the process of elimination and substitution for each row vector

not for each column vector (u[i,(l)],u[i,(2)],

..., u[i,( N ) ] ) T

( i = l , 2,

. - a ,

M).

Furthermore, a sweep of vector components is prior to that of vectors and hence, we can say that the vector pipeline processing is unnatural. AD1 method and ADE operation In the history of simulation schemes, the appearance of AD1 method certainly was epoch-making, and provided a principle of successful solution of multidimensional problems. In fact, we have got a class of ‘economical

4.6.

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295

schemes' with unconditional convergence/stability, as well as simplicity, that is, the same order of computation work as for explicit schemes. It is followed by many kinds of 'splitting-up operator' methods. All of them retain implicitness in each one-dimensional processing. Also for parallel processing, it is a keystone whether such algorithms can be effectively realized. Since they contain a serial procedure for one-dimensional processing, it is natural to take parallelization by dimension. A procedure of one-time iteration of AD1 method on a processor array is as follows ( r is a n acceleration parameter): forall j : = 1 to N pdo begin i : = l to M do

for

Mi,(j)l:=u[i, (j-l)l+u[i, (i+l)I+(r-2)*u[i, procedure to solve

(A1 ;

1, (A1

(r+2)*u[i, (j)l-u[i- 1, (i)l-u[i+

M),

=", (j)l+f[4(j)l (i=1,2, u[O,(i)l=u[M+ 1, (i)l =o end ; forall i : = l to M , j : = 1 to N ptransfer

h [ ( i ) i, I : = h [ i ,(Al, u [ ( i ) ,jl:=u[i, forall i : = l to M pdo procedure to solve

(A1 ;

(r+2)*u[(i),j l - u [ ( i ) ,j - 11- u [ ( i ) , j + 11 =2*r*u[(i), j]-h[(i), j ] ( j = 1, 2, ~ [ ( iO]=u[(i), ), N+1]=0 ; forall i:= 1 to M , j : = 1 to N ptransfer

- - -,N ) ,

(a)

u[i,(A1:= u[i, (171 ; where u [ ( i ) j, ] on the right hand side of (*) is, of course, what is found through the former half part of the procedure. The characteristic point of this algorithm is that one-dimensionalized processing are taken for two dimensions respectively, and they are done alternately. It is expressed as the Alternating Direction Implicit method. In processing this method by a one-dimensional array of processors, it is necessary to deal with both sets of rows and columns. In order to write such procedure explicitly, we have given two types of expression for variables u and h with parentheses ( ). Their declaration is given as follows: var u : array [O. . M + 1, (1. .N)], [( 1. . M ) , 0 . . N + 11 of real ; h : array [l..M, (l..N)], [ ( l . . M ) ,1..N] of r e a l ;

296

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Each (k)-th processor deals with both row vector (u[-,(k)]) and column vector ( u [ ( k )-1). , In the last program, the former half is for row vectors and the latter for column vectors. For transition from the former to the latter, it is necessary to edit the set of row vectors (here (h[., ( j ) ] ) into ) the set of columns ( ( h [ ( i )-1)). , For return from the latter to the former, it is again necessary to edit columns (here (~[(i), - 1 ) ) to rows ((u[.,( j ) ] ) ) . We call such edition ADE (Alternating Direction Edition). As seen in the above program, ADE’s are realized by the command ‘ptransfer’. Realization of ADE is considered in 4.8. Our ADENA computer is just produced through consideration of smooth ADE. 4.7.

Fast Poisson solver The concept of ADE is essential for parallel computation based on onedimensionalizarion for multi-dimensional problems. We shall here give another example among fast Poisson solvers. It is a direct method for the simple Poisson equation with constant coefficients, and is based upon FFT. Let’s find the solution to the problem ( 1 ) in the form

( i = l , 2 , * * - , M j; = l , 2 ,

..., N ) .

The equation to be satisfied by coefficients u[r, j ] is then

which are the set of equations with tri-diagonal coefficients with respective r , where

(9)

We can really see a n alternating one-dimensional processing. FFT of row data ( f [ ,.j ] ) ( j = 1 , 2 , , N ) by (9) give respective rows (h[-,j ] ) ( j = 1 , 2 , .-.,N ) . Find columns ( [ u [ r ,.I) from respective equations (8) with the computed right hand side h[r, -1, each ( u [ r , being found independently. F F T of rows (v[., j ] ) by ( 7 ) finally give the desired solution (u[.,j ] ) , each again being found independently. Its realization by a processor array are given as follows:

...

a])

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291

begin forall j : = 1 to N pdo procedure of FFT to find h[., (j)] from f[.,(j)]; forall r : = l to M , j : = 1 to N ptransfer h [ ( i ) j, l : = h [ i , (j)l ; forall r : = 1 to A 4 pdo procedure to find ~ [ ( r )-1, ; forall r : = l to A4 , j : = 1 to N ptransfer ~ [ r (A1 , := ~ [ ( r )i ,l ; forall j : = 1 to N pdo procedure of FFT to find u [ - , (j)]from v[., ( j ) ];

end;. This is an example of a fully implicit scheme which is solved fully in parallel.

Realization of ADE As seen in the examples of 4.5 and 4.6, ADE is essential for some parallel algorithms. But, if ADE takes much time, such parallel computation may be of n o value. For the one-dimensional array of processors, each being able to access only its neighbours, ADE needs many times the data transfer to send between distant processors, and thus consumes much time. In the case of vector pipeline computers, application of list vectors allows processing of both directions. Instead of ADE, other operations appear: to gather data with a stride of addresses to make a vector (GATIHER) and to decompose a vector to scatter its components in a storage (SCATTER). For example, the list vector 4.8.

J[j]=j*(M+2)

( j = O , 1, 2,

- a * ,

N+l)

is introduced to have a vector whose components have contiguous addresses, being row-weise, (10)

(U[J[jll, u [ J [ i l + l l , U[J[j1+21,

-..,W [ i l + M + 1 1 ) ( j = l , 2,

*

-

*,

N)

and also another column-weise vector whose components have addresses with the stride M+2, (19)

(U[i+J[OIl, U[i+J[llI, U[i+J[ZII,

*

- -,U[i+J[N+

111) (i=l,2, - . . , M ) .

We must here notice a fundamental problem of vector processors. Suppose that there are 16 memory banks and M+2=16, and each vector of (10)

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Fig. 4.

Allocation of vector components (P=16(N+1))

with the length 16 are stored on the same transversal level of all banks, its components being scattered contiguously from Bank 1 to Bank 16. Then, each vector of (11) lies in a corresponding bank as a column vector (see Fig. 4). Suppose further that data can be accessed by the interleave scheme of 16 ways. Then, each vector of (10) are written/read only in a memory cycle, and the first condition of high speed processing mentioned in 2.5 is fulfilled for such vectors. However, access of columns (11) needs 16 memory cycles since each of them is stored in a corresponding memory bank. It breakes high speed processing. Memory conflict In order to moderate the difficulty of memory conflict to access to a bank as mentioned in the last subsection, some kinds of allocation schemes have been proposed. An example (Budnik and Kuck [1971]) is a skewed array, which is produced by shifting by one box in cyclic for every row under the first row of Fig. 4 successively. (See Fig. 5 . ) In such allocation, both accesses of rows and columns can be done in the fully interleave way, if N g M . By way of compensation, memory access control becomes fairly complex, because first components are stored in different banks row by row and in different positions in banks column by column.

4.9.

Fig. 5 .

A skewed array

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299

The same problem also occurs for the processor array having common memory banks. Furthermore, we must solve some important problems: which banks processors can access, or which allignment network is necessary. We shall not here go into detail for such problems. Instead, we shall introduce the ADENA computer, which is invented from the point of view that ADE is directly realized by the hardware itself.

5. ADENA Computer 5.1. ADENA-I We shall first introduce ADENA-I, since it is a prototype of ADENA-I1 for practical use. It is useful especially for two-dimensional simulations. We now have a real machine of 16 slave processors, which was constructed by ourselves. ADENA-I is a multiprocessor array system of SIMDiMIMD type, and its main part is composed cf one-dimensional array of N processors and a two-dimensional array of N x N square form of buffer memory units, each having a small capacity. The processor array can take two positions, say, row and column position, as seen in Fig. 6. Giving its position to each processor, we will use the symbol

P[ , ( j ) ] on the row position, P [ ( i ) , ] on the column position

(i, j = 1 , 2 ,

.-.,N )

for identifying processors. Processor P[ ,( j ) ] can access all buffer units on the corresponding horizontal line (solid line in Fig. 6), which we give the expression B[i,(j)] (i=l,2, N). me.,

m I

I

I

!

!

!

.

I

.

,

.

,

I

2

!

!

!

!

Fig. 6. Structure of ADENA-I on the row position :.__. ; . . . .processor on the column position 0 . . ..buffer memory unit

0 .. ..processor ,.-.

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300

On the other hand, P [ ( i ) ,] can access all buffer units o n the corresponding vertical line (broken line)

B [ ( i ) ,j ]

(j=l,2,

* - *)

N)

.

The k-th processor is, indeed, just of one and only changes its position:

P[ ,(k)]or P [ ( k ) ,1. The (i, j)-th buffer unit also is just of one and takes two situations: B [ i , (j)]and B [ ( i ) ,j]. So, one characteristic of ADENA-I is that it may take two 'situations' and it takes one of them on each time and changes its situation according to kinds of processing. The system description of the main part of ADENA-I may be given as follows: type PE=processor element ; BU=buffer memory unit ; situation=(row, column) ; ADENA-I=record case situation of

(P:array [ ,(1. . N ) ] of PE ;

row:

B : array [ l . . N ,( l . . N ) ] of BU) ;

(P:array [ ( l . . N ) , ] of PE ;

column:

B : array [ ( l . . N ) ,1 . . N ] of BU);. If parallel processing is done with definite situations, the number of PE's accessible to each BU on every instance is only one, so that buffer memory access conflict never occurs. It turns out that there is always a bi-direction data path between any two processors, P[ , ( j ) ]and P [ ( i ) , I:

P[ ,(.ill

-

BIi, (A1= B [ ( i ) ,j l

c-*

f"(i), 1 .

Needless to say, two situations of ADENA-I may correspond to two ways of expression of data array: u[i, ( j ) ]and u [ ( i ) j, ] as appeared in 4.6. Processor P[ , (j)]finds a row vector (u[., (j)])and Processor P [ ( i ) , ] does a coulmn vector ( u [ ( i ) ,-1). Consider the concerned problem: ADE opeartion. For simplicity, we assume that for a square array of data u[i,j] (i,j= 1, 2, -,N ) , there are just N processors. If every buffer unit were to have such a vast capacity that all simulation data could be stored in the buffer array, every component of ( u [ i ,j]) would be, we assume, stored in a corresponding buffer B[i,j ] , and hence the ADE statement

--

forall

i,j:=l to N ptransfer u [ ( i ) ,il := u[i,(A1 ;

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301

does not need any real transfer of data and only means a change of situations, so that it produces no loss of time. With a large number of processors, N , the number of buffer memory units becomes so great, N 2 , that each unit is not allowed to have a large capacity. Indeed, we must reduce the number N 2 itself. Such a condition produced a n alternative, ADENA-11. As mentioned later, the scheme of allotting data array to a buffer array can not be taken on ADENA-11. Considering such a circumstance, we consider here that the buffer array is only used for data transfer and has the minimum capacity. So, we assume that simulation data are shared by private storages of processors. The k-th processor has usually both a row vector ( u [ - ,( k ) ] )and a column vector ( u [ k ) ,-1) in its private storage. The above statement of ADE is then realized as follows: every Processor P[ , ( j ) ] sends u[i,( j ) ] to B[i, ( j ) ] ( i = l , 2, ., N ) respectively, and then every P [ ( i ) j, ] receives u[(i),j ] from B [ ( i ) ,j ] ( j = 1,2, ,N ) respectively. This data transfer is sure to be done smoothly, but it is accomplished by a sequence of write in and read from buffer units with a quantity of work, O ( N ) . If such work were to appear explicitly in ADE, it would result in much overhead of data transfer. Such overhead can be, however, reduced largely by concurrent processing of data transfer and real computation. Such a scheme is dealt with in the following subsection. Here we notice a limitation due to architecture. The array of processors in ADENA-I is one-dimensional, but each processor can not access to private storages of neighbouring processors as before. Therefore, we assume, for example, that the sentence

..

- -.

seen in 4.6 is not allowed. That is, we make a rule that in parallel processing the index in ( ) must be the same integer variable. Under such a condition, it is known that the above sentence must be practiced on the column position and the result is transfered to get the row data h[., ( j ) ] . Such consideration is a little troublesome, but it never produces loss of computaion time. 5.2.

R-S or S scheme The algorithm of AD1 method in 4.5 is realized in ADENA-I as follows: forall j:=1 to N pdo procedure to solve

(r+2)*u[i, (j)I-u[i-1,

=Mi, (i)l+f[i,( i ) l

(j)l-u[i+l, (j)I-u[i+l, (i= 1,2,. * -,M ) ,

4 0 , (ill = u[M+ 1, (A1 = O ; forall i : = l to M , j : = 1 to N ptransfer

u[(i),i l : = u [ f , (i)l ;

(ill

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forall i : = l to M pdo begin procedure to solve

(r+2)*u[(i),j I - u [ ( i ) , j - l I - u [ ( i ) , j + 11 =2*r*u[(i), j ] - - h [ ( i ) ,j ] ( j = l , 2, u [ ( i ) ,O ] = u [ ( i ) ,N + 1 ] = 0 ; for j : = 1 to N do

...,N ) ,

h [ ( i ) ,j l : = u [ ( i ) , i - l I + u [ ( i ) ,j + lI+(r-2)*u[(i), end ;

il

forall i : = l to M , j : = 1 to N ptransfer Mi, (A1 := h [ ( i ) , i l

;.

Iteration of a number of times of the last program becomes such a sequence as --t

ptransfer

--t

procedure to solve + ptransfer

+

procedure to solve

-

+

ptransfer

.

If the ‘procedure to solve’ is the Gauss elimination method, it is composed of two processes: elimination (REDUCT) and determination (SUBSTITUTION). Both need O ( M ) or O ( N ) work of computaion. On the other hand, ‘ptransfer’ is accomplished by two processes: to write data in buffer memory array (SEND) and to read data from (RECEIVE), and both again count O ( M )or O ( N ) work. If the time to write/read of a word (a memory cycle) is less than or equal to that of a one time floating point operation, Receive and Reduct, or Substitution and Send can be done concurrently so that Receive and Send are imbedded in Reduct and Substitution respectively. Thus, times for data transfer disappear effectively, and no overhead is produced. We call such a way of concurrent processing “R-S scheme”. Contents of computation are, of course, not always composed of pure elimination and substitution. For example, the corresponding computation of the fast Poisson solver in 4.6 is F F T processing. Even in that case, we can consider the entrance and exist part as R and S respectively, and the R-S scheme is available. From the point of view of both hardware and software, the R-S scheme, however, produces some problems of realization. So, we now choose another way of concurrent processing we call S scheme. It is more simple and is to imbed a pair of Send and Receive only into Substitution. It aims to reduce the quantity of hardware and to get high speed transfer. ADE appears also in algorithms that seem at a glance not to need it. This is due to the condition mentioned in the last subsection. Consider, for

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303

example, a typical explicit scheme, the point Jacobi method (2) of Section 2. Its realization on ADENA-I is written as follows: forall

i : = l to M , j:=1 to N ptransfer u[ ( i ) ,il:=u[i, ( i l l ; i : = l to M pdo

forall for j:=1 to N d o

+

v [ ( i ) i,l :=u [ ( i )i, 11 u [ ( i )j, + 11 : forall i : = l to M , j:=1 to N ptransfer U [ i , (ill := U [ ( i ) , il ; forall j:=1 to N pdo for i : = l to M do

u[i,(j)l:=O.25*(u[i- 1, (j)l+u[i+ 1, ( i ) l + u [ i , ( j ) l + f [ i ,

;.

Here, the value u[i, ( j ) ] corresponding to u [ i , j - l]+u[i, j + 11 is computed beforehand on the column position so that all the terms of the last equation may have the same index ( j ) . ADENA-I has greater capability of edition than processor array of ILLIAC-IV type, and hence more generality. But it may not glow to a practical machine since the hardware of buffer array is too much for large number of processors, N . Thus, we must choose a n alternative with less hardware burden. It is, indeed, answered from consideration of 3-dimensional simulations desired strongly in scientific and engineering computation. We call it ADENA-11. ADENA-I1 We again follow such a fundamental scheme of parallel processing as in 2-dimensional simulations. That is, each processor plays a role of solving a one-dimensional problem produced from a n original problem by decomposing. For simplicity of exposition, we consider only a case of simulation on a cubic region with N 3 mesh points. In order to practice parallel processing of a set of one-dimensional subproblems, we put a n N x N two-dimensional array of processors, { P ( s ,t ) } ,and use it on 3 positions for 3-dimensional processing. It also is essential to put a 3-dimensional array of buffer memory units, {B(r,s, ( t ) ) } for ADE. We take a new way of connection among processors and buffer units: 5.3.

m,t )

t--f

B[r, s, (t)l

++

p ( t , r)

.

This relation means that there is a t-th buffer memory board {B(r,s, ( t ) ) ;r, s= 1,2, N ) between a pair of a t-th row of processors, { P ( s ,t ) ; s= 1,2, -,N } and a t-th column, {P(t,r); r = l , 2, -,N } , and they allow data transfer

- .,

-

--

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B b C

P(t,r) P(S,t)

Fig. 7. Structure of ADENA-I1

M

b C

0

B

b’ A

b

C

0 Fig. 8. States of ADENA-I1

305

Parallel Computation

similar to that in ADENA-I. That is, every processor P ( s , t ) standing on the row position can transfer/receive to/from anyone P ( t , r ) on the column position. While the same set of processors stands on both positions against a buffer array in ADENA-I, sets of processors on the row and column position against a t-th buffer board are different in ADENA-11, one being a r-th row and the other a t-th column. As a whole, we put N sets of such complexes similar to ADENA-I. As a 2-dimensional array of processors, the same array stands on both positions. It must be noticed that there is not necessarily a buffer unit allowing direct data transfer between any pair of two processors, P(s,t) and P(1,m). This reflects on the quantity of hardware. We here have O ( N S )buffer units against N 2 processors, while in ADENA-I, O ( N 4 ) against the same number of processors. For realizing a 3-dimensional simulation, we make the complex of processors and buffer units take 3 states against the coordinate system [ i , j , k ] fixed in the cubic region, as seen in Fig. 8. We call them a-, b- and c-states respectively. Here, the square OABC is the 2-dimensional array of processors, and the solid one stands on the row position, while the broken one stands on the column position. The square OLMN is a 2-dimensional array of buffer memory units connected to the array ab and a'b' of processors with the same number of the row and column order. We give expressions for processors and buffers so as to fit their numbers directly to the space coordinates, accepting different expressions according to their states: a-state:

P[ , ( j , k)lt*B[i, ( j , (k))l=B[i),j,( ( 4 1

b-state:

P [ i ) , , (kl B [ ( ( i )j, ) , k I = B [ ( i ) ) , j ,(kl t--) P [ i ) , ,(kl

c-state:

P [ ( i ,i),I P [ , (i,k)l t* B[i,((A,k)l=B[(i, W), kl

5

5

where the underlined processors stand on the row position. The processor on the a-state and the column position and that on the b-state and the row position, for example, have the same expression P [ i ) , ,( k ] . So, the number of situations taken by processors against the fixed space coordinate is only 3. According to such situations, we allow three types of data expression:

Though such expressions seem to be troublesome, they allow us to write down programs by high level languages for one-dimensionalized parallel processing effectively.

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306

In such architecture, the follwoing ADE's are easily done on the a-, band c-states respectively (of course, their inverses also are possible) :

a: b: C:

As a n example of application, we consider the MacCormack method [ 19691 for the fundamental equation of compressible fluid with no viscosity.

The equation is of the form aU aF -+-+-+-=O,

at

ax

dG ay

aH az

where U is a n unknown vector with some conservative components and F , G and H are functions of U with vector values. The MacCormack scheme (its original form) is to proceed one time step (n--n+l) through the following 3 fractional steps: U(ntl)/s=L1 U" ,

U(nt2)/3=L u(nt 1 ) / 3

9

Unt' =L3U(nt2)/3.

The difference operator L, is one of 'predictor and corrector' type and is defined by the following formulae: V(nt1)/3[i,j,k]=Un[i, j , k ] - a ( F ( U n [ i , j ,k])-F(Un[i-l,j,k])) 1 U(nt1)/3[i, j , k ] = - [ U n [ i , j , k]+ V C nl i) l 3 [ i , jk] , 2

,

-a(F( VCnt1)/3[i+ 1, j , k ] ) - F ( VCnt1)/3[i,j, k]))] , where a=At/Ax=constant. Both L, and L, also are defined similarly. For simplicity, we suppose that the concerned region is a cubic with a side lengh I , and that F , G and H are defined beforehand. Further, we put At/Ax= At/Ay= At/Az= At/(l/N+ 1)=a (constant). Necessary variables are first declared as type vector=[l. . 5 ] of real ; var U : array [O..N+l,(l..N, l . . N ) ] , [ l . . N ) , O . . N + l , ( l . . N ] , [(l..N, l . . N ) , O . . N + l ] of vector ; V: array [ l . . N + l , ( l . . N , l . . N ) ] , [l . . N ), l . . N + l , ( l . . N ] , [ ( l . . N , 1. . N ) , 1. . N + l ] of vector ;.

Parallel Computation

5.4.

307

Various application modes of ADENA-I1 We first notice that the architecture of ADENA-I1 allows data transfer between any two processors with a n aid of another processor, at most. For example, between P(s, t ) and P(Z,rn), we can use P(t,Z) or P(rn,s) as a

T. NOGI

308

mediator:

where a buffer unit certainly exists between two processors connceted by --t. Therefore, the load of data transfer is independent o n the distance of indexes, (s, r ) and ( I , m ) . Systematic use of such fact allows ADENA-I1 to simulate ADENA-I1 for 2-dimensional simulations. Then, principal problems are ways of data expression and realization of 2-dimensional ADE. In order to deal with 2-dimensional arrays on ADENA-11, we take such a way to correspond as

where the left hand sides are data expressions by a user, while the right hand sides are those produced by a compiler. Indexes are related as follows: i-(q,r):

q=(i-l)divN+l,

j-(s,t):

s=(j-l)divN+l,

r=(i-l)modN+l, t=(j-l)modN+l.

Then, a 2-dimensional ADE, for example, forall i, j : = 1 to N*N ptransfer u[(i), i l := u[i,

Wl

is realized on ADENA-I1 as follows: forall q, s, t , r : = 1 to N ptransfer 4 ( q , 4, tl[rl:=u[q, (s,t)"1 ; forall q, s, I , r : = 1 to N ptransfer u [ q ) ,s, (rl[tl:=u[(q, 4, tl[rl

;.

Since data transfer are here done through mediators {P(s,t ) } , there produces some overhead. Therefore, we must accept a slight decrease of efficiency of parallel computation for 2-dimensional simulations. Finally, we will touch on the fact that ADENA-I1 can be used as a processor array of ILLIAC-IV type. So far, we have taken a natural correspondence between the processor number (r, s) and the space coordinate [i, j, k]. Processors are, however, not connected directly with each other, and hence we are free to consider their rearrangement logically. We will show a way of rearrangement that each processor may have four neighbours connected through buffer units. For illustration, we take a n example of 4 x 4

309

Parallel Computation

(err alsed

(natLral)

>

Fig. 9. Arrangement for realizing ILLIAC-IV type machine

array. In Fig. 9, the left is a natural one and the right is a n arranged one. The right one is constructed from the left such that every pair of numbers with a n odd sum is converted to a pair of numbers with the inverse order. Then, we can always find a buffer unit on solid bonds. Furthermore, it is clear that the arranged array can be expanded as a one-dimensional array with connection between neighbours. Therefore, we can expect that ADENA-I1 processes also such data as

u[(i)l

or

4 ( i , A1 ,

as mentioned in Sections 2 and 4, successfully.

6. Some Parallel Algorithms This section is concerned with some sophisticated algorithms fitted to parallel processing.

Nested disection ordering Some ways have been proposed to get ordering of unknown variables to reduce band width of a coefficient sparse matrix or to accept only a little filling of components for the Gauss elimination method. We know, for example, Cuthill-Mckee’s method [ 19691, Reverse Cuthill-Mckee’s method by George (Cuthill [ 19721) and Nested disection method (George [ 19731). From the point of view of parallel processing, we can consider the last method to be best among those. We shall touch on it briefly. Consider a square region R , (Fig. 10) on which a problem is solved by some finite difference method or finite element method. Divide first the region into R : , R; and a separation zone So (Fig. l l a ) , and give first numbers of ordering to nodes in R:, next to ones in R? and finally to ones in So. The produced coefficient matrix usually takes the form seen in Fig. l l b . Further, try to divide the region by added separation zones, S: and S:, as seen in Fig. 12a. Then, the corresponding shaded parts in R: and R? show the same pattern 6.1.

T. NOGI

310

Fig. 10.

Fig. lla.

Fig. Ilb.

Fig. 12c.

Fig. 12b.

Fig. 12a.

as that of division of the whole R , (see Fig. 12b). We will here change ordering to count first nodes in Ri ( i = l , 2, 3 , 4 ) successively, and the next ones in Sf,S: and So. The matrix then shows the pattern of Fig. 12c. We can say that the given matrix is of the form

Kl1

0

Kz[ K,I

:I' K ,5

K2 2

K52

K25

K33

K44

K53

K54

KG~

When the original matrix is symmetric, we have K , , = K , , . composition LLT is then found by

The Cholesky de-

Parallel Computation

311

where L,, has the same dimension as K,,, every L,, is a lower triangular matrix and {L,,} are found by solving the following system of equations: K,,=L,,L;

(i=l,2,3,4),

K,,=L,,L,T,

( i = l , 2, 3, 4) ,

4

K 6 , = - Z L5iLT6=L66Lc5 . i=l

Solution of LLT=b is carried out by taking two steps, first to solve Lu=b and next to solve LTu=u. These procedures are similar, and hence we consider only the first step. The equation Lu=b can be written explicitly as

or ( i = l , 2, 3,4) ,

L,,u,=b, 4

L65v5 =b6-

2 LS~U,.

i=l

For parallel processing, it is convenient to use 4 slave processors and a host processor. Every i-th processor first finds Lit and L,,, sends the latter to the host, and then finds v, and u, to send to the host. On the other hand, the host receives L,, (i=1, 2, 3 , 4 ) and determines L,,, and then, after getting u, ( i = l , 2, 3,4), finds u, and u,, and finally to broadcast u, to all slaves, which find u,'s by solving the equation L,*u,= u, - LSiU5. The whole procedure of the above algorithm can be carried out with no data transfer among slaves. Thus, there are no serious problems.

Combination of finite element method and splitting-up method It has been proposed to split up some approximate schemes by finite element methods for the one-dimensionalizing solution process (Marchuk and Kuzin [1983]). For simplicity, we shall introduce a similar method applied

6.2.

T. NOGI

312

to a difference scheme by Sofronov [1965] which has, indeed, inspired the former. Suppose that a parabolic equation

8% % A-+ at

ax2

a2u B-+

axay

a2u c-+ ayz

au D-+

ax

au E-

ar

in the square O < x , y < 1 is given by some change of independent variables from a n original parabolic equation in a n arbitrary quadrilateral region. It usually has terms of mixed derivatives and of lower order. Write it in the following form:

1-s d2U +-IBI---+D-+E-,

2

ar;

au

ax

au ay

where s=sgn B, l,=x+y and 12=x-y. Fractional steps for splitting-up of the one-step operator are here taken not only to the x and y direction, but also to the I, and I, direction, as follows:

(3)

(4)

Parallel Computation

313

Fig. 13.

It is here necessary to put the condition

for allowing stable double sweep algorithms with sufficiently small h to solve the first equations in (1) and (2). Such a condition is more severe than the parabolic condition of the given equation. Under the condition, the scheme is unconditionally stable when 1 O
and

in (3), the concerned ADE must prepare (u:$49

and

&$4+19

‘ . . 9

1/1 ul+(N--J),N)

T. NOGI

314

The ADE is as follows: forall

i : = l to l + ( N - j ) ,

j : = 1 to N - 1 ptransfer

w [ i , ( j ) ]:=zP4[(i), j + i- 11 ;

forall i : = l + ( N - j ) to N , j : = 1 to N - I

z[i,( j ) ]: = u l J 4 [ ( i ) , j + i -

ptransfer

N],

Incomplete HV-decomposition We will introduce a new method produced recently by the author, which is useful for parallel processing. It is based upon Stone's idea [I9681 which also opened the way for some ICCG methods. It replaces the part of incomplete Cholesky decomposition used in the ICCG method, etc. by another decomposition we call the incomplete HV-decomposition, which allows processing like the AD1 method. If we say that the AD1 method uses a decomposition of a sum type, H+ V, the new method would correspond to a decomposition of a product type, HV. We here consider only its application to a five point difference scheme for the Poisson equation, but it is probably applicable for finite element approximations, too, with much incompleteness of decomposition. Consider the problem of the difference equation

6.3.

ajkUjk-Cjk-1Ujk-l-CjkU3lr+l-bj-lkUj-tX-bjkUj+lk=fjk

in a square region, with assigned data on its boundary. With a natural ordering of unknown variables, we get a coefficient matrix for the equation to be solved which is of a block tri-diagnoal form. For simplicity, we will write it in short as A=(O,

0I

-cjk-i,

-b,-ik,

ajkr - b 3 k

10,

-c5kt

0)

7

where each triple shows three components on a j-th row of a k-th block row. Now, we try to express it as a product of two block tri-diagonal matrices, H and V , where H is a pure tri-diagonal matrix and V is composed of blocks of diagonal matrix:

I

H= O,O, o -h, rfk,

(

rj-lk

-*I

O,O,0) , rjilk

The product H V i s certainly a block tri-diagonal matrix, but has some non-zero elements filled, to which correspond zero elements of A . In fact, we have

Parallel Computation

315

Expecting the effect of filling in H V to be small, we will consider that H Y corresponds to a decomposition of the modified A with some added terms, whose application to a function produces the following with parameter dk's, all being less than and near to 1. In fact, A u is replaced by

By placing the last formula equal to HVu, we have the following equations to be used for finding H a n d V :

We can solve the last equations approximately, and find

where

T. NOGI

316

Iteration process is given as usual: HV(Un+' - U") =f- AU"

(n=O, 1, 2,

* *

.)

.

Each solution of the last equation is found by solving two sub-systems, HZ)n+I=f-Aun

,

V(/(Cln+l-Un)=Z)"+l

,

which are easily solved by direct methods, for example, the Gauss elimination method, since both are sets of tri-diagonal problems. Its parallel processing also is done with success on the machine powerful for ADE, like the ADENA machine. Numerical experiments for Poisson equations have shown that the above method has higher speed of convergence than Stone's method, and that its CG-acceleration has comparable or greater speed than the ICCG method, or more than. Thus, we can say that our method is very good from both the point of view of convergence and parallel processing.

7. Conclusion We have reviewed parallel computations with a veiw over simulation schemes, algorithms, languages and machines. We will here pick up some emphasis points; 1) multi-dimensional problems are replaced by splitting-up schemes with implicitness along each one-dimensional axis of co-ordinates, 2) each one-dimensional subproblem is solved by a sequential direct method, 3) ADE is essential in parallel processing for such schemes and algorithms, 4) array data are to make their situations of being processed clear with indexes contained in ( ), 5) ADENA is very fitted to treat such array data and allow ADE, and 6) ADENA also promises to support some new parallel algorithms. Today, only vector pipelines are accepted widely. We are, however, of the opinion that they seem unnatural in use for simulations or others, and ADENA must become a n alternative of vector pipelines or other parallel computers of ILLIAC-IV type. We are ready to accept the criticism that this is a distorted view, but we hope that ADENA against vector pipelines may suffer n o destiny of Goethe against Newton. [P.S.] It has turned out most recently that the incomplete H V decomposition is not but a generalized splitting-up operator, that is,

Parallel Computation

317

where

A,=(O, 0,o I -6, and

A,=(O,

-C,k-l,

0

The iteration scheme

(

B '+Zu

3 '3 aZ+l

,

(u: a scalar parameter)

is well k n o w n (see, for example, M a r c h u k [1975]) a n d is used successfully. Our numerical experiments using t h e CG acceleration rule for determining {T,,} also show that t h e last scheme h a s fast convergence superior t o other methods. This m e a n s that the H I / decomposition does not a n y gain from its generality b y D , a n d only D=uZ is sufficient.

References P. Budnik and D. J. Kuck, The organization and use of parallel memories, IEEE Trans. Comput. (Short Notes), C-20 (1971), 1566-1569. 0. Buneman, FORTRAN program distributed at APS Topical Conf. Numerical Simulation of Plasma, Publ. LA-3990, p. D2-1, Los Alamos Sci. Lab., 1971. E. Cuthill and J. Mckee, Reducing the bandwidth of sparse by symmetric matrices, ACM Proceedings of 24th National Conference, New York, 1969. E. Cuthill, Several strategies for reducing the bandwidth of matrices, in D. J. Rose and R. A. Willoughby (Eds.), Sparse Matrices and Their Applications, Plenum Press, New York, 1972. P. M. Flanders, D. J. Hunt, S. F. Reddaway and D. Parkinson, Efficient high speed computing with the distributed array processor, processor, High Speed Computer and Algorithm Organization, Academic Press, London, 1977, 113128. A. George, Nested dissection of a regular finite element mesh. SIAM J. Numerical Analysis, 10 (1973), 345-363. D. Heller, A survey of parallel algorithms in numerical linear algebra, SIAM Review, 20 (1978), 740-777. R. W. Hockney, A fast direct solution of Poisson's equation using Fourier analysis, J. Assoc. Comput. Mach., 12 (1965), 95-113. R. W. Hockney and C. R. Jesshope, Parallel Computers, Adams Hilger, Bristol, 1981. R. M. Hord, The ILLIAC-IV the First Supercomputer, Computer Science Press, Maryland 1982.

318

[121 t131

[201

~ 7 "281

T. NOGI T. Hoshino, T. Kawai, T. Shirakawa, J. Higashino, A. Yamaoka, H. Ito and K. Sawada, PACS, A Parallel Microprocessor Array for Scientific Calculations, ACM, Vol. 1, NO. 3 (1983), 195-221. J. S. Kowalik, (ed.), Parallel MIMD Computation: HEP Supercomputer and Its applications, The MIT Press, 1985. D. J. Kuck, Parallel processing of ordinary programs, Advances in Computers, 15 (1976), 119-179. R. W. MacCormack, The Effect of Viscosity in Hypervelocity Impact Cratering, AIAA Paper, No. 69-354, 1969. G. I. Marchuk, Methods of Numerical Methods, Springer, 1975. G. I. Marchuk and V. I. Kuzin, On the combination of finite element and splitting-up methods in the solution of parabolic equations, J. Comp. Phys., 52 (1983), 237-272. W. L. Miranker, A survey of parallelism in numerical analysis, SIAM Review, 13 (1977), 524-547. A. R. Mitchell, Computational Methods in Partial Differential Equations, John Wiley & Sons, 1967. T. Nogi, Parallel Machine ADINA, in Computing Methods in Applied Sciences and Engineering, V, eds. R. Glowinsky and J. L. lions, North-Holland, 1982, 103-122. -, The ADENA Computer, in International Symposium on Applied Mathematics and Information Science, Kyoto University towards Multidimensional Flow Models, Mathematics and Computers, Kyoto University, 1984, 7/9-16. D. J. Paddon, (ed.), Supercomputers and Parallel Computation, Clarendon Press, Oxford, 1984. A. H. Sameh, Numerical parallel algorithms in High-speed Computer and Algorithm Organization, Academic, New York, 1977, 207-228. U. Schendel, Introduction to Numerical Methods for Parallel Computers, Ellis Horwood Limited, 1984. I. D. Sofronov, A difference s c p m e with sweep of diagonal directions for solution of the heat equation, Z. VyEisl. Mat. i Mat. Fiz., 5 . No. 2 (1965). (in Russian) J. F. Traub, Iterative solution of tridiagonal systems on parallel or vector computers, in Complexity of Sequential and Parallel Numerical Algorithms, ed. by J. F. Traub, Academic Press, 1973. Y. Wallach, Alternating Sequential/Parallel Processing, Lecture Notes in Computer Sicnece 127, Springer, 1982. 1N. N. Yanenko, The Method of Fractional Steps, Springer, 1971. V. Zakharov, Parallelism and array processing, IEEE Trans. Comp., C-33 (1984), 45-78.

Department of Applied Mathematics and Physics Faculty of Engineering Kyoto University Kyoto 606, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 319-355 (1986)

A Theoretical and Computational Study of Upwind-Type Finite Element Methods By Masahisa TABATA Abstract. Upwind-type finite element methods are studied from both theoretical and computational aspects. Eight schemes are discussed from the points of nonnegativity preserving property and crosswind diffusion. They are also compared by numerical results for a convection dominated model problem. Key words: upwind-type finite element method, monotone scheme, crosswind diffusion, convection-diffusionequation

Introduction In this article we study upwind-type finite element methods from both theoretical and computational aspects. Upwind finite element schemes have been required in the numerical analysis of phenomena accompanied by diffusion and transportation effects, since the standard finite element schemes produce very poor results when the transportation effect is dominant. Some examples of such phenomena are heat or pollutant transportation problems with small diffusion effects. The other examples are flow problems described by, for example, the system of Navier-Stokes equations with a high Reynolds number. In these exapmles the analysis of convection-diffusion equations is fundamental. Heat or pollutant transportation problems are governed by the very equation. In the resolution of Navier-Stokes equations, the vorticitystream function formulation reduces them to a coupling of a convection-diffusion equation and a n elliptic equation. In the velocity-pressure formulation, the linearized equations, which are nothing but convection-diffusion equations, are often solved iteratively in order to treat the nonlinear convection terms. Thus establishing a good algorithm for the convection-diffusion equation is essential in the analysis of the phenomena mentioned above. In the following we restrict ourselves to the approximation of convection-diffusion equations. The efficiency of the standard finite element approximation to a convection-diffusion equation differs very much depending on the Peclet number, the ratio of convection to diffusion. A stationary pure diffusion problem, whose Received June 14, 1985.

320

M. TABATA

Peclet number is zero, can be transformed to a minimization problem, to which the standard Ritz-Galerkin finite element approximation produces nice numerical results. When a convection term is added, the problem is no longer equivalent to a minimization problem in general. (Under some additional conditions, the problem can be transformed to a minimization problem. As for a scheme based on this transformation, see section 8.8) The standard finite element approximation, however, still works well as long as the Peclet number is not so high. When the convection term grows dominant, the problem becomes almost of hyperbolic type, and it produces highly oscillating solutions. Such oscillations are caused by the fact that a standard finite element approximation leads to a centered approximation scheme. For the past several years upwind-type finite element schemes have been developed to overcome the deficiency mentioned above. Those schemes can be divided into two groups. The one consists of schemes which maintain the non-negativity preserving property, or equivalently a discrete maximum principle. The other consists of schemes which do not always satisfy the nonnegativity preserving property. In the second group much attention is paid to diminish spurious diffusions. In the following we do not intend to exhaust all upwind finite element schemes but concentrate on the next eight schemes. We choose four representative schemes from each group. From the first, schemes by Tabata [28], Baba-Tabata [l], Kanayama [19] and Ikeda [15], and Tabata [34], and from the second, schemes by Heinrich et al. [lo], Hughes [12], Hughes and Brooks [13], and Mitchell et al. [25]. They are analyzed theoretically and compared numerically. A way of diminishing spurious diffusion effects is to use the partial upwind approximation. We also present a new development of this approximation by the use of a monotone matrix theory. Some non-negativity preserving schemes can be improved to diminish spurious diffusions. The present paper is composed of nine sections. In section 1 we observe how solutions of convection-diffusion equations depend on Peclet numbers by considering a two-dimensional model problem. In section 2 we apply the standard Ritz-Galerkin finite element approximation to the problem considered in section 1 and see the limit of application to high Peclet number problems. In section 3 we explain two points from which we will discuss and compare upwind-type finite element schemes in this paper. The one is a non-negativity preserving property and the other is the effect of spurious diffusions involved in a scheme. In section 4 numerical results of each upwind scheme to a test problem are shown. Three flow directions, parallel and skew to mesh directions, are considered for a high Peclet number problem. In section 5 one-dimensional results are reviewed. In section 6, we seek a sufficient condition for a scheme to be non-negativity preserving. Using a decomposition theorem of a matrix into a product of positive-type matrices by Bramble and Hubbard [4],we show the PI-Galerkin finite element scheme is non-negativity preserving

Upwind-Type Finite Element Methods

32 1

when the Peclet number is less than some positive number depending on the flow direction. This theorem is also used in section 8.1 to derive a n improved partial upwind scheme T1. In section 7 we introduce three coefficients from a computation of truncation error with respect to a local orthogonal coordinate system. These coefficients are used to evaluate spurious diffusion effects of upwind schemes in the next section. In section 8, for each of eight upwind finite element schemes, after the derivation is reviewed, we discuss the dependence on the flow direction of che spurious diffusion as well as the nonnegativity preserving property. In section 9 we give some concluding remarks. All computations in this paper were performed on HITAC M180 at the University of Electro-Communications.

1. Convection-Diffusion Phenomena Let 52 be a bounded domain in R" ( n = 1, 2, 3). We consider the stationary convection-diffusion problem, (1.la)

-vAu+b.grad

(1.lb)

u=f

in

u=g

on 352,

Q

where Y is a diffusion constant, b is a given flow velocity, and f and g are given functions. The unknown scalar function u stands for heat or vorticity, for example. In problem (l.l), the term (1.2)

c

- Y ~ u =- Y

a2UIaxi2

i=1

describes a diffusion effect and the term (1.3)

b-grad u=

describes a convection effect. (1.4)

2bJx)au/ax,

i= 1

The dimensionless number

P,= lblliv ,

called Peclet number, indicates the ratio of the latter to the former, where 1 is a representative length of 52. The Peclet number is employed as a representative parameter of convection-diffusion state. By a n example let us observe the dependence of solutions of problem (1.1) on Peclet numbers. Example 1. Solve problem (1.1) with the data

(1.5) in the domain

b=(l,O),

f=1,

g=o

322

M. TABATA

9 = ( 0 , l)x(O, 1) .

(1.6)

We consider four values of u, u=1 , 0.1 , 0.01 , 0.001

(1.7)

,

which correspond to (1.8)

P,=l,

10,

100 , 1000.

Figs. 1.1-1.4 show elevations of u for Peclet numbers (1.8), where the origin is at the front left corner". When P,=l ( u = l ) , the effect of diffusion is dominant and due to the homogeneous Dirichlet condition u is nearly equal to zero. When P,=10 (u=O.l), the effect of convection appears clearly and the position where the maximum of u attains is drifted downstream. When P,= 100 (u=O.Ol), the phenomenon is dominated by the convection and boundary layers appear in the neighbourhood of 3s except x,=O. When P,= 1000 (u=O.OOl), example 1 almost reduces to a hyperbolic problem subject to an initial condition u=O at x,=O. Therefore the solution is nearly equal to u=xl except small boundary layer regions. Thus, the state of solutions of convection-diffusion problem (1.1) changes so much depending on Peclet numbers. Our aim is to find finite element

1.1

1.2

1.3 Fig. 1.1. 1.2. 1.3. 1.4.

1.4 Elevation Elevation Elevation Elevation

of u: v = 1 of u : v=O.1 of u: u=O.OI of u : v=O.O01

* Precisely speaking, these solutions are not the exact ones but numerical solutions obtained by an upwind finite element scheme TI of section 8.1.

Upwind-Type Finite Element Methods

323

schemes which produce good approximate solutions for every Peclet number. 2. Ritz-Galerkin Finite Element Approximations The standard Ritz-Galerkin finite element approximation to problem (1.1) is as follows. We begin by transforming problem (1.1) to a n equivalent weak form: (2.1)

Find a function u E V ( g ) such that a(u, v)=(f,v) for all v € V(0),

where

V(g)={v€H’(S); v = g on aQ} and (2.2)

a(u, v)=

s,

{U

grad uagrad v+b-grad u v}dx .

Let X , be a finite dimensional subspace of H1(Q). The standzrd RitzGalerkin finite element approximation is : (2.3)

Find a function u h € V,(g) such that

a h , v,J =(f,v,)

for all

8, E

V,(O),

where (2.4)

V,(g)={v,cX,;v,=g at every nodal point on as}

In the above, h indicates a representative size of elements. numerical procedure, a cell Peclet number defined by

In the

r= lblhb

(2.5)

!is used as a key parameter. Now we apply the standard finite element approximation to example 1. At first we choose a PI-finite element space as X,, that is, piecewise linear finite element space. Fig. 2.1 depicts a used subdivision of D into a union of triangles, where each side is divided into 20 intervals of same length (h= 1/20). Figs. 2.2-2.5 show elevations of PI-finite element solutions of example 1 for diffusion constants (1.7), which correspond to r=0.05,

0.5,

5 ,

50

.

When y=0.05 and 0.5, the PI-finite element method produces good results, but when r = 5 and 50, it produces oscillating solutions. It is noted that, although

M. TABATA

324

example 1 is symmetric with respect to x,=0.5, the oscillating finite solution is not symmetric. Secondly, we take up a Q,-finite element space as Xl,, piecewise finite element space. Fig. 2.6 depicts a used subdivision of Q into of squares of side-length 1/20. Figs. 2.7-2.10 show elevations of

2.2

2.1

2.3 I

2.4

Fig. 2.1. Subdivision of the domain for P,-finite 2.2. Elevation of P1-Galerkin finite element 2.3. Elevation of PI-Galerkin finite element 2.4. Elevation of P,-Galerkin finite element 2.5. Elevation of PI-Galerkin finite element

1

2.5

elements solution: v = l solution: v=O.1 solution: v=O.Ol solution: v =0.001

element bilinear

a union &,-finite

325

Upwind-Type Finite Element Methods

2.6

2.7

2.9 2.10 Subdivision of the domain for Ql-finite elements 2.7. Elevation of Ql-Galerkin finite element solution: 2.8. Elevation of Ql-Galerkin finite element solution: 2.9. Elevation of Q1-Galerkin finite element solution: 2.10. Elevation of Q,-Galerkin finite element solution:

2.8

Fig. 2.6.

v=l v=O.l u=O.Ol v=O.OOl

element solutions of example 1 for diffusion constants (1.7) or cell Peclet numbers (2.6). Like the Pl-finite element method, the Q,-finite element method produces good results when r=0.05 and 0.5, but again oscillating results when r= 5 and 50. As observed in the above numerical results, the standard Ritz-Galerkin finite element method does not work well for convection-diffusion problems with high Peclet numbers. Thus, in place of it the other methods were required and upwind-type finite element approximations have been developed.

M. TABATA

326

3. Non-Negativity Preserving Schemes and Spurious Diffusion For problem (1.1) the maximum principle holds: under the condition

f20, the smooth solution u of problem (1.1) satisfies

(3.1)

min {u(x);x E Q U dQ}=min {g(x); x E dQ}

.

Remark 3.1. As for the smoothness of u, if u belongs to W2,p(Q),p > n , we can obtain (3.1) by Bony [3]. We introduce two operators L and M defined by (3.2)

Lu= -vAu+b-grad u ,

(3.3)

Mu=u

for smooth functions u in Q u dQ. We regard Lu as a function defined in Q and Mu a function defined on an. Then the maximum principle is written as follows: LuzO implies (3.4)

min {u(x);x ~ Q u d Q } = m i n { u ( x ) ; x ~ % 2 } .

Equivalently, we have the following monotonicity property

(3.5)

Lu22Lu, and

Mu22Mu, imply

u2)u, ,

and the non-negativity of solution:

(3.6)

LuzO and

Mu2_0 imply u z 0 .

Now we consider discrete versions of the properties mentioned above. Let L, and Mh be operators approximating L and M, respectively. We replace 8 and dQ by 52, and dQh which are finite sets of nodal points. For a function u, defined on Q,uaQ,, Lhuh defines a function on Q, and Mhuh defines a function on 852,. Then a discrete approximation of problem (1.1) is:

(3.7)

Lhuh=fh

on Q,

Mhuh=gh

on dQ,

,

where fh and gh are approximations of f and g, respectively. maximum principle corresponding to (3.4) is: L,u,zO implies

(3.8)

min {uIL(P); PES,UdQ,}=min {u,(P); PEdQ,}

A discrete

.

The monotonicity property and the non-negativity of solution are replaced by

Upwind-Type Finite Element Methods

(3.10)

Lhuhzo

and M,u,zO

imply

327

u,zo,

where we assumed the unique solvability of problem (3.7).

Proposition 3.1.

(3.11)

Suppose the condition

L,1=0

and

M,=Z.

Then, the properties (3.8), (3.9),and (3.10)are equivalent. Proof. We only prove that statement (3.10)implies (3.8),since the other parts are trivial. Let u, be a given function satisfying L,u,lO. We put vh=U,L--a,

where a = m i n {u,(P); P c d S } . Then v, satisfies the conditions of (3.10) by virtue of (3.11). Therefore, V , is non-negative by (3.10)and we obtain (3.8). All the approximation operators L , and M h which will be considered in this paper satisfy the condition (3.11). Hence, statements (3.8)-(3.10)are all equivalent. We would like to use numerical schemes which satisfy the non-negativity preserving property (3.10), or equivalently the monotonicity (3.9), for two reasons. The one is that such schemes produce no oscillating solutions, and the other is that solutions with negative parts may become meaningless from a physical point of view. For example, consider the case where u stands for heat or pollutant density. Then, negative values of u have no physical meanings. It is not difficult to derive monotone schemes when Peclet numbers are low, that is, diffusion constants are large. Both P I - and Q,-Galerkin finite element schemes of the previous section will be proved monotone when v = 1 or 0.1 in section 6. However, as were shown in Figs. 2.5 and 2.10, numerical solutions derived from those schemes when v=O.OOl are very oscillating. Therefore, P I - and Q,-Galerkin finite element schemes are not monotone for all Peclet numbers. A way for obtaining a monotone scheme is to use a larger diffusion constant than the exact one. Such a n example is a classical artificial viscosity scheme, which uses a diffusion constant max(v, 161hj2) in place of v . It produces blunt solutions corresponding to the larger diffusion constant Ib/h/2 when the cell Peclet number y of (2.5) is high. The problem of bluntness of solutions becomes more serious in higher dimensional cases, where crosswind

328

M. TABATA

diffusion effects occur. Such spurious diffusion effects appear for other sorts of upwind schemes, since they usually contain some kind of additional diffusion. In the following sections we will compare upwind finite element schemes from the two points: ( i ) Are the schemes non-negativity preserving? ( i i ) How large are the cross wind diffusion effects? Schemes will also be investigated by observing numerical results to Example 2.

Solve problem (1.1) with the data

(3.12)

Y=

(3.13)

10-6 ,

b=(cosO, s i n 6 ) ,

=’ 1:

(3.14)

8 € ( - ~ / 2 ,7r/2),

when x 2 2 x , tan 8+0.2 otherwise ,

(3.15)

g=o.

The domain 8 is given by (1.6), which is divided like Fig. 2.1 or 2.6. noted that the function f is discontinuous across a streamline (3.16) Since (3.17)

x 2 = x , tan 0+0.2 Y

It is

.

is very small, the solution u of example 2 is almost equal to u ( x )=

x,/cos 8

to

when x , z x , tan 0+0.2 otherwise,

except small boundary layer regions. Remark 3.2. Hughes and Brooks [13] employed as a crosswind diffusion test an example of problem (1.1) where the boundary data g is discontinuous and the right-hand side data f=O. We note that whether f is equal to zero or not gives rise to much difference in the behavior of numerical solutions across the streaiiiline (3.16). (See sections 8.2 and 8.3.)

4. Comparison by Numerical Results Before analyzing upwind finite element schemes we show their numerical results to example 2. Schemes we consider are the following eight, denoted by schemes T1, HH, HU, HB, BT, KI, MG, and T2. Their derivations and analyses will be given in section 8. Scheme T1: scheme choosing a n upwind element by Tabata [28]. Scheme HH: scheme of Petrov-Galerkin approximation by Heinrich,

329

Upwind-Type Finite Element Methods

Huyakorn et al. [lo]. scheme based on a numerical quadrature by Hughes [ 121. scheme of streamline upwind type by Hughes and Brooks ~31. Scheme BT: scheme based on the use of barycentric domain as a control volume by Baba and Tabata [l]. scheme based on the use of circumcentric domain as a Scheme KI: control volume by Kanayama [ 191 and Ikeda [ 151. Scheme M G : scheme derived by modifying scheme H H by Mitchell and Griffiths et al. 1251. scheme derived from a symmetrization by Tabata [34]. Scheme T2: In example 2 for the flow directions B=O, n/4, 7118, we observe their numerical results of convection dominated states. Schemes T1, BT, KI, and T2 use triangular elements shown in Fig. 2.1. Schemes HH, HU, HB, and MG use quadrangular elements shown in Fig. 2.6. Some schemes are identical when Y vanishes. Numerical results obtained from those schemes when Y = are so close to one another that we cannot distinguish them in figures. In such cases their results are shown in a same figure. The values of the exact solution are almost equal to the ones given by (3.17) at the interior nodal points and vanish on the boundary. The figure of the exact solution is drawn by interpolating these values. In all the figures of elevations, the origin is at the front left corner. When B=O, the flow direction coincides with a mesh direction. Fig. 4.1 depicts the exact solution. Schemes T1, KI, and T2 are identical (when Y vanishes) and their solution is shown in Fig. 4.2. Their solution is equal to the exact one except on the streamline x,=0.2, where the computed value is half of the exact one because of the integral average in obtaining the righthand side (f,$,J. Schemes HH, HB, and M G coincide, and their solution Scheme HU: Scheme HB:

4.1

4.2

4.3

Fig. 4.1. Elevation of the exact solution u : B=O 4.2. Elevation of uh by schemes T1,KI, and T2: 0 = 0 4.3. Elevation of uh by schemes HH, HB, and MG: 0=0

M. TABATA

330

4.4

4.5

4.6

4.7

4.8

4.9

A

4.11

4.10 Fig. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10. 4.11. 4.12.

Elevation Elevation Elevation Elevation Elevation Elevation Elevation Elevation Elevation

of of of of of of of of of

4.12

uh by scheme HU: O = O uh by scheme BT: B=O the exact solution u : O=n/4 u,, by scheme T1: O=r/4 u h by scheme HH: O=n/4 uh by schemes HU, KI and T2: O=n/4 u h by scheme HB: 8=n/4 uh by scheme BT: O=n/4 u h by scheme MG: O=n/4

33 1

Upwind-Type Finite Element Methods

4.13

4.14

4.15

4.16

4.17

4.18

A

Fig. 4.13. 4.14. 4.15. 4.16. 4.17. 4.18. 4.19.

4.19 Elevation of the exact solution u: 0 = a / 8 Elevation of uh by scheme T1: 0=n/8 Elevation of uh by scheme HH: 0=x/8 Elevation of uh by schemes HU, KI and T2: 0=11/8 Elevation of uh by scheme HB: 0=7r/8 Elevation of uh by scheme BT: 8=n/8 Elevation of uh by scheme MG: 0=n/8

M. TABATA

332

is shown in Fig. 4.3. Near at x,=0.2 and x z = l the phenomenon of overshoot is observed. Scheme H U produces a n oscillating solution shown in Fig. 4.4. Fig. 4.5 presents the result of scheme BT, where the effect of spurious crosswind diffusion is seen. When 8 = ~ / 4 ,the flow direction coincides with a mesh direction. Fig. 4.6 depicts the exact solution. The result of scheme T1 is shown in Fig. 4.7, which is equal to the exact one except on the streamline (3.16) like the case 8=0. Fig. 4.8 depicts the result of scheme HH. Schemes HU, KI, and T2 are identical (when Y vanishes), whose result is shown in Fig. 4.9. Figs. 4.10, 4.11, and 4.12 present the results of schemes HB, BT, and MG, respectively. In Figs. 4.8, 4.10, and 4.12 we can see the solutions take negative values. In Figs. 4.9 and 4.11, especially in Fig. 4.9, the effect of spurious crosswind diffusion is observed clearly. When 8=n/8, the flow direction is skew to the mesh directions. The exact solution is shown in Fig. 4.13. Figs. 4.14 and 4.15 depict the results of schemes T1 and HH. Schemes HU, KI, and T2 are identical, whose result is presented in Fig. 4.16. Figs. 4.17, 4.18, and 4.19 show the results of schemes HB, BT, and MG, respectively. In Figs. 4.14, 4.16, and 4.18 the effect of crosswind diffusion appears. In Figs. 4.15,4.17, and 4.19 the phenomenon of overshoot is seen. It will be shown that schemes T1, BT, KI, and T2 always satisfy the discrete maximum principle (3.8). Hence, their solutions never oscillate, but some of them suffer pretty large crosswind diffusion effects. Scheme T1 produces good results with small crosswind diffusion. Schemes HH, HU, HB, and MG are not non-negativity preserving as shown in the numerical examples. Their solutions are overshooting or oscillating. Some of them (scheme H H when 8=n/4 and B=r/8, scheme HB when O=x/4) produce pretty good results with small overshoot. We will analyze the dependence on the flow direction of the crosswind diffusion for each scheme in the subsequent sections. 5.

Results in the One-Dimensional Case

Here we review briefly results of the one-dimensional case. Some of them are used in deriving upwind finite element schemes in higher dimensions. The reduction of (1.1) to the one-dimensional case is: (5.la) (5.1 b)

+bdu/dx=f

- ud2u/dx2

@)=go

in Q = (0, 1) u(l)=g,

7

*

--

Let N be a positive integer. We denote by x i , i=O, -,N, the nodal point ih, where h = l / N . The central finite difference approximation to (5.1) is: (5.2a)

-uA,ui+biDhu,=fi

(5.2b)

uO=gO

, 9

i=l, UN=gl

***, 9

N-1 ,

333

Upwind-Type Finite Element Methods

where Ahui=(ui+l

-2ui+ui-l)/h2 ,

U,-,)/(2h) ,

DhUi=(Ui+l-

and uiare unknown values at xi. The stability condition for (5.2) is max

(5.3) where

{lril; i = l ,

-.., N-1}(2

,

ri is defined by ri=b,h/u .

(5.4)

Inequality (5.3) means that the cell Peclet numbers should be less than or equal to 2. A full upwind finite difference approximation to (5.1) is: (5.5)

-YAhUi+bidhUi=fi

,

where S,=d,(b) is a n upwind difference operator defined by (5.6)

b,(b)u, =

ti

b,LO , bi
u,-ui-,)/h when ~ ~ + ~ - u J / when h

This scheme is proved to be unconditionally stable.

(5.7)

d,=D,-sgn

Since it holds that

(b)hA,/2 ,

equation ( 5 . 5 ) is equivalent to -u(l

+ 176l/2)&u, +biDhUt=fi

.

Hence we can regard ( 5 . 5 ) as a n order h2 approximation of equation (5.la) with a diffusion constant u + u f t in place of u, where ~ r i = ~ I r i 1 ./ 2

(5.8)

In order to minimize the additional diffusion, the partial upwind scheme (5.9)

-VA,U,+ (1- a i ) b i D h ~ l t + ~ t b i d= h f, ~t

is considered, where a i € [ O , 11 is a parameter. greater than or equal to the critical value aCi=max {O, l-2/lril}

Scheme (5.9) is stable if ai is

.

When at=aci,the additional diffusion uei corresponding to uri is (5.10)

v,,=Y

max {0, Iri1/2-1}

.

Il’in’s scheme [16] whose solutions converge to the exact one uniformly in u as well as in x is

M. TABATA

334

(5.11)

-(vTt/2) coth (r,/2)Ahu,+b,D,,u,=f*

*

The additional diffusion voLof this scheme is (5.12)

v0*=vI(r,I2)coth (rJ2)- 11

*

Scheme (5.9) coincides with (5.11) if we take az=aol, aY,,=sgn(bJIcoth ( ~ , / 2 ) - 2 / r J.

(5.13)

When b is constant and f = O , Il’in’s solution is exact at the mesh points. In this sense a,, is the optimal value. This value is referred to in choosing the partial upwinding parameter for higher-dimensional problems. Obviously we have (5.14)

acz< aOt< 1

and

vcl< p o l


The additional diffusion in the optimal case is greater than vet, which can be explained as follows. When a,=a,,, the value u, is determined only from the upwind value u , - ~or u , + ] . If a,>a,,, equation (5.9) gives a relation between u % - ~u,, , and u L b l . This fact leads to a better approximation of solution in boundary layer regions than the case a=a,,. But from the point of the additional diffusion the choice of a,, is best.

6. Monotone Matrices When the subdivision of the domain is uniform like Fig. 2.1 or 2.6, finite element schemes can be written in the finite difference manner. We use the stencil notation C-I,l

co.1

C1,l

C-l,-l

co,-1

cCl l’ ,o- l

(6.1)

1

1

’

which defines a difference operator L,,, 1

(Lhuh)(x)=

1

cJ,kuh(xl+ih,

1,k=-l

xL?+kh)

.

Each scheme in section 4 can be written in the form (6.1), especially c- 1 , 1 = c,, - 1 =0

for schemes T1, BT, KI, and T2. Let N be the number of nodal points in SZUaSZ. to a system of linear equations

(6.2)

Ahuh=Fh

1

Equation (3.7) reduces

335

Upwind-Type Finite Element Methods

where A , is a n N x N matrix, F,=[f,, ghlr is a n N-vector. (Note that the values of u, on dQ, are usually treated as unknown in the finite element procedure aiming at genera1 purpose programs.) Under condition (3.11), scheme (3.7) satisfies a discrete maximum principle if and only if A , is a monotone matrix, that is, A, is invertible and each entry of A;' is non-negative. A well-known subclass of monotone matrices is the set of positive-type matrices. Let B be a real-valued matrix. We define a n index set J ( B ) by J ( B ) = { i ; C b,,>O}

.

j

A matrix B is said to be of positive type if (i) for i f j , ( i i ) Crbij>=O for every i , (iii) for i 6 f J ( B ) , there exists a connection in B from i to J ( B ) , that is, there exists a set of indices k ( l ) , ., k ( m ) and a n entry j E J ( B ) such that

+.

him, b k ( l ) k ( 2 ) ,

-..,

bk(m-l)k(m),

b,,d,fO

*

For example, the five-point finite difference approximation to - A or the PIGalerkin finite element approximation to - A in the subdivision of Fig. 2.1 leads to

0 -1

01

(6.3)

Therefore, the derived matrix is of positive type, and the scheme satisfies the discrete maximum principle. The stencil forms to the terms &/dx, and &/dx, of the PI-Galerkin finite element approximation in the subdivision of Fig. 2.1 are 0

2

(6.4) L--1

Note that positive off-diagonal entries appear. Therefore, even if the diffusion constant Y is large, the matrix derived from the PI-Galerkin finite element approximation to -vAu+b grad u is not of positive type. Thus, the class of positive-type matrices is too small for our purpose. The product of two monotone matrices is monotone. In particular, the product of two positive-type matrices is monotone, though it is not in general of positive type. Bramble and Hubbard [4] gave a sufficient condition for decomposing a matrix into a product of positive-type matrices.

M. TABATA

336

Theorem 1 (Bramble and Hubbard). Let B have unit diagonal entries with If B is written as the matrix sum B=I-H,-H,, where ( i ) (H,),,,=O, ( i i ) I - H , is of positive type, (iii) ( I - H1)-'HZ2 0, (iv) f o r each k @ J ( B )there exists a connection in H , from k to J ( B ) ,

xi b,,ZO, J ( B ) # @ . then

( I - H l ) [ I - ( I - H1)-'HZ] is a decomposition of B into a product of positive-type matrices and B is monotone. In condition (iii) of Theorem 1 and the expression (6.5) below, the inequalities are considered entrywise. Condition (iii) can be derived if H,H: L H i

(6.5)

is satisfied as was noted by Ruas [27], where (Hz')ii=max (0, WZ)J ,

(ffi)il=max (0, -(H2)tf).

Now the normalized matrix derived from the P,-Galerkin finite element approximation of problem (1.1) is in the stencil form 0

(6.6)

-6

I-Hi-Hz= (-c-s)~

+( c +2 s ) ~ -

(C+s)r

24 -6+(2c-s)y -6+(~-2~)y 0

where (6.7)

y=[blh/v ,

1

X

1/24

b=Ibl(cos 0, sin 0)=lbl(c,s) .

Applying Theorem 1, we seek a bound for y which ensures the monotonicity of the matrix (6.6). Suppose that 0 E [0, ~ / 4 ] . Let E be a non-negative number. We set 6 + (c- 2 s ) ~ 0

(6.8) (c+s)~

6 + ( - C+ 2 s ) ~

(6.9)

The matrix I - H , is of positive type if

0

Upwind-Type Finite Element Methods

(6.10)

337

62(3c+~)r.

Inequality (6.5) is satisfied if

(6.11)

{(6+ (c- 2s)r)/24}(c+s+ +/24 2 (c+s)r/24

.

After a simple calculation we obtain that if

r5rc(B),

(6.12)

there exists a non-negative number

(6.13)

rA0) = {- 15(c+s)

E

satisfying (6.10) and (6.11), where

+31/33 +30cs}/{(2c-s)(c -2s)} .

We extend the definition of ~ ~ (for 0 )8 € [-a/4, 3x/4] by (6.13) and for [ - 5x14, - ~ / 4 ]by

(6.14)

=

t9e

re(-d - 8) .

Then we can show that matrix (6.6) is monotone under condition (6.12) for any 8 E [-5x/4, 3 ~ / 4 ] . Fig. 6.1 depicts the graph of rc(8),8 E [-x/2, x/2]. The maximum 2.828 is attained at 8= -n/4, and the minimum 0.857 is attained at O=x/4. From (6.12) we have

(6.15)

p 2 Iblh/rc(@)

.

Substituting into (6.15) and (6.13)

Ibl=l,

6.1

h=1/20,

8=0,

6.2

Fig. 6.1. Graph rc(0)for the PI-Galerkin finite element scheme 6.2. Graph rc(0)for the Ql-Galerkin finite element scheme

338

M. TABATA

we get

~ 2 0 . 0 4 5.

Therefore, when ~ = or 1 0.1, the P,-Galerkin finite element scheme of section 2 is monotone. Next we consider the Q,-Galerkin finite element approximation. The stencil form is

-4+(-c+s)y

-444sy

-4+(c+s)r

-4+(-C--s)r

32 -4-4sy

-~+(c-s)T

(6.16) Therefore, if

(6.17) then L,, is of positive type and monotone, where (6.18)

Fig. 6.2 depicts the graph of yc(0), O€[-n/2, 7421. The maximum 1.414 is attained at O=k;r/4, and the minimum 1 is attained at O=O, -t?r/2. Substituting into (6.15) and (6.18)

lbl=1

,

h=1/20,

o=o ,

we get ~20.05.

Therefore, when 2 is monotone.

Y=

1 or 0.1, the Q,-Galerkin finite element scheme of section

7. Estimation of Crosswind Diffusions When a scheme is given by stencil form (6.1), we may compute the truncation error by using the Taylor expansion. Let Pibe a n interior nodal point. We introduce a local orthogonal coordinate system (tl,E,) such that El-direction coincides with flow direction b(P,). Expanding u a t Pi in this coordinate system, we obtain

(7.1)

L ~ U = -,,AU+

ibl{au/at,-Rh)+o(v)

,

where

(7.2) We use the coefficients A i j for evaluating additional diffusions of a scheme.

Upwind-Type Finite Element Methods

339

Central approximations like the P,-and Q,-Galerkin finite element schemes are of order h2. Therefore, the coefficients A,, of those schemes vanish. As for all the upwind schemes in section 4, the coefficients A,, and A,, satisfy

and

(7.4)

A2220

for every flow direction. Inequality (7.3) shows a n additional diffusion to the flow direction, which makes the scheme stable as analyzed for one-dimensional problems in section 5 . Inequality (7.4) shows the crosswind diffusion. In fact, the term au/aE, - hA,,a2u/aEZ2

is parabolic and u diffuses to E,-direction as the flow marches to El-direction. Therefore, A,, causes the spurious crosswind diffusion in the numerical solutions. As a n example, we compute the coefficients A,, of a n artificial viscosity method for the Galerkin finite elements treated by Kikuchi [ 2 2 ] . We consider the Q,-elements. Then the stencil form is given by (6.16), which is not monotone when r is large. Suppose the case 101(rr/4. If a n artificial viscosity term (7.5)

--Y

max (q- I,O)A,

is added to L,Lof (6.16), then the obtained stencil form becomes of positive type. Thus, we get a monotone scheme. By a simple calculation we have A,,=A,,=max (c-l/r, 0) ,

A,,=O

.

Similarly we obtain All=Azz=max (lsl-l/r, 0) ,

A,,=O

,

when 7~145101 5 4 2 . Fig. 7.1 depicts the graphs of A,, when

r tends to infinity.

Fig. 7.1. Coefficients A*j : artificial viscosity method for Ql-elernents

M. TABATA

340

8. Analysis of Each Scheme In this section we take up in due order the eight upwind schemes presented in section 4. After the derivation is reviewed, we discuss the monotonicity and the spurious diffusion effects. We show the stencil forms to example 2. The entries are calculated down to three places of decimals. 8.1.

Scheme T1 Scheme T1 was proposed by Tabata 1281 in full upwind form. Here we present the partial upwind approximation, which is proved to be still of positive type by using Theorem 1. Let P, be any nodal point and b, be the flow velocity at P,. We choose a n upwind element Ti of Piwith respect to b, as Fig. 8.1. Then, b-grad u at P, is approximated by (8.1)

b(P,)d,u, =b(P,)grad u,( Ti).

Note that grad U, is constant, since we use the piecewise linear finite element. This approximation is combined with the PI-Galerkin approximation of -vAu by using a lumping technique. (See Tabata 1281, 1291.) When the subdivision is given in Fig. 2.1 and OE[O, a/4], the stencil form of (8.1) is

bd,=Ibl

0

0

0

-c+s

c 0

0 xl/h

[

-s

.

0

The original full upwind scheme is L,= -vA,+b(P,)13,

and the partial upwind scheme is (8.2)

L,= -vA,+(l--a,)b(P,)D,+-aib(Pi)G, ,

where 0

0

s

By a similar argument in section 6, we obtain a sufficient condition 0 5 Q is x , ( O )

for the operator L, of (8.2) to be monotone, where a,(@ =max {O, [scy - ( c + 3s)

+d s 2 c 2 y 2+2sc(c+ 3s)y+c2 -2sc+9s2]/(2scy)}

,

Upwind-Type Finite Element Methods

34 1

and c, s, and r are defined in (6.7). The maximum value To of r for which a,(O)=O is (2/(--s) (8.3)

ro(B)=

2/(c+3s) \2/(s+3c)

when # € [--lr/2, --lr/4] , when #~[--lr/4, 0 J , when # € [ 0 , n/4] , when

#€[

xi21

742,

.

Fig. 8.2 depicts the graph of ro(8). Fig. 8.3 shows the graphs of additional diffusion coeffcients A,,. A , , and A,, vanish at B=O, n/4 and i 4 2 , while A,, attains the maximum 0.354 at 8= --lr/4. The stencil forms to example 2 when 8=0, x/4, z/8 are

(8.4)

0 0.707

.h=[

-0.707

Lh=

-0.541 [-1.383

0 xl/h,

0 0 0.924 0 x l / h * 0

This is the unique scheme in all the eight schemes which reproduces the exact

. h2

0

R1Z

8.1 8.2 Fig. 8.1. Upwind element Tiof nodal point f" with respect to b(P,) 8.2. Graph of ~ ~ ( 0 )

M. TABATA

342

Fig. 8.3. Coefficients A i j : scheme T1

solution of example 2 when the flow direction coincides with any one of the mesh directions. The spurious diffusion is small for 8 E [0,7~/2], but not so for 8 € [ - 7 ~ / 2 ,01. The worst direction is 8= -7r/4. Then, the stencil form is

Liz=

lo -0.707

L O

-0.707 1.414

O0 I x l / h 3

0

0.1

which is similar to (8.14). Therefore, a pretty large spurious diffusion like Fig. 4.9 will appear in this case. However, scheme T1 produces nice results for 8 € [0, x/2] as was observed in Figs. 4.2, 4.7, and 4.14. Scheme HH Scheme H H proposed by Heinrich et al. [ 101 is based on the Petrov-Galerkin approximation, where the test functions differ from the shape functions. Suppose the domain is divided into a union of rectangles like Fig. 2.6. Let X,, be the usual Q,-finite element space. A finite element solution is sought in an affine space V J g ) of (2.4) in X,L. We denote by $hc the base function at 8.2.

Pz=(X:r

GI, 2

$h&(XI, x 2 ) =

JJ $ ( ( x 2 - - x ; ) / N

3-1

where $ is the one-dimensional base function when Isl<1 otherwise. The test function at P,is

where

343

Upwind-Type Finite Element Methods

W; a )=$W +ads) , u(s) =

when Is1 < 1 otherwise .

--s(l-Isl)

{O

The two parameters a; are chosen depending on the flow direction (&, b;) at p,, (8.7)

aZ,=coth (rj/2)-(2/rJ) ,

rj=b;h/v

.

Then scheme HH is: (8.8) Since sgn (a;)=sgn

E such that Find a function U ~ V,(g)

A 6 ) = (Oh%) f,

4u,,

for all i

.

(4),we have 9 ( ( x J- x ) ) / h ; a;)Z$((xj--Xi)/h)

when x , exists in the upwind direction of x; with respect to b;, and

ID((xJ-xE)/h;a;) 5 $((xj-xI;)/h) when x , exists in the downwind direction of x ; . Therefore, the centroid of (llh6 exists in the upwind direction of P, and scheme (8.8) is of upwind type. The choice (8.7) of a; is derived from the consideration of the one-dimensional problem (5.1). Scheme (8.8) reduces to Il'in's scheme (5.11) when applied to one-dimensional problems with constant functions b and f. The stencil forms to example 2 when 8=0, 7r/4, 3r/8 are

I r

Lh=

-0.667 -0.167

0.059

(8.10)

L,=i -0.177 -0.589 ~

0.070 (8.11)

The stencil form L h ( O ,

-0.456 -0.544 Y)

1

0.167 0

--0.167

(8.9)

0.667 0 X l / h , 0.167 0

-0.059 0.943 -0.177 -0.070 0.871 0.130

o

-0.059 0.0591

0 -0.0321 0.032

1 X

is discontinuous at (0, 0),

lim lim L h ( O , v)flim lim L,'(B, v ) 8 1 0 "10

and neither is equal to

8 1 0 "10

1/12

.

M. TABATA

344

Fig. 8.4. Coefficients AtJ:scheme HH

1

Fig. 8.5.

Coefficients A C j :schemes HU, KI, and T2

lim lim L,(O, Y I O

Y)

8--0

of (8.9). Let u be the exact solution (3.17). When 8=0, L,u does not approximate f of (3.14) at the boundary or internal layer regions, which causes the overshoot. Note that if the source term f is null, then (8.9) can precisely transfer the stationary discontinuous state to the flow direction 8=0. This scheme is not monotone and we see overshoots in Figs. 4.3, 4.8, and 4.15. Fig. 8.4 depicts the graphs of A i j . The coefficient A,, vanishes identically for every 8. The coefficient A , , is discontinuous at 8=0. Scheme HU Scheme HU proposed by Hughes [12] is based on a numerical quadrature method. In computing entries of element stiffness matrices a numerical quadrature whose quadrature points depend on the flow direction is used. Suppose the domain is divided into a union of rectangles like Fig. 2.6. Let X , be the Q,-Galerkin finite element space. Equation (2.3) reduces to a system of equations (6.2). The ( i , k)-entry of matrix A , is written as

8.3.

Upwind-Type Finite Element Methods

where

a:;= and

(8.12)

s. s. Y

a:; =

345

grad $,,.grad $hidx

b grad $,,$,,dx

.

In scheme HU, the integral (8.12) is replaced by a one-point integral formula mes (e)b(Oe)(grad#hk$ht)(Ee)

9

where O c = ( O ; ,0 ; ) is the centroid of the element e and P=(E;, E l ) is a quadrature point defined by

E;=O;+haf/2 ,

j=1,2

.

Here a; € [ - 1, 11 are the values given in (8.7). Let P,,k f i , be a nodal point of the element e found in the upwind direction of P,. Then the quadrature point P moves from 0 toward to Pi. Therefore, we have

and ( A h ) i kbecomes a significant entry in the i-th row of A,. Hence, scheme HU is of upwind type. The choice of the quadrature point is derived from the one-dimensional results. Scheme H U reduces to Il’in’s scheme (5.11) when applied to one-dimensional problems with a constant function f. The stencil forms to example 2 when 6=0, 7r/4, 7r/8 are

-0.250

0.250 0

-0.250

0.250 0

(8.13)

(8.14)

0 Lh=[-0.707

I

L

0 1.414

O

0

(8.15)

Note that the stencil form L,(B, Y) is discontinuous at (0,0),

M. TABATA

346

lim lim L,(8, Y)#lim lim L,(O, Y) e-o

Y

,

I0 8-4

The left-hand side is equal to (8.4) and the right-hand side is given in (8.13). When O=O, the quantity of the second line of (8.13) is less than that of (8.9), which causes more oscillating solutions than scheme HH. When the source term f is null, scheme (8.13) precisely transfer the discontinuous state to the direction 8=0 as was remarked by Hughes and Brooks [13]. When 8 = ~ / 4and 71/8, L,Lis of positive type and scheme H U is monotone. Then, the effect of crosswind diffusion is pretty large. (See Figs. 4.9 and 4.16.) Fig. 8.5 depicts the graphs of A t j . The coefficient A , , attains the maximum 0.354 at 8= k n / 4 . Scheme HB Scheme HB proposed by Hughes and Brooks [13] is considered to be an artificial viscosity method. The distinguished point is that the additional diffusion is not uniform for all directions but restricted only to the flow direction. In section 5, we saw that Il'in's scheme has the additional diffusion voi of (5.12). Therefore it is regarded as a n order h2 approximation of the operator 8.4.

L = -( Y

+

+bd/dx

YOi)d2/dX2

Scheme HB is derived as a n approximation of the operator

[k,,]=k

[" cs

"1

s2

Here k is a non-negative constant defined by

Fig. 8.6. Coefficients A t j : scheme HB

Upwind-Type Finite Element Methods

347

+

k =(alrl a2r2)12

and a,, rj, c, and s are given in (8.7) and (6.7). Suppose the domain is divided like Fig. 2.6. Let X , be the Q,-Galerkin finite element space. Scheme HB is derived from (2.3) by using a bilinear form corresponding to the operator (8.16). Fig. 8.6 depicts the graphs of A i j . The coefficients A , , and A , , are identically zero for every flow direction. Therefore there exists no crosswind diffusions of order h. The stencil form to example 2 when B=O coincides with (8.9). Those when B=z/4 and i ~ / 8are (8.17)

(8.18)

I

0.059

0.118

LL= -0.354 -0.412

0.943 -0.354

-0.039

0.250

-0.648 -0.333

0.871 -0.005

-0.117

I

0.118 x l / h , 0.059 -0.115

1

-0.032 X l / h . 0.052

Scheme HB is not monotone and overshoots are observed in Figs. 4.10 and 4.17. Scheme BT Scheme BT introduced by Baba and Tabata [ l ] is derived by using the barycentric domain as a control volume. This approximation was developed in order to obtain a n upwind scheme satisfying both the discrete maximum principle and a discrete conservation law. When the flow is incompressible, the balance of inflow and outflow of u through any surface is preserved. This approximation satisfies a discrete version of such balance. Here we suppose that div b=O and the subdivision is uniform. For the general case we refer

8.5.

Fig. 8.7. A barycentric domain

Fig. 8.8.

Coefficients A',: scheme BT

M. TABATA

348

to Baba and Tabata [l]. Let a domain be divided into a union of triangles. Then, the barycentric domain D, around a nodal point Pi is the polygonal domain whose vertices are the centroids of the elements around P, and the midpoints of the sides connected to P,. Fig. 8.7 shows a barycentric domain when the subdivision is given in Fig. 2.6. In scheme BT, b grad u is approximated as follows.

( b grad u)(P,)=div (bu)(P,)

=j

div (bu)dxlmes D,

Di

(8.19)

=\

bun dslmes D, JD,

- C 1rij bun,, dslmes D, -c j

r,l{P:3u(P~)-P~,u(PJ)}/mes

D&

Y

j

where j runs over all indices such that P, is a n adjacent nodal point to P,. In (8.19), rm,is the side of D, which intersects the segment P,P,, n,, is the unit outer normal to r,,,rt3is the length of rt,, and

0) , P,:=max (ntJb(ptJ)9

P,=max

where P,, is the midpoint of the segment P,P,. volume D,to a control volume D,,that is,

(-n&',,),

0) ,

If u outgoes from the control

- b(P,,) > 0 , then the upwind nodal value u(P,) is used in the last line of (8.19). On the contrary, if u enters from D , to D,,the upwind nodal value u(P,) is used. Therefore this approximation is of upwind type. Scheme BT is obtained by combining the approximation (8.19) with the PI-Galerkin approximation to - A . If the triangulation of the domain is of (weakly) acute type, it is known that (8.20)

grad $,,-grad $hLSO

for i+j by Ciarlet and Raviart [7]. Therefore the derived matrix is of positive type and scheme BT is monotone. The stencil forms to example 2 with B=O, n/4,and n/8 are

-0.333 (8.21)

0

349

Upwind-Type Finite Element Methods

0

(8.22)

(8.23)

L,=

-0.488 [-:.436

-0.053 0 0.977 0 0

0

Fig. 8.8 depicts the graphs of A & , . There is n o direction where the coefficient A,, vanishes. The maximum 0.449 is attained at 9= - 0 . 1 3 2 ~ and -0.368~. In Fig. 4.5, the effect of pretty large corsswind diffusion is observed.

Scheme K I Scheme KI developed by Kanayama [19] and improved by Ikeda [15] is derived by replacing the barycentric domain of scheme BT by the circumcentric domain. Since this approximation is well-suited to the PI-Galerkin approximation of -A, the partial approximation is naturally applied. Let a domain be divided into a union of triangles. Then the circumcentric domain D, around a nodal point P, is the polygonal domain whose vertices are the circumcenters of the triangles around P,. Every element is supposed to be a rectangular triangle or a n acute triangle in order to get the circumcenter in the closure of the element. Fig. 8.9 shows a circumcentric domain when the subdivision is given in Fig. 2.1. The term b-grad u is approximated by (8.19), where D, is replaced by the circumcentric domain around Pi. The corresponding central difference type approximation to this upwinding is 8.6.

(bD,u)(P,)= C r~,n~,.b(P,,){u(P,)+u(P,)}/(2 mes Dt) . 3

Iwaki [17] showed that the PI-Galerkin approximation to -Au can be written as ~

-AiLu(PI)=

[C r,jlu(Pt)--u(Pj)}lPtPjllm, 7

3

where j runs over ail the indices such that Pj is a n adjacent nodal point to P,. Here m, is one-third of the total measures of triangles around P,, and rij is the side-length of the circumcentric domain D, which intersects the segment P,Pj. Therefore the use of the circumcentric domain fits well with the P,Galerkin approximation. Noting this point, Ikeda [ 151 considered the partial approximation such that

350

M. TABATA

8.9

8.10 Fig. 8.9. A circumcentric domain 8.10. Coefficients Air: scheme MG

where

a,, = max 10, 1- 21, mes D,/(rn,P,P,In,lb(P,,) 1)) i We denote this by scheme KI. Scheme KI is monotone. When the subdivision is given by Fig. 2.1, approximation (8.24) reduces to the partial upwind approximation established by the finite difference method. The stencil form to example 2 when O = O coincides with (8.4) and those when O=x/4 and n/8 coincide with (8.14) and (8.15), respectively. The graphs of A,, are shown in Fig. 8.5, which coincide with those of scheme HU.

Scheme MG Scheme M G is a variation of scheme HH. Mitchell et al. [25] showed that the cross diffusion coefficient A , , as well as A , , vanishes if a;, j = 1 , 2 , of (8.7) are replaced by

8.7.

a;=c,

&=s.

We denote this scheme by MG. Fig. 8.10 depicts the graphs of A i j . The stencil form to example 2 when 0=0 coincides with (8.9). Those when 8=n/4 and x18 are

1

(8.25)

0.042 L,= -0.231 L-0.415

(8.36)

-0.016 -0.534 -0.338

I 1

0.064 0.667 -0.231

-0.002 0.064 X l / h . 0.042

0.139 0.667 0.048

-0.005 0.014 x l / h . 0.026

This scheme has no crosswind diffusion of order h. It is not monotone and produces overshooting solutions shown in Figs. 4.3, 4.12, and 4.19.

Upwind-Type Finite Element Methods

35 1

8.8.

Scheme T2 Scheme T2 proposed by Tabata [34] is derived from a symmetrization of the problem. Suppose that the flow is irrotational, rot b=O and that the domain is simply connected. Then there exists a velocity potential

0 such that b=grad @

.

Problem (1.1) can be transformed to a minimization problem in a weighted function space : Find u E V ( g ) such that

(8.27)

J(u)=min { J ( v ) ;U

E

V ( g ) },

where V ( g ) = { v E H ’ ( Q ) ;v=g on o?Q},

J ( v ) = a ( v , v ; w)/2-(f, v ; w ) , grad u-grad v w dx ,

and w is a weight function defined by

(8.28)

w(x)=exp ( -@(x)/Y)

.

A finite element approximation is obtained from (8.27) by replacing V ( g )by a finite element space V,(g) of (2.4). The obtained matrix is symmetric. Hence if the flow is irrotational and the Peclet number is not high, we can solve the convection-diffusion problem in the framework of symmetric approximation. When the Peclet number is high, a modification is required since the argument of the exponential function (8.28) becomes too large. We consider the PI-Galerkin finite element space. Let $,< be base functions. Then the weak form derived from (8.27) is equivalent to a system of equations

c 4$*p

(8.29)

3

In the support of

$hi

$hi:

w)u,=(f, hi: w )

-

the velocity potential @ is approximated by

M. TABATA

352

since bi=b(P,) is equal to grad @(Pi). Substituting (8.30) to equation (8.29) we have (8.31)

vwi

c grad

$hj

exp ( - ( x - P , ) b J v ) d x uj

grad $h,

e,j

=wift

c\ e

Se $hi

exp ( - ( x - P , ) b , b ) d x

,

e

where w , = w ( P , ) . Let x*$ and x , , be the points where the minimums of x - b , are attained in the support of $hi and in element e, respectively, that is, x’*< is the extreme point in the support of $ h i to the upwind direction with respect to b,. Multiplying both sides of equation (8.31) by a constant

, exp ((~*,-f‘~)W)h, we obtain

=f,c e

Pei

1

$hi

exp ( - ( x - x e , ) b , b ) d x

3

e

where

Ppi=exp ( - ( x , ,-X*t)bm/Y) *

We prepare a subroutine which returns the value of integral on the fundamental triangle T with vertices (0, 0), (1, 0) and (0, l),

1

4 exp ( - (c,el+ C , E , ) ) ~ E

,

where q is a polynomial of first order and c,, i = l , 2, are non-negative constants. The two integrals of (8.32) are computed by using this subroutine. Equation (8.32) is scheme T2. The quantity Pei ( 5 1 ) measures the contribution to the coefficients and the right-hand sides from a n element e. For a n upstream element e where x , < = x * { , Per is equal to unity. If e is a downstream element, Pet tends to zero as Y tends to zero. Thus, scheme T2 is of upwind type. The reduction to one-dimensional problems of scheme T2 leads to Il’in’s scheme. (See Tabata [34].) If the triangulation of the domain is of (weakly) acute type, then scheme T2 is monotone from (8.20). The stencil forms to example 2 coincide with those of scheme KI when Y = O . For non-zero Y , this scheme uses not only upstream nodal values but also downstream nodal values like Il’in’s scheme. Therefore, a better approximation of the solution in layer regions can be expected.

Upwind-Type Finite Element Methods

353

9. Concluding Remarks

Through the discussions in the previous sections we can derive the following conclusions. ( i ) Monotone schemes like T1, BT, KI, and T2 are stable for every Peclet number. They can be applied to any problems and their solutions are non-oscillating. The number of nodal points used in these schemes reduces to a few when the Peclet number tends to infinity. Therefore if the flow is skew to mesh directions, that is, if there exist no nodal points to the upwind direction of a nodal point, crosswind diffusion appears. Scheme T1 has the advantage that its crosswind diffusion vanishes when the flow direction coincides with any one of the mesh directions, while that of the other schemes do not always vanish even if the flow direction coincides with a mesh direction. The crosswind diffusion of scheme T1 is small when it is used properly for the flow direction. ( i i ) Crosswind diffusions of schemes HH, HB, and MG are small or null modulo order h. These schemes use eight or nine nodal points even when the Peclet number tends to infinity. In general, higher order approximate schemes have the shortcoming that they produce overshooting solutions when the exact one is not smooth. Such a n overshoot appears even if the flow direction coincides with a mesh direction. The behsvior of scheme H U when Of0 resembles that of schemes KI and T2. When O=O, scheme H U produces an oscillating solution. For the computation of convection dominated phenomena, it is desirable to distribute nodal points along streamlines. In many flow problems, however, the flow velocity is also unknown. Therefore it is not possible to distribute nodal points a priori in such a way. In some finite difference literature [20, 371, body-fitted curvilinear coordinate systems, say (El, E2), are employed in flow problems around a body. Near the body, where boundary layers may appear, streamlines fit well with curves of constant E,-values. Such a technique will be required for obtaining good numerical results also in upwind-type finite element methods. As for the evolutional problems au/at+ 6 . grad u-uAu= f

,

a similar analysis to that of the stationary problems is possible by using lumping techniques in approximating the term &/&. (See Fujii [8].) Some results are shown in Tabata [31] and Ikeda [15]. For the use of characteristics in finite element methods we refer to Bercovier et al. [2]. For nonlinear problems there is much left to be done. In Tabata [32] an extension to finite element methods of Engquist-Osher nonlinear upwind technique [26] for finite difference method is presented by the use of the circumcentric domains. Upwind finite element schemes for Navier-Stokes equations

354

M. TABATA

w e refer t o Bristeau e t al. [ 5 ] , Thom a sse t [37], and their references. A bifurcation analysis of an upwind sc he m e f o r Navier-Stokes e q u a t i o n s is discussed in Girault and R a v i a r t [ 9 ] .

References [ 1 ] K. Baba and M. Tabata, On a conservative upwind finite element scheme for convective diffusion equations, RAIRO, Anal. NumCr., 15 (1981), 3-35. [ 2 ] M. Bercovier, 0. Pironneau and V. Sastri, Finite elements and characteristics for some parabolic/hyperbolic problems, Appl. Math. Modelling, 7 (1983), 89-96. [ 3 ] J. M. Bony, Principe de maximum dans les Cspaces de Sobolev, C. R. Acad. Sci. Paris, 265 (1967), 333-336. [ 4 ] J. H. Bramble and B. E. Hubbard, New monotone type approximations for elliptic problems, Math. Comp., 18 (1964), 349-367. [ 5 ] M. 0. Bristeau, R. Glowinski, B. Mantel, J. Periaus, P. Perrier and 0. Pironneau, A finite element approximation of Navier-Stokes problems. Approximation Methods for Navier-Stokes Problems, Lecture Notes in Math., 771 (ed. R. Rautmann), Springer, 1980, 78-128. [ 6 ] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [ 7 ] P. G. Ciarlet and P. A. Raviart, Maximum principle and uniform convergence for the finite element methods, Comput. Methods Appl. Mech. Engrg., 2 (1973), 17-31. [ 8 ] H. Fujii, Some remarks on finite element analysis of time-dependent field problems. Theory and Practice in Finite Element Structural Analysis (eds. Y.Yamada and R. H. Gallagher), Univ. Tokyo Press, 1973, 91-106. [ 9 ] V. Girault and P. A. Raviart, An analysis of upwind schemes for the NavierStokes equations, SIAM J. Numer. Anal., 19 (1982), 312-333. [lo] J. C. Heinrich, P. S. Huyakorn, 0. C. Zienkiewicz and A. R. Mitchell, An ‘upwind’ finite element scheme for two-dimensional convective transport equation, Intern. J. Numer. Methods Engrg., 11 (1977), 131-143. [ l l ] J. C. Heinrich and 0. C. Zienkiewicz, The finite element method and ‘upwinding’ techniques in the numerical solution of convection dominated flow problems. Finite Element Methods for Convection Dominated Flows (ed. T. J. R. Hughes), ASME, New York, 1979. [12] T. J. R. Hughes, A simple scheme for developing ‘upwind’ finite elements, Intern. J. Numer. Methods Engrg., 12 (1978), 1359-1365. [13] T. J. R. Hughes and A. Brooks, A multi-dimensional upwind scheme with no crosswind diffusion. Finite Element Methods for Convection Dominated Flows (ed. T. J. R. Hughes), ASME, New York, 1979,19-35. [14] T. Ikeda, Artificial viscosity in finite element approximations to the diffusion equation with drift terms. Mathematical Analysis on Structures in Nonlinear Phenomena (eds. H. Fujita and M. Yamaguti), Kinokuniya, Tokyo, 1980,59-78. Maximum Principle in Finite Element Models for Convection-Diffusion 1151 -, Phenomena, Lecture Notes in Numerical and Applied Analysis, 4, NorthHolland, 1983. [ 161 A. M. Win, Differencing scheme for a differential equation with a small parameter

Upwind-Type Finite Element Methods

355

affecting the higher derivative, Math. Notes, 6 (1969), 596-602. 1171 T. Iwaki, Comparison of FEM and triangular F D M in heat conduction problems, Theoretical and Applied Mechanics, 23, Univ. Tokyo Press, 1975, 279-288. 1181 C. Johnson and U. Navert, An analysis of some finite element methods for advection-diffusion problems. Analytical and Numerical Approaches to Asymptotic Problems in Analysis (eds. 0. Axelsson et al.), North-Holland, 1981, 99-116. H. Kanayama, Discrete models for salinity distribution in a bay: conservation laws and maximum principle, Theoretical and Applied Mechanics, 28, Univ. Tokyo Press, 1980, 559-579. [201 T. Kawamura, H . Takami and K. Kuwahara, New higher-order upwind scheme for incompressible Navier-Stokes equations, in preprint. 1211 F. Kikuchi, Discrete maximum principle and artificial viscosity finite element approximations to convective diffusion equations, ISAS Report No. 550, Inst. Space Aeron. Sci., Univ. Tokyo, 1977. 1221 F. Kikuchi and T. Ushijima, Theoretical analysis of some finite element methods for convective diffusion equations, Finite Elements in Fluids, 4 (eds. R. H. Gallagher et al.), Wiley, 1982, 67-87. 1231 P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Element Methods in Partial Differential Equations (ed. C. De Boor), Academic Press, 1974, 89-123. [241 R. H. MacNeal, An asymmetric finite difference notwork, Quart. Appl. Math., 11 (1953), 295-310. 1251 A. R. Mitchell, D. F. Griffiths and A. Meiring, Finite element Galerkin methods for convection-diffusion and reaction-diffusion, Analytical and Numerical Approaches to Asymtotic Problems in Analysis (eds. 0. Axelsson et al.), North-Holland, 1981, 157-174. 1261 S. Osher, Nonlinear singular perturbation problems and one sided difference schemes, SIAM J. Numer. Anal., 18 (1981), 129-144. 1271 V. Ruas, On the strong maximum principle for some piecewise linear finite element approximate problems of non-positive type, Rapports de Recherche, INRIA, No. 43, 1980. M . Tabata, A finite element approximation corresponding to the upwind finite differencing, Mem. Numer. Math., 4 (1977), 47-63. -, Uniform convergence of the upwind finite element approximation for semilinear parabolic problems, J. Math. Kyoto Univ., 18 (1978), 327-351. -, Some applications of the upwind finite element method, Theoretical and Applied Mechanics, 27, Univ. Tokyo Press, 1979, 277-282. -, L”-analysis of the finite element method, Numerical Analysis of Evolution Equations (eds. H. Fujita and M. Yamaguti), Kinokuniya, Tokyo, 1979, 2562. -, Conservative upwind finite element approximation and its applications, Analytical and Numerical Approaches to Asymptotic Problems in Analysis (eds. 0. Axelsson et al.)., North-Holland, 1981, 369-381. 1331 __ , Symmetric finite element approximation for convection-diffusion problems, Theoretical and Applied Mechanics, 33, Univ. Tokyo Press, 1985, 445-453. A numerical algorithm for a n upwind‘-type finite element method using 1341 -, exponential functions, to appear in Theoretical and Applied Mechanics, 34.

356

M. TABATA

[351 R. Teman, Navier-Stokes Equations-Theory and Numerical Analysis, NorthHolland, 1977. [361 F. C. Thames, J. F. Thompson, C. W. Mastin and R. L. Walker, Numerical, solutions for viscous and potential flow about arbitrary two-dimensional bodies using body-fitted coordinate systems, J. Comput. Phys., 24 (1977), 245-273. [371 F. Thomasset, Implementation of Finite Element Methods for Navier-Stokes Equations, Springer, 1981. Department of Information-Mathematics The University of Electro-Communications Chofugaoka, Chofu-shi Tokyo 182, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 357-367 (1986)

Positive Solutions to Some Semilinear Elliptic Equations in L1(R") By Masaharu A R A Ia n d Akira NAKAOKA Abstract.

A semilinear elliptic equation

Au=&)l.(u)

-m

in the whole space R" is treated. The coefficient a(x)is assumed to be positive and bounded and p(u) is assumed to satisfy the conditions (@1)-(@3) in the text. We give a necessary and sufficient condition on a(x) for this equation to have a unique non-negative solution in L1(Rn)for any non-negative f ( x ) in L1(R"). The result is extended to the equation

.

Au=@(x, u ) - ~ ( x )

Key words: semilinear elliptic equation, positive solution 1. Introduction

We shall study a non-negative solution in L 1 ( R n to ) a semilinear equation ( E)

Au=a(x)cp(u)-f(x)

(xE

R")

where we assume that O
y(u)= 1-e-"

as a n auxiliary tool of the study of the asymptotic behavior of some chemical reaction system. (This equation will be denoted by (E,).) They gave a sufficient condition on a ( x ) for the equation (E,) to have a unique non-negative solution in L 1 ( R n f) o r any positivef(x) in L 1 ( R * ) :

Theorem 0. If the Lebesgue measure of { x ;a ( x ) < c } with some positive constant c isfinite, then the equation (E,) has a unique non-negative solution in L 1 ( R n ) for any non-negative f in L 1 ( R n ) . They also gave a sufficient condition on the pair { a , f}for the non-existReceived March 29, 1985. Revised December 25, 1985.

M. ARAI and A. NAKAOKA

358

ence of positive solutions in L1(Rn). But there is a large gap between the two conditions. On the other hand, BBnilan, Brezis and Crandall [ l ] study a positive soiution in Maricinkiewicz space iW'(n-2)(Rn) and in L1(Rn) to the equation (E) with constant a. We note that they show, among other things, that if cp(u) satisfies the conditions (0-1) and (0-2) below and a is constant then the equation (E) has a unique non-negative solution in L1(Rn)for any non-negative f in L1(R"). Their study in Mn/("-*)(Rn) is extended by Gallouet and Morel [2] to the case where the non-linear term depends also on x , but the solutions are not necessarily in L1(Rn). The aim of this paper is to give a necessary and sufficient condition on a ( x ) for the equation (E) with ~ ( usatisfying ) the conditions (0-1)-(0-3) below to have a unique non-negative solution in L1(Rn) for any non-negative f in L'( R"). In what follows we assume that

(A-1)

U(X)€

Lm(Rn),

(A-2) f ( x ) is non-negative and of class L1(Rn), and that

(0-1) y(u) a is monotone increasing function of class c~(R,) , (a-2) cp(O)=O and c p ' ( O ) f O , (0-3) y'(u) is bounded in

E+.

We mean a solution u(x) to be non-negative and of class L1(Rn)satisfying (E) in the sense of distribution. Let k , ( x ) be the inverse Fourier transform of and K , be the convolution operator kc*,which is the inverse of (c2-A). Our main theorem is Theorem 1. Assume (A-l), (A-2), ( 0 - l ) , (0-2) and (0-3). The equation (E) has a solution for any f ( x ) if and only if

(A)

inf (K,a)(x)>O

for some

c>O

.

X

Remark 1. If the condition (A) is valid for some positive c, then it is so for any positive c . We use the notations I/ 11 and 11 I(_ to denote the L1 norm and L" norm, respectively. For a condition P ( x ) of x , E(P) denotes the totality of x satisfying P ( x ) , e.g. E ( u > E ) = { x E R " ;a ( x ) > & } . B ( z ; r ) and B ( z ) denote the closed balls in Rn with radius r and center at z and at the origin, respectively. For the measurable set S in Rn, mS denotes its Lebesgue measure. Remark 2.

As for the problem in a bounded open set, see the review

Semilinear Elliptic Equation in L1(R")

3 59

article by P. L. Lions [3] and the abundant list of references there. One would be able to regard the equation (E) as a model of the following situation: a system of a reaction of a n enzyme of consistency a(x) and a substance of consistency u(x), which is constantly supplied by the amount of f ( x ) per unit volume per unit time. If the reaction obeys the MichaelisMenten law, the substance is diffusible and is not a n enzyme and the system reaches a stable state, then the stable state would be given by the equation (E) with

y(u)=u/(u+K)

( K : positive constant) .

2. Preliminaries

As is well known, the function k,(x) has the following properties: Lemma 2.1. (K-1) k,(x) depends only on 1x1 and k,(xj>O (K-2)

\

.

k,(x)dx=l/c2 .

(K-4) k,(x)lM(n, c) exp [ - c l x l / l / 2 ] I x l 1 - " , where M(n, c ) is a constant depending only on n and c. To study the equation (E) we use the method of iteration: (2.1)

Au,

- C'U,

= ~ ( x ) P ( u , _ ~-)c ~ u , --J(x) ~ ,

u,(x)=O

,

which is equivalent to (2.2)

u, =Ke(c2u,-1-ay(u,- 1 ) ) +KJ , uo=O

.

In what follows we choose c so large that c 2 u - a ( x ) ~ ( u ) is a monotone increasing function of u, which is possible by (A.l) and (0-3). Then it is easy to see

Lemma 2.2. 0 < u,(x)

U , + ~ ( X ) E L'(R").

Arguments similar to those in [4] show

Lemma 2.3. The equation (E) has a solution and in this case the solution is given by (2.3)

u(x)=lim u,(x)

.

if and onZy if { Ilu,l\}

is bounded

M. ARAI and A. NAKAOKA

360

Lemma 2.4.

The solution to (E) is unique if it exists.

Lemma 2.5. Let (pl and ‘pz satisfy the assumptions (@-1)-(@-3) and p l l y z . If the equation (E) with v=pI has the solution then so does equation (E) with p=pz.

Proof. Let {uj,rn}be the sequence defined by (2.2) with p replaced by pj ( j = l ,2). Then we have ul,rn-uZ,rn=K,{a(pz(uz,rn-l)-pl(uz,m-~)}

+ ~ e ~ l ~ 2 ~ ~ , m - t - ~ ~ ~ ( ~ ~ , m - ~ ~ ~ -9 ~ ~ 2 ~ ~ , m - ~ - ~

which is positive by the assumption of the induction so that Lemma 2.3 yields the result.

3. Proof of the “if” Part By virtue of assumptions (0-1)and (0-2), ‘p(u) is bounded from below by a function p1 satisfying

(0-4) p(u)=C,u for small u and p(u)=C, for large u (Cl, Cz: positive constants) with p replaced by

‘pl and

(0-5) monotone concave and smooth.

By virtue of Lemma 2.5 we may assume (0-4) and (0-5) without loss of generality in this section. Note that these assumptions imply

(3.1)

[cp’(O)-’p’(u)]/y(u)

is bounded in

and

(3.2)

0 5 ‘p’ ( u ) uIp(u) I p’(0)u

I

Define K , by (2.2) and u by (2.3), which is not yet known to be in L‘ or not. By virtue of Lemma 2.3 it suffices to show that { ~ ~ K , I I } is bounded. Lemma 3.1.

and

I t holds that

Semilinear EIliptic Equation in L1(Rn)

361

(3.5)

Proof. Integrating (2.2) over R”,we have

c-“

\

\

15

f ( x ) d x- a(x)p(u,-,(x))dx = u,(x)dx-

s

u,-,(x)dx

,

which is non-negative by Lemma 2.2 so that we have (3.3). The identity (2.2), Lemma 2.2 and (3.2) show (3.4). Integration of (3.4) over R n gives (3.5). Q.E.D.

Lemma 3.2. Zf g E L”(R”) then ( K , g ) ( x )is continuous. If g E L1n L” then (K,g)(x)tends to 0 as x tends to the infinity. Proof. I (K,g)(x’)-(K,g)(x)I x since k , is in L’. For any positive E , we have (3.6)

I(K,g)(x)/S

1

5

Ik,(x’)-k,(x)ldxllgl1, tends to zero as x’+

k(x-y)lg(y)ldy+

E(l&lZE)

s

k,(x-Y)Ig(Y)ldY

The second term in the right hand side of (3.6) is estimated by term of the right hand side of (3.6) is estimated by

s

k,(x -Y) I g(Y)I dY B(z;IXI/Z)~

+

s

*

E(ISli8)

E/c~.

The first

~ c ( X - Y ) ~ Y l l ~ I l 9. a

B ( X ; I z 112) n E i 181Z E )

whose first term is bounded by the value of k , ( y ) at lyl = lxli2 times llgll which tends to zero as x tends to the infinity and whose second term equal to 5 B ( 1 2 1 , 2 ) ,,~l,i,,,k,(y)dyllg1I, which tends to zero as x tends to the infinity since mE( l g l 2 E ) is finite. Q.E.D.

Lemma 3.3.

If (A) holds, then q-sup (Kc(c2-ap’(u))(x)<1 z

.

Proof. Note that q l l . We put g=c2-up’(u), g,=c2-p’(0)a(x) and Now we have g2(x)=a(x)(cp’(0)--cp’(u(x)). Then g , , g 2E L” and g = g , + g , . sup ( K , g , ) ( x = ) 1-y’(O) inf (K,a)(x)< 1 z

X

by the assumption (A). Next, since g , E L 1 n L - by virtue of (3.1) and (3.3), Lemma 3.2 shows that lim (K,g,)(x)=O .

M. ARAIand A. NAKAOKA

362

Hence, if 9-1, it must be the maximum, say at x = x o . Then

so that a(x)p’(u(x))=O for almost all x, which with (3.1) and (3.3) shows a ( x )€ L 1 . Then Lemma 3.2 and the assumption (A) contradict each other.

Q.E.D. Proof of the

“if”

part of Theorem 1. Lemma 2.2 and the concavity of

p(u) implies c2- a(x)p’( u,(

x)) Ic2-a(x)p’( u(x)) .

Multiply (3.4) with this inequality side by side and integrate it to obtain

4.

Condition (A)

The condition (A) describes a global property of a(x). In this section we give local versions of the condition (A), which will be convenient to prove the “only if” part of Theorem 1. We introduce the following two propositions (B) and (C) on a ( x ) :

(B) For any E > 0, there exists a triplet of sequences ({xp}, b p } ,

{ell})

such that ( i ) x p E R n + m as p + m , ( i i ) ap>O and ap+m as p-’m, (iii) e p is a measurable subset of B ( x p ;a p )whose Lebesgue measure tends to zero as p tends to the infinity and satisfies

Semilinear Elliptic Equation in L’(R”)

(4.1)

B(x,; a,)\e,cE(a<e)

( C ) There exist positive constants (4.2)

E,

363

.

and ro such that

m ( B ( x ;r,) n E(a > e,)) 2 1

for any x € R”

.

We denote by -(B) the negation of (B).

Theorem 2.

The propositions (A), -(B) and ( C ) are mutudy equivalent.

Proof will be decomposed into a series of lemmas. Lemma 4.1.

(A) implies -(B).

Proof. Assume (B). Then we have

which is estimated by 2e for large p . Lemma 4.2.

-(B) implies ( C ) .

Proof. Assume that (C) does not hold. sequence { x , } in Rn such that

Then for any e > O , there exists a

m(B(x,; p ) n E(a > 4 )< 1

.

Now consider the balls with radius p‘ (Oe)) tends to zero. If the sequence {x,’} accumulates at a finite point, say x o , then B ( x o ; p u / 2 ) c B ( x , ’ ; p ufor ) large p . Let a,=p/4, x , be a point satisfying Ix,-x,I <pa/4 and e,=B(x,; a,)n E ( ~ > E ) . Then a triplet of sequences ( { x , } , {a,}, {e,}) satisfies the condition (B). If a sequence {x,’} has n o accumulation point, (B) is already valid. Lemma 4.3.

( C ) implies (A).

Proof. Assume ( C ) . Then (K,u)(x)is estimated from below by k , ( x - y ) d y 2Eok0(X)II I I =r 5.

*

Proof of the “only if” Part of Theorem 1 By virtue of the assumptions ( 0 - 2 ) and (0-3), ~ ( uis) estimated by C,u with

M. ARAI and A. NAKAOKA

364

some positive constant C, so that by Lemma 2.4 it is sufficient to show the existence of somef to which the equation (E) with p ( u ) = C , u has n o solution. Replacing C,a with a , it is sufficient to show

Assume

Lemma 5.1.

inf (K,a)(x)= O

(5.1)

z

.

Then there exists (not necessarily non-negative) f ( x )E L1(Rn)such that the equation (5.2)

(-A+a(x))~(x)=f(x)

has no solution in L 1 ( R n ) . Indeed, let f be as in the above lemma and f*=max (kf, 0). Then at least one of the equations (5.2) with f replaced b y f , has no solution so that we have the "only if" part of Theorem 1. Proof of Lemma 5.1. Assume (5.1). Then, by Theorem 2 and the diagonal argument, (B) holds with E replaced by l/p. Let v,(x) be the characteristic function of the ball B(x,; a,) multiplied by the inverse of its measure so that I1upll=l. Then the L' norm o f f , ~ [ Z - ( ~ ~ - a ) K , ]isu ,estimated by (5.3)

s

I v,(x) --c2(K,v,)(x)Idx+ B I Z p ; a,)

+(UP)

s

5

l~,(X)-CZ(K,~,)(X)l~X B ( z p :& , ) C

I.,

(Kcv,)(4dx+c2 B ( z p ;a,)

(K,up)(x)dx

(K,u,)(x)dx ,

f C Z j

B(zp;apC)

where we choose c so large that c 2 2[lullm. Since v , ~ c 2 ( K c u , ) ( xin) B(x,; a,), the first term of (5.3) is equal to

=cz\

(K,v,)(x)dx B(zp;ap)C

so that the right hand side of (5.3) is estimated by (5.4)

3c2

\

+

+

(K,v,)(x)dx ( l/p)c-2 rne,/mB(a,) BIZ,; u p C )

.

Lemma 5.2 below shows that the first term of (5.4)tends to zero as p tends to infinity. Thus j p tends to zero while IIvpl/=l so that the operator [ I (c2-a)K,] from L1 into L1 is not bijective and so the operator

Semilinear Elliptic Equation in L1(R”)

-A

+u =c2

-

A - (c’- a ) = [ I -

365

(cZ- u)K,..(c~ - A)

from {u € L’; Au E L 1 }into L1 is not bijective. Let u be a solution of the equation

(-A+u)u=O.

(5.5)

Then we have (cz-A)u=(c?-a)u and (5.6)

for c2>ess supa(x). Integration of (5.6) shows jlaull=O, which with (5.5) shows Au=O and so u=O. Thus -A+a is injective so that it is not surjective, which completes the proof. Lemma 5.2.

For any z € R n and R > 0, we have

Z(z, R ) = \

k,(x-y)dy
dx\ B(z;RIC

B ( z : n)

where a(R) is independent of z and lim,,,o(R)=O. Proof. The proof of the case of n = l is similar to and simpler than that of the case of n 2 2 . So we prove only the case of n 2 2 . By virtue of (K-4), we have

I( z, R) I Const.,R n+ I

I -yll-ndy.

e-clz-Yl/J2

Change the variables to x’=c(x-y)/(&-R), y’=c(x-y)/(z/TR) and omit the primes to obtain (5.7)

sB(.f;c

Z(z, R ) ~ C o n s tRn+’ .

where we put c ‘ = c / d / T .

e-Rlvllyll-”dy , dx \ B ( % ; c f )

Since the second integral in the right hand side of 0 , x ) and take the polar coordi-

(5.7) depends only on 1x1, we put x=(O,

.-.,

nates in x and y to obtain (5.8)

I ( z , R)
ede\8+e-Rcdr, 8-

where s + = r cos 6 + l / d z - r z sin213 and sin Bo=c’/r. The last integral in the right hand side of (5.8) is estimated by R - l c R a - . Change the valuable 0 to s=s-. Then sin 0de=r-1[(r2-c’z)/(2s2)- 1/2]ds< (r2-c’2)/(2rs2)ds

M. ARAIand A. NAKAOKA

366

6.

Additional Remark The arguments in the preceding sections can be applicable to the equation

Au=@(x, u ) - ~ ( x ) ,

(El'

where @(x,u) is a function on R n x K such that (6.1)

(6.2)

,

u)

and

@(x,O)=O

0 I@Jx, u)

and

@,Ax,u) I0 ,

OI@(X,

a(x)=@,(x, O ) € L " ,

(6.3)

and there exists a positive constant C such that

(6.4) Theorem 3. by ( E)' holds.

[ @ A x ,O ) - @ J X , u)l/@(x,u) Ic

.

Under the assumptions (6.1)-(6.4)Theorem 1 with (E) replaced

Note that (6.4)implies

(6.5)

@ ( x ,u) >a(x)(1- e - c u ) / C

.

Replace @(x,u ) in (E)' with the right hand side of (6.5) and then replace Cu and Cf with u and f, respectively, to obtain (Eo). Hence, by Lemma 2.4, as for the existence of the solution, it suffices to consider (Eo).

Examples. Let a ( x ) and b ( x ) be positive bounded functions. ( i ) @ ( x ,u ) = a ( x ) ( l - e e - b ( z ) usatisfies ) the conditions (6.1)-(6.4) with C= sup b ( x ) . ( i i ) @ ( x ,u)=a(x)u/(l+b(x)u)satisfies (6.1)-(6.4)with C = 2 .

Semilinear Elliptic Equation in L'(R")

367

References [ 1 ] P. BCnilan, H. Brezis and M. G. Crandall, A semilinear equation in LL(RN), Ann. Scoula Norm. Sup. Pisa, 2 (1975), 523-555. [ 2 ] T. Gallouet and J.-M. Morel, Une equation semi-IinCaire elliptique dans L1(RN), C.R. Acad. Sci. Paris, 296 (1983), 493-496. [ 3 ] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. [ 4 ] M. Mimura and A. Nakaoka, On some degenerate diffusion system related with a certain reaction system, J. Math. Kyoto Univ., 12 (1972), 95-121.

Masaharu ARAI Faculty of Economics Ritumeikan University Kyoto 603, Japan Akira NAKAOKA Department of Mathematics Kyoto Institute of Technology Kyoto 606, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 369-383 (1986)

On the Vlasov-Poisson Limit of the Vlasov-Maxwell Equation By Kiyoshi ASANOand Seiji UKAI Abstract. The Vlasov-Maxwell equation is shown to converge to the Vlasov-Poisson equation at the limit of the infinite light velocity. The initial layer and the asymptotic expansion of solutions as the light velocity tends to infinity are discussed in detail. Key words: Vlasov-Maxwell equation, Vlasov-Poisson equation, VlasovPoisson limit, initial layer, asymptotic expansion

1. Main Results

The change of the density distribution of charged gas particles is described by two types of equations: The Vlasov-Maxwell equation and the VlasovPoisson equation. The latter describes the motion of plasma when the magnetic field generated by the plasma is small. The purpose of this paper is to study the relation between the two equations. Roughly speaking, the solution of the Vlasov-Maxwell equation converges to the solution of the VlasovPoisson equation when the light velocity tends to infinity. Letf,=fi(f, x, u ) be the density distribution of charged gas particles of the type i ( i = l , 2 , -.-,N ) at time t 2 0 and position x E R S with velocity uERS. Let E = E ( t , x) and B = B ( t , x) denote the electric and magnetic fields generated by the charged gas particles. The Vlasov-Maxwell equation is,

(1.1)

supplemented by Received May 29, 1985. Revised July 20, 1985.

K. ASANOand S . UKAI

370

(1.2)

VZ-E=4nKf,

V,-B=O.

and x are the scalar and vector products in R3,V, is the gradient in v, while c is the light velocity and ai=qilm,, qi and mi being the electric charge and mass of a single particle of i-species. Further, Jf and K f are the current and charge densities generated by f=(fi, f 2 , fN); Here

x and V , in

..-,

(1.3)

Notice that (1.2) can be deduced from (1.1) for t > O if it is satisfied at t = O , i.e., if the initials f o = ( fi,o,f2,0, -,f,,,) and (Eo,B,) satisfy

--

(1.4)

V,.Eo=4nKfo,

VZ.Bo=O.

This is a physically reasonable compatibility condition and will be assumed throughout the paper. On the other hand, the Vlasov-Poisson equation is,

(1.5)

This equation is formally obtained from (1.1) and (1.2) by setting B=O or c = w . With the notations and function spaces defined by (1.19)-(1.23) below, our main results are stated in the following three theorems.

Theorem 1.1. (1.6)

Let 123, a 2 0 , p 2 0 , P E R , and suppose ft,o€fC,p,p

(1CiSN)

9

(Eo,Bo)CH'

.

Suppose further that (1.4) is fulfilled. Then, there exist positive constants C, T , r and the following holds. ( i ) (Uniform existence) For each C E [ l , cu), (1.1) has a unique classical solution f=(fl,f 2 , f ) and u=(E, B ) on the time interval [0,TI such that

-..,

Vlasov-Poisson Limit of the Vlasov-MaxwellEquation

IUI 1 , T I I uo I L + Clf o 1 1 . 0 , p. B.

(1.10)

371

*

( i i ) (Continuity in c ) A s functions of c, f and u satisfy 1

[A.llF,b

f~

nM w , m); c m ,TI; w , ~ - J )

j=O

Thus, the solutions exist on the interval [0, T ] independent of c € [ l , m) and are continuous in c. Further, they have limits at c=oo which we call the Vlasov-Poisson limits. More precisely, we show Theorem 1.2 (Existence of limit). Let f,u be as above. ( i ) They can be extended to [ 1, 001 as functions of c so that [~.31:,~ [A.4IL

f € B O ( [ l001; , CNO, TI; f C : ; , p - J )

7

v E >o

9

u € B O ( [ la, 1 x [O, Tl\I(w, 0)l; BL-2(R3)) ,

holds good, and in particular, they converge as c--rm in the topology indicated here. W e write the limits as f"=(fT,fy,

- - - , f F ),

u"=(E", B")

.

( i i ) Bm=O while (f",E") is a unique solution to the Vlasov-Poisson equation (1.5) on [0, TI satisfying (1.11)

(1.12)

(f"7

Ern)€Co([O,TI; f f b , p , p X A L t ' )

3

I f m l l . o l p , B I T ~ ~ l f o l L , o . ~ 7, B

IIE"llo,T+

lV.J"lz-i,T<

Clfolz,o.p,p

*

Observe from [A. 3-41 that as c+m, f converges to f" uniformly in I on [0, TI while u converges to u" uniformly on [6,TI for any 6>0 but not uniformly on (0, TI. Physically, this implies the development of initial layer. Also, comparing (1.7) and ( l . l l ) , we see that the limit f" which is a solution to (1.5) belongs to a better function class thanf, the solution to ( l . l ) , and similarly for E" and E. Finally, we shall discuss a n asymptotic expansion which is somewhat complicated due to the presence of initial layer. To simplify the notation, we introduce the operators L , A , A defined by

(1.13)

L ( f , E ; f ',E') = (ad,+ v.V,f,+a,E'.V,f,+a,E.V,f,'),N_, Af=(-4~Jf, 0) , Au=(V,XB, -V,XE).

,

K. ASANOand S . UKAI

372

The first of these is a n N-dimensional vector while the remaining two are 6dimensional. We seek the expansion of the form k

f=f"+ (1.14)

cc-Y

j=O

,

k

u=u-+C c-'uj , j=O

where f", u" are as in Theorem 1.2. The coefficients f j = ( f : ' , f ; , -..,fi,), u j = ( E f ,Bj) still depend on c. We wish to determine them as solutions of the following equations whose derivation will be described in S 5 . Note that the 0-th term of the expansion (1.14) is ( f " + f o , u"+uO). The equation for (fo, uo) is the nonlinear equation,

L(f0,EO;f",E"+EO)=O , acUo-CAUo=PIAf "+ / I f o , f"c=O=o, u"c=o=P,uo ,

(1.15)O

where P I is the projection defined in S 3 (see the remark below (3.3)), and u, is the same initial as in (1.1). The equation for ( f j . u j ) , l < j < k - 1 , is the linear inhomogeneous equation,

j-1

+~1 x W

Fj = - (LY, C (I? * 0,fi -' 7=1

*

V&-')

+L Y ~ Ux Bo- 0,fi

-I):=

,

and the equation for (fk, uk)is the nonlinear one just obtained by substituting (1.14) into (1.1) and taking account of ( 1 . 5 ) and ( 1 . 1 5 ) ~ ,OCjLk-1:

where F k is a given function of (fj , uj), 0 1j < k- 1 and their derivatives, and V , c. The expansion (1.14) is verified by the Theorem 1.3 (Asymptotic expansion). and 1.2, let O
Under the situation of Theorems 1.1

Vlasov-PoissonLimit of the Vlasov-Maxwell Equation

313

( i ) The equations (1.15) can be all solved successively, uniquely on the same time interval [0, TI as in Theorem 1.1 and uniformly for each c E [ l , w ) , with the solutions,

fj€[A.l];$, , f k E [A. 11;~/-8 uj E [A.2IL-f,

(1.16)

O<j
t

O<j
where [ A . l]+,petc., are the function classes in Theorem 1 . 1 . With these ( f j , uj), (1.14) holds. Note from (1.16) that we can take r = O for O < j < k - l and that (f', u') are all uniformly bounded in c. ( i i ) As c+w, we have, (1.17)

(fj,

More precisely, then,

f

if (f

j,

u j ) are

E [A.31k~j-j9

O<j
u+O,

extended to [ 1, O<j
031

as functions of c with 0 at c= co, fke[A.3]:;j-k ,

uoE [A.4IL,

u j € B O ( [ l ,m]x[O, T ] ; B L - ~ - 2 ( R,3 ) ) l < j < k

.

(iii) Suppose, in addition to the assumption of Theorem 1.1, that 0 2 2 and Then, we can strengthen the convergence (1.17) as

uoE Hi.

(1.18)

If 11L-3-2,,,,p.p-j-2,
7

0 5j
,

l<j
,

9

where d=C(I f o l L , , p , p + IIuollL), C being a positive constant depending only on I , 0 , p, ,E. A similar estimate is also available for j = k . Remark. After completion of this work, we learned that Degond [5] had proved a similar result under slightly different conditions. However, his asymptotic expansion is derived rather formally. The rest of the paper is devoted to the proof of Theorems 1.1-3. Since most of materials and tools for the proof are provided in [ l , 2, 71, we will give a n outline only, supplementing some technical results missing there and needed here. In the next two sections we discuss linear problems associated with (1.1) while in S 4 we solve ( l . S ) , and finally in S 5 we will prove Theorems 1.1-1.3. Now we shall state the definition of the function spaces used in Theorems Intro) the Sobolev space of order 1, with the norm 1.1-3. H 1 = H L ( R a is duce the weight function

K.ASANOand S. U K A ~

374

Let 8 be a (possibly closed) domain in Rm and Y be a Banach space with Cc(8; Y ) (resp-ML(8;Y ) ) will denote the space of Y-valued the norm I.IY. functions h(y) on 8 which are strongly continuous (resp. strongly measurable) on 8, together with derivatives a:h, l a l s l . We set B c ( 8 ;Y ) = C L ( 8 ;Y ) n M c ( Q ;Y ) . When Y = C m , we drop Y; e.g., Bc(R3)=BL(RS; C m ) ,etc. Note that if Q is compact, then B c ( 8 ;Y)=Ci(Q;Y ) . B L ( Q ;Y ) and M c ( Q ;Y ) are Banach spaces with obvious norms. We write the norm of Bc(R3)as [I. Ill, and when I=O and Q=[O, TI, we set,

~ Bc(R3),we set, Further, when Y is H L ,H h , p ,or

(1.21)

These norms will also be used when 8=[1, m]x[O, TI in which y = ( c , t ) moves. The space C{([O,T I ; Hj,P,a)which plays a crucial role in this paper is defined by the set of functions f=f(t, x, u ) such that, (1.22)

$,,,p-Tt,&a:8:'f ~fll,u,p,,B,).,T=

€ Co([O,T I ; L2(RB)) , O
~ f ( ~ ) ~ L , o , p - ) . C 9, , b

Olrlj, (j=o)

la1

+ la'l < I

,

.

We have seen in [l] that to treat the factor u x B of (l.l), we must take r>O. Finally, we also need the spaces, (1.23)

f f = { u € B o ( R 3 ) ( V , u €H L - l ),

Hb={u I (1+Ixl)"~",u(x)€L2(R3),lalll} ,

with due norms. 2. The Linear Vlasov Equation In this section we solve,

375

Vlasov-Poisson Limit of the Vlasov-MaxwellEquation

where u=(E, B ) , f, are given functions. Suppose, first, that u is in the class [A. 2],, 123, and f oeC;+I(R0)(compact support). Then, (2.1) can be solved easily by the aid of the characteristic equation associated with it. We write the solution as

f = U ( t ,s; 4 clfo

(2.2)

9

where U ( t ,s; u, c ) is the evolution operator to (2.1) (see [l]). Since f, is now assumed to be of compact support, f is also of compact support in x, u (cf. Lemma 2.6 of [l]). Noting this, we see readily that f € M ’ ( [ l ,m); Co([O,T I 2 ;Hk;;,!-,)n C’([O, T I 2 ;Hi;,!,j?j-I))

(2.3)

,

for O < j < l . Proceeding as in the proof of Lemma 2.5 of [I], we further get, I f ( t , ~ ) l n , , , p - 7 1 t - s ~ , p ~ ~ b ~ f - s ~ l f o l ,r , o , pO. ~< k < l

(2.4)

,

where u, ,020 and (2.5)

b,, c, being positive constants depending only on 1. in H:,p,p,we have thus proved the

Since Citl(RO)is dense

Let 123, 0 2 0 , p>O and / 3 R~ . Suppose [A. 21, for u a n d f , € Let y, T be as in (2.5). Then f of (2.2) is a unique solution to (2.1) in [ A. 1 1 ; ~(modified by [O, T I 2 , # o , p - 7 1 t - s l , p in place of 10, TI, #o,p-7r.p7 see (1.19, 22)). Also, (2.4) holds which is a uniform estimate in c .

Lemma 2.1. H:,p,p, O
Recall that u=u(c)=u(c, t ) is a function also of c, and write (2.2) as f ( c , t ) = U ( t ,s; u(c), c)fo

Let l
, (with

s fixed)

.

It follows from (2.1) that

f ( c , t)-f(c’, I ) =

s:

U ( t ,r ; u ( c ) , c ) ~ ( cc’, , r)dr ,

with g(c, c’, r ) = - a { E ( c , r)-.E(c’, r ) +ux( B ( c,r)/c-B(c’, r)/c’)}.V,f(c’, r )

Applying (2.4), we get,

.

K. ASANOand S. UKAI

376


\'

(IIE(c, r)-E(c', r)llo+ IlB(c, r)lln/c+ IIB(c', r)lldc')dr

.

8

Consequently, if U E [A. 4J2,f(c, t ) is continuous in c and converges as C ~ uniformly on [0, TI, both strongly in HZ,p-Tlt-sl,B-l. By virtue of (2.4) and the interpolation theorem, this is also the case in H&,-rlt-sl,-B-c, for any E > O . Thus, f(c, t ) is strongly continuous on [l, w]x[O, TI in this space and the l i m i t f ( a , t ) exists. This proves the first part of the following lemma. Lemma 2.2. Under the situation of Lemma 2.1, suppose, in addition, that U E [A. 4J2. W e have, ( i ) With the modification indicated in Lemma 2.1, it holds that

fc [A.31:,8 . ( ii )

f(00, t ) is a unique solution to

(2.6)

4 J + v.V,f+E(

a) .V,f

=o ,

f It=,=fo .

) (iii) Moreover,f(oo, t ) E Co([O,T]2;H : , p , B satisfying

(2.7) with b=bo{(p+ IPI

lf(a,t ) l k , a , P , ~ I e a ' t - s ' I f o l r , o , p , B

9

+ l)lE(w)lL,T+ Ij+o.

For the proof of (ii) and (iii), we notice that (2.6) is a special case Of (2.1) with u=(E(oo),O). Then we can prove a n analogue to Lemma 2.1 for (2.6), in which we can take r = O because the term v x B is absent. On the other hand, it is clear that f(o0, t ) solves (2.6) as seen by passing to the limit in (2.1). This indicates that U ( t ,s; u, c ) of (2.2) has a limit which gives the evolution operator to (2.6). We write this operator as V(t,s; E ( m ) ) . Thus, (2.8)

f(m, t)= W , s; E(a)lfo.

Then, (2.7) gives a n estimate of its operator norm in H:,p,B. 3. The Maxwell Equation

Assuming that f is a given function, we seek a solution u = ( E , B ) to the inhomogeneous Maxwell equation, (3.1)

W

Vlasov-Poisson Limit of the Vlasov-Maxwell Equation

377

where A , A are as in ( 1 . 1 3 ) . First, it is well known that A generates a unitary group erAon H 1given by ;i-etAu 0--eztA(E)Li0(E),

(3.2)

U ~ = ~ (B,) E ~ (column , vector) ,

where t i o = . F u o is the Fourier transform of uorE being the dual variable to x, and A(E) is a 6 x 6 matrix given by

0

Ez

-E3

A ( [ ) is the symbol of the differential operator -iA. The matrix A(E) has eigenvalues 0, [ E l , -lei, each of multiplicity 2. Denote the corresponding eigen projections by Po([),P+(E),P-(E), and set Pl(E)= P + ( Q+P-(E). We have, Po(E)~o=~((SO ( B* Eo )-~~, ,F ) g=5/1EI

(3.3)

P,(E)=I-Po(E)

.

.

All these projections are also symbols of the singular integral operators of CaldBron-Zygmunt type (see [ 6 ] ) . We denote these singular integral operators by Po, P,,P - , P , . They are orthogonal projections in H 1 and it holds that

+

+

etA=Po+ etAP+ etAP-=Po etAP, .

(3.4)

Moreover, we have the expressions,

\

etAP+uO=(271)-3'2 e"z.'+t'~')Pi(~)lio(E)dE .

(3.5)

R3

Applying Lemma 5 . 1 of [ 2 ] to this, we have,

Lemma 3.1.

Z f u , ~Hi with 1 2 2 , then,

l l e t A P l u o ~ [ ~ - ~ ~ C ~ ~ + l ,t l ~ - lI l€uRo l,~ , ~

(3.6) where

(i)

- Ill,

I IllO denote the norms of B 1 ( R 3 )Hh , respectively.

( ii) Let uoE H 1 with 1 2 2 .

Then,

e t A u o E B o ( ( - m , a); H 1 ) n B o ( [ - a ,a ] ; B 1 - 2 ( R 3 ,) )

(3.7)

IletAP1u011,-2-0

(t+m)

.

Let us return to ( 3 . 1 ) . Its unique solution is given by

K. ASANOand S . UKAI

378

where g=g(c, t ) = A f ( c , t ) . Let 123, 020, p, 7, T>O, P E R with p-yT2p/2. Assuming [A. l]r,,9 and [A. 31f.p for f , we can easily see that I

[A.51:

,n

g c 3=0 M W ,

4;co([o,T1;H:-WnB0([1, m ] ~ [ o ,T];H;-? , VE>O.

Furthermore, we have, I ~ l ~ , s ~ ~ l f l ~ . o ,7 p , p , ~ . ~

and g ( ~ , f ) € M o ( [ O , T ] ; H ~ ) n B o ( [ O , T ] ; HVe>O ~ - r ) ,. Using this and noting that etn and P,, j=O, 1 are bounded operators on H i with the norm 1, we apply Lemma 3.1 to (3.8) to conclude the

Lemma 3.2. Let 1, u etc., be as above and assumefe [A. 1,3]f,@. Let u(c, t ) be given by (3.8). We have, ( i ) If u, € H [ , then K ( C , t) € [A. 211n [A. 4]', and

I~ l c T, I I uoIi +aTlfl[, 0 . p . 8.7, T

(3.9)

*

( i i ) Decompose K ( C , t) as K(C, t)=K"(f)+Ko(C,

t)+K'(C,

f)

,

where

Then it holds that

and that U"(t)=(Em(t), O ) = K ( W , I)

,

VS*(-4xJf(w, s))ds ,

V,XE"=O

.

Vlasov-Poisson Limit of the Vlasov-Maxwell Equation

4.

379

The Vlasov-Poisson Equation

The equation (1.5) contains the Poisson equation,

(4.1) where g=Kf.

Vz-E=4ag,

,

V,XE=O

This has a solution of the form,

G(x)=x/lxl'

.

We note that V,G(x) € C"(RS\{O}), is of homogeneous degree - 3 and has the mean value zero on S2. Thus V,C is a singular integral operator of CaldkronZygmund type. By easy calculation, we get, II~~llOlCOl~l2 IVxGI,ICilgl, k20 7

9

.

Knowing this, we readily have the Lemma 4.1. Suppose gECo([O,T ] ; H 1 - ' )with 1 2 3 and let E be given by (4.2). Then E E Co([O,T I ; @) satisfying [ l E l l ~ ~ ~ ~ ~ C. l g l ~ - i , ~ Recall the operator L of (1.13) and consider the equation, (4.3) This is the same equation as (2.6) with s=O, so with V of (2.8), we find its solution as

f = w,0; EU-0 . Suppose Ee C 0 ( [ O T , ] ; A 1and ) f o E H i , p , pwith 1 2 3 , u20, p>O, ,4ERS. Then Lemma 2.2 says that

f € C0([0,T I ; H A , p , p )

7

3 80

K. ASANOand S. UKAI

satisfying (2.7) with k = l and s=O;

Further, we readily see that

Kf € C0([0,TI; H E )I IKf l E , T < ~ l f l l , O , P , p , T

.

Now we discuss the Vlasov-Poisson equation (1.5). It is a coupled equation of (4.1) and (4.3). Therefore, if g is a fixed point of the equation,

(4.4)

g = K V t , 0;

W f o,

then (1.5) is solved by

f = V(t,0; Gg)fo ,

(4.5)

E=Gg=GKf

8

Using the results obtained so far and by the successive approximations, we can show that for any f o € H&,,p with p>O or with p=O and /l>3/2, there exists a positive constant T and (4.4) has a unique solution g € Co([O,T I ; H 2 ) . Now we can have the

Theorem 4.1. Let fo€Hj,,,p, 1 2 3 , u20, p>O, P E R . Then, there is a constant T 2 0 and (1.5) has a unique solution of the form (4.5), satisfying

f c Co([O,TI; f G , p , p ) ,

E € Co([O,TI; A'+'),

If(~)ll,.,p,~~ebflf~ll,o,p,B where b is that of (2.7). Moreover, T depends only on Ifols,o,p,p.

Finally we shall solve the modified Vlasov-Maxwell equation appearing in (1.15):

(4.6)

L ( f , E ; f,8 ) = h (or L(f,E + c - ' u x B ; Lltu-cAU=Af,

f,E+c-lvxB)=h) ,

(f,u)Ic=,=O . Here f,ii, h a r e given functions. Rewrite this in the form of the Volterra type integral equation,

or

f ( t )= combined with

s:

U(t , s ;ii, c){-a(&)

+c-lv x B(s))- V,f

(s)

+h(s)}ds,

381

Vlasov-PoissonLimit of the V7asov-MaxwellEquation

u(t )=

s:

ecct-s)AAf(s)ds .

Applying Lemmas 2.2 and 3.2 to this, we can readily prove the Lemma 4.2. Let 1 2 2 , a 2 0 , P E R and p ,

with p-yT>p/2, and let

.

li E [ A.211n [A.41

n [A.3];, ,

f ,h E [A.1

r, T>O

7hen there exists a unique solution to (4.6) such that

5. Proof of Theorems 1.1-1.3 Theorem 1.1 (i) has been proved in [ l ] using the successive approximation (the contraction mapping principle). The proof was given for o = O but is valid also for a>O. This and Lemma 2.1 then prove Theorem 1.1 (ii). In order to prove Theorem 1.2, we shall recall the successive approximation mentioned above; (fo, uo)=O and for n 2 1 ,

f "=U(t,0 ; un-1, c)fo , U n = ectAUo+

s:

ectc-s)A/lfn-l

(s)ds

Thanks to (2.4), (3.9) applied to (5.1), we can find 7, T , C>O such that

If

(5.2)

nll,o,p,B,pT+

IUn12*TS

c

holds for all n. Then by Lemmas 2.1, 2.2, 3.2, it follows that (5.3)

f n € [A.l]f.an[A.3]f,B,

u n € [ A . 2 I i n[A.4I1,

n20

.

Using these, we repeat the argument of [ l ]to see that

f,-f un+u

in B O ( [ l00); , CNO, T I ; HZp-A) , in B O ( [ lm); , Co([O,T I ; H1-l)),

strongly as n+m, with some limit (f,u). By (5.2) a n d the interpolation theorem, this convergence is also true if 1-1, 19-1 are replaced by I - E , B-E, for any E > O . This and (5.3) then imply that for any 6>0,

un+u strongly in B O ( [ lm , ] x[a, T I ; B2-2(RS)). Now the first half of Theorem 1.2 follows since the limit (f,u ) obviously coincides with the solution of Theorem 1.1, and the latter half comes directly

382

K. ASANO and S. UKAI

from Lemmas 2.2, 3.2 and Theorem 4.1. The asymptotic expansion in Theorem 1.3 is obtained as follows. let f" be that of Theorem 1.2 and assume the expansion,

First,

k

f=f-+Xc-fjj.

(5.4)

3=0

Substitute this into (3.8) which holds for our (f,u) by going to the limit in (5.1). Then we have,

where

riO=ecLAuo+jt ec(L-s)AA( f "(s)+f O(s))ds , 0

e c ( t - s ) A A ~ ( s ) d s ,l < j i k

uj=\'

.

0

Using (3.41, we decompose ao further as l i o =u"+ uo

,

s: 1'

u"(t)=Pouo+

PoAfm(s)ds,

uO( t ) =ectAf',UO+

ec(c-s)A (P,/2frn(s)+/2f0(s))ds .

0

By Lemma 3.2 and since our f " is that of Theorem 1.2, u" defined above is just that of Theorem 1.2. Also, recalling that (3.8) is a unique solution to (3.1), we see that uj solves (formally) the Maxwell equation in (1.15)1, O i j < k . Substitute (5.4) and (5.5) into (1.1) to deduce the equation for f j in (1.15)j. Now the proof of Theorem 1.3 can be completed by the help of Lemmas 2.2, 3.2 and 4.1, and by proceeding as in the proof of Theorems 1.1, 1.2 and 4.1. The detail is omitted.

References [ 1]

K . Asano, On local solutions of the initial value problem for the Vlasov-Maxwell

equation, 1984, Preprint. On the incompressible limit of the compressible Euler equation, 1985, Preprint. [ 3 ] K. Asano and S. Ukai, On the fluid dynamical limit of the Boltzmann equation, Lecture Notes in Numer. Appl. Math., 6, Kinokuniya/North-Holland,1985,l-19. [ 4 ] C . Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Internal report no 101, Centre de Math. Appl. E.P.P., 1983. [ 2]

-,

Vlasov-Poisson Limit of the Vlasov-Maxwell Equation

383

[ 5 ] P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and

convergence to the Vlasov-Poisson equations for infinite light velocity, 1984, Preprint. [ 61 S. Mizohata, The theory of partial differential equation, Cambridge Univ. Press, 1973. [ 7 1 S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, 1984, Preprint. Kiyoshi Asano Institute of Mathematics Yoshida College Kyoto University Kyoto 606, Japan Seiji Ukai Department of Applied Physics Osaka city University Sugimoto, 3, Sumiyoshi-ku Osaka 558, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equationspp. 385-418 (1986)

A Discrete Model for Spatially Aggregating Phenomena By Tsutomu IKEDA Abstract. The objective of the present paper is to nuinerically study the behavior of a solution of a mathematical model for spatially aggregating phenomena of population. For this purpose, we propose a discrete model, which preserves important properties of the continuous model with the aid of its nonlinear artificial viscosity term. Using this discrete model, we study the behavior of a solution and the stability of pulse-like stationary solutions. Key words: spatially aggregating population model, standing pulse-like solution, monotone finite difference method, nonlinear artificial viscosity, asymptotic behavior

1. Introduction

In the present paper, we study a finite difference approximation for the following nonlinear degenerate diffusion equation involving a nonlocal convection term P(m,r)

(

ULx, d = ( U 7 n ) z z ( xf)+ , U(x, t )

{ s'

U(Y, W Y -

2-T

SXtr

U(Y, W Y } )

X

2:

in R X (0,

m)

subject to the non-negative initial condition (1.1)

U(x,0)=Uo(x)20

for x € R .

Here, m > 1 and O S r s m are parameters, and U(x, r ) 2 O denotes the population density at position X E R and time t > O . We assume that Uo€C(R) and the support of Uo, which is denoted by supp [UO], is compact. The equation P(m,O) agrees with the porous medium equation, which appears in the theory of fluid flow through a porous medium (Bear [4] and Scheidegger [29]). In the case of r > O , P(m, r ) represents a mathematical model for spatially aggregating phenomena of population, proposed by Mimura and Yamaguti Received July 2, 1985.

386

T. IKEDA

[22]. The second term on the right-hand side of P(m, r ) ecologically shows a n aggregating mechanism of individuals, which is motivated by the notion of “centripetal instincts” (Hamilton [14]). In fact, the term provides a mechanism that moves individuals at position x to the right (resp. left) direction when

JZ

Jz-r

The first term ( corresponds to the transport of population through a nonlinear diffusion process called density-dependent dispersal (Gurney and Nisbet [12] and Gurtin and MacCamy [13]). The diffusion speed mUm-’ decreases with the density U and vanishes at position x where U(x)=O. Consequently, P(m, r ) is provided with a homogenizing process and a dehomogenizing process. We may expect that a delicate balance between these two processes gives rise to a spatial pattern, which shows a n aggregation of individuals. We first review mathematical works related to the Cauchy problem P(m, r ) subject to (1.1) and the stationary solution of P(m, r ) . The distinctive feature of the Cauchy problem P(m, r ) , which is caused by the degeneracy of diffusion at U=O, is that a n initial distribution with compact support spreads out at a finite speed and loses its initial smoothness (Aronson [2], Kalashnikov [18] and Oleinik et al. [27] for P(m,O)). A solution of P(m, r ) is therefore defined in a generalized sense (Aronson [l], Caffarelli and Friedman [5] and Gilding and Peletier [lo] for P(m,O)). For a class of Cauchy problems including P ( m , r ) , Nagai [23] has shown the unique existence of a generalized solution, and he has proved the finite propagation property. A stationary solution W of P(m, r ) , which ecologically exhibits a spatially aggregating pattern of individuals, is defined to be a non-negative valued function belonging to L1(R)n L-(R) that satisfies

in the distribution sense. The trivial function W=O is always a stationary solution of P(m, r ) , and we naturally are concerned with the non-trivial stationary solutions. In the present paper, a non-trivial stationary solution W of P(m, r ) is called a standing solitary pulse (abbr. a n ss-pulse) if supp [W] is connected and W>O on the interior of supp [W]. The porous medium equation has n o non-trivial stationary solution; while P(m, a)has ss-pulses (Mimura and Yamaguti [22]):

Theorem 1.1. For each q > O , P(m, a)has an ss-pulse W such that

A Discrete Model for Spatially Aggregating Phenomena which is unique up to coordinate translation Moreover,

(11 - 11

387

denotes the supremum norm).

-

where 11 denotes the usual norm of L1(R), and diam (W) denotes the length of SUPP[WI. 17 For the general case of O
where { W k } k s(nA : an index set) is a set of ss-pulses such that

(1.9)

dis(supp[Wj], supp[W,])Zr

for j E A , k c A and j f k .

(Here, dis ( A , B ) denotes the distance between subsets A and B of R . ) [7 Theorem 1.3. (1.10)

(i)

P ( m , r ) has no ss-pulse W such that

r2 11 W11_2-ms2.

( i i ) A n ss-pulse W of P ( m , m ) is an ss-pulse of P ( m , r ) (1.11) Remark 1.4. (1.12)

r 2 2 d i a m (W)2=F(m)2IIWII,”-2

if

. 0

For each a Z F ( m ) ,P(m, r ) has a n ss-pulse W such that r2 IIW11,2-m=a2 and

diam ( W ) s r

by Theorems 1.1 and 1.3. In the case of m=2, we have shown in [17] that P(2, r ) has a n ss-pulse with compact support, which is unique for each r> 42, up to coordinate translation and multiplication by a positive number. 0 In the case of r=m, Nagai and Mimura [25] have shown the asymptotic behavior of the solution of P(m, 00): Theorem 1.5. A s t+m, the solution o f P ( m , 00) subject to (1.1) tends to an

T.IKEDA

388

ss-pulse W , which is uniquely determined by the initial distribution Uoso that

(1.13)

s"_ s'_

{ ~ O ( Y ) -

W(Y)}dYdx=O.

Another interesting property of P(m, r ) is the appearance of interfaces. Because of the finite propagation property, the half plane R x (0, m ) is divided into two regions { ( x , t ) ; U ( x , t ) > 0 } and { ( x , t ) ; U ( x , t ) = O } by interface curves. The behavior of interfaces has been studied in detail for the porous medium equation (for instance, Aronson [3],Caffarelli and Friedman [ 6 ] and Knerr [19]). Recently, Nagai and Mimura [26] have derived a n equation that describes the motion of the interfaces to P(m, a ) , and have shown the asymptotic behavior of the interfaces. The objective of the present paper is to study the behavior of the solution of P(m, r ) by numerical methods. As reviewed in the above, we have obtained all stationary solutions of P(2,r). However, we do not know whether P ( m , r ) has a n ss-pulse W such in the general case of m f 2 . The first subject of that 2
Conjecture. Let m f 2 . For each dZdiam ( W,) >r. [7 Theorems 1.2 and 1.3 and Remark 1.4 assure that P(m, r ) ( O < r < a ) has standing pulse-like solutions, which are stationary solutions given by (1.7),as shown in Figure 1.1. However, the stability of these standing pulse-like solutions has not proved. The second subject is to numerically study the behavior of the solution of the Cauchy problem P(m, r ) , and to numerically examine the stability of standing pulse-like solutions of P(m, r ) . To achieve the above objective, we need to derive a discrete spatially aggregating population model P J m , r ) . The discrete model in required (Pl) to give non-negative solutions since the unknown U ( x , t ) stands for the population density. According to the purpose of the discrete model P J m , r ) is also required

x

Figure 1.1.

A standing pulse-like solution of P(m, r).

A Discrete Model f o r Spatially Aggregating Phenomena

389

(P2) to have the stationary solutions with connected compact support, which correspond to those of P(m, r ) given in Theorems 1.1 and 1.3, (P3) to satisfy discrete analogies of Theorems 1.2 and 1.5. In Section 2, integrating P(m,r ) , we transform P(m, r ) to another Cauchy problem Q(m, r ) . In Section 3, we propose a discrete model Qh(rn, r ) , which is obtained as a finite difference approximation for Q J m , r ) . The solution of Qh(m,r ) is given by differentiating that of Qh(m,r ) . The key of Q,(m, r ) is to introduce a nonlinear artificial viscosity ah[ ;m, r ] . We show in Sections 3 to 5 that a h [ .; m, r ] permits P,(m, r ) to fulfill the above requirements. In Section 4, we deal with the case of r = m , and show that Ph(m,r) satisfies discrete analogies of Theorems 1.1 and 1.5. In this section, we need to discuss the translation of a stationary solution of P,(m, m). For the continuous model, any translation W c ( x ) = W ( x - c )(CER) of a n ss-pulse W ( x )also is an ss-pulse, and for each initial distribution Uo such that \ ~ U o ~ ~ l = ~one \W~~l, of Wc's satisfies (1.13). The discrete model P,(m, m) is of conservation form, and if Wh(x)is a n ss-pulse of P,L(m,m), then W h , j ( x ) =W,(x-jh) ( h : the spatial mesh size,jE Z) also is a n ss-pulse. However, for almost all initial distributions U: of P J m , m) such that \\Ui\\l=\\Wh\\l,none of Wh,j'ssatisfies the discrete analogy

-

J-m

J-m

of (1.13). We need to show that for each c>O, there is a one-to-one correspondence between R and the set of ss-pulse Wh of Ph(m,a) satisfying II Whlll = c . In Section 5, we deal with the general case of O < r < 00, and show discrete analogies of Theorems 1.2 and 1.3. We also examine the conjecture in this section. In Section 6, using the discrete model, we examine the stability of standing pulse-like solutions of P(m, r ) . A rigorous proof of the stability of the discrete model has not been obtained. However, the discrete model seems to be stable under the condition (3.13) on the time increment given in Section 3. Moreover, under (3.13), the discrete model seems to approximate both the value of the unknown function and the interface curves. (For the porous medium equation, Graveleau and Jamet [ l l ] and Tomoeda and Mimura [30] have proposed finite difference shcemes where the degeneracy of diffusion is taken into account; and schemes proposed by DiBenedetto and Hoff [8], Mimura et al. [21] and Hoff [15] approximate the interface curves as well as the value of the unknown function.)

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390

2.

Reduction of the Problem

Let U(x, t) be a solution of P(m, r ) subject to the initial condition (1.1). Then, the total distribution j y mU(x, t)dx is kept constant, that is,

Through the change of the unknown function (2.2)

for x € R and t Z O ,

U(y,t)dy-T~~UoII, 1

u(x,f)=[' -m

the problem P(m, r) with (1.1) is transformed into the following Cauchy problem : (2.3) uc(x,t)=(u:),(x,

t)-u,(x,

t){u(x+r, t)+u(x-r, t)-2u(x, t ) } in R x ( 0 , 00)

,

where UoE C(R) is non-negative on R and supp [UO] is compact. The one-toone correspondence between solutions of P ( m , r ) and (2.3) has been shown by Nagai [23]. For each a)O, let (2.5) Xm,,,(a)={v~C1(R);v,zO on R, lim v(x)=-a

and limv(x)=a}

2+--Cu

.

Z-m

With v ~ X m o n o ( we a ) , associate a function (2.6)

J"[v;m , rl(x) =v,(x)"-

1'

V,(Y){V(Y +r)+v(y - ~ ) - ~ U ( Y ) } ~ Y

-m

By integration by parts, J"[v;m, r](x) is rewritten as (2.7)

~"[m v ;, r ~ x=) v,(x)"+

\'

~v(y)v,(y+r ) - v,(y)v(y--r)~y

-m

-

+

{v(x)v(x+r ) -a'} { ~ ( x )' a'}

If u(x, t ) is a solution of (2.3) with (2.4), then u ( - , t)EXm,,,(IIU0111/2) for t > O . Using J"[. ; m , r ] , we rewrite (2.3) as

Qh r)

uc(x, t)=J"[u;m , r],(x, t ) in R x ( 0 , a).

3. A Discrete Model for Spatially Aggregating Phenomena In this section, we propose a discrete model Q , ( m , r ) for spatially aggregating phenomena of population, which is obtained as a finite difference

A Discrete Model f o r Spatially Aggregating Phenomena

391

approximation for Q ( m , r ) . The population density is obtained by differentiating the solution of Q,(m, r ) . Let h be a positive number, which denotes the spatial mesh size. We use the following notation ( a : a non-negative number):

(3.1) X ” ( a ) = { v , € C ( R ) ;v, is linear on each interval (ih,i h f h ) ( i E Z ) , lim v,(x)=-a and lim v,(x)=a} , z+-m

z-m

(3.2) X h = { C h € L m ( R )C; h is constant on each interval (ih,ih+h) ( i E Z ) } ,

v*=v,(ih)

(3.3)

(3.4) (3.5)

for

v , € X h = U Xh(a) and i E Z , a20

Vv,=(the derivative of v , X~h ) € x h , {VIvh=(the value of Vv, on (ih,ih+h))

for v , € X h and i € Z ,

Xkono(a)={vhE X h ( a ) ;Vv,zO o n R} and X2,,,,= U Xkono(a). a20

In general, it is not easy to derive a discrete model that fulfills the requirements (Pl)-(P3) described in Section 1. To see this, let us consider a simple case r = m . In this case, Q(m,a)has no nonlocal interaction:

Qh, a)

u,=

(u;)~+(u~)~ in R x ( 0 , a) I

It is natural to use the central finite difference approximation 1

t(V‘~h)m-(V*-luh)m~ (u, E X&m0)

for the first term (u;)~. However, by the same reason as that in the case of the Burgers equation ut=(u2),, the central finite difference approximation does not apply to the second term (u2), if u, is approximated by the forward difference. We consider the application of schemes for the Burgers equation to Q(m, 00). The requirement ( P l ) is not fulfilled by a scheme derived from a nonmonotone scheme (the Lax-Wendroff scheme ([ZO])for instance). On the other hand, a scheme derived from a monotone scheme may fulfill (Pl). In fact, a n application of the Lax-Friedrichs scheme ([7]):

392

T. IKEDA

fulfills ( P l ) under a condition on the time increment r,,. the Enguist-Osher scheme ([28]): For u:

e X;on,,(T 1

And application of

, find { u ; } ; = , c X ~ , . , ( ~ llVu:lll)

/lVu:lll)

such that

1

(u:+'- ~ , " ) = - { ( V ' U ~ ) ~ - ( V I - ~ U ~ ) ~ } h

+ 1 If-(u:+

1)

-f- (u,"1+f+(u,"1-.f+(U?-

111

9

where f ( u ) = u 2 , f+(u)=f(min {u, 0}) and f-(u)=f(max { u , 0}), also fulfills (Pl) under a condition on rn. However, neither scheme has a stationary solution w such that supp[Vw] is compact. This fact is easily shown by expressing (L-F) and (E-0) in the divergence form:

For (L-F) the flux is given by F:+112=G:,llz+h2Vtujt/4r,, and F:,,,,=G:+l12+ {f-(u:+l)-f-(u:)-f+(u:l 1)+f(uzI))/2 for (E-0) where G:+1,2=

(ViUhn)llL+

1

-{(u1)2+ 2

We return to the general case of 0 5 r 5 00. by

(u:.kl)z}

.

We define b,[ * ; r] : X~,,,,,+X"

A Discrete Model f o r Spatially Aggregating Phenomena

393

where ai[v,; m, r ] denotes the value of a nonlinear artificial viscosity

-

on the interval (ih,ih+h). Xh(0) defined by

-

Using Jh[ ;m, r ] , we introduce L,[ ;m, r] : Xiono-+

Now, our discrete spatially aggregating population model is:

t

1

-(u;tk1-u;)=L,,[u;t; m , r ]

for n=O, 1, 2,

-

a ,

rn

where the time increment rn is determined so that

two propositions where r,=h2/max {2m( ~ ~ V U ~ /hllVu:lll}. ~ J ~ ) ~The - ~following , assure that Q,(m, r ) has a solution for any u: E Xi",,,, that is,

(3.14) if u;E X&,no(c), then u:+'E X&,.Jc) Proposition 3.1.

If

0,

E X ~ o n o ( cthen ) , Gh=vh+rLh[uh;m, r ] E Xh(c),and

(3.15) Proof.

{ 4 , ( x )- v , ( x ) } d x=0

.

Since L,[v,; m, r ] € P ( O ) , 8,€ Xk(c). By integration by parts, a,

-m

under the condition (3.13).

i=-m

T.IKEDA

394

Proposition 3.2. Let V,E X:o,,o(c). Then, 9,=v,+rL,[v,; X&"Jc) under the condition

t

2r max llV~,]\:-~,

(3.16) Proof.

(3.17)

h 2

- Ilb,[v,; rlllm} S h 2

m, r ] belongs to

.

We put

h p,[w,] = (V,w,Jrn-l +a,[w,; m, r ]=max ( V i ~ h ) m - l , 16" w,; rl I

t

for W, E Xk0,,,, and i E Z. Then, L,[w,; m, r] is rewritten as

With the aid of the nonlinear artificial viscosity (3.11),

h 2

for i E Z .

pt[w,]--lbb,[w,;r]I~O

(3.19)

The expression (3.18) of L , [ - ;m, r] yields

ai+ 6=(vi, -

= :

+

{pi+Jv,l

't

+-h

Jv, ; m , rl -L J v , ; m. rll h 1 2 bttl[vh;r l } (Vr+lv,)f y W - ~ ~ P J ~ , I N V A J

- v,) rWt

+-

h p i - I [ v h ] - - b t - l [ v h ; r ] (Vb,-lu,)

2

1

for i E Z .

The coefficients of Vitlvh and V,-,v, are non-negative by (3.19), and that of Viv, is non-negative under the condition (3.16). We thus complete the proof. 0 A stationary solution w h of Q,(m, r ) is defined to be a function belonging to Xkon0such that (3.20)

Lh[wh;m,r]=O

on R

.

Since Q,(m, r ) is of conservation form, if w,(x) is a stationary solution, then w,(x-ih) is a stationary solution for i E Z . The function w,=O always is a stationary solution of Q,(m, r ) . A spatially aggregating pattern is obtained by differentiating a non-trivial stationary solution of Q J m , r ) . For the continuous model, the standing solitary pulse means the stationary distribution of the population density. However, for brevity, a non-trivial

395

A Discrete Model f o r Spatially Aggregating Phenomena

stationary solution w h of Q,(m, r ) is also called a standing solitary pulse (abbr. an ss-pulse) if supp [Vw,] is connected. The key of Q,(m, r ) is the nonlinear artificial viscosity a,[-; m, r ] . We have shown that a h [ - ;m, r ] permits Q,(m, r ) to fulfill the requirement ( P l ) (Proposition 3.2). We shall see in the forthcoming sections that a,[.; m, r ] permits Q,(rn, r ) to fulfill the other requirements. 4.

Spatially Aggregating Population Model Q,(m, a)

In this section, we study the discrete spatially aggregating population model Q,(rn, a). After observing how a,[ ;m, a]works and proving a comparison theorem, we show that Q,(m, co) satisfies discrete analogies of Theorems 1.1 and 1.5. We introduce two functions U ( K , v) and J(u, v), defined on the half plane D = { ( u , v ) E R 2 ;u ~ u } ,

-

(4.1) (4.2) Then, bt[vh;a], a,[v,; m, a]and JJv,; m, a]( v , Xkon0) ~ are reduced to

1

m, w]=a(v,, v , , ~ ), b2[vh; w l = ~ ~ + v, ~ + a,[v,; ~

(4.3)

(i

>’

JI[vh;m, 031=J(vt, vt+J- - IIVvhlll ,

respectively, and Q,(m, a)is rewritten as (4.4)

~;+~=u;+p,J(up,~p+~)-p~J(u,”-,, up)

where p,=rJh.

for

i € Z and n20

,

We divide the half plane D into the following three regions:

(4.5)

(Figure 4.1). By (4.1), (4.2) and (4.5), a(u, v ) is continuous in D , and is of class C1 on DoU D , U D a(u, v)=O on Do and a(u, v) > O on D\d, ,

,

396

/!q/ T. IKEDA

0

D

0

DD-

D-

(a)

1<m
(c) m > Z

(b) m - 2

Figure 4.1. Regions DO,D+ and D - . (4.7) J(u, v) is continuous in D , and is of class C’ on DoU D , U D -

,

where the symbol - denotes the closure. We easily see that J(u, v) and the partial derivatives Ju(u,v) and Ju(u,v ) are expressed in the following form:

(4.8) ( J ( u ,v)=v2 on

(4.9)

IJ,(~,

6 , and

m v-u

*--l

v)= - h ( k )

‘J,(u, v)=O on D ,

J(u, v)=uz on

6- ,

on D , ,

+u

and Ju(u, v)=2u

on D- ,

(4.10)

‘J,(u, v)=2v

on D ,

and J J u , v)=O

on D-

.

Observing ( 4 . 3 , (4.9) and (4.10), we see (4.11)

J,sO on D and J,
(4.12)

J,ZO on D and J,>O on D,UD+ ,

h

-la+Pi,

2

h yIp+rI}

for

aspsr,

and obtain the following comparison theorem:

Theorem 4.1. Let v: € X;o,o(c) and then, under the condition

vie X ~ , , , ( c ) . If v:(x) 5 v:(x)f o r x € R,

397

A Discrete Model f o r Spatially Aggregating Phenomena

both O;=v;+rL,[v,W; m, 001 and v^i=uk+rL,[vi; m, w] belong to A'&,no(c), and

(4.15)

$ ~ ( x ) S O ~ ( x ) for

xGR.

and put 8t=v;+

Proof. For 0<0<1, let vt=(l--8)v;+Bvi€X~,,,(c), rLh[v;; m, a].By (4.3)and (4.141,

(4.16)

t

h

I

2 r m a x m ~ ~ V U ~ ~ ~llbh[vl; ? - ~ ,m]II.. Sh2 2

for 0 1 8 5 1

.

Hence, OiE X,&(c) for O S 8 5 1 by Proposition 3.2. Putting p=r/h and using (4.4)and (4.7), we rewrite at-@ as

+(v:-v:)

s:

U+PJM,

U:+J-~JJV:-~,

vwe.

The coefficients of ~ : + ~ - - - 2 / ~ + ~and vt-l-u!-l are non-negative by (4.12)and (4.11), respectively; while (4.13)implies that the coefficient of vt-vp is nonnegative under the condition (4.16). Thus, we obtain (4.15). 0

Stationary patterns We discuss in this subsection the stationary solution of Q,(m, 00). stationary solution w, € X;,,,,(c) of Q,(m, w), (3.20)is replaced by

4.1.

J(wI,w,+,)=c2

(4.17)

for ~ E .Z

With each c > 0 and h > 0, we associate the number

E ( c , h ) = m i n { q ~ R ; ( v , ~ ) ~ .6 + }

(4.18)

(See Figure 4.1.) By (4.9,

(4.19) (4.20) (4.21)

[ ( c , h) > - c

and [ ( c , h)= -max

{v E R ; ( - c ,

E(c, h)+c as h+O

7) E B-},

,

( v , c ) E D + if E ( c , h ) < v < c , if - c < 7 j l < - e ( c , h ) .

{(- c , ? ) E D -

Moreover, there exists a number a,=a,(c, h) ([(c, h)
For a

T.IKEDA

398

(4.22)

t

( a , p ) ~ D +if a,Sa
.

We begin with necessary conditions for a non-trivial stationary solution. Proposition 4.2. Assume w, E X&,n,(c) to be a non-trivial stationary solution of Q,(m, a).Then, w, is an ss-pulse of Q,(m, 00) and supp [Vw,] is compact. Moreover, diam (Vwh)22h, and (a) (w,,w,+J E 6- and (wK.w K + JE 6+, (b) (w,,wi+Je D ofor Z- c for all i, then (w,, w,+,)E D- for some j by (4.21) and (4.22), andJ(w,, w j t l ) = w ~ < c 2 . This contradicts (4.17). Suppose that diam (Vwh)<2h. Then, w , = - c and W , , ~ = C for some i, and J(w,, witl)= ( 2 ~ / h ) ~ + ( c ~ + c>~c2. ) / 2 This contradicts (4.17). Suppose that (w,,w,,,) 64 6-. If (w,,w,,J € Do,then

>-1 (w,,,+ h

h 2

C) - I w,t,-cI

+-21

( C 2 + w 1 + l 2=c2 )

by (4.8) and (4.5). This contradicts (4.17). If (w,,w,,,) € 6,, then J(w,, wItl)= w,,,2
Zf - c < a < E ( c ,

h), the following problem

(4.23) Find B=B(a)€ R such that J ( a , p)=c2, ( a , b)€ D o and a < / 3 < c hasa unique solution B=p(a). Zf E(c, h ) S a < c , (4.23) has no solution. Moreover, (a) B(a) increases with a ( - c < a < f ( c , h ) ) , (b) P(a) is continuous in the interval ( - c , &c, h ) ) , (c) limal-cB ( d = -E(c, h) and limurt(e.h) P(a)=c. Proof. Let - c < a < f ( c ,

h). Then, ( a , c ) € Do,and

399

A Discrete Model for Spatially Aggregating Phenomena

J(a,c)-c2=

(ci

a)m

+

+

(az-

c2)

c-a h 1 - cz)=0 >- Ia+ cI + - (ay2 h

2

2

;

For ,8 satisfying (4.24), we have ( a , P ) E D oby (4.8). Hence, (4.23) has a solution P. The uniqueness of a solution follows from (4.7) and (4.12). If E(c, h)$aJ(az,p ( a , ) ) - c 2 by (4.11). Hence, /l(az)>P(aI) by (4.12). We thus obtain (a). Let us show the property (b). If {a,,} is a monotone sequence of real numbers a,, € ( - c , E(c, h)) that converges to a , then P(a,) converges to some 8 € [ a ,c ) by the property (a). By (4.7),

while while w,(ih)i; j>i;

Proof. Proof. Let Let w, w, €€ X&,(c) X&,(c)

be and be aann ss-pulse ss-pulse of of Q QJJm m,, a a) ), , and let let

LL==m maaxx{{ii€€ZZ;; w,=-c} w,=-c}

and and K K==m maaxx((iiEEZZ;;w w,,<
T. IKEDA

400

By Proposition 4.2, - c < w L , , ~ - ~ E ( ch,) , ((c, h)sw,
00) has infinitely many ss-pulses But these ss-pulses are determined by 4.3, w,’s ( i > Z + l ) are continuous func-

Q,(m,

ss-pulse w,(x;s) E X&,,,,(C) of Qh(m,00)

w,(iR;s)= - c for i S Z , w,(Zh+h; s)= -c+(c-C(c, h))(Zh+h-s)/h

(4.26)

,

where Z=max {icZ ; i h s s } . This gives a one-to-one correspondence between R and the set of all ss-pulses of Qh(tn, 00) belonging to X&,,,o(c). Moreover, wh(~;s,)~w,(.;sp) for s 1 5 s 2 ,

km

(4.27)

as s+sl

{w,(x:s,)-wh(x;s))dx-0

by Proposition 4.4 and Lemma 4.3. Figure 4.2 displays ss-pulses wh(*:ih/20) E 1

1

Wh

h ’

to

lo

-1

-1 (a) m

-

1.5. h = 0.5

(b) m

(c) m

Figure 4.2.

Stationary solutions of

=

-

2.0.

h = 0.5

3.0. h * 0.5

Qh(m, 00).

A Discrete Model for Spatially Aggregating Phenomena

XLono(l)of Q,(m,

a)( i = O ,

- - .,20).

40 1

We thus obtain

Theorem 4.5. A stationary solution w, of Q,(m, m) is an ss-pulse and supp [Vw,] is compact. For each v, E X&,,,, Q,(m, m) has a unique ss-pulse w, such that (4.28) Remark 4.6. A stationary solution w of Q,(m, m) is a translation of a n arbitrarily fixed stationary solution belonging to X,,,,( IlwllJ2). For Q,(m, m) however, a n ss-pulse w,(. ;sI)is not a translation of another ss-pulse w,( ;s2) X&,n,([~w,(~; sI)]Il/2) except when ( s 2 - s l ) / h €Z (Figures 4.2). We note that a h [ . ;m, 001 permits Q,(m, m) to have these ss-pulses. If the right-hand side of (3.11) is replaced by others, the requirements ( P l ) and (P2) are not always fulfilled. For instance, let us replace a,[ ; m, m] by

-

-

( a : a non-negative number). If a < 1, Q,&(m, m) violates (Pl). If a > l , Q,(m, 00) violates (P2). When we replace a h [ . ;m, m] by a linear artificial viscosity, Q,(m, a)may fulfill ( P l ) , however, Q,(m, m) violates (P2). 171 Theorem 4.7.

A n ss-pulse w, of Q,(m, 00) satisfies

(4.29)

Proof. Let c=IIVwhll1/2. (4.30)

1

By (4.17), we have

+

1

(Vtw,)m =c2- -( wi2 w , , , ~ )--a(wi, wttl)(wt+l- wJ 2 h

< c2 ,

which implies the first estimate in (4.29). By (3.20), we have 1 (V, W h P -(Vi - 1%) = - -(w, + 1- w, - I)( wi t I w, - I ) 2 1 1 - -4 w , , wt t 1)(w,+ 1 - Wt) -aov,- 1, WO( w, - wt - 1) h h

+

(4.31)

+

.

We first assume that diam ( V w , ) L 3 h . Proposition 4.2 and (4.31) imply (4.32) L e t j be a n integer such that IIVwhllm=V,wh.

Then, (w,, w j t l ) € & and w , 5 0 5

402

T. IKEDA

wjtlby Proposition 4.2 and (4.32).

Since max {lwj\, Iwjtll}Sw,+l-wj=

hllvwhllm,

(4.33)

[IVW,ll,m=(VrW,)m=Cz--

1 (wj2+Wj+12)2C2-h2 11VW,[l2 2

.

This implies the second estimate in (4.29). We next assume that diam ( V W , ) ~ 2h. The second estimate in (4.29) follows from the fact that IIVw,ll,lc/h. 0 Theorem 4.8. Fix an arbitrary positive number c and an arbitrary real number s. For each h>O, let W,E X&,”,,(c) be a unique ss-pulse of Q,(m, c*)) such that wh(s)=O. Then, lim,,o (Vw,)(x) =(Vw)(x) for X E R , where w is the unique stationary solution of Q(m, m) such that

(4.34)

w E Xmon,,(c) and

w(s)=O

.

Moreover,

(4.35)

limhL0 diam (Vw,)=diam (Vw)

Proof. Let Z(h)=max{jEZ;w,=-c} each w,. We first show

(4.36)

.

and K ( h ) = m a x { j E Z ; w j < c } for

lim,, diam (Vw,) =F ( ~ ) C ‘ -, ~ / ~

where F(m) is the same given by (1.6). For Z(h)
by Proposition 4.2, where ql=w,/c.

Hence,

+

diam (V w,) =h { K(h)-Z(h)}h

(4.38) As h-d, the last term of (4.38) tends to c ~ - $?, ~ /{ 1~- ~ 2 } - 1 ’ m d ~ = ~ 1 - 2 ’ mby F(m) (4.29) and

(4.39)

lirnhlow,~,),,=-c and lirnhlowK(,)=c.

((4.39) follows from (4.20).) Thus, we obtain (4.36). Let W,=Vw,. We may assume that diam (W,)23h. By (4.29) and (4.32), W,’s are uniformly bounded and have a uniformly bounded variation. Hence, by Helly’s theorem, we can select a subsequence {W,,} from {W,} that con-

403

A Discrete Model for Spatially Aggregating Phenomena

verges to a bounded function W E L 1 ( R ) . Clearly,

(4.40)

O S W ( X ) ~ C ~for / ~ x € R and

[lWlll=2c.

For proving that W ( x ) is a stationary solution of P ( m , a), it suffices to show

(4.41)

W(x)-+(S2

W(y)dy)(

-m

1'

--m

W(y)dy--Zc)=O

for almost all x € R ;

in fact, it follows from (4.41) that

for all +€C;(R).

(4.43)

Let w , € { w , . } = { - c + s ~ , W,,(y)dy}.

By (4.1) and (4.29),

1 --a(w,, ~ , + , ) ( w , , , - w , ) ~ h c ~ +for ~/~ i E Z h

,

Hence, for i h S x s i h + h ,

(4.45)

+-21 ( w $ + c ) ( w $ - c ) + -21

= Wh(Xln

(witl+c)(w,+l-c)+

O(h)

=J( w,, w,+ - c2+ O(h)= O(h) . Letting h 1 0 in ( 4 . 4 9 , we obtain (4.41). Clearly, w ( x ) = -c+ 1-: W(y)dy is the unique stationary solution of Q ( m , m) satisfying (4.34). By the uniqueness of the stationary solution of Q(m, m) satisfying (4.34), the original sequence {W,,}={Vwh}converges to W = V w . The equality (4.35) now follows from (4.36), (1.5) and (1,6). 0 Asymptotic behavior We now proceed to the discussion on the asymptotic behavior of the solution of Q,(m, m). Let {u;};=~ be a solution of Q,(m, m). (The time incre-

4.2.

T. IKEDA

404

ment r,, should satisfy the condition (4.47) below.) By Theorem 4.5 and Proposition 3.1, Q,(m, 00) has a unique ss-pulse W, such that m

(4.46)

for n z O

{u;(x)-w,(x)}dx=O

Let wi and w; be translations of w, such that condition

.

W;~U;~W;. Then,

under the

(4.47)

Theorem 4.1 implies (4.48)

for n 1 0 .

wh2u;zw;;

For each integer n z O , we define Z;t€Xh(0)by Z;(ih)=Cf=_,h(u;-w,) for i E Z. By (4.48), there exists a positive integer N such that (4.49)

Z:=O

for

Since {u;} is a solution of Q,(m, (4.50)

I i l Z N and n z O

m) and

w, satisfies

w,=w,+LJ(w,, W ~ + J - ~ J ( W w,) ~ - ~ ,for i E Z and r E R , h

11

it follows that (4.51) Z:+'=Z:+hp,{J(u:, where pn=r,/h.

U:+~)-J(W&,w ~ + ~ ) }for i € Z and n z O ,

Let ~;t.~=(l-O)w,+Ou; for 0 5 0 5 1 , and put

which are non-negative by ( 4 , l l ) and (4.12), respectively. We rewrite (4.51) as

where I,,,,: Xh(0)+Xh(O) denotes the linear finite difference operator (zh,n$k)(ih)

=p&:($%- 1 -$i)

f p~ln@ ($i: + 1 -$i)

.

A Discrete Model for Spatially Aggregating Phenomena

We here assume that there exists a pair and a positive number r satisfying

t

(4.54)

+,(ih)>O for l i l < N

(Zh,n$d(WS -r$i

I$,, r} of a

and $,(x)ZO for Iil < N .

405

function #,E X h ( 0 )

for ~ E R ,

We define a sequence { $ ; t } ~ - ~of functions $;E X h ( 0 ) by (4.55)

# ; = ( l + ~ ) - ~ $ , for

n20.

BY (4.49) and (4.54), K$,Z/Z,l on R for some large number K>O. K$;I-Z;t. Then, (4.53) and (4.54) yield @;

(4.56)

Let @=

+' I@;+ ( L n 0 3 ( i h ) +

Ki#;+'-$a --(L,,$Xih)} 20: +(Zh,n@Xih) = (1-pna," -pn/3:)@: +p,a:@:-l +p,,/3:~@,"~~

-

for lil
@ ; = K ( l + ~ ) - ~ $ ~ - Z , " 2 0 for

The same argument to @;=K$;+Z; (4.58)

K(l+~)-"#,+Z20

l i l < N and n 2 0 .

yields for

l i l < N and n 2 0 .

By (4.49), (4.57) and (4.58), llz;IllmSK(l+r)-nll$h/lm

(4.59)

for n Z 0

.

Consequently, we obtain Theorem 4.9. Let {u;};=, be a solution of Q J m , m) and w, be a unique ss-pulse of Q,(m, m) satisfying (4.46). Then, limn,- u ; ( x ) = w , ( x ) f o r X E R under the condition (4.47). Proof. It suffices to seek a positive number 7 and a function #, satisfying (4.54). Through a suitable translation of coordinate, we may assume that w,+w,+, O for i > O . By Proposition 4.2, (wt,w , + , ) g 6+for i < O and (w,,w,,,)e 6- for i > O . Hence, by (4.11) and (4.12), there exist positive numbers 0 and a. such that

f (4.60)

-J,(u;*e, v;;f)2a0

i'

-{-JU(Cve,

Put p=r,/h

for O
v,n;B,)+Ju(~;*e, v,.;!)}za,

for O<@,

i=O and n20 ,

J , ( v ; * ~v;;T)zao ,

for O<@,

i > O and n z O

((3.13)) and a=p@a0. The above estimates lead to

.

T. IKEDA

406

(4.61)

pna;2pOa,=a

1:

for i
-{p,a;+pc(,P;}2pOao=a

i=O and n 2 O .

for

p,$; 2 pea, =a

for i>O and n20.

Clearly, there exists a positive number p such that

We define functions q&=1,

wo=l,

Q,-Qi-l=qwi-l

wh and Qh belonging to Xh(0) by

$h,

Q0=O,

,

w , - ~ , - ~ = q Q , and

$,-$$-,= -pqQ,$,

(4.63)

for O < i S N , for

$,=w,=Q,=O

Q,=-Q-,

,

w,=w+

i>N, and $,=&,

for

i
where p and q are positive numbers such that (4.64)

l - p ( l + q ) 2 N - / 3 / ( a q 2) ~ ~and

l+pq254.

It is easy to see that (4.65) Q , = - Q - , > O ,

wi=w-,21

and $,=$-,>O

(4.66)

wi>IQz,I

for

(4.67)

$i>$itl

for O s i < N ,

(4.68)

Q,S(1+q)w,-ls(l+q)2t-1

for O < i d N ,

]ilSN,

for O < i z N .

Let us show that the pair {$h,r=apq2/2} satisfy the third condition of (4.54). By (4.63), (4.61) and (4.64), (lh,,$,)(0)=p,(a,"+p,")($l-$O)=-E(n(aOn = - p,(a,"

+p,")PqQl$l

+Po")pq2/(1+pq2) 5 - -21 apq2= -740 .

Let O < i < N . We rewrite (Zh,n$h)(ih) as

A Discrete Model for Spatially Aggregating Phenomena

407

For - N < i < O , rewriting (l,,,,$,)(ih) as

we can similarly show (lh,n$h)(ih)S -r$$. 5.

0

Spatially Aggregating Population Model Q,(m, r ) (0 < r < 00)

We study in this section the stationary solution of Q,(m, r ) for O < r < By integration by parts, J,[v,; m, r] is rewritten as

A stationary solution

W,

of Q,(m, r ) satisfies Jh[wh;m, r]=O on R

(5.2) 5.1.

00.

.

Decomposition of a stationary solution We show in this subsection a discrete analogy of Theorem 1.2. We let s(r, h)=min { i h z r ; i E Z} .

(5.3)

Proposition 5.1. Assume W , to be a stationary solution of Q,(m, r ) . If Vw,=O on an interval [nh, nh+h] ( n E Z), then there exists an integer i such that

(5.4)

, Vw,=O

n h + h s i h + s ( r , h) , on the interval [ih, ih+s(r,h)]

Proof. Let i=min { k E Z ;n h f h - s ( r , h ) s k h , w,=w,} and j =m ax { k E Z ; k h s n h + s ( r , h), wr=w,}. I f j hzi h+ s ( r , h), then we obtain (5.4). Suppose that j h < i h + s ( r , h). Then,

T. IKEDA

408

which contradicts (5.2). [7

Theorem 5.2. A function w, € X&,nobecomes a non-trivial stationary solution of Q,(m, r ) if and only if W , is expressed in the f o r m w,(x)=

(5.5)

where

c wP)(x)

for

k ~ , 1

x€R,

{w?)},~,,is a set of ss-pulses wi*) of Q,(m, r ) such that

(5.7)

dis (supp [ V w i L ) ]supp , [VwLL’)l)2s(r, h)

f o r k € A , k’ E A , k f k ’

.

Proof. Assume w h to be a non-trivial stationary solution of Q,(m, r). be connected components of supp [Vw,] (A: a n index set). For each Let k € A, define W p ) by (5.8) Then, c,=

W i k ) ( x ) = ( V w h ) ( x )for x € S ,

(U2)ll W?)lll< (U2)

IIvwhll19

w:~)(x)=\’

(5.9)

and

Wik)(x)=O for x4: S,

.

and

Wi*)(y)dy-c,€ X&o,,(c,) .

-m

Each supp[Vwp)] ( G S , ) is connected, and {WL*)}~~,, satisfies (5.5) and (5.6). By Proposition 5.1, {wp)},..,, also satisfies (5.7). Let us show that each wp) satisfies (5.2), by noting that (5.10)

rn, r] is determined by Vi+yh’S (ljlhSs(r, h ) ) .

Jt[vh;

(See (3.6), (3.7), (3.11) and (5.1).) Let x € ( t h e interior of S,). VwLk)(x+y)=Vwh(x+y) for 1yI 5 s ( r , h) by (5.7), and (5.11) J , [ w p ) ; m, r](x)=Jh[wh;m, r](x)=O

Then,

for x € (the interior of S,)

by (5.10). Let x @ S,. Then, Vwp)(x)=O, and by (3.10) or (5.1),

A Discrete Model for Sparially Aggregating Phenomena

Jh[wLk); m, r](x)=O

(5.12)

for

x

409

S, .

Now, each wr) is a n ss-pulse of Q(m, r). Assume {w?)}~.,,to be a set of ss-pulses of Q,(m, r) satisfying (5.6) and (5.7). Then, the function w, given by (5.5) belongs to X:,,,, and satisfies Jh[wh;m, r]=

(5.13) by (5.7) and (5.10). 5.2.

C Jh[wiX); m, r]=O

ksA

Hence, w, is a stationary solution of Q,(m, r).

0

Standing solitary pulses We now proceed to the discussion on a discrete analogy of Theorem 1.3.

Proposition 5.3. Assume w, € Xkon0(c)to be a stationary solution of Q,(m, r). Let u be an arbitrary positive number, and put q=hum-2. Then, z,(x)=umw,(xu2-m)

(5.14)

€ X&ono(Cum)

is a stationary solution of Q,(m, rum-2). Proof. Put

It follows from the definition of z, that and ( V Z , ) ~ = U ~ ~ ( V ,W , ) ~

Vz,=u2Vw,

(5.15)

J iq

(5.16) ih+r

(Vw,)(x){w,(x-r)-w,(ih)}dx

for i E Z

,

ih

Siha

ihafha

(5.17)

=-urn - -u2m-2

{w,(~+r)+wh($-r)-2w,($)}dy ih+h

(d7w,)(x)dx=u2m-zhb,[w,; r]

for i E Z ,

\ih

a,[z,; m, ra](Vz,)(iq)=max (5.18)

ra]l -(Vzq)m-l}(Vzg)(iq) Ib,[w,; ~ ] I - ( V W , ) ~ - ~

=uZrnah[w,; m,r](Vw,)(ih)

for i E Z

By (5.15)-(5.18), z, is a stationary solution of Q,(m, rum-'):

.

T. IKEDA

410

. (5.24) M ( m , h)=max {diam (Vz,); IIVzhllrn=l,zh is a n ss-pulse of Q,(m, a)} Theorems 4.7 and 4.8 imply (5.25)

M ( m , h)+F(m) as L O . We put

Let W, be a n ss-pulse of Q,(m,

m).

(5.26)

q = h IIVw,II,'-m'2 ,

z , is a n and define a function z , ( ~ ) = ~ ~ V w , ~ ~ ~ ~ ' ~ w ~It ( is ( hshown / q ) x ) that . ss-pulse of Qh(rn, m) by the same method as that of Proposition 5.3. More over,

A Discrete Model for Spatially Aggregating Phenomena

411

Now, a discrete analogy of Theorem 1.3 is stated as Theorem 5.5.

(i)

Q,(m, r) has no non-trivial stationary solution w, such

that

( i i ) An ss-pulse W, of Q,(m, a)is a n ss-pulse of Q,(m, r) if

Proof. ( i ) Assume W, to be a non-trivial stationary solution of Q,(m, r). Assume that Vtwh= IIVw,[lm and Vt--Iwh< I[Vw,[[.. for some ic Z. Then, we obtain

In the case of Viwh< llVwhllrn for all i E Z , we also obtain the same estimate as (5.31). Hence, Q,(m, r) has no non-trivial stationary solution satisfying (5.29). ( i i ) Let w, be a n ss-pulse of Q,(m, m). Then, by (5.28) and Proposition 5.4, W, is a n ss-pulse of Q,(m, r) under the condition (5.30). 0

T. IKEDA

412

Examination of the conjecture We demonstrate in this subsection the results of numerical examinations of the conjecture given in Section 1. We discuss the existence of a n ss-pulse W and the length diam ( W ) of supp [ W ] solely. By the same method as that of Proposition 5.3, it is shown that if W is a n ss-pulse of P(m, r ) , then, for each o>O, Z ( x ) = ~ ~ W ( x ois ~a -n ~ss-pulse ) of P(m, rum-e) and (ra"-z)zIIZll-2-"=r2~~W11~-". Values of 1.5 and 3.0 were chosen for the parameter m. We fixed h=0.1, and prepared various ss-pulse w h of Q,(m, m), by using (4.25). Each of these ss-pulse was used as a n initial value of Q,(m, r ) . As stated in Proposition 5.4, w, is a n ss-pulse of Qh(m,r ) if r 2 d i a m (Vw,).

5.3.

( a ) m = 1.5

Figure 5.1.

(F(1.5)=4.206...)

(b) m = 3.0

(F(3.0)=2.587...)

Examination of the conjecture for the existence of an ss-pulse.

A

3-

3-

2-

I

( a ) m = 1.5

Figure 5.2.

2.

(F(1.5)=4.206-..)

(b) m = 3.0

(F(3.0)-2.587..')

The length diam ( W ) of the support of an ss-pulse W.

A Discrete Model for Spatially Aggregating Phenomena

413

For some r and w, such that r < d i a m (Vw,), supp [VK;] (u;: the solution of Q,(m, r ) with ut=w,) tended to some finite interval, and Vu; tended to some non-negative function W,sO as n 00. In this case, we judged that Vu; and diam (Vu;) (n: sufficiently large) approximate a n ss-pulse W of P(m,r ) and diam (W), respectivly; and we made a dot at position ([[Vu;[lm,r ) in Figure 5.1. Figure 5.2 shows diam (W)/r ( W : a n approximate ss-pulse) plotted against r [IWII_l-m'z. Figures 5.1 and 5.2 indicate that the conjecture is correct for the existence of a n ss-pulse W and the length of supp [ W ] ,respectively.

6. Behavior of the Solution of P ( m , r ) In this section, we express our views, which are supported by various numerical studies using Q,(m, r), on the stability of standing pulse-like solu-

Figure 6.1. An approximate solution of P(2.0,3.9) (h=0.3, U " ( x ) = l + cos ( 4 1 8 ) for 1x1< 18, Uo(x)=O for 1x1 > 18).

Figure 6.2. Approximate solutions of P(2.0, 3.9) (h=0.3, U"(x)= max {O, cos (bxx)) for 1x1 <18, Uo(x)=O for 1x1 >18).

T. IKEDA

414

(1)

X

a = 12.0

tu _g

(2)

a = 15.0

(3)

a = 21.0

X

Figure 6.3. Approximate solutions of P(2.0, 3.9) (h=0.3,1 UO is given .by (6.1)).

(1)

a = 10.8

(2)

a = 13.5

.. ..-.

Figure 6.4. Approximate solutions of P(2.0, 1.8) (h=0.3, Uo is given by (6.1)).

A Discrete Model for Spatially Aggregating Phenomena

415

tions of P(m, r ) and the behavior of the solution U(x,t ; U a ) of the Cauchy problem P(m, r ) with the initial condition U(x,0; Vo)=Uo(x) ( O < r < a).We denote by #( V) the number of connected components of supp [ V ] .

Stability of standing pulse-like solutions. Our view on this subject is that each standing pulse-like solution of P ( m , r ) is stable in a sense. Figures 6.1 to 6.6

rl

(1)

a = 15.0

(2)

a = 21.0

-

U

I

.

_-

X

Figure 6.5. Approximate solutions of P(1.5, 3.9) (h=0.3, U o is given by (6.1)).

(1)

a = 12.0

(2)

a = 14.7

(3)

a = 16.0

X

. -

Figure 6.6. Approximate solutions of P(3.0, 3.9) (h=0.3, U ois given by (6.1)).

416

T. IKEDA

( 1 ) Uo(x)

i s g i v e n by ( 6 . 1 ) w i t h a = 33.0

( 2 ) U0(x)=2/9

(-13<x<-7).

U0(x)=l/9

(5<x<11).

U o ( x ) = O (otherwise)

Figure 6.7.

Approximate solutions of P(2.0, a).

show approximate solutions of P(m, r ) obtained by using Q,(m, r ) . Even if Uois of class C1and unimodal (Figure 6.1), U ( . , t ; Uo) may tend to a standing pulse-like solution W( ; U o )such that #( W ( -; U 0 ) ) 2 2 ,as t+m.

-

Behavior of the solution of P(m, r ) . From numerical results, we certainly infer that for each U O ,U ( - ,t ; U o ) tends to a standing pulse-like solution W ( . ; U O ) as t 4 m . However, the behavior of U ( . , t ; U o )is complicated. To see this, let us consider the case where the initial distribution is given by (6.1)

U,O(x)=l/a for Ixl
and

U,O(x)=O for l x l > a

with a positive number a (Figures 6.3 to 6.6). It has been observed that #(U(., t ; U,O))s#(U(.,t ; U;,))

for

t>O

if O < a g a ' .

If a > O is small, then the behavior of U( -,t ; U:) is simple and

#(U:)=#(U(., t ; U,"))=l ((1) of Figures 6.3 to 6.6). follow :

t>O

For some a>O, however, #(U(., t ; U:)) changes as

l1

# ( U ( . , t ; U:))= n + 1 > 2

nzl

where O < f , < t , < t l < t ,

for

(Figures 6.2 to 6.6).

for O < t < t , , for t 2 < t < t 3 , for t > t , ,

A Discrete Model for Spatially Aggregating Phenomena

417

F o r an arbitrarily fixed T>O, t h e mapping from an initial distribution U0 to U ( -; T ; U o )is continuous. However, t h e mapping from Uo to W ( - ;Uo)= limt, U(-,r ; Uo)is n o t continuous. I n fact,

# ( W ( - U,"))=l ; # ( W ( . ;U,O))L2

i

for for

O
a>a,,

where Ui is g i v e n b y (6.1) a n d a o = s u p {a>O; # ( W ( . ;U:))=l}. infer t h a t for s o me fixed E > O ,

1 s u p T(E, a ) ; -a,
1

=oo

Moreover, w e

,

where T ( e , a ) = i n f { T > O; I l U ( - , t ; U g ) - W ( - ; Ui)II,<e for all t > T } < m , We also note that the behavior o f a solution U, of P ( m , r ) ( O < r < o o ) differs entirely f r o m t h at of P(m,a), e v e n if U , tends to an ss-pulse a t t--t 00 ((1) a n d (2) of Figure 6.3 a n d Figure 6.7).

Acknowledgement. T h e au t h o r would like t o tha nk Prof. K . Tom oe da of Osaka Institute of Technology for reading manuscript a n d offering valuable advice. References D. G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461-467. -, Regularity properties of flows through porous media: A counterexample, SIAM J. Appl. Math., 19 (1970). 299-307. Regularity properties of flows through porous media; The interface, Arch. 131 -, Rational Mech. Anal., 37 (1970), 1-10. J. Bear, Dynamics of Fluids in Porous Meida, American Elsevier, New York 1972. L. A. Caffarelli and A. Friedman, Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. SOC.,252 (1979), 99-113. Regularity of the free boundary for the one-dimensional flow of gas in a [ 6 1 -, porous medium, Amer. J. Math., 101 (1979), 1193-1218. M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21. E. DiBenedetto and D. Hoff, An interface tracking algorithm for the porous medium equation Trans. Amer. Math. SOC.284 (1984), 463-500. B. H. Gilding, Properties of solutions of an equation in the theory of infiltration, Arch. Rational Mech. Anal., 65 (1977), 203-225. B. H. Gilding and L. A. Peletier, The Cauchy problem for an equation in the theory of infiltration, Arch. Rational Mech. Anal., 61 (1976), 127-140. J. L. Graveleau and P. Jamet. A finite difference approach to some degenerate nonlinear parabolic equations, SIAM J. Appl. Math., 20 (1971), 199-223. W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous populations, J. Theoret. Biol., 52 (1975), 441-457.

418

T. IKEDA

1131 M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1979), 35-49. [I41 W. D. Hamilton, Geometry for the selfish herd, J. Theroet. Biol., 31 (1971), 295311. [15] D. Hoff, A linearly implicit finite difference scheme for the one-dimensional porous medium equation (preprint). [16] T. Ikeda, Discrete asymptotic behavior for a nonlinear degenerate diffusion equation, Computing Methods in Applied Sciences and Engineering VI (eds. R. Glowinsky and J.-L. Lions), North-Holland, Amsterdam, 1984. [ 171 --, Standing pulse-like solutions of a spatially aggregating population model, Japan J. Appl. Math., 2 (1985) 111-149. [18] A. S. Kalashnikov, On the occurrenfe of singularities in the solutions of the equation of non-stationary filtration, 2. VyEisl. Mat. i Mat. Fiz, 7 (1967), 440-444. [19] B. F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. SOC.,234 (1977), 381-415. [20] P. D. Lax and B. Wendorff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237. [21] M. Mimura, T. Nakalti and K. Tomoeda, A numerical approach to interface curves for some nonlinear diffusion equations, Japan J. Appl. Math., 1 (1984), 93-139. [22] M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Adv. Biophys., 15 (1982), 19-65. [23] T. Nagai, Some nonlinear degenerate diffusion equations with a nonlocally convective term in ecology, Hiroshima Math. J., 13 (1983), 165-202. [24] T. Nagai and M. Mimura, Some nonlinear degenerate diffusion equations related to population dynamics, J. Math. SOC.Japan, 35 (1983), 539-562. [25] -, Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics, SIAM J. Appl. Math., 43 (1983), 449-464. [26] -, Asymptotic behavior of the interface to a nonlinear degenerate diffusion equation in population dynamics to appear in Japan J. Appl. Math., 3 (1986). [27] 0. A. Oleinik, A. S. Kalashnikov and Chzou Yui-lin, The Cauchy problem and boundary value problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk., 22 (1958), 667-704. [28] S. Osher, Nonlinear singular perturbation problem and one sided difference schemes, SIAM J. Numer. Anal., 18 (1981), 129-144. [29] A.E. Scheidegger, The physics of flow through porous media, Univ. Tronto Press, 1974. [30] K. Tomoeda and M. Mimura, Numerical approximations to interface curves for a porous media equation, Hiroshima Math. J., 13 (1983), 273-294.

Department of Mathematics Faculty of Science Ehime University Matsuyama 790, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 419-429 (1986)

A Note on the Blowing-up Problem of a Certain System of Nonlinear Parabolic Equations By Nobutoshi ITAYA Abstract. The author discusses the so-called blowing-up problem in partial differential equations for a certain kind of nonlinear parabolic system of differential equations on the basis of the results obtained before by him for nonlinear single parabolic equations. Firstly, a sufficient condition is sought for the non-blowup of the solution of the above-mentioned nonlinear parabolic system. The maximum principle plays an important role here as in the author’s earlier papers. The obtained assertion is exemplified by some examples. Secondly, a theorem is given which asserts that the solution of this kind of nonlinear parabolic system blows up under some conditions on the form of the system itself and on the initial value. Finally, some examples illustrating the blowing-up of the solution are given for reference. Key words: non-blowup, blowing-up

S 1. Introduction Previously the author has discussed the blowing-up problem of single nonlinear parabolic equations from a somewhat different view-point than Fujita’s (cf. [2]) and has obtained some results ([3], [4], [ 5 ] , 161). In this paper we shall study this problem for a certain system of nonlinear parabolic equations. For simplicity’s sake we consider only the case in which the spatial dimension is 1, which is not always essential. The notation is almost the same as in [4] and [ 5 ] . The system of nonlinear parabolic equations to be considered is as follows:

where #, $, @, and 4 are functions of the C2-class defined on [0, m)x[O, a) such that # and $ > O , @ a n d 420, and pach of the four functions is monotonically increasing in u and u, resp. (therefore, #(., v)z#(O,O)>O, $(., u ) 2 $(O, 0) >O). The initial-boundary conditions are the following: Received April 2, 1985.

N. ITAYA

420

v( x, 0 )= vo(x)( 2 0 )€ H t f ; ; u(0, t)=u(l, t)=v(O, t)=v(l, t ) = O

( t 2 0 );

Hereafter, we shall write H2+",H2,+",etc. instead of H;f,", WTT;), etc., resp. Without proof we state:

Theorem 1. For some T G ( 0 , a),there exists a unique solution ( u , v) for (1.1)-(1.2)belongingtoH2,'"xH$+" (N.B.: u(x, t ) , v ( x , t)zO). [The proof can be carried out in a conventional way.] Moreover, we say that the solution (u, v) for (1.1)-(1.2)blows up, if either u or v blows up.

S 2.

NOU-BIOWUP

Under the assumption that T belongs to (0, 00) and (u, v)€ HF" x H F a satisfies (1.1)-(1.2), if we have a priori estimates of lul',") and IulC) from above, then it follows that the solution (u, v) does not blow up, that is, there is a global solution for (1.1)-(1.2) (see [ 5 ] ) . In this section we try to search for a sufficient condition on u, and vo under which the solution (u, v) does not blow up. Now, we define

I$u,

v)-sup {B(u', v'): O i u ' 5 u , O S v ' 5 v }

.

Let p o , q,, and T be positive constants, and (u, v)E H ~ * HFi-" x satisfy (1.1)(1.2). Then, U( x, t ) and V ( x , t ) defined by

I

" ( :)z , V ( x ,t ) = v( x, t)+% x- 2( :y

V ( x ,t ) - u ( x , t ) + L x - 2

1

satisfy the following system of equations,

Blowing-up Problem of a Parabolic System

421

and the conditions,

If B(u, v ) - p o and B(u, v)-qo are nonnegative on Z X [0, T I , then we have, by the maximum principle, inequalities

(2.4)

Therefore, the above reasoning leads to the assertion that, if, from the beginning, uorvo,poland qo satisfy

then, taking into account the inequalities B j B and B S B , we have the relations (2.4) by the method of reductio ad absurdum (see [3], [ 5 ] ) . Moreover, we note that at the same time the following inequalities hold,

Defining p 1 and q1 by

(2.7)

we rewrite (2.4) and (2.5), having, as a result, resp.,

N. ITAYA

422

(2.9)

The latter relations (2.9) are equivalent to

I

12

0 5 IuoI (O)
(2.9)'

8 12

-

B(PI, 41) I

OSIvoI'o)<41--BB(Plr

Now, we define E by

8

12

( p , q ) € R 2 :p , q > O , p - - - & p , q ) > O ,

8

41)

9

.

q-$g(p,q)>O

If E is not empty and, further, p , q, uo, and v o satisfy

(2.11)

then, defining p o and go anew by

(2.12)

we have the following a priori estimates from above of the solution (u, v) for (1.1)-(1.2), 12

05U(X,

(2.13)

t)51uol'o'+-po=P,

8

12

O ~ V ( X ~, ) j l v o l ( o ) + s q o = q ,

Thus we have:

( O l r 5 T ).

Blowing-up Problem of a Parabolic System

423

Theorem 2. If the set E (see (2.10)) is not empty and, further, p , q, uo, and

v0 satisfy (2.11), then there exists a unique temporally global solution (u, v) for (1.1)-(1.2) which, as restricted to Zx[O, TI, belongs to H;++"xH~T++" for an arbitrary T x ( 0 , m) (also, see [3], [4], [ 5 ] ) . Therefore, the solution (u. v) does not blow up. Moreover, it holds that (2.13)'

0 5 u ( x ,t ) S p ,

O S v ( x ,t ) i q ,

(OZt<)..

.

Next, we give some examples.

Example 1. For $(u, v)=&u, v)-1 and $(u, v)=$(u, v)=uv, ( B ( u ,v)=B(u, v)=B(u, v)=ii(u, v ) = u v ,

Thus, if ( p , q) belongs to E and it holds that

(2.15)

then the solution (u, v) for (1.1)-(1.2) does not blow up.

Example 2.

For $(u, v)=&u, v ) = l and $(u, v)=$(u, v)=u2+v2,

( B ( u ,v)=B(u, v)=B(u, v)=B(u, v)=uZ+v2,

Thus, if ( p , q) belongs to E and it holds that

(2.17)

Further, we have

N. ITAYA

424

then the solution (u, v) does not blow up.

Moreover, we have

uv b ( u , v)=B(u, v) =-=u l+v (2.19)

.B(u, v)=B=(u, v)=-=vuv

l+u

( p , q ) : p , q>o, p---->OY l2 P4 8 l+q

p q >O} 8 l+p

q - - -l2

We divide our problem into 3 cases. 1) In the case of 8 > 12,

(2.20)

Therefore, if uo and v, satisfy

then there exist p and q such that

(2.21)’

Thus, the solution (u, v) for (1.1)-(1.2) does not blow up. Moreover, it holds that (2.21)”

Blowing-up Problem of a Parabolic System

425

2) In the case of P = 8 ,

(2.22)

Now, we remark the inequality

which implies that, if u, and v, satisfy (2.24)

IuoI (0)- /ool(0) < 1 ,

then the solution (u, v) does not blow up, since there exist p and 4 such that

3) In the case of P > 8 , we can treat the problem in the same way as in Example 1.

S 3. Blowing-up In the preceding section we have endeavored to obtain a sufficient condition under which the solution (u, v) for (1.1)-(1.2) does not blow up. However, it is very difficult to obtain a necessary and sufficient condition on u, and v, for the blowing-up of the solution (u, v). In this section we shall show that there are some cases in which the solution (u, v) blows up. Now, let $(u, v) and &u, v) satisfy

-(->=”(’>, a 1

av

$5

au

6

and (u, v) be a global solution for (1.1)-(1.2). Dividing both sides of the upper and lower equations of (1.1) by $ and 6,resp., we have

(3.2)

By the equality (3.1), there exists a function @(u, v) (u, u 2 O ) such that

N. ITAYA

426

(3.3) For example, @(u, v) defined by (3.4) (3.2), we derive from

satisfies (3.3). Hereafter, we adopt this function as @. (3.2)

a

(3.5)

v)=(u+v),,+{B(u, v)+B(u, 241 .

-@(u, at

Here, we note that @ has the following properties,

d dw

I:;

-@(w,

(3.6)

-@(w,

w)=-+-

1

$(W,

w)

w ) = -~ $(w.

>o, #(W, W )

{ $ L ( w ,w)+$,(w,

41

W)z

and that, therefore, @ ( w , W ) is monotonically increasing and concave. NOW, multiplying both sides of (3.5) by K

X

s(x)=-sin-x 21 1

(3.7)

I!(

s(x)dx= 1

and integrating them from 0 to t in r and from 0 to I in x , we have a n equality

(3.8)

1'

@(u(x,t ) , v ( x , t))s(x)dx

For the left-hand side of (3.8), firstly, by the definition (3.4) of @(u,v), we easily have a n inequality

21' @(u(x, t ) , v ( x , r))s(x)dx, 0

(OSt< 03)

Blowing-up Problem of a Parabolic System

427

Secondly, by the inequality

(3.10)

@(u+v, u+v)LO(u, v)

(N.B.: Ou, @,>O)

,

the concavity of @, and Jensen's inequality, we have

Hence, if there is a suitable function F(w) such that

B(u, v)+B(u, v ) L F ( u + v )

(3.12)

,

then all that remains for us is to make similar arguments to those in [4] and [ 5 ] , and to show that in some cases there arise contradictions to the assumption that (u, v) is a global solution for (1.1)-(1.2). Thus, we have:

Theorem 3. Let $, $, 4, and q be the same as in the preceding sections, $ and $ satisfying (3.1). (i) Zf there exists a function F ( w ) which satisfies (3.12) and has a form F(w)=Clw~-Ccz(where C,, C,, and /3 are constants such that C , >0, C z 2 0 , and /3> l), then, under a certain condition on uo and vo,the solution (u, v) for (1.1)-(1.2) blows up. (There are also cases to which Theorem 2 is applicable.) (ii) Zf B(u, v) and B(u, u ) satisfy C,(u+u)+C,zB(u, v)+B(u, v ) ~ C 3 ( u + v ) - C c , (where C3, C,, and C, are constants such that C,>O, C,LO, and C,LO), then, under a certain condition on uo, vor C,, and @, the solution ( u , v) blows up. (There are also cases to which Theorem 2 is applicable.)

As for the proof of the above theorem, refer to [4], [ 5 ] , and the examples to be given below. The inequality (3.11) is necessary to demonstrate (ii) of the above theorem. Remark. (i) For $(u, v)=$(u, v), the equality (3.1)implies $ u = $ u = $ z I , which shows that $(u, v) has a form $(u, v)=$,,(u+v). (ii) For $ ( u , U ) = $ ~ ( U ) Therefore, (3.1)is satisfied. and $(u, v)=$,(v), q5u=$,=0. Example 1. In the case of $(u, v ) = p (const. >O), $(u, v ) ~ (const. p >O), and $(u, v)=q(u, v)=u2+v2, it is obvious that

(3.13)

@(u, v)=p-'U+p-'v,

{ B(u,

u)=,i-1(u2+v2)

B(u, v)=p-'(uZ+v2)

.

By (3.8)and (3.9), we have, (a0=p-l+p-l)

(3.14)

aoJ(t)=ao

,

N.ITAYA

428

2

5'

(p-luo+,E-lvo)(x)s(x)dx-k2

0

+a,

5: 1' dt

5'

J(r)dr

0

( u 2 + v 2 ) ( xr)s(x)dx ,

0

Thus, if it holds that (3.15) then the solution (u, v) blows up. Example 2 . In the case of #(u7 v ) = l + u , J(u, u ) = l + v , +(u, v)=a(l+u)v, and $=b(l+v)u ( a , 6, const. >O), we have

(3.16)

@(u, v)=log(l+u)(l+v)

,

B(u,v ) = v ,

B(u, v)=u

.

By (3.8) and (3.11)7it holds that (3.17)

@ ( J ( t )J(t))=2 , log ( l + J ( t ) )

1:

21' {log ( l + u , ( x ) ) ( l + v , ( x ) ) } s ( x ) d x + (c-k2)J(r)dr, 0

).

( k = F , c-min { a , b}

Hence, if c is larger than k 2 = r 2 / l 2and uo+vo#O, then the solution (u, v) blows UP. Example 3.

The system of equations

(3.18)

[ u ( x , O)=uo, v ( x , O)=vo (uo, v o 2 0 , and € H Z t u ) ,

u(0, t)=u(Z, t)=v(O, t)=v(Z, t) = O,

etc.]

is Petrowsky-parabolic as seen from the view point of classification. However, w - u + z , and z-u-v satisfy

Blowing-up Problem of a Parabolic System

429

(3.19)

[ w ( x ,O)=u,+u,

(>=O), z ( x , O)=uo-uo, w(0, t)=w(l, t)=z(O, t)=z(l, t ) = O , etc.]

.

Therefore, w e can treat t h e system of equations (3.18) in almost the sa m e way as i n the preceding cases.

References A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. H. Fujita, On the blowing up of the solutions of the Cauchy problem for ut= Au+ulta, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124. N. Itaya, On the counter-blowup effect of the viscosity coefficient in nonlinear parabolic equations, Collected Papers in Commemoration of the 50th Anniversary of the Foundation of K6be University of Commerce. (In Japanese). -, A note on the blowup-nonblowup problems in nonlinear parabolic equations, Proc. Japan Acad., 55, Ser. A. (1979), 241-244. -, On some subjects related to the blowing-up problem in nonlinear parabolic equations, Lecture Notes in Numer. Appl. Anal., Vol. 2, Mathematical Analysis on Structures in Nonlinear Phenomena, Kinokuniya, 1980, 27-38. -, Re-discussion on the blowing-up problem in nonlinear parabolic equations, Jimmon-ronshO of KBbe Univ. Comm., 16, No. 4 (1981), 150-157. (In Japanese). 0. A. Ladyzhenskaya, et al., Linear and Quasi-linear Equations of Parabolic Type, Nauka, 1967. (In Russian). J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, 1983. A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, RIMS, Kyoto Univ., 13, (1977), 193-253.

Kbbe University of Commerce 4-3-3, Seirybdai, Tarumiku Kbbe 655, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 431-444 (1986)

L’iquation de Kadomtsev-Petviashvili approchant les ondes longues de surface de l’eau en ecoulement trois-dimensionnel Par Tadayoshi KANO Abstract. A mathematical justification for Kadomtsev-Petviashviliequation as an approximate equation for long waves of water surface of three dimensional flow. Introduction On Ctudie dans cet article les ondes longues d’ampleur finie de surface de l’eau e n Bcoulement trois-dimensionnel qui se distinguent d’ondes de surface en eau peu profonde. On donnera une justification mathkmatique pour l’kquation de KadomtsevPetviashvili [ 7 ] , [14]comme une Bquation approchCe des ondes longues, ce qui correspond, pour ainsi dire, ti une justification mathkmatique pour 1’Bquation de Korteweg-de Vries [12] et celle de Boussinesq [2] dans le cas de 1’6coulement deux-dimensionnel [lo].

S 1.

Equations non-dimensionnelles des ondes longues de surface de I’eau

Les ondes de surface de I’eau en Ccoulement trois-dimensionnel sont rCgies par les Cquations aux dCrivCes partielles suivantes par rapport a @=O(t, x, y , z ) , le potentiel de vitesses, et a r = T ( t , x, y ) qui dCfinie la surface libre de l’eau par z - f ( t , x, y)=O:

(1.1)

.a

@zl+@y,+@zr=O

dans Q ( t ) ,

(1

@‘,=O,

(1.3)

Ot+-

(1.4)

rc+ (wz +rpy)O~=o

1 2

z=o, (Dz2

Received February 1, 1985. Revised May 21, 1985.

+@,2+ D S Z )+gz=O -

T. KANO

432

oh Q(r)={(x,y , z ) : ( x , y)GR2, O < z < r ( t , x , y ) } , t > O , est le domaine rempli de

I’eau. Dans 1111, on a montr6 I’existence locale par rapport a u temps d’une solution unique pour (1.1)-(1.4) dans une Cchelle d’espaces de Banach S = Up,,, B, de fonctions analytiques, en connaissant le potentiel initial @(O, x, y , z ) et la forme initiale r(0,x , y ) d e la surface de I’eau. On l’a dkmontrk dans une forme non-dimensionnelle OG intervient un parametre non-dimensionnel d=h/I: le rapport de la profondeur moyenne de I’eau h a la longueur I des ondes de surface. On a demontrk, de plus, que notre solution Btait indkfinirnent differentiable par rapport a ce parametre 8 , obtenant ainsi une justification mathematique du developpement de Friedrichs en cas de l’bcoulement trois-dimensionnel, voir aussi [5], 161, 191 et [191. En s’appuyant sur ces resultats, on va expliciter maintenant les equations non-dimensionnelles adequates et le developpement de Friedrichs correspondant pour 1’Ctude des ondes longues de surface de I’eau. On e n deduira par la suite, comme Bquations approchees pour les ondes longues de surface de I’eau, I’Bquation deux-dimensionnelle de Boussinesq et, en particulier, 1’6quation de Kadomtsev-Petviashvili [7], [14]. Pour ce faire, introduisons tout d’abord l’ampleur non-dimensionnelle des ondes de surface de l’eau: soit zl la difference de la profondeur z de I’eau en mouvernent de la profondeur h de I’eau en repos:

.

z=h+z,

(1.5)

Soit, d’autre part, a le dbplacement moyen de la surface z = r ( t ) de la surface de I’eau en repos. Alors, en posant z,=az,’, on dBfinit I’ampleur non-dimensionnelle yj par r=h+av.

(1.6)

Appliquons a (1.1)-(1.4) le changement d e variables: ( t , x , y ,

Z)H

(tl’, xl’, y,’, z,’) defini par

(1.7)

Posons

-=a, h I

a h

--=€

.

Alors que I’on a btudik, dans [ll], le probleme non-dimensionnel tel que I’on

L’kquation de Kadomtsev-Petviashvili des ondes superficielles

433

ait 6-0 lorsque R+w tout en laissant a pouvoir 2tre finie, on va ktudier dans cet article des ondes de surface de I’eau telles que 6z=(h/R)z et E=a/h soient de mbme ordre comme infinitksimaux lorsque R+w, a+O, e n suivant les suggkstions d’Ursell [21] et de Stokes [20]. Pour la simplicit6 de calcul, on pose carr6ment (1.9) dans ce qui suit. I1 faut noter que l’on Btudie e n fait le probleme pour 6 E [0, 11, et non pas seulement pour 6<1. En posant la condition ci-dessus sur les rapports entre E et 6, ou plus prkciskment entre la profondeur moyenne de l’eau: h, la longueur des ondes dans les directions de (x, y ) : R et I’ampleur des ondes: a , on pr6cise un rapport entre la nonlinkaritk d’kquations rkgissant les ondes de surface de l’eau et la dispersion des ondes. V u (1.6), le potentiel non-dimensionnel de vitesses qui correspond a la surface zl’=v sera p=cp(tl’, xl’, yl’, z,’) dkfini par (1.10)

O

-=@’= CR

,

-t’+Ecp

c= 4 3 , g: la pksanteur

.

- -.

Alors, e n kcrivant tl’, xl’, comme t , x, -,les kquations non-dimensionnelles par rapport a {cp, v } dkfini par (1.10) et par 1 - r = ~ = i + ~ ~ h

(1.11) sont maintenant les suivantes: (1.12) (1.13) (1.14) (1.15)

dans

62(cp,z+cp,,)+cp,z=0

p,=O, %+,

(x,y)€R2,

62

((DZZ+

9:)

1

+v+-v22 2

%+62(vz50z+vyPY)

z=O,

=O

- 6-2Pz=0

1

Q(t)

, tzO,

z=l+62v.

D’apres la transformation de Nalimov [15]: ( t , x, y , z)H(f, x, y , z / f ) ,

{cp,

v } est transform6 k {B, ?j}dkfini par

(1.16)

$= - t + P J ,

r=l+65j

sur Q 1 = { ( x , y , z): (x, y ) E R 2 , O O , oh {$, T } est l’image de {O’, P}. Or, {$, T } satisfait aux 6quations suivantes comme on l’a vu dans [ll], le paragraphe 2:

T. KANO

434

(1.17)

(1.18)

&=O,

z=o,

(1.19)

S 2.

Le diveloppement sur la surface de I'eau: z= P ( t )= 1+ @ p ( t ) Soit @=q(t,x, y; 6) le potentiel de vitesses sur la surface z = f ' = l + 6 2 T :

(2.1)

d t . x, Y ; 6) =lo(&

x, Y, 1+62T(t, x, Y; 6); 6) .

Compte tenu du dkveloppement de Friedrichs dans [ 111, on a le dkveloppement suivant sur la surface z = f ' = l + 6 2 T pour {@, T } dans B,, quel que soit p<po, pour It1 < 4 p o - p ) :

Preuve. Montrons d'abord que {&t, x, y, z; a), ?(t, x, y ; 6)} dkfini par (1.16) a le dkveloppement suivant sur z = l d a m B,, quel que soit p<por pour It1
L’kquation de Kadomtsev-Petviashvili des ondes superficielles

I

(2.3)

435

” 62

jqc+d$+ 3 42$+62((q$,),+(q$~”y)~)=0(~4)

.

D’apres les estimations a priori pour les dBrivCes normales du potentiel

9 aux bords z = i l obtenues dans [ l l ] et compte tenu de (1.16) et (1.21), on a: (2.4)

J1(

t

1) = -PA$( 1) -a4 2qd$(i)

+-3l

~ 2 $ (

1)

}

+O ( P )

dans B,, quel que soit p < p o , pour It1
@c=$t(l),

(2.5)

~,=6,(1)

et

(~,=$,(l),

comme dans le paragraphe 6 de [ll].

QED.

De (2.3), on dtduit le systerne suivant par rapport a {ii=q,, V=@,, q } : u,+sz(uu,+~~,)+q,=0(64)

(2.6)

i

fit+

w u u , +mu)

9t +621(q4,+

,

= o(64) ,

(aV),}+Ez+Vy+- a2 (~,,,+~,yy+v,,,+V,,,)=o(64) 3

,

dans B,, quel que soit p<po, pour It1 < a ( p o - p ) . Dans le cas de I’tcoulernent deux-dimensionnel, on a dCmontrt dans [lo] que, parmis les ondes de surface de I’eau rCgies par ce systeme, il existaient certaines qui Ctaient approchCes par solutions de 1’Bquation de Korteweg-de Vries a une erreur d’ordre 0(64) pres dans B,. Mais, dans le cas prtsent, la rnkthode ne marche plus. Une thkorie correspondante, en cas de 1’Bcoulement trois-dimensionnel, sera fournie par une justification mathkmatique pour 1’Cquation de Kadomtsev-Petviashvili comme une Bquation approchBe des ondes longues de surface de I’eau, un peu plus tard, dans le paragraphe 4. Remarque 2.1. par exemple:

1”) Si on pousse le dkveloppement (2.2) plus loin, on a,

T. KANO

436

2”) En particulier, (2.2) montre dkja Gj,+p=o(62).

{ 7 ,+A? =O(67 ,

Ou encore, on a

dans B,, quel que soit p<po, pour It1
vtt-A p =

et

O(P)

dans B,, pour tout p < p o et pour It1
7

satisfait a 7jtt-AT=0, avec les donnkes initiales satisfaisant a

oh

(‘1-7)(0)=O(S2),

S 3.

(?,-s,)(O)=0(S2)

.

Equation deux-dimensionnelle de Boussinesq

On considere dans ce paragraphe, un dkveloppement de {v, p } a u moyen de valeurs de potentiel a u bas-fond de I’eau (cf. n ” 8 dans [lo]). En supprimant le signe “prime”, le potentiel 0,(l.lO), a le dkveloppement suivant, vu 6 2 ( @ , , + @ y y ) + @ r r = 0 :

(3.1) Oh

9 = w ,x, Y , 0;6)

(3.2)

et A=d2/dx2+d2/i3y2,v z , IzI 51. En se servant de ce dkveloppement dans B,, nous obtenons le dkveloppement suivant dans B,, v p < p , , pour It1
1 +T (c@2”c@9 +f

(3.3) ft+

( f g Z i+ z

62

- -f

2

2(&J,

+gZAg,+42,A@, - ( A W )=O(64) ,

62

-Tr 2 ( r , d @ , + r y d @ ) y + r A 2=q

D’ou, on a le dkveloppement suivant pour {p, - p } dkfini par

(3.4)

-cp=cp(t, x, Y ; 6)=v(j, x, Y9 0;6)

.

0 ~ 3 4 )

L’e‘quationde Kadomtsev-Petviashvili des ondes superficielles

437

dans B,, Vp<po, pour It1 < a ( p o - p ) . Or, comme dans le cas deux-dimensionnel, le problkme de Cauchy pour I’Bquation homogkne suivante dkduite de (3.7) est rksoluble dans S= U p > o B,:

A=d2/dx2+d2/dy2. En effet, si on applique a ( 3 . 8 ) I’opBrateur (Z-(62/2)A)-1,on e n obtient une Bquation hyperbolique quasi-linkaire et le thkorkme abstrait non-IinBaire de Cauchy-Kowalevski nous permet de rdsoudre le problhme de Cauchy localement e n temps dans s. L’Bquation (3.8) est en fait une Bquation approchte de (3.7) au sens que I’on ait:

pour les m&mes donnkes de Cauchy dans Bpo, quel que soit p<por et pour It1
1’Bquation originale de Boussinesq d’une manikre rigoureuse de point de vue mathkmatique e n discuttant la relation entre elles (voir surtout “Remark” a la fin d e n ” 7 de [lo]). On ne peut pas Btendre ce rtsultat tout de suite au cas present pour (3.8) et on renvoie les Btudes approfondies de cette Bquation a u temps qui vient.

T. KANO

43 8

Retenons toutefois une remarque que les solutions de (3.8) admettent des estimations a priori suivantes: (3.11)

E,(t)=E,(O)

3

et

-

n g t 2 ( H 2 ( R Zet) ) 11 [I est L2-norme dans R2, pour $(r) € g t 0 ( H 4 ( R 2n) )gt1(H3(R2)) C dans (3.12) Ctant indtpendante de t et de S € [ O , 11. Ce qui nous permet de montrer, comme nous l’avons fait dans I’appendice de [lo], I’existence de solution globale par rapport au temps du problbme de Cauchy pour (3.8) dans I’espace de Sobolev ci-dessus (voir [lo], (A.2)).

S 4.

Equation de Kadomtsev-Petviashvili

On montre dans ce paragraphe que I’Cquation de Kadomtsev-Petviashvili donne une bonne approximation pour les ondes longues de surface de I’eau dont la longueur dans la direction de I’axe y est beaucoup plus grandel) que celle dans la direction de I’axe x. Ainsi, on montre, pour ainsi dire, que dans le cas prCsent de I’kcoulenent trois-dimensionnel, cette Bquation de Kadomtsev-Petviashvili joue le r d e de I’Cquation de Korteweg-de Vries dans le cas de I’Ccoulement deux-dimensionnel. Ce qui convient bien, nous semble-t-il, aux circonstances et motifs de Kadomtsev et Petviashvili pour proposer cette Cquation en 1970 dans [7]. Alors que nous avions supposC identiques la longueur des ondes dans la direction de I’axe x et de I’axe y pour passer aux problbmes non-dimensionnels dans le paragraphe 2, nous considCrons dbs maintenant des ondes dont la longueur I dans la direction de I’axe x, et la longueur L dans la direction d e I’axe y . De plus, supposons que, lorsque L et I tendent a I’infini et a tend vers zCro, ( I / L ) 2 , et n/h soient d e mEme ordre comme infinitbsimaux. Ou encore, pour la simplicitC, on pose comme dans le paragraphe 1: I1 est Cvident que x et y sont alternatives.

L'e'quation de Kadorntsev-Petviashvilides ondes superficielles

439

(4.1) I1 n'y a pas de raison concrtte pour ICgitimer la relation (4.1) entre L, R , h et a pour trouver une tquation approchke deux-dimensionnelle des ondes longues de I'ampleur finie de surface de I'eau. I1 n'y a non plus de pr6historique d'un peu plus d'un sitcle comme pour soi-disant "paradoxe des ondes longues"2) depuis une "controverse" entre Airy [l] et S. Russell [HI. Ce n'est qu'une d6couverte heureuse pour nous a u cours de ces Ctudes des ondes longues trois-dimensionnelles de I'ampleur finie. Revenons a notre sujet. Les Cquations non-dimensionnelles (1.17)-( 1.20) sont maintenant comme suit:

$,=O,

(4.3)

z=o,

Ensuite, on voit:

(4.7)

Vu (2.3) et (2.5), on obtient maintenant le systtme suivant au lieuide (2.2):

1

.

q c + zcp2++D=o(ao

(4.8)

77,

2,

2

+gzz+az@y,+-8'3 - +a2(7jr@,),=O(8') cpzzz,

Voir [21] et aussi I'Introduction de [lo].

T. KANO

440

(4.10)

17+u=2mf

7-u=2n,

alors, on voit que (4.11)

m(t)=O(l)

et

n(t)=O(P)

dans B,, Vp
(v+cj,)(0)=0(1)

,

pour la solution { $ ( t ) ,v(r)}avec des don(v-cjx)(0)=O(62) dans B,,,

.

Ceci dit, on dCduit de (4.9) deux Cquations suivantes par rapport a rn et a n:

(4.14) -

- - (6m2 m , ) = - 2

a2 rnzxsx-- 6 2 6 2

m,,+0(6‘) ,

dans B,, Vp<po, pour It(
(4.15)

6

2

), + $ M w y = 0 , ~

~

~

pour M = M ( t , x, y ; 6). Pour ce faire, on dCmontre le thCorBme d’existence du problkme de Cauchy pour (4.15) avec les m6mes donnkes de Cauchy que pour m(r), (4.12). Notation 4.1.

L’espace de Banach Bp,*:

uEB,*,

H

L'iquation de Kadomtsev-Petviashvili des ondes superficielles

441

u=u(x,y): analytique par rapport ( x , y ) et I-pkriodique par rapport a x , muni de norme:

(4.16) OG

1

(4.17)

C k ( y ) = j r e-zatzku(x, y)dx , -7r

et Ikl au lieu de 271.14 et 27rlkl pour la sirnplicitC8). et on Ccrit On a alors le lemme suivant:

Lemme 4.2. I1 existe une constante positive a telle que le p r o b l h e de Cauchy pour (4.15) avec la donne'e de Cauchy M ( O ) ~ B p 0admette ,, une et une seule solution M ( t ) appurtenant a BP,., Vp<po, pour It1
Soit

(4.18) le dkveloppement de M e n skrie de Fourier. Puisque Co(t,y)=O, les coeffidoivent satisfaire a I'Bquation suivante cients Ck(t,y ) , k= i1 , k 2 , 1 3 , pour que M soit une solution de (4.15):

--

+

62

Ck,t ikC, - i-

6

a2 3 k8C,- i -C,,,,= - i - 6' 2 lCICj. 2k 2 itl=k i.l#O

D'ob d * -Cc,+i dt pour

ek=ek(t, E), c'est-a-dire

(4.19)

- i - 632 ~ ~ e x p { - - i ( k - - 6k28 + + - - ~ e (t-s) 6 2k 82 2

)

De meme, & et k au lieu de 2nf et 2zk dam ce qui suit.

2 }j+l=k

A l.C,C,(s)ds.

T.KANO

442

Soit IIM(0)llp,,=Eo< R0/2, et on cherchera une solution M ( t ) telle que l l W t ) I l p , x < Ro, vp<po. Puisque l’on a l’inkgalitk suivante: Jp[

,

c

C IC,C,15C- j.- i - l = k I~IJp[C,l.Jp[C,l

j+l=k

9

on obtient:

compte tenu du fait e-(p-p’)lkl.(p-p’)lklZCte uniformkment. D’oG le lemme d’apres le thkoreme abstrait non-linkaire de QED. Cauchy-Kowalevski 181, [ 161 et [ 171. En prenant, si nkessaire, a plus petite, on va comparer M ( t ) et m(t) pour I l l
Proposition 4.3. Les ondes longues de surface de I’eau rkgiespar (4.2)-(4.5), pkriodiques4) par rapport a x appurtenant a BP,a,son? approche‘es par les solutions de I’e‘quation de Kadomtsev-Petviashvili au sens que I’on ait: (4.20)

11dt)- ( M ( t ) + N ( t ) )Ilp,.= a 6 4 )

9

02 M ( t ) est la solution de (4.15) satisfaisant M(O)= =(l/2)(7+gz)(0) cBPO,n et N ( t ) est la solution de Z’kquation non-homog2ne4) de Kadorntsev-Petviashvili :

V p < p o , pour It1
Preuve. D’apres la dkpendance continue de solution par rapport au second membre, on a d’abord, quel que soit p<po,

‘) Suggestions de T. Nishida. Si on prend I’tquation de Kadomtsev-Petviashvili homogtne aussi pourN(r), on ne peut obtenir que iiN(t)-n(t)ll,,,=U(62)au lieu de (4.23).

L’tquation de Kadomtsev-Petviashvili des ondes superfcielles

443

De meme, on a

d’aprks (4.22).

QED.

E n rtsumk, nous a v o n s ainsi donnk une justification mathkmatique pour l’tquation d e Kadomtsev-Petviashvili comme une kquation approchke des ondes longues, a u sens (4.1), d e surface d e I’eau. Ce qui correspond a une justification mathtmatique d e l’tquation d e Korteweg-de Vries pour I’tcoulement deux-dimensionnel dkmontrke dans [lo]. Voir aussi des considkrations dans [31 et VI.

Note ajoutke Aprks avoir termink la r6daction d e cet article, I’ktude trks intkressante d e H. Segur et A. Finkel [22] nous a ktk signalte par E. Date. Seulement, ils appellent “shallow water waves (ondes en e a u peu profonde)” les ondes d o n t le nombre d’Ursell U=50L2h-3est d’ordre 0 ( 1 ) , alors que nous les appelons les ondes longues pour les distinguer d’ondes dont le nombre d’Ursell U>>1 pour L>>1, pouvant rester finie, voir [lo].

<,

Bibliographies

[31 [41 [51

161 [71 t81

191

G . B. Airy, Tides and waves, B. Fellowes, London, 1845, 241-396. J. Boussinesq, Thkorie des ondes et remous qui se propagent le long d‘un canal rkctangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pure. Appl., 2” skrie, 17 (1872), 55-108. P. J. Bryant, Two-dimensional periodic permanent waves in shallow water, J. Fluid Mech., 115 (1982), 525-532. R. S. Johnson, Water waves and Korteweg-de Vries equations, J. Fluid Mech., 97 (1980), 701-719. K.-0. Friedrichs, On the derivation of the shallow water theory, Appendix to: The formation of breakers and bores, by J. J. Stoker in Comm. Pure Appl. Math., 1 (1948), 1-87. K . - 0 . Friedrichs, Asymptotic phenomena in mathematical physics, Bull. Amer. Math. SOC.,61 (1955), 485-504. 6.6. K a A O M U e B H , B. M. neTBHaLLIBHJIH, 0 6 y C T O f i W B O C T H YeAHHeHHbIX BOJIH B cna6o Aucneprupyrouuix cpenax, AOKJI.AH CCCP, 192 (1970), 753-756. T. Kano et T. Nishida, Sur les ondes de surface de I’eau avec une justification mathkmatique des equations des ondes en eau peu profonde, J. Math. Kyoto Univ., 19 (1979), 335-370. T. Kano and T. Nishida, Water waves and Friedrichs expansion, Lect. Note in Num. Appl. Anal., vol. 6, “Recent topics in nonlinear PDE, Hiroshima 1983”,

444

T. KANO

ed. M. Mimura and T. Nishida, Kinokuniya-North Holland, 1984, 39-57. [lo] T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves, a paraitre dans Osaka J. Math., 23 (1986). [ l l ] T. Kano, Une thtorie trois-dimensionnelle des ondes de surface de l'eau et le dtveloppement de Friedrichs, paraitre dans J. Math. Kyoto Univ., 26 (1986). [12] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and a new type of stationary waves, Philos. Magaz., 39 (1895), 422-443. [13] L. Lagrange, Mtcanique analytique, tome 11, MmeV" Courcier, Paris, 1815. [141 A. M. JeOHOB, 0 ABYMePHbIX YpaBHeHHRX KOpTeBera-Ae Bp~lsaB HeJIHHefiH08 BHYTPeHHUX i BOJIH, AAH CCCP, 229 (1976), 820-823. TeOPMN IIOBePXHOCTHbIX € [15] V. 1. Nalimov, A priori estimates of solutions of elliptic equations in the class of analytic functions and their applications to the Cauchy-Poisson problem, Soviet Math. Dokl., 10 (1969), 1350-1354. [16] L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Diff. Geom., 6 (1972), 561-576. [17] T. Nishida, A note on a theorem of Nirenberg, J. Diff. Geom., 12 (1977), 629-633. [18] J. Scott Russell, Report on waves, Report of the fourteenth meeting of the British Association for the Advancement of Science held at York in September 1844, John Murray, London, 1845, 311-390. [19] J. J. Stoker, Water waves, the mathematical theory with applications, Interscience, New York, 1957. [20] G. G . Stokes, On the theory of oscillatory waves, Trans. Camb. Philos. SOC.,8 (1847), 441-473. [21] F. Ursell, The long-wave parodox in the theory of gravity waves, Proc. Philos. SOC.Cambridge, 49 (1953), 685-694. Bibliographie ajoutke: [22] H. Segur and A. Finkel, An analytical model of periodic waves in shallow water, A.R.A.P. Tech. Memo. 84-12, (July 1984). [Stud. Appl. Math., 73 (1985), 1832201. Dkpartement de Matlikmatiques, Universitk $Osaka, Toyonaka 560, Japon

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 445-458 (1986)

Application of an Iteration Scheme to the Analysis of Incompressible or Nearly Incompressible Media By F u m i o KIKUCHI Abstract. This paper presents an iteration scheme for a class of parameter dependent problems including those of incompressible or nearly incompressible media. We present the scheme, outline of the convergence proof, and some numerical results by the mixed finite element method. Key words: iteration scheme, incompressible (or nearly incompressible) media, Stokes equation, parameter dependent problems, augmented Lagrangian method, mixed finite element method

1. Introduction

Let 9 be a bounded domain in R2 or R3 with boundary dQ. In the analysis of a n incompressible or nearly incompressible medium, we have the following type of parameter dependent problems for a n unknown vector function ii and a n unknown scalar function p :

in which A, div, and grad are the usual differential operators, s i s a given vector function, and E is a physical parameter with small non-negative values. Physically, Ti stands for the velocity or the displacement of the medium, p means the pressure, and is the applied body force. For completely incompressible media, E is equal to zero. When the medium is fluid, the first of the above partial differential equations is called the Stokes equation [ 131. Parameter dependent problems similar to the above appear in various fields, and it is quite important to develop effective computational techniques to solve them. The finite element method is now widely used as a general discretization method for differential equations. It is to be noted that the analysis of the limiting case where E = O is often of special interest [6]. To

7

Received April 15, 1985.

F. KIKIJCHI

446

this end, the most natural discretization method is the mixed finite element method, in which both Z and p are independent unknown quantities. The penalty approach is also widely used to deal with the limiting case. The basic idea is that the solution of (1) for small E > O must be close to that for E=O. If E > O in (l),we have p = - e - ' div Z a n d hence we can obtain a n equation in .-. u only: (2)

1 -AZ--grad(divZ)=f

.-.

in 5 2 ,

4

u = 6 on 852.

E

This is the fundamental relation of the penalty approach, in which p is eliminated and the number of unknowns is reduced. In this method, E is rather an artificial number, since the only interesting value of E is zero. It seems quite natural for us to use finite element models directly based on (2). Bercovier [2] showed that the solution of (1) for E = O can be actually approximated by the solutions of (1) or (2) for small E > O . Moreover, he discussed the penalty finite element method as the discretization of (2). It becomes clear in his analysis that the scheme must be based on appropriate mixed finite element models to have nice approximation properties, although p is a subsidiary quantity in (2). Otherwise, the finite element solutions by (2) behave quite badly. In actual implementation, the explicit use of the mixed method may sometimes be avoided by the use of the so-called selective reduced integration, but the implicit role of the mixed method is in a sense essential. Arnold [ l ] considered a model parameter dependent equation, which has essentially the same properties as (1). In this case, E is a physical parameter, and it is practically meaningful to obtain solutions for E other than zero. He stressed the importance of the robustness of finite element approximations with respect to the parameter, which he showed may be realized by appropriate mixed finite element models. Kikuchi [7] also considered a similar equation and its approximation, and generalized the results of Arnold by means of the asymptotic expansion. In this paper, we present a n iteration scheme to solve the above type of problems. First, we consider a parameter dependent problem in a product Hilbert space as a generalization of problem (1). Then we introduce the iteration scheme with two theorems on its convergence. The basis of the present method is the usual penalty method, but, by using a n iteration technique, we can now obtain solutions for a fairly wide range of E . Although detailed results will be reported in [8], we will show the outline of the convergence proofs as well as some basic properties for completeness. In particular, the iteration process is convergent for E = O . Its relation to the augmented Lagrangian method is also noted: see Fortin and Glowinski [5]. Then the mixed finite element method is introduced and it is shown that the iteration process is also applicable to the arising approximate equations under certain hypotheses. Finally, we give some numerical examples for (1)to see the validity

447

Iteration Scheme to Nearly Incompressible Media

of the present approach. In particular, we observe the dependence of the iteration process on the parameter, and we may see that our approach has nice convergence characters when the above-mentioned hypotheses hold. It is to be noted that our approach is also effectively applicable to numerical analysis of the incompressible Navier-Stokes equations, where the use of some iteration processes is inevitable due to the nonlinearity of the problem, see Mizukami v11. This work is in part supported by the Grant-in-Aid from the Ministry of Education.

2. Preliminaries The norm of a Banach space X is denoted by 11. l.y, and if X is a Hilbert For two Banach spaces X space, its inner product is designated by and Y , L ( X , Y ) implies the Banach space of all linear bounded operators from X into Y. The norm of L ( X , Y ) is conventionally written by I I . I I L ( X , Y ) but The null space and the range of a n operator is often abbreviated as /I.II. T € L ( X , Y ) are denoted by N ( T ) and R ( T ) , respectively. For T € L ( X , Y ) and a subset Z of X , TIZ implies the restriction of T to 2. Let V and W be real Hilbert spaces. We consider two bounded bilinear forms: (a,

,

a ( * , a ) : V x V-R'

(3)

b(*,

,

We assume that a(

where R' is the set of all real numbers. tive in the sense that

(4)

V X W-R'

a):

a(u, u ) 2 0 ;

VU€

a,

0 )

is non-nega-

V .

We can define A E L( V , V ) ,B E L( V , W ) ,and B* € L( W, V )by (5)

(Au, u),=a(u,

V)

;

(Bu, ;Ow=(u, B*A),=b(u, 2 ) ;

(6)

vu, V

€

V ,

VuG V , I € W ,

We will only consider the case where

(7)

BfO

Let

If,

E~

f

N(B)#{O)

*

be a (small) positive constant, and consider the problem: given E E [ O , E ~ ] find , {ut, &} E V x W such that

g } E V x W and

(8) or, equivalently,

44,V ) + b ( V , &)=(f,V ) , ; b(u,, PI- 44,P)w =(9,P)w ;

VV€

VP €

V, w,

F. KIKUCHI

448

(9)

Au,+B*A,=f,

Bu,-d,=g.

We will regard the above as a parameter dependent problem with the parameter E chosen from [0, 4. From the second relation in (9), g must belong to R(B) for E=O. Then R, is necessarily in R ( B ) for E # O . For E = O , A, is indefinite in its component in N(B*). If R ( B ) is closed, then R ( B ) is the orthogonal complement of N(B*) in W , and hence it is sufficient to look for 1, in R(B) even for E=O. Hereafter, we will consider the above problem as a generalization of problem (1). As for its solvability conditions, the following two are wellknown: see e.g. [ll, [2], [3], and [ti].

[HI] There exists a positive constant k , such that

[H2] There exists

(11)

Q

positive constant k , such that

IIB*~II"Bk,ll~I[,;

VA€NB)

I

It is clear that R(B*) is closed when [H2] holds. Then, by the closed range theorem [14], R ( B ) is also closed, and (11) is equivalent to (12)

IIBullwLkzllullv ;

vu.s R(B*) .

Moreover, [ H l ] is equivalent to the following when [H2] and (4) hold, as is proven in [8]: there exists a positive constant k , such that

(13)

Q(U,

u)+llBul12,Lk311ullT. ;

VuE V .

Note that k , is expressed by a continuous function of k,, k,, and [IA[I. Sometimes, (13) is easier to deal with than [Hl]. Under the two hypotheses above, we have the following results as are proven in [l], [2], and [6]. Lemma 1. Assume that [Hl] and [H2] hold, and consider problem ( 9 ) with { J ,g } € Vx R ( B ) and 8 € [o, E ~ ]given. For E = O , there exists a solution { u ~A€} , which is unique in V X R(B) (but WIQYnot be so when it is considered in Y X W ) . Similarly, for E E 10, E,,], there exists a unique solution in V X W , again denoted by {us,As}, where A6 belongs to R(B). In both cases. {us,R6} satisfies

(14) where C is Q positive number dependent continuously on k,, k2,E I[B*II only, and hence is independent of E € [ O , E ~ ] .

~ I[AII ,

and 11B11=

For E ~ O the , second relation in (9) gives 1,=~-~(Bu,-g). Substituting this into the first equation in (9), we have a single equation in u, only:

449

Iteration Scheme to Nearly Incompressible Media

+

Au,+ E-'B*Bu,= f E-'B*g ,

(15) or, equivalently,

a(u,, v)+~-'(Bu,,Bv),=( f, U ) , + E - % ( U ,

(16)

g) ;

V U €V .

These expressions correspond to (2), and often appear in the description of actual physical problems; see Lions [9], Arnold [l], and Kikuchi [7]. They are also commonly used in the penalization of (8) or (9) for E=O. One of the merits of such a n approach is that RE can be dealt with as a subsidiary quantity. Furthermore, R , obtained by the penalty method automatically belongs to R(B) since small positive values of E are used, and hence we need not handle the indefiniteness of RE for E = O . 3.

An Iteration Scheme Fix

E*

(17)

€

[0,

and rewrite (9) as Au,+B*R,=f,

Bu,-E*R,=(E-E*)R,+g,

for which we can consider the following simple iteration scheme: (18) ( i )

=O

( i i ) for i = O , 1,2, (19)

E R(B) ,

.-., decide {u:),

+B * p =f ,

€ V XR(B) recursively

Bu!t)--*J!t)

= ( € - € * ) l i i - ' ) +g

by

.

The existence and the uniqueness of each {@, A:)} in V x R ( B ) follow from Lemma 1. Clearly, {u!O), R , ' O ) } is the approximate solution obtained by equating E to E* in (17). If E*>O, (19) may be rewritten by

The left-hand side of the first equation of (20) is exactly of the same form as that of (15) employed in the penalty approach. Thus we can deal with {R:i)}:=o as subsidiary quantities, and hence we can take full advantage of such a n approach. It is easy to check that {@, A:')} is rewritten by i

(21)

= C (E-c*)!u(f) j=O

,

=

i (E-c*)jJ(j)

,

3 =O

where the coefficients { u ( j ) ,A")} E V X R ( B ) for j=O, 1,2,

--

belong to V XR(B)

F. KIKUCHI

450

and are uniquely defined by (22)

Au(O)+B*J(O)=f

Bu(0)-EE*;j(O)-

A u ( j )+B*J(j)= O

Bu(j)-€*A(/)

--9

7

021).

=A(j-1)

Note that the coefficients do not depend on i. Similar type of expansions at E*=O are analyzed by Lions [ 9 ] , Temam [13], and Kikuchi [7].

We have the following results for the above expansion.

Theorem 1. Assume that [Hl] and [H2] hold. Then, for any {f,g } € V X R(B) and any E * E [0, E ~ ] , each of the coeficients { u ( j ) ,R ( j ) } for j 2 0 in (21) is determined from (22) uniquely in V x R ( B ) with the estimation (23)

llu(j)II"+ IIJ'."

I l f ll"+ Ilsllw)

llw 5 Cj+Y

9

where C is the positive number appearing in (14). Moreover, {uLt), Q)} for each E [0,4 satisfies (24)

Il@) - 4 I l v + I I P - A c l l w 5

Cit21E- €*I

ll"

""llf

+ llsllw )

for i 2 0 , and it converges to rhe solution {uE,A C } € V x R ( B )of (9) as i-+m if E C [ O , E ~ ]is subject to (25)

IE-E*I

< l/C .

Remark. From (25), {&), converges for E = O if E * < l/C, which is one of the most important cases in applications. In particular, E* need not be too close to zero even when we want to obtain solutions with E almost equal to zero, since the convergence is assured so long as (25) holds. This fact is practically important since we are likely to have numerical instability if we use too small a n E * . Proof. Applying Lemma 1 to (22) with E equated to E * , we can assure the existence and the uniqueness of each { u ( j ) ,A(')} ( j 2 0 ) in V x R ( B ) with the estimations

II u(O)ll"

+ llA(o)llw i C(llfll"+ 11-911w)

IlU(j)ll"+

9

~ l ~ ~ ~ ~ l l w i c l l A ~ (j21) ~ - l ~ 1~ l w

from which (23) follows. From (22), we find that (21) satisfies for i 2 0

Au!l)+B*l,")=f,

BULL)-E *1,( ~ ) = ~ + ( E - - * ) ; I ! I ) - ( ( E - - * ) ~ + ~ R ( I )

which, together with (9), gives A(uii' -U,)+B*(A!"-I.~)=O, B ( u , ' i ) - ~ e ) - E ( J ! t ) - -) =; (- ( € -

€* 1t + l J ( t )

,

Iteration Scheme to Nearly Incompressible Media

451

Applying Lemma 1 to the above with (23) taken into account, we can conclude (24). It is now clear that {u:), A:)} converges to {u,, At} as i+m if (25) holds, and the proof is complete. / I / / Notice here that the above iteration scheme reduces to a special case of the augmented Lagrangian method when a ( . , - ) is symmetric and e = O : see Fortin and Glowinski [ 5 ] . Suggested by the convergence proof of such a method, we can obtain another theorem on the convergence of the present iteration scheme.

Theorem 2. Assume that [Hl] and [ H 2 ] hold. Then the sequence {{uLi), {ue,As} of (9) in v x R(B) as i+m i f

ALi)}};=o based on (18) and (19) converges to the unique solution (26)

OI€52€*

(l€--E*I

S € * ).

Remark. Theorem 2 is in a sense complementary to Theorem 1. In fact, the constant C in Theorem 1 is difficult to evaluate in actual problems, while such a quantity does not appear in Theorem 2 : we can a priori make the scheme convergent for the considered E by choosing E* to satisfy (26). Especially, the iteration is convergent at e=O for any ~ ~ 2 0On. the contrary, when E* is smaller than l/C, Theorem 2 gives less information than Theorem 1: particularly, it yields nothing on convergence rate. Note also that the starting approximation & - I ) to A, need not be zero for the convergence of the iteration, provided that 1L-l) lies in R ( B ) . Furthermore, if O < E < ~ E * , A:-') need not be in R ( B ) , as may be seen by carefully checking the proof.

Proof. Define {u('), p")} E V x R(B) by u(')=u,'t)-uu, and p(')=A:')--AC i 2 0 . Moreover, put p ( - l ) = -A, E R ( B ) . Then we find for i20 that A ~ ( i ) f B * ~ ( i ) = o , Bv(L)-

E

for

*p ( t ) = (&-€*)p(t-l).

For E*=O, the conclusion of the theorem is trivial.

For E * > O , we have

and hence

), gives Furthermore, we find that Au(f)+(l/€*)B*Bu(i)=-( 1- ~ / e * ) B * p ( ~ - lwhich

F. KIKUCHI

452

From the above two relations, we obtain

l (26) and ( A u ( ~v)(, ~ ) ) , Z O from (4). That is, {IIp(')Ilw};==o since ( l - ~ / e * ) ~ S from is a non-increasing non-negative sequence and hence has a limit. Then the ~)), left-hand side of (a) converges to zero, and hence both ( A U ( ~ ) , U ( and [lBu(i)llwconverge to zero as i k c o . By (13), this fact implies that u ( t )- 4(t)-u,+O

in

V

(i-tco)

.

Now the relation A U ( * ) + B * ~ ( ~ )gives = O that B*p(*)+O in V ( i - a ) , which assures by [H2] that p(i)=,?~i)-,+O

in

R(B)

(i-co)

,

since p ( i ) belongs to R ( B ) for each i20 (for O < E < ~ E * , (a) directly assures that p*'L)+Oin W ) . Thus {uJi), ,?J$)}+{uc,&} in V x R ( B ) (ik-..), and the proof is complete. ////

4. Application to FEM The techniques presented in the preceding section are easy to apply to the finite element method. In the standard finite element method, we prepare a family of spaces { V h xW h j k e nwhere r the index set A is contained in [0, h,] for a positive constant h, and has zero as a n accumulation point, and, for each h E A , V h and W h are respectively finite-dimensional subspaces of V and W. Usually, h implies the representative element size, and we are interested in the asymptotic behaviors of the numerical solutions as h 10. Let P , € L( V, V h )and QhE L( W , W h )be the orthogonal projection operators, and define A , E L( V f i ,V h ) ,B, E L( V h , W h ) ,and BZ € L( W h ,V h )by

It is clear that A,=P,AI V h , B,=Q,BI V h ,and Bf=P,B*I W h . If we use the standard Galerkin method, the finite element approximation {uhL,,?fie} € V " x W h to the solution {uc,,?€} of (8) or (9) is determined from the condition

Iteration Scheme to Nearly Incompressible Media

453

or, equivalently,

where E , f, and g are the same as those appearing in (8) or (9). Clearly, Q,g must belong to R(B,) for the above to be meaningful for E=O. Therefore, for the validity of the present problem for all g c R ( B ) , it is necessary that (31)

IHo1h

Q , R ( B ) c R ( B h );

VhE A

Note that the above is equivalent to: N ( B ? ) c N ( B * ) for all h E A . The present finite element approximation is of mixed type since it employs two unknowns uhe and However, as in the continuous case, we can eliminate A,, from (30) for E > O : by using the relation Ah,=(l/E)(B,u,,-Qhg), we have

Therefore, we can also utilize the penalty approach for solving the discrete problem (30) with E = O . As is pointed out in [l], [2], [lo], [12] and in many others, such process is in fact effective if Q , has certain special structures. In particular, it can be easily implemented when suitable selective reduced integration formulas are found. It is also possible to apply the iteration scheme considered in the preceding section to solve the discrete problem above. In order to assure the validity of the discrete problem, it is quite natural to assume the discrete analogs of [Hl] and [H2]. [Hl], ( 34)

uh)zk?llUhll?J;

[H2],

(35)

There exists a positive constant kT independent of h E A such that VuhEN(Bh)

*

There exists a positive constant k$ independent of hE A such that

IlB?Ahllvzk$llkllw ;

VAhE R(B,)

.

As in the continuous case, (35) is equivalent to (36)

IIBh4hllwzk$ll4llv ;

v U , E R(B?) *

Furthermore, we can show that [Hl], is equivalent to the following when [H2], and (4) hold: there exists a positive constant k? independent of h E A such that

F. KIKUCHI

454

We can express k r by a continuous function of k r , k$, and IlAll. Under hypotheses [HO],, [Hl],, and [H2],, we can obtain essentially the same results as in the continuous case with respect to the uniqueness and existence of the approximate solution as well as the convergence of the iteration scheme. Especially, in the statement of the results, there appears a positive constant corresponding to C of Lemma 1 , which can be taken to be independent of h E A .

5. Numerical Results We apply our method to finite element analysis of the following problem related to the two-dimensional Stokes equation:

(38)

+

-AZ+gradp=O

,

-ddivZ--p=O

in f2 ;

z=zo on

af2,

where 9 is the square domain shown in Fig. 1 , G={ul, us} is the velocity, p is the pressure, and Zo denotes the prescribed value of ii on the boundary all. We consider the so-called cavity flow problem, where the boundary condition is inhomogeneous and is shown in Fig. 1. Note that Z0is discontinuous at the upper left and right corners of f2. We test two types of triangular finite element discussed by Crouzeix and Raviart [ 4 ] ,in which the approximate velocity ii,={uRI, u,~}and the approximate pressure p h are given as follows.

Element-1. Over 9, uhl and un2are continuous piecewise quadratic functions, while p, is a piecewise constant function. That is, in each triangle, uh1 and u,, are quadratic polynomials, while p h is constant. Element-2.

Over f2, u , ~and u,, are non-conforming piecewise linear func-

x2= 1

u1'UZ'O

r I

~

u1= 1 , Uz= 0 --

R

C

XI= 1

Fig. 1. Q and boundary conditions.

Iteration Scheme to Nearly Incompressible Media

455

tions: they are linear polynomials in each triangle, and are generally discontinuous along the sides of the triangles except at the midpoints. The functional form of p h is the same as that of Element-1. Note that p h is discontinuous, and hence the operation for Q h can be performed elementwise. The selective reduced integration technique is available for both elements, see Bercovier [2]. As for the stability and error analysis of these two elements, see Crouzeix and Raviart [4]. It is especially to be noted that Element-2 has nice properties corresponding to [HO],, [Hl],, [H2], etc., although it is non-conforming. We also consider two types of uniform meshes (Mesh-1 and Mesh-2) shown in Fig. 2. For Element-1, u , ~ ,is set equal to 1 at the upper left and right corners of 9, where K , is discontinuous. Numerical solutions are obtained for various combinations of E and E* by the use of the double precision arithmetic, and the numbers of iterations for convergence are counted. The employed stopping condition is

(39)

That is, the iteration is stopped when (39) is first satisfied, and the corresponding i’s are given in Table 1 for Element-1 and Table 2 for Element-2, respectively. Note in the second relation of (39) that p i i ) and are constant in each element. Convergence is generally rapid when E is close to E * . As we have discussed in the analysis, convergence is always attained for E = O , although it becomes slower as E* becomes larger. We can see that even the choice of is available in practice to analyze the case of E = O , and we need not use too small E* which may bring us numerical instability. Moreover, it appears that the numbers of iterations are in general insensitive to the choice of elements or meshes. To see the dependence of the numerical solutions on E , we give computed velocity vectors in Figures 3 through 6 for ~ = l 0.1, , 0.01, and 0: they are all

MESH-I

MESH - 2

Fig. 2. Triangulations employed for computations.

F. KIKUCHI

456

Table 1. Number of iterations for convergence: Element-1.

X

h

1 1

1 10-1 10-2 10-8 10-4 10-5

50

6 7 69 6 9 6 9 69

0 (-:

10-1

1

E*

2 1 53 7 3 76 7 6 7 6 76

1

10-3

10-2 2

-

-

1 1 3 1 4 1 4 1 4 14

1 1 4 1 5 1 5 1 5 15

1

-

2

_

1

-

10-4 2

-

-

1

-

-

-

-

-

1 6 6 6 6

1 6 6 6 6

-

-

1 4 4

1 4 4 4

4

10-5 2

- -

1

2

-

-

-

-

3

3

1 2

1 2

-

4

4 1 3 3

1 3 3

divergent)

Table 2. Number of iteration for convergence: Element-2. 1

€*

X

I 38 48 4 9 49 49 49

10-1 10-2 10-5 10-4 10-5

0 (-:

J2

10-1

10-2

10-3

10-4

10-5

2

1

2

1

2

1

2

1

45 59 6 0 61 61 61

1 1 1 1 1 1 1 1 1 11

1 1 2 1 3 1 3 1 3 13

_

-

_

_

_ -

5

6 1 4 4 4

5

1

1 6

5 5 5

6 6 6

1 3 4 4

3 1 3 3

2

-

4 1 3 3

1

2

-

-

3 1 2

3 1 2

divergent)

u,=1

E =I

-

& 0.1

Fig. 4. Computed velocity vectors (~=0.1,Mesh-2).

Iteration Scheme to Nearly Incompressible Media

E = 0.01

x2

457 E=O

U,=l

xL U1'1

...*

.

C

C

"

.

.

. . . * *. . . . . , , . ' " ' .

Fig. 5 . Computed velocity vectors (e=O.Ol, Mesh-2).

Fig. 6. Computed velocity vectors (e=O, Mesh-2).

based on t h e use of Element-1 a n d Mesh-2, a n d the vectors a r e shown only a t vertex nodes. N o t e here that the choice of E* has no influence o n the results so long as the iteration is convergent. W e c a n see that a vortex appears when E becomes closer t o 0, a n d qualitative difference m a y b e found between solutions f o r smaller E ' S a n d those for larger ones. Moreover, the numerical solution f o r ~ = 0 . 0 1is quite close t o that for E = O .

References D. N. Arnold, Discretization by finite elements of a model parameter dependent problem, Numer. Math., 37 (1981), 405-421. M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO, Anal. Numtr., 12 (1978), 211-236. F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO, 8 (1974), 129-151. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, RAIRO, 7 (1973), 33-76. M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1983. V. Girault and P.-A. Raviart, Finite Element Approximation of the NavierStokes Equations, Lecture Notes in Math., No. 749, Springer Verlag, BerlinHeidelberg-New York, 1979. [ 7 1 F. Kikuchi, Accuracy of some finite element models for arch problems, Comput. Methods Appl. Mech. Engrg., 35 (1982), 315-345. An abstract analysis of parameter dependent problems and its applications 181 -, to mixed finite element method, J. Fac. Sci., Univ. Tokyo, Sec. IA, 32 (1985), 499-538.

458

F. KIKUCHI

[ 9 ] J. L. Lions, Perturbations Singulieres dans les Problemes aux Lirnites et en ContrBle Optimal, Lecture Notes in Math., No. 323, Springer Verlag, BerlinHeidelberg-New York, 1973. [lo] D. S. Malkus and T. J. R. Hughes, Mixed finite element methods-reduced and selective integration techniques: a unification of concepts, Comput. Methods Appl. Mech. Engrg., 15 (1978), 63-81. [ll] A. Mizukami, Finite element analysis of the steady Navier-Stokes equations by the multiplier method, to appear. [12] J. T. Oden and N. Kikuchi, Finite element methods for constrained problems in elasticity, Intern. J. Numer. Methods Engrg., 18 (1982), 701-725. [13] R. Temam, Navier-Stokes Equations, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1977. [14] K . Yosida, Functional Analysis, Springer Verlag, Berlin-Heidelberg-New York, 1965.

Department of Mathematics College of Arts and Sciences University of Tokyo 3-8-1, Komaba, Meguro-ku Tokyo 153, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 459-479 (1986)

On a Local Existence Theorem for the Evolution Equation of Gaseous Stars By Tetu MAKINO Abstract. The equation of the hydrodynamical evolution of an adiabatic gaseous star, which consists of the compressible Euler equation coupled with Poisson's equation, is waiting for mathematical treatises. Although many interesting studies from the numerical point of view supported by the development of big computers can be found among astrophysicists, rigorous mathematics have not caught up with them. The author has heard nothing about even the proof ofthe existence of solutions for the associated Cauchy problem. Now this article is devoted to establishing the existence of local solutions under a suitable condition on the initial data. The discussion is along a standard line except for the crucial difficulty of integrating the Euler equation for solutions with compact support, which does not appear in the hydrodynamics on the Earth. However the result is too restrictive so that the author hopes that he has presented a tentative treatise initiating the further development of mathematical theory on this interesting equation. Key words: Cauchy problems, quasi-linear hyperbolic systems, hydrodynamics, self-gravitating systems, astrophysics

1. Introduction

W shall investigate t h e C au ch y problem

(1-i)

(3) (4-0) Received April 5 , 1985.

p Ir=o=pO9

460

T. MAKINO

v, It=o=vot

(4-i) Here K , 7 ,

K

are positive constants.

(i=l, 2, 3)

.

The unknowns are the functions p=

p(t, x), ~ = ~ (v2, v u,)=v(t, ~ , x), p = p ( t , x ) = K p ( t , x ) ~ $=$(t, , x) of t>=O and x= c ( ~ lx2, , x 3 )€ R S , while po=po(x) and vO=vo(x)are initial data. We wish to establish the existence of a solution p(t, x), v(t, x), p ( t , x), # ( t , x) on a domain [0, T ] x R 3 = { ( f , x ) / O 5 t 5 Tx, € R 3 } for given po and vo.

Equations (l), (2), and (3) describe the evolution of a star regarded as an isentropic ideal gas with self-gravitation. The variable p means the density of the gas, p the pressure, @ the Newtonian gravitational potential and u the velocity. Equation (1-0) is the equation of continuity, and (1-l), -2), -3) express the conservation of momentum. In this paper the system (1) consisting of (1-O), -l), -2), -3) will be called the Euler equation. Equation (2) is the equation of state, 7 being the adiabatic exponent. We keep in mind that r = 5 / 3 if the stellar material is treated as a monoatomic gas, 7=4/3 if the radiation pressure is taken into account and supposed to be excellent, and other values of r have significances of their own (see [l],Chap. 11, Section 12 or [ 6 ] , Section 53). Equation (3) is Poisson's equation, tx standing for the constant of gravitation. The solution of (3) of physical interest is that given by Newtonian potential:

(3)" For a detailed discussion in astrophysical contexts of the Cauchy problem (1)(4) we refer to P. Ledoux and Th. Walraven [6]. An existence theorem of the problem will be established along the following line: First, taking a n arbitrary p(r), not necessarily a solution, we integrate Poisson's equation (3) by (3)" to get the potential $ ( t ) (Section 4); next we integrate the Euler equation (1) together with (2) with respect to the now given potential $ ( t ) under the initial condition (4) to obtain a new density distribution p ( t ) (Section 3); if p(r) coincides with the first p ( t ) then it should be a solution of the problem (1)-(4). The final work will be done by applying the fixed point theorem (Section 5 ) . A model of this procedure can be found in the study of Vlasov's equation by S . Ukai and T. Okabe, [81. When we progress in this way, we meet with difficulty at the integration of the Euler equation. The standard mathematical treatment of the Euler equation is to transform it to a symmetric hyperbolic system to which Friedrich's theory is applicable. We are acquainted with such a study by S . Klainerman and A. Majda [ 5 ] . In their study, and in most other studies by mathematicians establishing rigorous theories on fluid dynamical equations, the density p of the fluid remains to majorize a positive constant uniformly throughout the whole space during the motion. However, in our astrophysical context the density is expected to have compact support or, at least, to

Evolution of Gaseous Stars

46 1

vanish at infinity, for, otherwise, the Newtonian potential (3)" would be divergent (Olbers' paradox). Unfortunately, if we trace [5] in order to obtain a symmetric system, we find that vanishing of ,o causes the coefficients to be degenerated or unbounded (see [ 5 ] , ( 2 . 2 ) ) . This is the difficulty of the Euler equation in astrophysical contexts. This paper is devoted to proposing a solution of this difficulty. It is a change of variables from p to a suitable variable w=w(p), by which one can reduce the Euler equation to a symmetric system whose coefficients behave gently even if p vanishes. In conclusion we can say that the change of variables

will lead us to the following result.

Theorem. Assume that the initial data po(x)and v o ( x )are continuously differentiable and that p o ( x ) is nonnegative everywhere and has compact support. If (1)

1
( p 0 ) ' r - 1 ) ' 2 € H 3 ( R 3 )and

vo€H3(RS)

or if (11)

1<~<3,

p o € H 3 ( R 3 ) , ( P O ) ( ~ - ~ ) / ~ € H ~and ( R ~ ) v0€H4(R3),

then the Cauchy problem (1)-(4)has a continuously differentiable solution p 2 0 , v, p=Kpr, @ which exists on a domain [0, TI x R3. Here T is a sufficiently small positive number depending on the parameters and the initial data, and H s ( R 3 ) , s = 3 , 4 , denotes the usual L2-Sobolev space of order s. The author should announce that there are no tricks other than the change of variables from p to w and that the rest is just along standard and somewhat easy lines. He expects that the sufficient conditions (I) and (11) will be weakened much more in the future by experts of the study of quasi-linear hyperbolic systems, and would like to describe the necessity of such improvements in Section 5.

Notations. In this paper we shall use the following notations. ) the Sobolev space of order s endowed with the 1) N s = H S ( R S denotes norm

where A is the Laplacian and 11u11= //u[1,is the usual L2-norm. 2) C m ( X ) ,X being R S or [0, TI x R 3 , denotes the space of all functions having continuous derivatives of order S m . B " ( X ) [ B m , B ( X )denotes ] the

T. MAKINO

462

space of all functions having bounded and [uniformly 8-Holder] continuous derivatives of order $m. By Sobolev, H2+"(RS)is imbedded into Bm*1'2(Rs) and l u l ~ ~ l l u ~where 12, for m=O, 1 , 2 ,

--.,

IuI,= sup lu(x)l

.

zf R3

For a function U E C(RS)we write R[uI=sup { l x l / ~ ( x ) f O.} Finiteness of R[u] means that u has compact support. 3) Given a function t-u(t) defined for O s t s T and valued in Hs(RS),we write

For m=O, 1, Cm([O,T I ; H") denotes the space of all H"-valued functions on [0, TI that have continuous derivatives of order $ m . L"([O,T ] ; H " )denotes the space of all Hs-valued strongly measurable and essentially bounded functions on [0, T I . 2.

Change of Variables Let us introduce a new variable

and rewrite the Euler equation (1) together with ( 2 ) . Multiplying (1-0) by d w / d p = e w / p , we get 2

Since d p / p = e w d w , if p = K p r , equations (1-l), -21, -3) divided by p can be 2 written as

Thus we get the Cauchy problem (7)

Evolution of Gaseous Stars UI,=,=

(8)

463

uo,

where

r--lw 2

0

0

v,

O r - l w

v1

0

0

0

21,

0

0

0

Vl

0

r -2- l w

0

Y,

0

0

0

v1

0

0

0

2).

A,=

0

0

2

0

-7-1

2 A,=

The system (7) represents (6) if U=Yw,vl,uz, v,) and G(t)=-c(a$/ax,, a@/ax,, &75lax3). Since the matrices A , ( U ) , k = l , 2, 3, are real symmetric and linear in U , we can apply Theorem I1 of T. Kato [4]. This powerful theorem leads to the following proposition.

Proposition 1. Let m be a nonnegative integer. Suppose that there are given positive numbers M o Z 1, M and T such that (9)

+

[ M , (cMO2+M)T]eearoT 52M,

.

Then (i) if G€L"([O,T ] ; H S t n ) n C ( [ OT,] ; H 2 + " such ) that I[lGlllSt,dM and U 0 € H 3 + "such that I I U o l ( 3 t n ~ Mare , given, then (7), ( 8 ) has a unique solution U € C ( [ O ,T ] ; H S t m ) n C 1 ( [TO], ; H Z + "such ) that (10)

II I UIII3 t n 5 2Mo .

) the solutions with respect to G=G(,), Gel) for the (ii) If U = U ( , ) , U C I are same Uo satisfying the conditions of ( i ) , then (11)

III U(1)- u,,,II 12 t n 5 ceCYoTTII IGm -G(0)II 1 2 t m . (iii) If wo(x)ZOeverywhere, then w(t, x ) z O everywhere for any t ; if wo(x)

T.MAKINO

464

has conipact support, then w(t, x ) does too f o r any t and R[w(t)15R[wO]+2Mot.

(12)

Here the constant c depends only on m and can be taken to be of the form c'+ cNr.

Proof of Proposition 1. The existence of a solution in a sufficiently short time interval is guaranteed by [4],Theorem 11. It suffices only to find the explicit bounds ( 9 ) , ( l o ) , ( 1 1 ) and (12). The following procedure is attributed to [4]and [ 5 ] . ( i ) Given U ( t ) € L " ( [ O ,T I ; H S t m )C([O, n T I ; H2+") such that ~ ~ ~ U ~ ~ / 3 + 2M0, the linear problem

o(t)

€ C([O, TI; Ptm n C1([O, ) TI; H Z t m )(by [ 4 ] , Theohas a unique solution rem I). Integration of the equation yields

s

1 d 11011'=s ( A ( U ) o ,0 ) d x + (G(r), o ) d x , 2 dt

--

where

A(U)=

1 1 2

L

--

Here , k denotes alax,. energy estimate

w.3

0

vk,k

0

Then, since l C ( U ) l ~ 5 ~ ~ A ( U ) ~ ~we z~C have l M othe ,

Estimates of the higher derivatives of 0 can be obtained in the same manner. To make sure we sketch the procedure. Let I J = ( I J ~ , Y ~ , L J ~be ) a multi-index such that I Y I = Y ~ + I J ~ + V ~ $ ~ + ~ . We write

0 ' ~=)~ ~ v = ( a i a ~ , ) ~ ~ ( a i a ~ ,X)) ~ . ~(aia~~)~~0(t,

465

Evolution of Gaseous Stars

Then 0‘”)€ C([O,T I ; HO)n C([O,T I ;H-I) solves

where 3

G(”)=DUG-

C (D”A,D,O-A,D’D,O)

k=l

=DUG-

2 [D”, A,]D,O.

k=l

Here and hereafter D, denotes a/ax, and [ , ] denotes the commutator. lvlh 1, [D”, A,]D,O consists of terms of the form D~D,A,.LvD,O,

If

l~+pis;~~-is+m.

Since D,A,, D , 0 E H 2 + r n ,we can apply the Sobolev’s estimation of products of functions, [7], Theorem 7.1: the L2-norm of each term is estimated by Const. IIDiAkllZtm. /lDkOllztrn5Const. M o 2 , provided a priori that l10(dl13trn5 l s any s.) Thus we have 2M0. (Note that IIAk(U)II,iConst.~ l U ~for \lG(”)l\ SIIDYGll+CzMo2 2 M oimplies . as long as ~ ~ U ( r ) ] ~ 3 + m i This

-

II U(f)ll

rn

+

+

5 [ M o (C3MO2M ) T l e C ~ M 5 2M0 ~T

for O s t s T provided that T is so small that the last inequality holds. over we note

Now we consider the set X of functions U(r)€ Lm([O,T I ; H 3 t m ) H 2 + ” )such that

More-

n C([O,T I ;

II1U I I13trn 52MO and

II U ( t + st)- U ( t )II2 t r n 5 LI 6“ . Let us denote the mapping U - 0 by 0. We have shown that 0 maps X into itself. To apply the fixed point theorem we make X into a complete metric space with the metric

T. MAKINO

466

4 U(I) U(0)1= III U(11- U(0)III 9

(for the completeness see [4], Lemma 2.14). We are going to verify that 0 is a contraction if T is sufficiently small. The perturbation SO= 0(1) - O,,, solves

where F= -

c[A

k ( U ( l > )- A k (

u(0))l

dO
-

Keeping in mind that A k ( U ) is linear in U , we see l\FllSC,llSUll 2C,MoI18UII. This implies

~

~

~

(

Ill~OlllS ~ C , Ml ,l l ~ ~ ~l l l e ~ . 3 ~ o ~ Therefore if T satisfies 2C,MoTeC3MoT <1

then @ is a contraction, so has a fixed point in X , which belongs to C([O, TI; H 3 + " ) nC'([O, T I ; H Z f mand ) solves (7), (8). This completes the proof of (i). ( ii) The equation satisfied by the perturbation 8U= U ( l )- U ( o )is

where 6G=G(,) -GC0)and vk,k

B(U)=

k

where

1

)

~

~

Evolution o j Gaseous Stars

F(’)=D’GG-Z

[D’, A,]D,GU-Z

467

[D’, BIGU.

For I Y 2 ~ 1, [D”,A,]D,GU consists of terms of the form D”D,A,.DPD,GU,

l~+/?I51+m,

M,JIGUII,,,. whose Lz-norm is estimated by Const. 11 D,A,IIz+,IlGUllz+,5Const. [ D ” ,B]GU consists of DaD,B.DWJ,

la+pl51+m,

whose Lz-norm is estimated by Const. 11 BII 2 + m IIGUllz+,5Const. MollbUII2t ., Thus we obtain IIF(’)II 5 IID’Wl +CPO 116U 112+m

-

This implies the estimate

I IGuII 12+m 5 C, I IIWI IZ+mTeCBMOT , which completes the proof of (ii). (iii) Let ( r , E ) be a n arbitrarily fixed space-time point in [0, T ] xRS. Since u € C1([O, T I ; H 2 ) c B 1 ( [ 0TI , x R 3 ) , the well-known fundamental existence theorem for ordinary differential equations (see e.g. [ 2 ] ,Theorems 3.1 and 4.1) guarantees that the initial value problem dx -=u(t, dt

x) ,

x

I,=,=[

has a unique solution x=cp(t) E Cl([O, T I ; R3). This is the particle path through the point E at the instant r. Equation (6-0) implies d r-1 -w(t, cp(t))=--

2

dt

Integrating the equation from

t=r

c ACax,%w .

to 0 backward yields

Looking at this expression, we know that if w(0, cp(0))=wo(p(O))~O then w(r, E ) 2 0 . If w(r, E)+O, then wO(cp(0))#Oso that Ip(0)l 5 R [ w o ] ;

I +\: 1:

IEI = Idr)l= d o )

5 /(o(O)l+

cp’(t)dt

lulmdt

6R[wo]+2Mot.

I

T. MAKINO

468

This completes the proof of (iii) and of Proposition 1 whole.

3.

Integration of the Euler Equation

Let us go back to the ( p , v)-equations. This section is devoted to the Cauchy problem

Here the function g="g,, g,, g 3 ) ,gt= -a$jdx,, is regarded as a given function, that is, we d o not think of Poisson's equation which would put the cause of g ( t ) back to p ( t ) itself. Thus in this section the unknown functions are p and u. Given initial data po, un and a function g ( t ) , we put U o = t ( ~uno,)and G(t)= '(0, s(t)),where

and go to the Cauchy problem discussed in the previous section. Under a suitable assumption we have the unique solution U(t)='(w(r),u(r)). It is expected that p ( t ) defined by

solves the Cauchy problem (1)-(2), (4) together with ~ ( t ) . More precisely, let the data pn, uo and g ( t ) satisfy the following conditions: (13) (141,

po(x)20

=

(pO)(r-1)/2~H3tm(~s)

r--1

(15 ) ,

for any x € R S and

,

R [ p 0 ] 5 R o < + w;

u n E H s t m (RS) and

I I ~ ~ o ~ ~ ~ - 1 ~ ~ ~ I 1 3 + m + I ;l u 0 1 1 ~ + m s M o

po € H2t" ( R 3 )

and

IIPollz+msM1;

Here m is a nonnegative integer, l < y < 3 , M n 2 1 , M,>O, M>O and T satisfies (9). The conditions (14), and (16), satisfy the condition of Proposition 1, (i). Hence there exists a solution U = L ( w ,v) € C([O, T I ; H S t m n ) C1([O, TI;H 2 + , ) of

Evolution of Gaseous Stars

469

(7), (8) such that JljUllls+mS2Mo.It follows from (13) that w(t,x)2_0and R[w(t)]SRo+2MoTfor O i t S T by Proposition 1, (iii). Let us go to p(t,x) defined by (5)" with this solution w(t,x). Since l < r < 3 , that is, 2/(~--1)> 1, the change of variables w-p=Const. P V * / ( ~ - ~ )is continuously differentiable in O i w < + m . Since w € C ( [ O , TI; H 3 ) nC1([O, T I ; H 2 ) c C 1 ( [ 0T]xRs), , we see that p ( t , x), as a composed function, belongs to the class C'([O, T I ; R 3 ) . Since (1-0) is derived by multiplying (6-0) by dp/dw= Const. w2'(7-1)-1 € C([O,T ] xR3), we see that p, v are CI-solutions of (1-0).Since r-l>O, p=Kpr €C1([O,T]xR3). Equations (1-l), -2), -3), which are derived by multiplying (6-l), -2), -3) by p € C', are satisfied by p, v. Summing up, we can claim that p, v is a C1-solution of (1)-(2), (4) on [0, TI x R 3 . Now we are going to identify and estimate the solution p in Sobolev spaces. To d o this we regard the equation (1-0) as a linear equation of the unknown p for which the coefficients v and Cav,/ax, have been known already. Since p c? CJ([O,T ]x R 3 ) has compact support, p € C([O,TI; HI) f l C([O, TI; HO). Hence, according to [4], Theorem I, p is the unique solution of the single linear equation, which belongs to the class C([O,TI; H2+")f l C'([O, TI; HI+") because the coefficients v and C dv,/ax, belong to C([O,TI; H Z t m ) and the initial value po to H Z t m . To find estimates of p we consider two cases separately. Case I. m=O and l < y < 2 . Then estimates of lllplllz and l l l p ~ l ~ - p ~ ~ ~ l l l ~ , pc0,,p(') being the solutions with respect to g=g(,,, g ( ' ) respectively for the ) via the following same P O , follow from (10) and (11) read for w and w ( ~-wc0)

proposition.

Proposition 2. Supposing l < y < 2 , put O=O(y)=min [l, 2/(~-1)-2]. Given w E H 2, we define

for

(0

w(x)< 0

.

Then (i) p € H 2 and

IIP!I?(CK-I/(r-l) = IIw II2 2 / ( 7 - 1 ) . (ii) if

,

correspond to w = w ( ~ )w,( ~respectively, ) then

Here C is a constant independent of

r.

This is proved by Sobolev's theorem on smoothness of composed functions, [7], Theorem 7.2. Let us postpone the proof until the end of this sec-

T. MAKINO

470

tion.

Combining the above estimates with (10) and ( 1 1 ) read for w and we get

W ( 1 ) --W(O)

< c K - l / (7-1) (2M0)2/( r- 1 ) IIPI12 = and llP(1, -Pto~112sCK-"'~-" (2M0)2/(r-1)-B[ cec'oT Tllg(,)-gto,[IJS

Case 11. m 2 l and l < r < 3 . In this case we obtain estimates of IIpllztm and IIp(l, -pto) ]I1+, by integrating (1-0) as a single equation for p. The standard argument provides the following proposition. Proposition 3. If m 2 l and if v € C ( [ O , T ] ; H S + "such ) that [l[vl[lstm62Mo and p o € H 2 + " such that IIp0112tmsM, are given, then (i) the solution p(t) of (1-0), (4-0)satisfies IIIPIIIP+m SM1eCaroT;

(ii) the solutions pto),pel) corresponding to

I IIpm -pto, III

1+m

U=U(~),u

( ~for ) the same po satisfy

5 CM1eC"oTTII I U(1) - Y(0) I I I 2 t m

*

Here the constant C depends only on m. Via this proposition the estimates ( l o ) , ( 1 1 ) read for u and vC1)- Z I ( ~ ) imply

I I IpIIIztm 5MlecroT and

II l p c1) -pto)II I

2 CMleCa'oTTcec'O TIlk7(1)- ~ t o ~ l l l z t mi

The discussion up to now can be summarized as the following lemma. Lemmal. AssumethateitherI:m=Oand l < r < 2 o r I I : m 2 l a n d l < r < 3 . Suppose that potuo and g ( t ) satisfy (13), (14), and (16)m,and additionally, (15), if m l l . Then there is a solution p, vEC1([O, T ] x R S )of (1)-(2), (4), which is unique in the sense that w defined by ( 5 ) satisfies ( 6 ) together with v. Here T is a positive number such that

T I ; H'+"), p(t, x)LO, Moreover p € C ( [ O , TI;H2+")nC1([O, (18)

+

R[p(t ) ]S Ro 2M0T

and the following estimates hold.

for 0 5 t 5 T

47 1

Evolution of Gaseous Stars

Case I (m=O, l < 7 < 2 ) .

where O=min[1,2/(7-l)-l] same po and vo.

and pto,, ,q1) correspond to g=g(,,, g(,) for the

CuseII(mz1, l < r < 3 ) . (19)m (20),

II IpII12+m 5 M,eezcm)noT, IIIP(1, - pto, I II l + r n 5c2(m)MlT2ecz‘m’dloT I I l-s(l) - - s o , I II 2 frn .

Here constants c,(m) and c,(m) depend only on m. Before ending this section we sketch proofs of Propositions 2 and 3 to make sure. Proof of Proposition 2.

Let us consider the function f : w+ f (w) defined by

Since 2/(7-l)-2>O, we see fcC2(--m, + m ) . According to Theorem 7.2 of [7], the composed function f(w(x)) belongs to H,2,,(RS). In order to estimate 11 f (w)llz explicitly we note

for n=O, 1 , 2 if I w I S l . Suppose IIwllz51 in themeantime. Then 1, therefore it follows from the above inequalities that

2 Hence f ( w ) € H 2 and I[f(wNl25C(->’ 7-1 arbitrary IIw[lz>O,we see

I

W

I

~

as long as jIwjJ251. For W E H with ~

~

~

~

W

T.MAKINO

472

therefore

’ is bounded as 1< y < 2.

It is easy to see that

This gives

a n estimate of l [ p [ l zin the’required form. A n estimate of llp(l, -ptu, /I2 can be obtained in the same manner by using the inequalities

which holds for n=O, 1 , 2 if

15 1 and I W ( ~ ) 151. We omit the details.

Proof of Proposition 3. Consider the equation

Integration of the equation leads to

therefore the energy inequality is

provided that luk,klmS IlvllaS2Ma. For a multi-index Y such that derivative p ( y ) = D y pof the solution p of (1-0) satisfies

5 2 + m the

~ I J

Evolution of Gaseous Stars

473

therefore IIP(t)llztm

5 Ilp(0)IIztmec~”ot.

The equation satisfied by the perturbation 6p=p(,) -pto, is

where

We can verify that the L2-norm of each term after the summation symbol is p ~ ~ l +that m l+rn>3/2, that is, rnzl). estimated by Const. IIDv,,, ~ ~ , + m ~ ~ 6 (recall Hence we have

This completes the proof of Proposition 3. 4.

Integration of Poisson’s Equation

Let us consider Poisson’s equation (3). Let Then the Newtonian potential

(3)”

$(x)=-K

s

P(Y) ~

IX-YIdY

,Q€ H 2 have

compact support.

T. MAKINO

474

is well-defined. Moreover, since p € B o , 1 / 2by Sobolev’s imbedding theorem, the potential $ belongs to C 2 ( R Sand ) satisfies (3) (see e.g. [3], Lemma 4.2). We put g= -‘(a$/ax,, a$/ax,, a$/ax,)= -grad $. Then g € B 1 ( R S )and admits the integral representation

(see [3], Lemma 4.1). follows.

Using this, we can estimate Sobolev norms of g(x) as

Lemma 2. Let m be a nonnegative integer. If p E H2+” has compact support, then g € H 3 t m and (211,

I1

I I ~ l 1 3 + m ~ ~ ~ ~ ( ~ ) ( l ~+ l ~ l e ~+ m ~* 1 5 ~ 2 )

Here the constant c&m) depends only on m . Proof of Lemma 2. easy to see

Let p € H 2 satisfy R [ p ] 5 l at the moment.

Then it is

Hence g E L2 and

If R=R[p]>O is arbitrary, then, by applying the above estimate to p R ( x ) = p(x/R), we obtain

Since g is shown to be a n L2-solution of the equation

we know that g E H 2 and

Using the above equation again, we get

Evolution of Gaseous Stars

475

This is a n estimate of the required form for m=O. It is easy to verify estimates for all m in the same manner by induction on m . This completes the proof of Lemma 2. Given a function p ( t ) E C([O,T I ; H 2 ) with compact support, we define g ( t ) by

Then it follows from the above lemma that if ,QEC([O,T I ;H 2 ) then g~ C([O, T I ; H 3 ) (Case I) and if p E L ” ( [ O , T I ; H z t m ) nC([O,T I ; HI+”) then g € L”([O,T I ;H3+”)nC([O,T I ; H Z t m (Case ) 11). 5. Proof of Theorem

We are ready to prove the theorem announced in Section 1. Case I . Suppose 1 < r s 5 / 3 and that po and uo satisfy (13) and (14)0. Choose a positive number T satisfying the following conditions:

(171,

[ M o + ( c l ( 0 ) M , 2 + M ) T ] e c ~ ~5 o2 ~Md0l ~ ,T

where

1 > ~ ~ = C ~ ( O ) K - ~ ~ ( ~ - ~ ) ( ~ M ~ 1+(2Ro)6/2] ) ~ ~ ( ~ -. ~ ) - ~ ~ ~ ~ ( ~ )

(25),

Let X be the set of all functions p E C([O,T I ;H 2 )such that (X-1)

p(t, x ) 1 0 ,

R[p[t)]SZR,

for O S t S T

and (X-2)

II IpI I l2 5 c 2 ( 0 ) K - ” ‘ ~ - ” ( 2 M 0 ) ’.~ ~ ~ - ~ ~

We make X into a complete metric space with the metric d(P(l,, P(o))=IIIP(I)

--P(o)11!2

.

The completeness is obvious. (Note that H 2 c B . ) A mapping (D from X into itself is constructed as follows. Let ,o€ X be given. Then we define $ ( t ) and g ( t ) by (22). Then according to Lemma 2, g E C([O,TI; H 3 )and

T. MAKINO

476

by (21),, (X-2) and (23),. Next we solve the Cauchy problem (1)-(2), (4) with respect to this g= -grad q5. Let p ( t ) be the solution given by Lemma 1. Then p € C([O,T I ; H 2 ) nC1([O,TI; H 1 ) ,p ( t , x ) Z O and the following estimates hold: R[p"(t)J 5 R,+2M,T52R0

by (18) and (24);

I IIpI 5 C ~ ( O ) ~ - ~ / ( I - ~ ) ( ~ M , ) ~ / ( I - ~ ) by (19),. These mean that p" remains in X. Thus a mapping 0:p--p from X into X itself is constructed. We are going to verify that 0 is a contraction mapping. Let p(,,, , o ( ~ )E X be given. Then the corresponding g(,,, g(l) satisfy

+

1119(1)-g
I112

by (21),. Since 1 < 7 5 5 / 3 , that is, 2 / ( ~ - 1 ) 2 3 , the estimate (20), holdsfor 0=1. Therefore

IIIp(l,-p(o, / 1 1 2

( C ~ ( O ) K - ~ / ( I - ~ ) ( ~ M , )( O ~ /) (I ~I -O lIlg(1) ~~)T- -S(O,lllz ~~~I 5 c2(O)K-l/ ( I - 1 )(2M0)2/( 7 - 1 ) -leC1(0)M O T TKC3(0)[1 (2Rd5//"II I IP(1)--P
+

I I12

= A IllP(1, -P(o,lllZ by (25),. Since A < l , the mapping 0 is a contraction. The fixed point p of 0 is a C'-solution of (1)-(4) together with o, p = K p r , q5 in the sense of Lemma 1. This completes the proof of Case I. Case 11. Suppose l < r < 3 , m Z 1 and that p o and oo satisfy (13), (14), and ( l & . Choose a positive number T satisfying the following conditions: (171,

[M,+ ( ~ , ( m ) M , ~ + M ) T ] e ~ l ( ~5 )2'M o ~, ,

where

(25)*

+

1> A = M,c2(m)ecz(m)~oT.c3( m - 1)[1 (2R,)5/2]T2

Evolution of Gaseous Stars

477

Let X be the set of all functions p~ L"([O,T I ; HZtm) n C([O, TI; HI+") such that p(t, x ) Z O

(X-1)

and

for O S t 5 T

R[p(t)]52RO

and

(X-2)

lllPIllztm52M1

*

The set X becomes a complete metric space with the metric P(o))=IIIP(1) - P t o ) l l l l + m

4 ~ ( l ) ,

*

The completeness of X is easily seen by [4], Lemma 2.15 and the imbedding H l t m c B (recall m 2 1). The mapping CJ: p-g+p is defined in the same way as in Case I. Let ,o€ X be given. Then according to Lemma 2,

n

g E L-([0, TI ;H 3 + 9 C([0, TI ; H Z t m )

and

IllglI I 1 3 t m 5 rc,(m)[l+ (2Ro)5'21I I IPI I I Z t m src,(m)[i

+ (2~,)5/21-~ M , = M

by (21), and (23),. By Lemma 1 we see that the solution p ( t ) belongs to 2 R(18) ~ C([O, T I ; H 2 + " ) nC1([o, TI; Hltm), p ( t , X)LO, ~ [ p " ( t ) ] s R ~ + 2 M ~ T 5by and (24), and

I Ilpll IZ+m

5Mlec2(m)BoT 5 2M1

by (19), and (26),. Thus p E X , that is, CJ maps X into Let pto),pel) E X be given. Then

III lg(1)-!~(o) II1 2 + m 5 ~ c a ( m -1)[1+ (2R0)5/21I I l ~ ( 1 )

X.

-P(O)

by (21) (note that 1 + m 2 2 ) .

II IF(

-

P"(o) II I

I I I 1+ m

Consequently

5 Cz(m)MlT2eC2(m)BoT II lg, -g ( oI)I 12+

+

5c2(m)M1T2ee2(m)BoT. ICCAIP? - 1111 (2Ro)5/21Illp(l) -pC0)

llll+m

= A lllP(1) -Pto)1111tm by (20), and (25),. Since A < 1, @ is a contraction. The fixed point gives a solution of (1)-(4). This completes the proof of Case 11. The proof of Theorem is completed.

6. Necessity of Improvement We have established sufficient conditions (I) and (11) on the initial data

T.MAKINO

478

for short time existence of solutions of the Cauchy problem (1)-(4). However we cannot content ourselves with these conditions by the following reason. The Lane-Emden equation

gives stationary solutions of the problem. In fact, if 6 / 5 < 7 < 2 , there exists a solution u ( r ) € Cz([O,R ] ) such that u(O)= 1, u(R)=O and u ( r ) > O for O < r < R , called the “Lane-Emden function of index l/(r-l)” (see [l],Chap. IV). Then

for R < l x l

‘0

together with v = O gives a spherically symmetric stationary solution of (l),(2), (3). As r - R - 0 the solution behaves as p = C(R - r ) ’ / ( r - l )1[+ P ( R - r , ( R - r ) r ’ ( r - l ) ) ] ,

where C is a positive constant and P a double power series with positive radii of convergence. Therefore, since r< 2, that is, l/(r- 1)> 1, p is a C1-solution. If r < 3 / 2 , then p € H 3 . However p ( r - 1 ) ’ 2 - Const. ( R - r ) l j 2 cannot belong to H ’ . In other words, this stationary solution is excluded from account as long as we content ourselves with the present conditions. We expect that the conditions will be weakened to allow this stationary solution. There is another reason for the necessity of further improvement, which follows from the fact that the uniqueness via the (w,v)-system is somewhat roundabout. The (w,v)-system is much stronger than the original ( p , v)-system. In fact, when we divided (1-1). -2), -3) by p, we undertook the regulation of v in the exterior of the support of p, that is, in spite of the absence of the matter the velocity is forced to vary in obedience to the gravitational field. This situation reminds us of “a grin without a cat” of the Cheshire Cat of Alice’s Wonderland. We should develop a mathematical treatment which establishes the uniqueness in terms of ( p , p v ) and so on. This is another reason for the necessity of further improvement. Acknowledgment. and helpful advice.

The author thanks Professor S. Ukai for his initiative

References [ 1 ] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Univ.

Chicago Press, 1939.

Evolution of Gaseous Stars

479

[ 2 ] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, [3

[4J [5 ] [ 61 [7] [8]

McGraw-Hill, New York, 1955. D. Gilberg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. S . Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. P. Ledoux and Th. Walraven, Variable Stars, Handbuch der Physik, Bd. LI, Springer, Berlin, 1985, 353-604. S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, 1973. S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov’s equation, Osaka J. Math., 15 (1078), 245-261.

Department of Liberal Arts Osaka Industrial University 3-1-1 Nakagaito, Daito Osaka 574, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations PP. 481-505 (1986)

Fundamental Solution of the Linearized System for the Exterior Stationary Problem of Compressible Viscous Flow By Akitaka MATSUMURA Abstract. The asymptotic behavior of the fundamental solution of a linearized system associated with the three dimensional exterior stationary problem of a compressible viscous flow past an obstacle is investigated. It is proved that if the flow at infinity is subsonic, the bahavior as Ixl-+w is essentially the same as that of an incompressible flow and if it is either transonic or supersonic, it is crucially influenced by compressibility along the Mach cone. Key words: asymptotic bahavior, fundamental solution, linearized system, exterior stationary problem, compressible viscous flow

1. Introduction

We consider the stationary flow of a compressible viscous and heat-conductive fluid in the domain 9 E R Z Sexterior to a bounded closed obstacle 0 with the smooth surface 8 0 . For simplicity, we neglect the external force and assume the fluid to be isotropic and Newtonian. Then the flow is described by the density p, the velocity vector u = ~ ( u , u,, , us) and the absolute temperature 0 whcih satisfy the system of five equations

(1.1)

-V.(pu)=O, -p( u V ) u- vp v * ( p@ p’lV u ) =0 , -peff(u-V)O- BpffV.u+V (rVB) K = 0 ,

-

+

-

+

-

+

the boundary conditions on the surface (1.2)

UlJO=O

7

BlJo=80

f

and the conditions at infinity

(1.3)

(p, 4 O ) l m = ( p ,

a,

a

9

where p is the pressure, p is the viscosity coefficient, p’ is the second coeffiReceived March 30, 1985.

A. MATSUMURA

482

cient of viscosity, K is the coefficient of heat conduction, e is the internal energy per unit mass, all of which are known functions of p and 8, p o = dp/dO, e8=ae/d0, V = ( a / a x , , a/dx,, d/ax,), 0 is the 3 x 3 matrix defined by Eltj= au,/axl+du,/axi ( l C i , j 5 3 ) , I is the 3 x 3 unit matrix, Yr is the dissipation ~ , p and B are given positive confunction defined by Yr=,uO: 0 / 2 + , u ' ( V - ~ )O,, stants, ii is a given constant vector. We can always choose the coordinate system so that ii='(c, 0, 0), c= liil. As basic assumption, we assume p, p',

(1.4)

p,

K,

K , got

p and e are smooth functions of p, 0 > 0 and P , p P ,p O > O ,

3pr+2p20

for p, O > O

.

Although the exterior stationary problem (1.1)-( 1.3) is unsolved in general, we can easily construct a solution in the case ii=O (Mach number at infinity is zero). In this case, since we expect static distributions of p and 0, setting u=O in (1.1)-(1.3) leads to the simple system u=o,

By virtue of well known arguments on the exterior problem for harmonic functions, we can construct the solution (p, 0 ) of (1.5) and obtain the asymptotic behavior

By making use of the property (1.6), as a direct exercise of the arguments in [4], we can furthermore confirm its asymptotic stability for the evolutionary system. This fact and conventional arguments for the incompressible flow (cf. [l, 21) suggest that in order to attack the more interesting case ufO, it seems to be very important to investigate the fundamental solution for the linearized system, espcecially its asymptotic behavior as (1.6), which is derived by linearizing the equations at the flow state (p,ii, f7) at infinity and representing the obstacle by dirac measures set at the origin. It also seems very important from a physical point of view that the behavior of such fundamental solution is expected to show the various influences of obstacle to the flow profile far from the surface and to clarify the relation between incompressible and compressible flows. The corresponding incompressible flow is described by the system

(1.7)

483

Fundamental Solution f o r a Viscous Flow

and its linearized equation at infinity is given by

where we set ,ii=p(,p, 3). The fundamental solution U = { U l j } l s t , j s 3of the system (1.8) and the corresponding pressure Q , = { q i } l s i s Sare well known as Oseen’s hydrodynamical potentials (cf. [ 5 ] ) :

+

Uij=( - 6 i j A aZiazj)@9 qi= -p(47r-1xilxl-3 ,

(1.9)

lo

(Ci2U)O

@ =c- I ( 4 4 -

( 1 -e-s)s-lds

,

where v=,ii/p and u=IxI-xl. This solution U exhibits a so-called “wake region” behind the origin about the x, axis. More precisely, the velocity tends to its limiting value ii as IxI--t+w at the rate of I x I - ~ in the wake region, and at the more rapid rate / x ( - in ~ any direction other than that of ii. In this paper, we shall investigate the fundamental solution of the linearized system of (1.1)-(1.3) and compare it with the incompressible one. The important difference from the incompressible case is that the flow profile depends on the Mach number M a t infinity defined by

(1.10)

M=ck

,

c,”=(pp+6p02/p2edp,8).

First it is proved that the singularity at the origin spreads over the positive part of the x, axis with rapidly decreasing weight. Next, it is proved that in the wake region the essential difference from the incompressible part behaves at most at the rate of \ x ( - ~ as ’ ~ \x(--t+oo. Therefore we cannot distinguish the influence of compressibility in the wake region. It is also proved that in any direction other than that of ii, if M t l , the essential difference behaves at the rate of XI-^ and if M > 1 (resp. M = l), it behaves at the rate of XI-^ except in the direction of the Mach cone x1=2/M2-11x’l (x’=(xz, x,)) and Ixl-”/‘ (resp. IX/-/‘/~) along the Mach cone. Therefore, when we neglect the terms of order I X I - ~ , the influence of compressibility emerges along the Mach cone only if the flow at infinity is either transonic or supersonic.

2. Linearization and Main Theorem Let

US

rewrite the equation (1.1) by the change of unknown variables ii-tu, 8+8) and collect the linear parts in the left-hand side:

( p , U, B)+(p+p,

-cp,,-pv-u=g,,

(2.1)

( P +P’)V(V. EA6 =g , ,

- cpu,, -V(Ppp+ Po61 +Fdu+

- ~ p t & , , -gP,V- U+

u ) =g

,

A. MATSUMURA

484

where p p = p p ( p ,8), ,Ti=&?,

8) and so on. If we set

the system (2.1) has the form (2.2)

A(D)W=F,

(2.3)

A(D)=-icZD,+B(D), ia ' D I J ' D D Z + ( L J + U ' ) D ~ Di i D 1 ip c D C'DD

(2.4)

9

We define the sound speed c, and Mach number M by

The fundamental solution of (2.2) is defined by the 5 x 5 matrix valued distrij14 G 9 " for the equation butional solution E={Eij}oli, A(D)E=bZ

E-0,

in 9', as IxI++oo.

To state the results precisely, we further set

where Y denotes the Heaviside's function. Here we note that since p - p = a w o + ~ w 4 + O ( [ w 1 2 ) ,E, exactly corresponds to the pressure part. Our main result is

Fundamental Solution for a Viscous Flow

485

Theorem 2.1. There exists a unique solution E of (2.6) in 9’which ’ satisfies the following. ( i ) The singular support of E lies over R:l={xE R3 I x,>O, x’=O}, that is, E € Cm(R3\ Rzl). ( ii ) E-E, E L:,,(R3). ( i i i ) E has a f o r m , E=E,+E,+E,+E,+E,+E,, which satisfies the following (iv)-(viii). ( iv ) E, satisfies Eo€C”(R3\{O)) and I E , ( ~ ) I S C , ( ~ + I X ~ ) - ~where ~XI-~ we represent rapidly decreasing function by C,( 1+ Ixl)-”. ( v ) E, satisfies E,€C”(R3\Rzl) and C,(l+lxl)-Nlxl-l for lE,(X)lS{CN(l+lxl)-Nlx’~-l for ( vi )

x,
E, satisfies Ez € C”(R3)and f o r 1x1 >> 1, x12u+lxl

vu>O,

O
3C, s.t.

+

~Ez(~)l=O(l~I-’),

O z Z ~ +IE,(x)15C,(1+IxI)-2 ,

where we call the region x, Lu+ 1x1 “wake region” and 0, denotes the angle which x and x 1 axis make. ( v i i ) E, satisfies E , €C ”( R S)and f o r Ixl>>l, if M<1,

3C s.t.

IE,(x)ISC(1+IxJ)-2,

if M > 1 (resp. M = l ) ,

+ I E , ( ~ ) l = O ( l x l - ~,/ ~(resp. )

B,=O, O
IC,

S.t.

IxI-~’~)

8 , € [ 0 , B , - ~ ] U [ O n r + ~ , ~ ] - - t IEe(x)15C,(1+Ix1)-29

where OM is the Mach angle defined by sin OM=l/M. (viii) ER satisfies ER € C0(R3) n C”(R3\Rz1) and 3C, ( ix )

S.t.

I E R ( X ) I S C ( ~ + I X I*) - ~

E, and E , have forms, E,-(0, U,O)=E,’+El’+E3’+Ee’+E,‘ E,=((uS, 0, O)+Eo”+E~”+Ee”+ER” ,

,

where the terms with sufix 0 , 1, e and R satisfy the same properties (iv), (v), (vii) and (viii) respectively, and E,’(x) satisfies E3€ Cm(R3)and f o r 1x1 >> 1 , vu>O,

x12u+lxl

O < ~ E ~ T , ’C, s.t.

-

B,LE+

IE,(x)/=0(1~1-~’~), ~E3(~)lSC8(l+l~ . l)-z

A. MATSUMURA

486

Remark. The essential differences from the incompressible part are represented by E,' and E," in (ix). For more precise property as to each component of E , refer to the lemmas in the following sections. 3.

Preliminaries

We define the Fourier transform and inverse transform for x € R3 and its dual variable E € R 3 by

s

a ( i ) = F [ ~ ] ( E ) = ( 2 ; r ) -e~ /az ~ ' b (x)dx, (3.1)

s

t ( x ) =F-l[u](x) = ( 2 ; r ) - 3 / 2 etz*Eu(E)dc.

The Fourier transform of the equation (2.6) takes the form (3.2)

A(E)s(e)=Z,

E € R3 .

So our purpose is to investigate F - ' [ A - ' ( e ) ] . Denote the eigenvalues of A(E) (resp. B(E)) and the corresponding projection matrices by I,([) and p,(E) (resp. A,(E) and P,(E)). Then, by (2.3), it obviously holds that

R",(E)=

(3.3)

--icEI+A,(E) ,

P,(E)=P,(E) ,

and 1, is one of the roots of the characteristic equation det IRI-23 =(A+v1[l2))"f(1)=0 ,

(3.4) where f ( A ) =R3

+ (i+ 7 )I El"+

+ +

(s71El4 ( a 2 jZ) IEl"A+ a"lEl

.

We define 1, by A , , ( E ) = - Y I E ~ ~ and {1,};=, by the three roots of f ( A ) = O . us list up the basic properties of 1, which are well investigated in [ 3 ] .

Let

Lemma 3.1. ( i ) Re A, O and 0 5 j 5 3 . ( i i ) 1, is semisimple for I E l > O , that is, rank (1,1--B)=3 ,

for

IEI > O .

(iii) There exists a positive constant r , such that for O < 1615 r l , { I , } ; = , are distinct roots and can be expanded by the Taylor expansion as I,(E) = C,"=, a,tk)/ E l k . (iv) There exists a positive constant r2 ( > r J such that for 151 2 r 2 , {A,};=, are distinct roots and can be expanded by the Laurent expansion as i , ( E ) = CiZ2b,tk)IEIk. By the property (i), we have det A(E)fO for IEI > O which implies that the equation A(D)W=O in 9 ' ( R 3 ) admits only polynomials as a solution. This

Fundamental Solution for a Viscous Flow

487

fact proves the uniqueness of the solution in 9' of (2.6). The property (ii) enables us to define the partial inverse A-'(E)P,(E) for V I E 1 > O by A-'(E)P0(E)=

(&J(E) - icEJ-IP,(E). Lemma 3.2.

F-l[(A,-icEl)-lP,]=

Proof. The direct calculation gives the relations

(3.5)

(3.6)

=U,, ,

(lzi,j 5 3 )

where w=E//1El. This completes the proof. This lemma shows F-l[?,-lP,] exactly corresponds to the incompressible part of the velocity profile. Let {xt(t)}:?=l be C"(R;) functions which satisfies

~ ~ - 1 ,x 2 - x 3 = O ,y1=x3-0, xz-l xI=xz=O, xs=l

(3.7)

xl+xZ+x3=

for

tSr12/4,

for r 1 2 5 t 5 r z 2 , for t 2 4 r Z 2 , for t E R 1 .

Then A-l is divided as A-'(E)=C;=, xt(lEIZ)A-'(E). Since the property (i) implies that x 2 ( l f l * ) A - 1E( ~C;(R3), ) we have Lemma 3.3.

F-l[xz(IEIz)A-'(E)]is in Cm(R3)and rapidly decreasing.

By the properties (iii), (iv), xrA-' ( k = l , 3) can be represented as 3

(3.8)

xk(lEIZ)A-YE)=C (~,(E)-icE,)-'xk(IE12)PI(E). 2=0

] k = l , 3 and l s i ( 3 . Thus it is sufficient to estimate F - 1 [ 2 , - 1 x 9 , for In the following, we shall analyze the case \El >> 1, \El<< 1 ( M < 1) and IEl<
4.

Analysis for It/>> 1 For IEl z r z , {At(E)}:=l can be expanded by the Laurent expansion as follows;

A. MATSVMURA

488

(4.4)

I A,(x)/5

l+lxl)-~v ( CNlx'l-l(l+ixl)-N

for x , < O , for x,LO,

Fundamental Solution f o r a Viscous Flow

where

* denotes some constant.

489

By elemental calculations, we first note that

-x3(a2/r+icE1)-'= -(az/r+icE,)-'+(l

--x3)(a2/y+icEI)-1

and (l-x3)(a2/r+icEl)-'€ C;, we have by (4.6) F-'[Qil=E,+Ri

(4.10)

.

For Qz, since x3(a2/y+ icEl)-'(iE) IEI

-'= (az,$+icEJ'(iE)(l+ IEI 'I-' +(x3- l)(iE)(1+ lElz)-'(az/r+icS,)-' +x3(aZ/ir+icE1)-'O(IEl =QzitQzz+Qzs

-3)

9

Qzz€ C; and F-'[Qz3]has the property of R , which we show next, we have by (4.6) and (4.7) (4.11)

F-'[x3(a2/r+icEl)-'( iE)

-7 = ( r / c ) (Y ( ~ , ) e - ( ~ ~*/V(e-lZllxl-') ~ r ) ~ l ) +R , 21

=A,(x)+R,,

where it is not difficult to check A , has the desired property (4.4). Now we show F-'[Qzslhas the property of R , . If we regard F-'[Qz3]as F-"(aZ/r+icE1)-'1 * F-"x30(1e1-3)l ,

it is easy to see F - ' [ Q z 3 ] € C"(R3\R:I). To see it to be in Co and rapidly decreasing, we only show ~ 3 ( a 2 / ~ + i ~ E l ) - 1 0€( L' because a t Q z 3 € L' ('la1 2 0) can be proved in the same way:

w3)

By using (4.8) and (4.9), remaining terms Q3, Q, and

Q5

are estimated in the

A. MATSUMURA

490

same way. So we omit the details. Thus completed. This lemma corresponds to the property is comparatively estimate for xs(x2-1P,+x3-1P3) So we only state the result which corresponds Theorem.

the proof of Lemma 4.1 is of El in the Theorem. The easier than that for xsxl-lP,. to the property of E, in the

Lemma 4.2.

11

0 'B, F-'[x3(a,-1P,+x,-'P3)]= B, B, B,

(0 where B,, B , , B , and b, are in C"( R3\{O})

IB,

+R , ,

and have the forin

(4.12) and R , represents fiinction which is in Cm(R3\{O}) and rapidly decreasing. 5.

Analysis for IEl<< 1 ( M < 1).

For \El S r , , {,?t(E)}:=l can be expanded by the Taylor expansion as follows:

(5.1)

where a=a26c,-2 and a'=y/2+5P2/2c,2. Let us start with the estimate for x,?l-lPl. Using (5.1), we can expand xlR",-lP,as follows: (5.2)

+

xIxl-lP,=xl{(a15I2+icE,)-' +b15I4(alEl2 icEl)-21(Qo+Q,) +xl(alE12 icEl)-'(Qz+ Q3+O( IEl')) ,

+

-P2 Q0=c~-'[

0

aP

P:

Ql=c8-4a/3C[ir 0

i P02 E i:E) , - i a cE

Fundamental Solution for a Viscous Flow

where

(5.3)

*, b and e denote some constants. F-I

49 1

If we take into account x,R”,-~P, € L’,

[(aI El 2 + icE,)-’]= (x/2)1 / Z a - 1 I I - I e -

( c / 2 a ) (121-z1)

7

and

F- [ P.f .( aI E I 2 + ice,)- 2 ] = (71./2)’/2ae-( ~ / 2 a(1)

(5.4)

f

1 -“I

)

we can show by (5.2), (5.3) and (5.4)

Lemma 5.1.

andfor 1x1>> 1, where a,, A , and R , are in Cm(R3)

and

It is easy to see that this lemma corresponds to the properties of E, and

Esin the Theorem. Let us turn to the estimates for X~(R”,-~P,+R”~-~P,). Since (5.5)

F-’[X~(R”,-’P,+R”,-’P,)] =2 Re F-1[~IR”2-1P2] ,

it suffices to estimate Re F-’[X~R”,-~P,] in what follows. The expansion

shows that the leading term is ic,lEl(l-Mo,) for M< 1 and therefore the case M < l is easier to treat than M z l .

Lemma 5.2. If M< 1, then F-1[~lR”2-1P2] € C”(RS)and IRe F - ’ [ X ~ R ” ~ - ~ P5C(l+lxl)-2 ,](~)I .

Proof. First, P2(5)is expanded as

A. MATSUMURA

492

Fl(E)

f FZ(E)

and estimate each term as follows.

+

+

F3(E)

9

For F,, since a f F , E L' for la1 2 3 , it holds

IF-'[F,](x)l ~ C , I X I - (~N 2 3 )

(5.9)

for

1x1 2 1

.

For F, and F4, since At(F3+F4)€ L1,we have (5.10)

IF-1[F,+F4](x)15Clxl-z

for

1x121

.

For F,, since Fl has a form F,= IEl-'(B+K(w)) ,

(5.11) where

represents a constant matrix and K ( w ) represents a matrix satisfying

(5.12)

K ( w )E C"(Sltl=,)

s

and

K(w)du=O ,

it follows from the arguments on singular integrals that (5.13)

+

I x I 2F-l [ F ,I =F-' [ A @ IE I - 1 K ( 0 )IE I -1) 1 , =F-"B8t+v.p. K(o)IEl-31 , =B+K(x/lxl)

.

Hence it holds (5.14)

jF-'[F,](x)i ~ C I X ] - for ~

1x121

.

Combining (5.8)-(5.14), we have F-l[xI~z-lP,] E C" and IF-'[&lP,](x)l 5 C ~ X Ifor - ~ 1x1 2 1 . This completes the proof of Lemma 5.2. This lemma corresponds to the property of E, for M<1 in the Theorem. Thus, up to the present, we have completed the proof of Theorem for M < 1 .

Fundamental Solution for a Viscous Flow

493

the first term degenerates on w,=M-' for M L 1 . So we have to take account of the second term d I E 1 2 as the leading term. The most crucial term in has a form Re F-l[xln",-lP,] Re F-'[x1(151')(iclEl(1- Mul)- lE12)-11

(6.2)

where we write c again in place of c,/a', and the other terms are comparatively easier to treat. By investigating (6.2) precisely, we show the following which corresponds to the property of E, in the Theorem. Lemma 6.1. for Ixl>>l, 8,=8,

If M > 1 (resp. M = 11, then F - ' [ X ~ ~ , - ~€PCm(R3) ,] and it holds +

\ R e F-l[~IR",-lP,l(x)l = O ( l ~ l - ~ ,/ ~ (resp. )

IXI-~'~)

and o
Proof. We only show the estimate for (6.2) with M > 1. The other term and the case M = l can be analyzed along the same lines. First, we introduce a polar coordinate system ( r , 8, 9) (OSr<+oo, 0 5 6 5 ~ 0, 5 9 5 2 ~ such ) that [ = ( r cos 8, r sin 8 cos 9, r sin 8 sin 9). Then (6.2) is rewritten in the form (6.3)

Re

idrn

K(r, w)eilxlrw.o, dwdr ,

\,w,=l

where K ( r , w ) = ~ ~ ( r ~ ) r ( i c ( l - M c6)-r)-I os and (6.4)

x = IXI6JZ 7 w,=(cos O,, sin 8, cos # x , sin 6, sin # x )

Since the integrand of (6.3) is a n even function of r, we may estimate the integral

We define 8, ( 0 < 8 , < ~ / 2 ) by cos Om=M-l, that is, 8,=n/2-8,. For a small positive constant O0 (
A. MATSUMURA

494 x4(t)=o,

(6.6)

x4(t)= 1 ,

for tS(0,-80)2 or t 2 ( 8 , + 8 0 ) 2 , for ( e - e , / 2 ) 2 ~ t ~ ( e + e , / 2 ).2

Then we divide Z as (6.7)

Z=Re

s

x4(02)K(r,w)e'"'fdwdr+Re

.

=11+Z2

s

(1-x4(02))K(r, w)eiz'Edwdr

If we note that the term ( l - - M w , ) does not degenerate on the support of 1-x4 and 1-x4E C-(S,f,=,), the integral Z, is estimated in the same way as in Lemma 5.2, that is, lZ2(x)lSC(l+~xl)-2. So it suffices to estimate the integral I,. We divide the estimate into four cases according to 8,; 0 S 8, < OM or x- 8, < 8, Sx, OH
w.w,~6,>0,

(6,: some constant)

on the support of x4(02)by choosing 8, small again if needed. e i z * f-

(6.9)

(eiz'f/ilxlw.

Noting

,

we have by integration by parts, (6.10)

Z,=Re

s

dw IO--U,I50,

\+m (x1K2(r,w)+x1K3(r,w)+xl'K4(r, w))ei"'edr -_

=zz+z3+z4,

where K1(r, w ) = -~4r/ilxl~~w,(r-iic(l--Mcos 8))2, K2(r,w ) = x4/ilxlw.w,(r -ic( 1 -M cos 0)) , K3(r,w ) =2x4r2/ilxlw.w,(r - ic( 1 --M cos 0)) . For Z,, using (6.9) and integration by parts again, it is easy to check (6.11) For Z2, we divide Z2 as (6.12)

I I 4 ~ x ) l ~ C l x l -. 2

Fundamental Solution for a Viscous Flow

495

In the integral 15,we have for any R>r,, (6.13)

Re

sirn

X1K2e*Z'Edr=Re

+R

XIKzetx'Edr

-R

--m

=Re

+R

K,eiz.Edr+ Re

-R

Ks(0)+K&O)

\

+R

(xl-

l)K,etx'~dr

-R

1

For K6(w),using (6.9) and integration by parts again, it is easy to see

s:

IK,(w)[S CR-'lxl - 2 + C l ~ l - ~( 1 +r2)-'dr .

(6.14)

For K 6 ( w ) ,we complexify r, say z, and apply the Cauchy's integral formula to K 5 ;

= K h ) +K,(w)

9

where C = { z [ Im z=O, IRe z [SR}u { z I IzI = R , Im Z ~ O } . It is not difficult to see the integral K 8 on Iz[= R has the estimate IK8(w)I5 C R - ' l x [ - 2 .

(6.16)

For the crucial term K,(o), it follows from residue calculus that for a suitably large R, (6.17)

K8(w)= -(27cix4/Ix 10. w x ) ( z e i ~ x ~z )2z l0r =~i e0( i - , v c o s a ) - - (27ci~Jlx 10. wz)e-clsl( l - N c o s a ) w . w1 s-( c( 1-M cos 8)I xlw-w,) , 9

which leads to the estimate, using (6.8), (6.18)

[ K8(w)1~ ( 2 ~ ~ ~ / ~ ~ [ x ~ ) e - 1~+~c lox l~( l "- ~M( cos ' - "0)) ~ ." ~ ~ ) (

Therefore, introducing the new variable t= 1-M cos 8, (6.19)

a,sasa,+ao

I KAw)Idm

SClx[-l 5Clxl-2

\:

e - e 8 ~ l s l t ( l + t l x ~ ) d (t f, o = l - M c o s (8,+B0))

.

Thus, combining (6.13)-(6.19) and taking R++M, we have (6.20)

l z 5 ~ x ~ l ~ c l x.l - z

A. MATSUMURA

496

By the same arguments as for Z5, I,, and Z3 have the same estimates l~E(x)l? 113(x)1 s c l x l - 2 .

(6.21)

Hence by (6.11), (6.20) and (6.21), we obtain

Im)Isc(19,)lxl-2

(6.22) where C(O,)-+m

as 0,-0,

or T-0,.

1

Thus the proof for Case 1 is completed.

Case 2. 6,
In this case, there such that

(6.23)

or

o.w,=O

and

l-McosO=O

at 0=(0,,#,)

(0,,27r-$,).

In what follows, we are only concerned with the estimate in a neighbourhood of (Om,9,). We construct a cut-off function x-(@) as x , ( O z ) which localizes $ about $m. By the same arguments as for Case 1, the crucial part to be estimated in ZI is (6.24) Noting that on the support of (6.25)

x4x6,

(w.w,)+=

-sin 19,sin I9 sin $#O ,

and (6.26) we have by integration by parts I,=Re

s

+Re

x6KQ(r, u))eiz,'+(xt- l ) ~ , K , e i x ' f d w d r

s

~,'Klo(r,w)eiZ.?+(xl- l)xs'K,,ei"'~dodr

1 8 f IQ$.1 1 0 fI , 1

1

where

-

K&r, w ) =x 4 ( w w,)+4/ilxl(ww,)42((ic(1- M cos 6 )- r ) , K l o ( r ,w ) = - 2 ~ , ~ / i l x l ( o . o , ) ~ ( i c ( l - - M c oOs) - r ) .

+

By the same arguments as for I,, we have lZgl 1ZIl15CI X I - ~and the essential

Fundamental Solution for a Viscous Flow

497

part in I8 and I , , is that about the residues at z=ic(l-Mcos 0). Thus, if we define

K,,(w)=2 q 4 ( w .w,)++e-clzl ( 1 - J I ~ o s o ) a . w~ / l x l ( w - J , ) + z K,,(") = -471x4$e-~lzl( l - J I c O s O ) w . w , /Ixl(w.w,)$b . 9

For I,, and I,,, noting that on the support of w-wX5-d,

in

D,,

xs',

w-w,.d,

in D,

for some positive constant 8, and introducing a new variable t in D, (resp. D2) by r= M cos 0- 1 (resp. 1--M cos 0), we obtain (6.28)

where to is some positive constant. For I,, and I,,, if we introduce new variables ( t , s) by t = l - M c o s 0 and s=w-w,, there exist positive constants I , and so and the function f(s, t ) € C"(R:,,)such that

Hence

5Clxl-2. Thus combining (6.24)-(6.29) we have

A. MATSUMURA

498

IZ(X)Irc(e,)ixi-e

(6.30) where C(O,)+co Case 3.

as 8,+@,

,

or x-8,.

8,=Onr.

In this case, it holds (6.31)

w-w,=O

and

1-Mcos 8=0

Noting that on the support of

at a=(@,,+,,,)=(Om,

a)

.

x4xa,

(6.32)

(w.4eS;O

I

and &X.f-

-(

etx.E

)s/ilxlr(w*w,)e

2

we perform integration by parts in I,. However in this case, a n integrand after integration by parts is not absolutely integrable (see Kl6 below). So we manipulate as

+X1X4K1B(rI

o)+XlX4’Kl7(r,

w)dwdr

9

where K14(r,w ) = -Xso.o,/i(xl(w.w,)e2(r-i(l- M cos 8)) , Kla(r, a)= -xS cos O/ilxl sin 8 (o-oz)s(r--i(l--Mcos 8)) , K,&, w)=-qsMsin ~/lxl(w.w,),(r-i(l-Mcos O ) ) , ,

Kl,(r, w)=-2~sO/(r-i(1-Mcos

0)).

Here and in the following we set c = l for simplicity. By the same argument as in previous sections, we may set x l = l in (6.33) and the term about K,, is estimated by CIXI-~. If we take limit € 3 0 after calculation of residues at .z=ic(l-Mcos @),we have (6.34) where

17=--2al~I-’

Fundamental Solution f o r a Viscous Flow

+27r

\4

K1,(w)d0-2lclxl-'

In: .

499

K,,(w)do+O( 1x1-2) ,

~z17+zl~+zl,+zz~+O~lxl-z~ In (6.37) we may set x4xa=1 because the terms multiplied by (1-x4) or (l-xa) are easy to treat. On Di we have for (s, t)=(8--8,, z--$), (6.38)

(sin 8 / ( ~ ~ ~ , ) B ) e ~ ~ x -~ ( o-~a + o O( ~ t2))e ~ 1 -- I ~X I (c(1 116) o ~a2&4 ~ ~t 0- (t'$ )

where we set a=sin 19, and b=cos 8,.

On D:,for (s, t)=(8-8,,

x-$),

(1-Mcos 8 ) sin B=Ma2s+O(s2+r4), WOJ,(~-Mcos 8)= - M ~ s ~ + ( M / 2 ) a ~ b s t ~ + O ( s. ~ + t ~ )

(6.39)

On D;,for (s, r)=(o.o,,

+--lc),

-

+

d8dL"= I(0 W,)@I -1dsdt =( 1 O(s+ t2))dsdt , M(w.w,) sin2O / ( ~ - W , ) ~ =-Mu2s+O(s2+t4) , o.o,(l-Mcos 8) = - M ~ a s ~ + ( M / 2 ) a ~ b s t ~ + O ( s.~ + r ~ )

(6.40)

Applying the estimates such as

5:"

1 x l - ~ ( ~ ) e - a l z l t 4(to, d tC ~ > O ) S O O ( I ~ I - ~ / ~ )

T-l/4e-mrdT

1

A. MATSUMURA

500

to Z17--Izo, we have for some to> O

(6.41) (6.42)

Z,7=4nalxl-l

Z18= -4nMa2

1:" \

,

jo

e-l"l("/16)Q3b2L4dt+O(I~I-7/4)

W41abt2

Se-21xlMr(a2bt2-as)d

~ d t + O ( \ x l - ' / ~,)

0

(6.43) (6.44)

~lQ=~18+o(~x~-7'4) t

I,, = O(1x1-714)

.

The first term of (6.41) is bounded by, setting cu=(M/16)aSb2,

(6.45) and the first term of (6.42) is bounded by, setting j=(1/4)ab,

Hence, combining (6.41)-(6.46), we have

(6.47)

IZ715C(x(-5'4.

To show Z7=O(lxl-5'4) we need the estimate from below: setting a=(M/16), and r=t41xI,

Thus, by (6.47) and (6,48), we obtain

(6.49)

l I ( x ) l = O ( I ~ l - ~ / ~ ) for Bs=BM

.

Fundamental Solution for a Viscous Flow

Case 4.

50 1

0, is around z - O M (=7~/2+0,).

In order to complete the proof of Lemma 6.1, taking account of the results in Case 1-Case 3, we finally have to obtain the estimate lZ(x)I$Cl~l-~uniformly with respect to 0, in a small neighbourhood of z-OM. In what follows, we show the uniform estimate for 0,=~/2+0,--~ with respect to a small €20 (the case E ~ can O be treated more easily than € 2 0 ) . In this case, it holds (6.50) Then $,(E)

o.w,=O

and

l-McosO=O

cos $,(€)=tan (O,--E)/tan 0 ,

at o=(0,,

,

$,(~)20

-t$,(~)),

.

is expanded as $ , ( ~ ) = 2 / 2 / a b ~ " ~ + O ( ~,~ ' ~ )

(6.51)

where we again set a=sin 8, and b=cos 0,. We suppose x4(Oz) and x6(@) to localize w in a small neighbourhood of o=(8,, 0) and define D, and D, by D,={w€Supp. of x4x5 I o * w , < O , D,={o€supp. of XnXa I o.o,>o,

l-Mcos0<0},

.

1-Mcos 0>0}

Then, similar to having the form of I , in (6.34), the essential integral to be estimated is given by (6.52)

I,= -2~lxl-'

\

KI,($)@+~K O=O,

\

K18(m)da-2r D2

\

K18(o)do D1

Since the integrands in (6.52) are even functions of $, we treat only the part of D, U D, with $ 2 0 and furthermore divide the part into four parts such as DXresp. D:)= {o€ D,(resp. 0,)I $ , ( E ) 5$}, D:(resp. D ; ) = { o € D,(resp. D,)1 0<$<(Clm(c)}

.

We also divide I , of (6.52), corresponding to (im(c)5$ and O < $ < $ , ( E ) as I,= Z;+I;. We only show the estimate for I : since I ; can be estimated along the same line. Using (6.36), the integration by parts for K,,(w) in D: gives (6.57)

= ~ 2 1 + ~ 2 2I

where we have already set x4x5=1 for the same reason as in (6.37). AS to

A. MATSUMURA

502

Zzl, introducing the new variable (s, t)=(O,-0, the integrands as (6.58)

j:

Z,,i2a=!:

$-Qm(c)) on D:,we expand

M( -a2s+- 3 abs2+O(sS)

2

{

x exp -Mlxl[.'

sin (Om-€)

4

- - a1z b ~ t 4 + O ( ~ 4 + t B ~ ) 4!

6

where cl=sin(28,-c)sin~ and so, to are some positive constants. To estimate (6.58), we prepare

Lemma 6.2.

Set c l = u z / ~ ,c,=sin(O,-E)

and c3=(l/2)abc2. Then

x exp{-M ( c l z / T ( y ~I )ll4+ x C,Y + c , r ) } d d y . Proof. It is easily checked by changing variables as ( r , y ) = ( ~ t ~ l ~s2IxI). l,

Applying Lemma 6.2 to (6.58), we can extract the crucial terms from (6.58) as (6.59)

Z21/21:=j:

1:" (

2

where

f ( s ,t)=exp {- IxIM(c,St+czs2+cSSt2)~ . On the other hand, as to IZ2,introducing the new variable (s, t)=(ao-o,/sin (O,-E), we expand the integrands as

$-$,(E))

,

Fundamental Solution for a Viscous FIow

(6.60)

Zz,/2n=

\ :" \

503

( M d s+2Ma2bZst2+2Mubs2+ 4 M ~ ~ b c , - ~ 1 / ~ s 1 0

+[xl-'O(s+r2))exp

+-4!1 db(9b2- l)st4+-21 bs3+O(s'+sta) 1 +d~(nst+2bszt+-u(9bz-l)sts 6

After we extract the crucial terms from IZ,/2x in the same way as (6.59) and add the estimate (6.59) o f Zz1/2n,we consequently have

We estimate each integral of ZZ3-Z3, by Lemma 6.1. For example, by Lemma 6.1 and integration by parts,

A. MATSUMURA

504

(6.64)

4Z2,=

s

--7 M 2 a 2 b 2 ~ 2 - ' ~ ~y1'4r1/Zgdrdy ~-r/4 4

_--

Mab[xl-7/4

-

y1/4r-l/2

4

g dTdY

s

+-78 M ~ ~ b c ~ - ' 1 / ~ 1 xy-'l2gdrdy 1-~/~

s

-_ M e u 3 b ~ z - 1 e 1 1 ~ ( y-"4r1/2gdrdy+O((xl-2) -5/4 16

AS w e see (6.62)-(6.64), w e can estimate each integral b y a form

(0, -+,$),

(0, -3,

3)

and

(0, 0, -+)

for Z 2 3 ~ Z 3respectively. z uniThus, summing them up, w e finally reach the estimate IZ:(x)l S-CIXI-~ formly with respect to a small € 2 0 . T h u s t h e proof of L e m m a 6.1, therefore the Theorem 2.1, is completed.

References [ 1 ] K. 1. Babenko, Stationary solutions of the problem of flow around a body of a viscous incompressible liquid. Soviet Phys. Dokl., 18 (1973),300-302. [ 2 ] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and

associated perturbation problems. Arch. Rational Mech. Anal., 19 (1965),363406. [ 3 ] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979),337-342. [ 4 ] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983),445-464.

Fundamental Solution for a Viscous Flow

505

[ 5 ] C. W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Akadernische Verlagsgesellschaftm.b.H., Leipzig, 1927.

Department of Applied Mathematics and Physics Kyoto University Kyoto 606, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 507-542 (1986)

Nonlocal Advection Effect on Bistable Reaction-Diffusion Equations By Masayasu MIMURA,David TERMAN a n d T ohr u TSUJIKAWA Abstract. Spatial localization of biological individuals is modelled by semilinear diffusion equations with a bistable reaction and a nonlocal aggregative advection. Two different types of pulse-like stationary solutions exhibiting phenomenologically aggregation of individuals are shown to exist. It is numerically observed that the larger pulse is stable, while the smaller one is unstable. Key words: spatial aggregation model, pulse-like stationary pattern, reaction-diffusion-advection equations

1. Introduction

We are concerned with a bistable reaction-diffusion equation with a nonlocal advection describing phenomenologically spatial aggregation of biological individuals. The equation is

Here u(t, x ) is the population density at time t > O and position x E R , and p(s) is the velocity of the population. We assume that p(s) is a n odd function in s E R which satisfies

(2)

for s € R + , and p(O)=O,

p(s)>O

where s=k*u=

SR k(x-E)u(f)df. k ( x )=

{

The kernel k ( x ) is assumed to satisfy

-k+(x)O

for x E R , for X E R - .

This sort of equation, involving nonlocal advective terms, has been discussed by Mimura [8], Alt [l] from a biological aggregation point of view. Let us briefly explain the advective term. The individual moves to the right if Received August 20, 1985.

508

M. MIMURA, D. TERMAN and T. TSUJIKAWA

jn k+(x-E)u(t, o d e < -m

sr

k-(x-E)u(t, E1d.t

,

and to the left if the inequality is reversed. That is, because of the advection [q~(k*u)u],,the individuals have the tendency to aggregate. One of the simplest examples of k(x ) is

(3)

k(x)=1-2H(x)

,

where H ( x ) is the Heaviside step function (see Figure 1).

xx Figure 1. The function form of k(x). The kinetics of the process is f ( u ) , which represents the supply due to births and deaths. We assume that f(u) is a cubic-like function satisfying f(O)=f(l)=O, f’(O)
(4) which has been fully investigated by many authors (see, for example, Fife [ 5 ] ) . It is known that (i) the constant states urO, u = l are both asymptotically stable, but u = a is unstable, (ii) all non-constant states are unstable, (iii) if the initial function u(0,x ) has compact support and is small in an appropriate sense, then u(t, x ) tends to 0 uniformly in x E R , and (iv) if ~ ~ f ( u ) d u > O and u(0, x ) is large in an appropriate sense, then u(t, x) is expanding and tends to 1 uniformly on any compact subset of R . In this paper we study (1) in the presence of aggregative advection. We are mainly interested in the effect of the nonlocal aggregative mechanism on the spatial localization of the individuals, not only for ecological interest, but also as a prototype of a class of reaction-diffusion-advection equations.

Nonlocal Advection Eflect on Reaction-Dirtision Equations

509

With the choices of y(s)=s, f ( u ) = u ( l - u ) ( u - a ) , O O there are two different types of pulse-like stationary solutions: a large pulse which is stable and a small one which is unstable (see Figure 2 ) . Note that there is only one pulse-like solution, which is unstable, in the case E=O (see (ii)). These numerical computations motivated us to prove the existence of different types of pulse-like solutions of the stationary problem (5)

O=u,,-[(p(E(~*U))UI,+f(U)

7

where E ( > O ) is a sufficiently small parameter. We first define a pulse-like solution u ( x ) of (5) as a nonnegative function, which satisfies that u€

CZ(R)rl L1(R)n L"(R)

,

and u satisfies ( 5 ) .

U

1 l2

(A) U

U

11.2

11.2

Figure 2A. Large and small pulse-like solutions of ( 5 ) with c = O . l : (A) a= 0.1, (B) n=0.2, (C) n=0.3, (D) a=0.4.

510

M. MIMURA, D. TERMAN and T. TSUJIKAWA

-

10

I

0

'

I

'

0.5

a

Figure 2B. Global solution branch with respect to the parameter a € (0, 1/3. Before stating our result, we define ,I*f( ) b y the following: Consider the problem

t

Wz,+2,IW,+f(W)=0 for x € R , W(--oo)=l, W(+oo)=O.

Then there uniquely exists , I * ( f ) such that there is a unique strictly monotone decreasing solution W ( x ;A*( f )). Moreover, ,I*(f)i$O for $ i f ( u ) d u s O . The proof can be shown in Fife and McLeod [6] for instance. For the particular choice f(u)=u(l-u)(u-a), ,I*(f) is explicitly represented by l * ( f ) = (1-2a)/2JT.

$A

Theorem. Suppose that f(u)du>O and p(s) is smooth and monotone increasing. Then there is an E ~ > Osuch that there exists a small pulse-like solution, u ( x ; E ) , of ( 5 ) f o rO<~<~,,satisfying

1.

dx; 4dx=O(l)

lim g(x; c)=uO(x)

for each x € R

,

El0

where u o ( x ) is the unstable pulse-like solution of ( 5 ) when E=O. the above conditions, i f (6)

SUP p(s) > 22*(f)

t€R+

holds, there exists a large one, U(x;E ) , for O < E <

c0 satisfying

In addition to

Nonlocal Advection Effect on Reaction-Dirusion Equations

511

,

E(x; e)dx=O(f) R

for each x E R .

limii(x;e)=l 610

Remark 1. For cp(s)=s, the assumption ( 6 ) in Theorem is apparently removed. For more general functions p(s), one could show the existence of pulse-like solutions, following the proof of Theorem which will be in the next section. We prove Theorem in two different approaches. In Section 2 we use singular perturbation methods and in Section 3 we use phase space methods. In Section 4 we give some concluding remarks concerning the equations. Throughout this paper we use the following function spaces. Let Z=(O, l), J=R+ or R-, u and p be positive numbers, and n be an integer. Let

Also K, (i= 1,2,

- - - ) which we will use in Section 2 mean positive constants.

2. Singular Perturbation Approach 2.1. The large pulse-like solution In this section, we construct the large pulse-like solution of ( 5 ) by using singular perturbation techniques. From Mimura and Ohara [9, Lemma 11 it then follows that a pulse-like solution of ( 5 ) satisfies

(7)

u(+w)=O

and

u,(+-m)=O.

We look for a symmetric solution which satisfies u(x)=u(-x) for x E R,, that is,

(8)

u,(O)=O

.

Then, by setting EX

and

v(y)=

5 12

M. MIMURA, D. TERMAN and T. TSUJ~KAWA

(51, (7) and (8) can be reduced to the following system:

(9) and (10)

u,(O)=O

,

v(O)=O

u(+oo)=O,

i

We shall show that if E is small, then (9) and (10) fall into the framework of a singular perturbation problem. In order to construct an approximate solution of (9) and (10) for small E , we formally set e=O in (9) and then obtainf(u)=O. We define an approximate function ao(Y)=

{1

for Y E ( 0 , P ) for Y E @ , a)

for some fixed p > O which will be determined later. the second equation in (9) and using (lo), we obtain 'O(Y)={i

for Y E ( % P ) for y E (p, w )

Substituting a,(y) into

.

We now have the approximate function (Uo(y),Vo(y)). Let us divide R + = [0, co) into two subintervals I-=[O, ?I!, and Z+=[,4, +co) and by using (So, V,,) construct a solution of (9) and (10) with E > O in each I- and I,. The boundary conditions are assumed to be, respectively,

+

(11)

u,(O)=O,

u(O)=O

u(P)=cU,

and (12)

u(P)=a,

u(+m)=0

9

v(P)=r

t

where a € (0, 1) and y > O are arbitrarily fixed constants. Since n,(y) does not satisfy the condition @)=a, we need to construct another approximate function in a neighborhood of y = p . Using the usual stretched variable f = ( y - p ) / ~ in (9), we obtain

Here, the subintervals I- and I, are transformed into [-PIE, 01 and R,. Consider the limiting case when e = O in (13). Then the second equation in (13) becomes vc=O. Hence, we take v(E)-fi,,(/3)=/3. Then (13) simplifies to a scalar equation of u only

Nonlocal Advection Efect on Reaction-Diyusion Equations ( 14)

O = ~ ~ ~ + ' p ( 2 P ) u e + f ( u ) for CER,

513

.

The boundary conditions (11) and (12) are assumed to be

(15)

u(-m)=l,

u(O)=a

respectively. Solutions of (14), ( 1 5 ) and (16) in R-=(-m,O] and R+=[O, cm) are obtained in the following.

Lemma 1. Consider the two boundary value problems with a parameter I (17,)

I

R

O= W&UW:+f(W')

for E E R , , w-(-m)=l , W-(O)=a, W + ( O ) = a , W+(+ca)=O .

Then there exists 6 > 0 such that for any fixed I € A 8 = { I I II-I*(f)I<6}, (17,) have, respectively, unique, strictly decreasing solutions W*(E;A ) satisfying IW-(c; I ) - l l E X:-c2)(R-)and fW+(E;I)I € X:+c2)(Rt),which are continuous with respect to I in the X:+(R,)-topology, where ~ + ( I ) = I t - d / n ~f'(O), r-(I)=Iz/Iz-f'(l), lit=inflEnar,(I) and li- =supledar ( R ) and A * ( f ) is the number defined in Section 1. Furthermore,

The proof is shown in Hosono and Mimura [7, Lemma 4.21. Let us consider (9) and (11) in I - . Using the transformations z=y/,8, u= E/P, we rewrite (9) and (11) as

(u,(O)=O,

u(l)=a ,

v(O)=O .

We introduce a parameter R=(p(2P)/2. Then, by the monotonicity of 'p and the assumption (6),p is defined by P=,B(I)=(p-1(2A)/2for R E A, with sufficiently small 6>0. Thus, we regard u and R as new parameters instead of E and /I in (18). Let us seek a solution (u, v)=(ii-(z; u, A ) , V-(z; u, A)) of (18) which takes the form

514

M. MIMURA,D.TERMAN and T. TSUJIKAWA

where O(z) is a C"-cut off function defined by for z€[--co,

+]

for 8-(t;1)=W-(C;1)-1 and p-(E;u, R)=$E,O(l+ur1)8-(r1;R)d~with E=(z-l)/u. Substituting (19) into (18), we obtain

Lemma 2. Suppose that ( 6 ) holds and 1-a is fixed suficiently small. Then there exist u, >O and 6, > 0 such rhat for any u € ( 0 , ul)and R € Adl, there are K , and K, independent of u and R satisfying ( i ) lim IIP(0;B , R)II,=O wiformly in 1E Ad,, 010

(ii)

IIPAtl;

0,

4 - P t ( t 2 ; u9~ ) I l r Y s - Y ~ ~ l l l f l - f 2 1 1 X s

f o r any tl, rz E X ,

and

(iii) P,(O;u, 1) has an inverse with l l P ~ ~ (u,0 ;l)IIY-rrYuIKz, where P,is the Frkchet derivative of P with respect to t . Proof. We note that P ( 0 ; u, 2 ) is represented by

+

+e q c+$0(2#8(1)(z+ o8-))(oez8-+e q ) +~ u v ' ( ~ B ( ~ UT-))P(R)(I (z+ +e W + j ( 1 +OD-),

P(.)(O;u, 1)=uze**8- 2ue,8; p y 0 ; 0, R ) = O

.

Since Lemma 1 implies that to show that (21)

-8[2ROi

8-and p--are uniformly

bounded in R , it suffices

+f(1+ 8-)] +f(1+O~-)+~(~,~(R)(Z+UV-))OO~

tends to 0 uniformly in 1 as u 10. Rewrite (21) as

Nonlocal Advectwn Effect on Reaction-DiffusionEquations

515

and

hold for Ka independent of 1, we find that (21) becomes zero in the Co(Z)topology as 0 LO. Thus, (i) is proved. (ii) is obvious. Next we show (iii). Note that

is described by

+SU-)dz

+2p‘( 2/3(1)(z+aP-)) (1

Pfd)= -@(#I) and

It is sufficient to show that for any F = t ( F ( r )P , )) € Y, there uniquely exists r € X , satisfying

such that

IItlIx,
M. MIMURA, D. TERMAN and T. TSUJIKAWA

516

(23)

IIP,"'-'IIcO(r)-c~(r)

22

*

Also, we find that for any fixed a close to 1, there exists K6 independent of a n d A such that j ' ( l + B ( z ) ~ - ( ( l - z ) / oA;) ) < --K, for any ~ € 1 .Therefore, applying the maximum principle for elliptic boundary value problems to Pp), we find that Pp) has a n inverse satisfying Q

(24)

IIP!')-' IIco(r) - o ~ (1 ~)5, K O ~

for KOindependent of u and 1. Thus (22) can be written as

t

r= - p ~ ~ ) - ~ p I r ) ~ + p ~ ) - ~ F ( r ) s=--pp-l p,( a ) r + p p ) - l p )

or

.

~ = pT ( ~ ) - l p ( r ) p r ( ~ ) - l p ( ~ ) ~ - ~ ( ~ )- F(T)) l(p~~)p~s)-l~(s) I

S

It follows from (23) and (24) that

-

<

I I ~ ~ ) - l ~ ~ ~ ' ~ ~ ~ ) - l ~ ~ s ) I2p(1)K,IIP!"IIc~(,,-cO(r) I c o ( ~ ) - c ~ ~ ~ )

When u, I,?-A*(f)l a n d 1-a are sufficiently small, ~ ] P ~ 7 ) ~ ~ c ; can t r ) ~be c~(I) made sufficiently small. This leads to

Thus, applying the implicit function theorem due to Fife [4, Theorem 3.41 to (20), we obtain

Lemma 3. Under the assumption of Lemma 2, there are u,>O and 6,>0 such that for any u f ( 0 , u s )and If A,,, there exists t(u, A) E X , satisfying ( i ) P(t(u, A ) ; u, 1 ) = 0 , ( i i ) lim IIt(u, A)llxo=O uniformly in and

010

(iii) t(u, 1) is uniformly continuous with respect to u and A in the X,-topology. Consequently, Lemma 3 yields a solution ( u - ( y ; E , p), v - ( y ; E , p)) of (9) and ( l l ) , which takes the form u-(Y; € 9 P)=fi-(y/P;

(25)

E/P,

P(2P)/2)

for

1-

.

Remark 2. (19) and (ii) of Lemma 3 lead to dulim ~ - - ( p ; CIO

dy

lim v-(p; E ,

E,

d WP)=--(O; dt

B)=p

(p(2P)/2)

uniformly in /3

.

Nonlocal Advection Effect on Reaction-Diyusion Equations

517

We next consider (9) and (12) in I + = [ @ ,+a). Using the transformation z = ( y - p ) / ~ we , rewrite (9) and (12) as

and u(O)=a ,

(27)

u(+Oo)=O

,

v(O)=y=v-(P;

E,

p) ,

respectively. Again we use a parameter I (=y(2/3)/2) instead of p. Let us seek a solution ( i P ( z ; E, I ) , V + ( z ;E , I)) of (26) and (27) which takes the form

t

(28)

u+(z;E , I ) = W + ( z I; ) + r ( z ; E , 2 ) P ( z ; E , I ) = v - ( p ( I ) ;E , / 3 ( I N + E V + k I)+s(z; E , 2 ) ,

s:

We note that the boundary conditions of where F + ( z ;,I)= W + ( eI)de. ; t = t ( r , s) become r ( 0 )= r (

(29)

+

00)

=s(O)

=O

.

Substituting (28) into (26), we have

where

+ +

+

+ +

Q (‘)( t ; E , 2 ) =Y,, p(2(V - cPt s))( W: r,) Wt, +2p’(2(v-+cVt + s ) ) ( E W+s,)(Wt + + r ) + f ( W ++ r )

and Q ( s ’ ( tE;, I)=s,--EY

.

Let Q ( t ;E , I ) be a mapping from X p = X;,,(R,)x k;,,,(R+) into Y,= X;(R,) x Xj(R+) for any fixed p ( O < p < r + ) . Lemma 4. There are E~ > 0 and 6, > 0 such that for any E E ( 0 , el) and R E A,,, there exist K7and K8 independent of E and I such that ( i ) lim IIQ(0;E , R)Ilu,=O uniformly in A € Aa3, €10

( i i ) llQt(tl;E , ;O-Q,(t,;

E,

;Oll~,-Yp~K~lltl-tzll~,for any t l , f,€ X ,

and

(iii) QJO; E , I ) has an inverse satisfying IIQ;’(O;

E,

R)IlYp-t,SK8.

Proof. Note that Q = t ( Q ( l ) ,Q ( * ) )is described by Q(.)(O;E , 4=p(2(v-f E ~ + ) ) W : + ~ E ( ~ ’ ( ~ ( ~ _ + E ~ and ~ ) ) Q(l)(O; ( W + )E ,~A)=O. - ~ I W(i) ~ directly follows from Remark 2 . (ii) can be easily obtained. We will prove (iii) in a way similar to (iii) of Lemma 2. QJO;E , 2 ) is given by

M. MIMURA,D. TERMAN and T. TSUJIKAWA

518

where Q:r)=&+p(2(v-+~7+))-+2~p'(2(u-+&+))W++f'( d dz

W+)

and d err)=240'(2(~-+E V + ) ) W : + ~ E ~ "+EV+))( ( ~ ( V _W + ) ' + 2 9 ' ( 2 ( ~+- EP+))W+dz

i

It is sufficient to show that for any G=t(G(r), G ( * ) E ) Y p , there uniquely exists t = t ( r , s) E X, satisfying and

for K B independent of E , R and G. We note that dldz has an inverse with [l(d/dz)-'11,$~~,,<(1+ U p ) and QP) satisfies ~ ~ Q ~ v ) ~ ~ for ~ p K,, ~ x inde~ < K l , is invertible, we divide QP) into pendent of E and A. To show that Qir) Q:)= QZ)+ QS)where Q J ; ) =d2 - + 2 i Z +df ' ( dz2

W + ( Z1)) ;

and

Q$) = ( 9 ( 2 ( ~ ~ - + ~ 7 + ) ) -d2 1 ) - + 2 2 ~ 9 ' ( 2 ( v - + ~ .~ + ) ) W + dz

Since #+(z)=(dW+/dz)(z; 2 ) (
~ KK, ,l independent l ~ ~ Q ~ ~of xE , ~R and Q , which This shows that ~ ~ P ~ ~ x ; ~ ,for implies that QZ) has an inverse. Thus, we find that if E is small, then Q:?) has an inverse with llQ!')-'llXo,~,;,,~Kl2 for K,, independent of E and A and then that &llQ!"-'Q6r)(d/dz)-111=~-=~< 1 follows. By an argument similar to the proof of (iii) of Lemma 2, (iii) is obtained. rn

Nonlocal Advection Effect on Reaction-Diffusion Equations

519

By a n argument similar to that of Lemma 3, we obtain

Lemma 5. There are c Z > O and 6,>0 such that f o r any Aa4, there exists t ( e , I ) E ip satisfying

E

€

( 0 ,E

~ and )

I€

QME,4; E , 4=0, ( i i ) lim I l t ( E , I)]li,=O uniformly in R E Aa,

(i)

€10

and

(iii)

t(e,

I ) is uniformly continuous with respect to

Thus, the solution (u,(y; E ,

P),

v,(y;

U d Y ; P)=u'((Y-P)IE; { v t ( Y ;€ 3 P)=v'((Y-PP)/E; €9

(30)

€9

€7

E,

P)) of

E

and I in the *p-topology.

(9) and (12) is given by

P(2P)P) P(2,@/2)

for y e I t

.

~ of (9), (11) We have now constructed the solutions ( u - , v-) and ( u + v,) and (9), (12), respectively. In order to construct a solution of (9), (10) in the whole interval R,, we match u- and u, a t y=P in the C'-sense. So we define Y ( E P) , by

Noting that e(du+/dy)and E(du-/dy) are uniformly continuous in E and P, we ! continuously so as to be defined for E = O . Setting E=O in (31), can extend F we have K(0, P*)=O and (a/ap)K(O, P*)+O for P*=y-l(2I*(f))/2 from Lemma 1. Then there is E ~ > Osuch that there exists a uniquely continuous function P(E) satisfying W ( E ,P ( E ) ) = O for E E [0, E J and limELO P(E)=P*. It is easy to see that U(x;E ) =

t

U-(EX;

€9

U + ( E X ; e7

P(4) P(4)

for x [O, P ( E ) / E l for x E [P(E)/E,

+

00)

is a solution of ( 5 ) . The nonnegativity of U ( X ; E ) can be easily shown by phase space methods. Thus, the large pulse-like symmetric solution is constructed. The small pulse-like solution I n this section, we will construct a small pulse solution of ( 5 ) . By putting u(x)= u ( t ) d [ ,rewrite ( 3 ,(7) and (8) as 2.2.

$:

(32) and (33)

u,(O)=O,

First we divide R += [0,

u(+Oo)=O,

v(O)=O .

+ m) into two subintervals J- = [0, h] and J , =[h, + m)

M.MIMURA,D. TERMAN and T. TSUJIKAWA

520

with a constant h to be determined later. interval are assumed to be, respcectively, u,(O)=O,

(34)

The boundary conditions for each

v(O)=O

u(h)=p,

and

(35)

u(h)=p

,

,

u(+co)=O

v(h)=r

y

for positive constants p and K to be specified later. We construct a solution of (32) and (34) in J - . By the transformation y = x / h , (32) is reduced to

(0 = u,- hu

The boundary conditions on (36) are

(37)

u,(O)=O

,

u(l)=p,

u(O)=O

.

In addition, we impose an extra boundary condition

0) ='I

(38)

'I.

for some This condition will determine h so that (36)-(38) has a solution. When E LO, (36)-(38) can be formally written as

I

1

0 =- u,, + f ( u )

(39)

h2

for y e 1

O=v,-hu u,(O)=O

,

u(O)='I,

u(l)=p ,

v(O)=O .

We first look for (h, u, u ) satisfying (39) for a given 9.

Remark 3. For any fixed h, the problem,

has a unique monotone decreasing solution u ( y ; h) satisfying u(0; h)=?*, where 7" is uniquely determined by s,"*f(u)du=O.

Lemma 6. Fix ,OE (0,7;*). For V E (p, 1) satisfying $;f(u)du>O, the first equation in (39) has a unique monotone decreasing solution _V-(y;7) for h = h o ( ~ ) = ( -2 tf f(f)de)-""u+ 0.

s; s

Nonlocal Advection Effecton Reaction-DiflusionEquations

52 1

Since the proof is easily achieved by using phase plane methods, we omit it. On the other hand, a solution of the second equation in (39) is given by

Y ~ Yp , h)=h ; 1,"U z ; pldz.

(c-,y-)

By using the solution of (39), let us seek a solution (h(7,r), g ( y ; E , p , r ) , _v-(y;E , p , r ) )of (36)-(38) in the form

where _V-(y;p , r)=_V-(y; 7, h,(p)+r). It is obvious to see r,(O) =r( 1)=s(O) =O

(42) and

r(O)=O .

(43)

Substituting (41)into (36), we have

(44) for t = t ( r , s), we have

and L("(t;E , 7 , r)=sr-(h,+r)r

.

We first treat the problem (44) and (42)in the absence of (43). Let L be a mapping from i= C;,(Z) x c:(I) into Y E CO(Z)x CO(Z).

Lemma 7. L ( t ;E , p , r ) has the following properties: ( i ) L is a continuous mapping from X into Y , and C~L~C~E, and L, are continuous with respect to ( t ,E , p , r). ( i i ) L(0;0 , p , O)=O. (iii) L,(O;0 , ~ .0) has a bounded inverse uniformly in p .

M. MIMURA, D. TERMAN and T. TSUJIKAWA

522

Proof. (i) and (ii) follow from Lemma 6. Consider (iii). We note that

L?) LAO;

0 7 %

0) = ji-h0(d

'

It is easy to see that where L!.)=(l/ho(7j)2)(d2/dye)+f'(_U-).

Using the fact # ( y ) = ( d c - / d y ) ( y ;7) is a nonnegative solution of L!.)@=O, one finds that Green function G ( y , 7 ) of L:r) can be explicitly represented by

where

So we know that L!r) has a bounded inverse uniformly in p , which directly leads to (iii). From Lemma 7, we can apply the usual implicit function theorem to (44). Lemma 8. Let B q = { ( ~ , p , ~ ) €1 R O <SE < q , l~--p*lO such that there exists a unique continuous mapping t ( E , 8 , T) from B,, into

. satisfying i ( i 1 t(0, 7 , O)=O,

( i i ) JWE, 'I, r ) ; E , T, r)=O and

(iii) t ( ~9, , ~ ) = O ( l ~ l + l ~ lin) X uniformly in 7 . Thus, we obtain the solution ( g - ( y ;E , p , r ) , _v-(y;E , 7 , T ) ) of (36) and (37). We next show the relation between E and 't such that r ( y ; E , 7, r ) satisfies (43), that is, r(0; E , 7jr, r)=O. By the invertibility of LLr) and d/dy, and (iii) of Lemma 8, (44) is written as

where N,(r) is the higher order term.

Nonlocal Advection Efect on Reaction-Difusion Equations

523

Lemma 9. There is ql>O such that r(0;E , 7, T ) and (ar/&)(O; E , 7, continuous mappings from Bql into R, which satisfy ( i ) r(0;0, v , O ) = O and ar 2 ( i i ) -(O; O,q, O)=-L!')-'f(q)fO is uniformly bounded in p . dr ho(ri)

T)

are

Proof. (i) is obvious from (i) of Lemma 8. Noting that t(E,v,r)=

O(l~l+Irl), we obtain (ii). The uniformity of (ar/ar)(O;0, p , 0 ) follows from the continuity of h,(p) and the uniform boundedness of L:r)-' in p . 4 Lemma 10. Let q1 be defined as in Lemma 9. There exists e4>0 such that r ( 0 ; E , 7, T)=O has a unique continuous solution r(&,7) for E c [0, e4) and '1 I q l = Ip I 1p-p"I
Lemma 11. Let E~ and q1 be defined as in Lemma 10. For any E E [0,c 4 ) and p € Iql, (36)-(38) has a solution M E , p ) , _u-(Y; E , v), _v-(Y; E , p))=(h,(p)+r(e,p ) , LWY;v)+r(y; E , p ) , VT'(Y; E , T)+s(Y; E , 7)) satisfying

1:;

lim llH-(Y; &,v)--_u-(Y, v)llc;o(n

lim

(45)

I\ u -(Y;

E,

=o

~ ) - Y - ( YE , ;p) IIc; ( I) =O

uniformly in

pc Zql

lim h(E, p)=ho(q) El0

where

r ( ~E ,; p ) =U(Y; E , 7,

and

T ( E , 7))

S(Y; E,

p ) =s(Y; E , p , r(e, 7))

.

By y = x / h , we may write ( u - ( x ;E , q), ~ ( xE ,; 7)) as

Remark 4. From (49,we find that lim €10

-(-y I 1 du-

2

dx

f(u)du

= z=h(r,ri)

uniformly in p

.

p

Next we consider (32) and (35) in J , . we rewrite (32) and (35) as

By the transformation y=x-h(E,

v),

M. MIMURA, D. TERMAN and T. TSUJIKAWA

524

and (47)

u(O)=p

,

u(+m)=O

,

v(O)=r=V-(h(E, 71); E , 9 ) .

Setting E=O in (46) and (47), we have

Lemma 12 (Fife [3, Lemma 2.11). The first equation of (48) has a unique monotone decreasing solution _u'(y)E X&(R+) satisfying

for K,, and K,, independent of y , where r o = d - f ' ( 0 ) .

Moreover, the solution of the second equation of (48) is given by Y ( y ;7;)= $,"_U+(z)dz+v-(ho(v); 0,111. Let us seek a solution ( g , ( y ; E , T ) , _v,(y; E , 7 ) )of (46) and (47) in the form

(49)

i:: -t

-+

Y ; € 9 7;)=-Uf(y)+r(y;EY 7;) y ; E , v)=Yk,d+_V+(y; 7I)+s(y; € 9 71)

,

where HE,v)=v-(h(e, 7 ) ;E , v)--v-(h0(d;0, v). It is easy to see that t = l ( r , s) satisfies

+

r(0)=r( co)=s(0) =0

(50)

.

Substituting (49) into (46), we obtain

where

and

M y t ; E , v)=su-r

,

Let M be a mapping from i r = X 2 , , o ( R + ) ~ i : , o ( Rinto t ) Y,-X:(R,)XXO,(R+), where r is any fixed (0 < r < 7 0 ) .

Nonlocal Advection Effect on Reaction-DiffusionEquations

Lemma 13. There exist cs > O and q2> O such that for any 7 € I q 2there , are K,, and KIBsatisfying ( i ) lim IIM(0; E , 7)Ily,=0 uniformly in V E A',,,

525 E

€ (0, E J

and

E l 0

( i i ) IlWt1; € 3 7)-Mt(tz; E , 7)llx",-Y,l~l,lltl--211;rfor

any tl, f z E Xr

and

(iii) MJO;E , 7 ) has an inverse with l\M;l(O;E , 7)11Yr-,;r
We note that M'"(0;

E,

9)=(P(2E(~+/+))_Uy++2EID'(2E(/+/t))_Vy+_U+

and M c S ) ( OE ;, p ) = O , which lead to (i). (ii) is obvious. We only show (iii). M,(O; E , 7 ) is given by

M ( 0 ; 7)= €9

where

d -7+JD(2E(/+Y+)) -+24(2E(_V+Y+))Y; dY

("

d -

-1

A= 2#(24_V+/+))CJ;

+f'(Y) *

dY

+2 9 ' ( 2 4 1 + y ) ) - d +~ E C ~ " ( ~ E ( / + _ V + ) ) ~ ~ V ., . _ U + dY

By a n argument similar to that of (iii) of Lemma 4, (iii) can be shown, so we omit the proof. Thus, we have

Lemma 14. There exist E ~ > O and q 3 > 0 such that for any E € (0, E ~ )and 7~ ZQ3,there is t(E, 7 )E & satisfying M ( ~ (7E) ;,E , 7)=0. Furthermore t ( E , 7 ) is uniformly continuous with respect to E and 7 in the X,-topology, and lim Ilt(E, 7)1l.Yr=O El0

uniformly in 7 .

Therefore, we have a solution ( u + ( x ;E , takes the form

t

v),

u + ( x ; E , 7 ) )of ( 3 2 ) and ( 3 5 ) which

u t ( x ; € 7 7)=_Ut(x--h(E,7 ) ; 7 ) u + ( x ; E , 9)=vt(x-h(E, 7 ) ; 7 ) €9

€9

f

Remark 5. By Lemma 14 and (49), we find that =-'j'f(u)du 0

uniformly in 7 .

526

M. MIMURA, D. TERMAN and T. TSUJIKAWA

Finally we determine 7 as a function of in the C1-sense. Put

E

to match u- and u+ at x = ~ ( E v) ,

Since @(O, v)= - $if(u)duby Remarks 4 and 5 , it follows from Remark 3 that @(O, v*)=O and [(a/av)@(O,~)]1~=,.=f(;l*)fO. Thus, it can be shown that there is + > O such that there exists a unique continuous function q ( e ) satisfying and limeLo P ( E ) = ~ / * . We define @(x;E ) by @ ( E , ~ ( E ) ) = O for E E (0,

It is easy to see that g(x; E ) leads to a symmetric small pulse-like solution. By (41), (49) and Lemmas 11, 14, the latter half of Theorem can be directly proved. Thus, the proof of Theorem is completed. 3. Phase Space Approach In this section we demonstrate how phase space methods can be used to prove the existence of pulse-like solutions of ( 5 ) for small E , assuming simply p(s)=s. In order to apply phase space techniques we transform ( 5 ) into a first order system of ordinary differential equations by setting

If u(x) is a solution of (3,then ( v ( x ) ,u(x), w ( x ) ) satisfies the system

1

vr=u u’= w w’ = - &[ (2u - I)w +2/42] -f( u) ,

where ’=d/dx. It follows from (7) and (51) that (v,K , w ) must also satisfy the boundary conditions

(53)

(v, u, w ) ( - c o ) = ( O , 0,O)

and

( u , u, w)(+m)=(Z,O,O) .

The constant I is unknown a priori. It must be chosen so that there exists a solution of both (52) and the boundary conditions (53). It is, however, possible to eliminate Z from (52) by setting

I

V(x)=v(x)--,

2

U(x)=u(x)

and

W ( x ) = w ( x ).

Nonlocal Advection Effect on Reaction-Diffusion Equations

527

Then (52) transforms to the system

1

(54)

V’=

u

U’= w

W’= -€[2VW+2U21--f(U) ,

which does not involve Z. We eliminate Z from the boundary conditions as follows. Suppose that (V, U , W ) is a solution of (54) for x > 0 which satisfies (55)

( V , U , W)(O)=(O,U,, 0)

and

+ m)=( V,, 0,O)

(V,U , W)(

for some positive constants U , and V,. Then, setting I=2V,,

one finds that ( ~ ( x )u(x), , w(x)) satisfies (52) and (53). The original problem of finding a pulse-like solution of (5) and (7) with v ( s ) = s has now been reduced to finding a solution of (54) and (55) for some positive constants U,and V,. In terms of the phase space, the boundary conditions (55) imply that we must find a trajectory which beings, at x=O, on the U-axis and ends, at x= 00, on the V-axis.

+

3.1. The large pulse-like solution For the construction of the large pulse-like solution, it is convenient to

set y=-x,

-

v = € v , D=u,

P=-w.

Then (54) becomes

v= - € U

1:

u=p

& 2 r P - - 2 E D -f(D )

where *=d/dy. In terms of these new variables we are looking for a solution of (56) for y < 0 which satisfies the boundary conditions (57)

(r,d, P)(-m)=(V,,0,O)

and

(r,8,w ) ( O ) = ( O ,

U,, 0)

for some positive constants U , and V,. For convenience we write V , U and W for 7, d and Since we will never refer to the old V , U and W , this shall cause no confusion.

w.

528

M. MMURA, D. TERMAN and T. TSUJIKAWA

Each point on the V-axis is a critical point of (56). To understand the behavior of the flow near the V-axis, we fix E > O and p > O , and linearize (56) about the critical point (V, U,W ) = ( p ,0,O). Doing so, we obtain the linear system Y=MY where Y = t ( V , U,W ) and

M=[!

-; a 2P,

for a= -f’(O).

The eigenvalues of M are

The zero eigenvalue, ,uo,corresponds to the fact that the entire V-axis consists of critical points. Let

be an eigenvector associated with the positive eigenvalue p+(p). A standard result from the theory of ordinary differential equations is then Lemma 15. For each €20,p > O , there exists a unique, up to translation, trajectory

YAY; P ) = ( V , ( y ; PI, VkY;P), WAY;B)) such that

lim YAY;B) =(B, 0,O) .

y-r-m

Moreover, Y,(y;/3) is tangent to e(E, /3) as y - -

00.

We may pick the translation so that for each

E,

fl,

Y,(O;P)
Recall that a is the middle zero o f f . Since p+(/3) and e(8, /3) depend continu,B) depends continuously on ously on E and ,B, we may also assume that Y,(O; E and p. If 6>0 and T>O are given, Lemma 16. Fix arbitrarily E = E ~ ~ O/3=Po>0. , then there exists m such thatfor any ( E , ~ ) E I ~ g)l~>O, ,~= l ~ -{~ ~(l E + l , ,B - ~<ml, ~l

Nonlocal Advection Efect on Reaction-Dzrusion Equations

529

The proof is obtained by the continuous dependence of solutions on parameters. Fix E > O and suppose there exists /I(E)>O such that Y,(y; P(E)) intersects , such that the U-axis. That is, there exists y ( ~ ) U,>O

Then & ( y - y ( ~ )@;( E ) ) satisfies (56) and (57). Therefore, the problem of finding

a large pulse-like solution, for a given E , has been reduced to proving that there exists @ ( E ) > O such that Y,(y;,B(E))intersects the W-axis. In what foHows we shall usually assume that E is fixed in which case we drop the €-subscript. When e=O, (56) becomes

V=O

i

U=W W=2VW-f(U).

(59)

Then V=constant, say V=p, and (U,W ) satisfies

t w?="

=2/?w- f(V )

Lemma 17. There exists a unique ,B* such that (60) has a solution ( w , ( y ;p*), @*I, W-(-y; p*)) satisfying

W,(y; /?*I) = ( W - ( - y ;

lim (U,(v;P*), W,(Y;B*))=(L 0) .

?I-+-

Since Lemma 1 directly gives the proof, we omit it. Since V=constant when E = O . it follows

In the three dimensional phase space, Y,(y; @*) traces out a curve which is shown in Figure 3. It is labelled by the symbol 'C,'. The singular large pulse, that is the limit of the large pulse solution as E 10, will consist of two pieces. One of these will be C,. Of course, C,cannot be the entire solution, since it does not cross the W-axis. The second piece of the singular large pulse is denoted by 'CZ'. It is defined by

C,={(V,

u, W ) 1 o<

v
.

We shall prove that if E i s sufficiently small, then there exists a solution of (56) and (57) whose trajectory in phase space lies arbitrarily close to the union

M. MIMURA, D. TERMAN and T. TSUJIKAWA

530

V

I

c1 W CZ

1

0

U

1

Figure 3. Singular orbit of the large pulse.

of the curves C , and C,. It is necessary to first discuss further properties of solutions of (60), for which we need to introduce some notation. Let r= -f'(1) and choose 8>0 such that (61)

f(U)>--L(U-l) 2

for

I-~
Define a€, e:, e; and e; by a'={(U, W ) I U < U * ( E ) O, < W<4,8*8(1+(U*(~)-U)/(1-8-U*(~)))} . e ; = { ( U , W ) 1 U=U*(E),O< W<4/3*S}, e ; = { ( U , W ) I l--d< U < U*(E),W=O}

and

e ; = { ( U , W ) I U < U * ( E ) ,W>O, W = ~ , ~ ? * ~ ( ~ + ( U * ( E ) - U ) / ( ~ - B - U * ( E ) ) ) } , where U*(E)( 2 1 ) is the largest root of the equation 2 e U 2 + f ( U ) = 0 .

Lemma 18. Fix ,k? E (0,2,8*) and let (U(x), W ( x ) ) be any solutions of (60). Then (U(x), W ( x) )can only enter ao through e!, and leave through ep or e i .

o=

Proof. Trajectories cannot enter ao through ey or ei because on e?, W>O, while on e!, W = -f ( v ) < O . Trajectories cannot enter uo at the corners, because at (1-8,0), W < O , at (1,4,¶*8),dW/dU=2,¶<4,9*, and (1,O) is a critical point. Finally, to show that trajectories cannot exist uo through e!, we let n=(4/3*, - 1 ) be a vector normal to e! pointing into

oo, and

let

531

Nonlocal Advection Effect on Reaction-DiffusionEquations

y=( W, 2PW-f(U)) be the vector field defined by the right hand side of (60). Then, if O<,B<2,B*,

n. u=4,B* W-2B W + f( U ) >f(U)>O

7

which is what we need to show. For convenience we set Z(y; P ) = (U0(y;P), W0(y;PI).

Lemma 19. ( i ) There exists a positive constant r such that f o r any [,8*-r, ,9*+r], then Z(y; 8)E uof o r some y. ( i i ) After entering uo, Z(y; P*+r) exists uo through e!. (iii) After entering uo,Z(y; ,8*-r) exists uo through e,".

fl€

Proof. Part (i) follows from Lemma 16 and the facts that uo is open and Z(y; ,9*) E uo for some y. Define yo, yl, y2, P,PI and P 2 by

and

It follows from Lemma 18 that P E ei and Pice:

for i = l

and 2

.

To prove (ii) we first show that (62)

pI
and

p2
.

Let W = @ ( U )be the curve traced out by Z(y; ,B*) in ( U , W ) space for U*(O) (= 1) and let M be

o< u<

M = { ( U , W )j O < U < l , O I W l @ ( U ) } .

We show that (63)

Z(Y;P*+r) @ M

for all y E R

.

This certainly implies ( 6 2 ) . As y--*--oo, Z(y; p) is tangent to the eigenvector

M.MIMURA, D. TERMAN and T. TSUJIKAWA

532

U Figure 4. Trajectories of Z ( y ; p), D: Z ( y ; ,P),D1:Z ( y ; p*+r) and Dz:

z(Y;P - r ) .

where p+(/3)was defined in (58). Since

and the curve W = @ ( U ) , say D, the upper boundary of M , is tangent to (O,O), we conclude that e + ( @ * + r ) does not point out into M (see Figure 4). Hence, there exists y , such that if y < y s , then Z(y;,B*+r)eM. The proof of (63) and therefore (62) is completed by observing that along D , the vector field determined by (60) with p=p*+r, points away from M . Hence, Z ( y ; B*+r) can never enter M through D . The vector tangent to D at some point ( U , W )E D is given by e+(p*) at

while the vector determined by (60) with /3=/3*+r

is

(0,mp+,=(w, 2(P*+r)W--f(U))

*

Since Ufi*+r=ob*, Wb,+r>Wb*and M lies below D, we conclude that (U,W ) points away from M which is what we wished to show. This completes the proof of (62). We now consider the stable manifold at (1, 0). For p > O , let Z s ( y ; @ ) = ( Us(y;,8), W3((y; fl)) be the unique trajectory which satisfies limy.++,Z ( y ; ,B)= (1,O) and W ( y ;fJ)> O for sufficiently large y . Note that Z ( y ; p*) and Z s ( y ;8") are the same trajectory. Hence, there exists yo such that Z ( y o ; p * ) = P = ( p l ,p z )€ e! and Z ( y ; @*)€ uofor y > y o . Clearly y,=sup { y I Z ( y ;@*+r) € uo}exists. By Lemma 18, Z ( y , ; /3*+r) € ey. Setting Z ( y , ; P * + r ) = Q l = ( q ; , qi), by choosing r sufficiently small, it follows that

Nonlocal Advection Eflect on Reaction-Diyusion Equations (64)

q:
and

533

q:
From (62) and (64) we conclude

That is, as they enter uo, Z ( y ;B*+r) lies above Z ( y ; p*+r). Then trajectories Hence, Z ( y ; p*+r) never cross and Z * ( y ;p*+r) tends to (1,O) as y + + a . must exit uo above the critical point (1, 0), that is, through e? (see Figure 4). The proof of (iii) is very similar to the one just given. Hence, we do not give the details. Before discussing the full three dimensional flow defined by (56) it is necessary to introduce some notation. Let s'={(V, u, W ) I

o< v<2p*, ( V , W ) € f f ' } ,

E ; = { ( V , U , W ) [ O < V<2,9*, ( V , W ) E e ; } (i=1,2, 3 1 , Ei={(V, U , W ) I V=O, ( U , W ) E > } , Eg={(V, U ,W ) I V=2p*, ( V , W ) f o " )

and E'=E;UEgUE;.

Recall that when e=O, V ( y ;8j-P is constant. Since go, e:, e,O and Z ( y ; ,B) are, respectively, the projections onto the ( U , W ) plane of So, E?, E: and Y o ( y ;,8), the following result follows immediately from Lemma 19. Lemma 20. Let r be as in Lemma 19. For any p E [,B*-r, P*+rl, Y&Y;8) belongs to SO f o r some y . Moreover, after entering So, Yo(y;,8*+r) leaves So through EY and Y o ( y ;,B*-r) leaves So through E:. This completes the analysis when E = O . The following result demonstrates that Lemma 20 remains true if Y o , So, E! and E: are replaced by ye, 9 , E: and E: for small E . Lemma 21. Let r be as in Lemma 19. There exists E~ > 0 such that f o r any [p*-r, ,B*+r], Y,(y; p) belongs to s' f o r some y . Moreover, after entering Ss,Y,(y;P*+r) leaves s' through E; and Y,(y;p*-r) leaves .Ye through E;.

E€

[O, 4 and

Proof. This follows from Lemma 16 and the fact that SE is open and [,B*-r, p*+r] is compact.

Throughout the remainder of this subsection, we fix E E ( O , E J . [p*-r, p*+rl, let

For /3E

M. MIMURA, D. TERMAN and T. TSUJIKAWA

534

and If Y,(y;@)belongs to pp= YE@&8).

s" for all ~ > 3we~ let ,

Ya=+m.

If Jp is finite, we let

Lemma 22. For each @ E[B*-r, @*+r], J p isfinite. That is, after entering SKar y=pg, Yc(y;@) must leave 9 . Proof.

Note that in 9,

p= - &U
and

U=W>O.

Since SKis bounded, each Y,(y;p) must either leave Sr for some y > j j ~ ,or approach a critical point which lies in the closure of 9. Because there are no such critical points, the result follows. Lemma 23. For each ,!? E [@*-r, b*+r], P, belongs to E c . Proof. We prove that a trajectory can only exit s' through E". A trajectory cannot exit SI through &, because on Ej, P = - E U < O . T o prove that a trajectory cannot exit SI through E;, we let n=(O,4B*, -1) be a vector normal to E; pointing into 9 ,and X = ( - E U , W , ~ V W - ~ E U ~ - ~the ( Uvector )) V<2/3*. Therefore, field defined by (56). On

e,

n-X=4@*W-2VW+2EUZ+f(U) >2&U2+f(U)>O, which implies that trajectories cannot leave S through &. It remains to consider the edges of 3 through E;. Along Zl={( V, U , W ) ; O< V<28*, U = ~ - C Y ,w=o},& )O}, , and Int ( E n s ) is a piece of flow of (56). These all imply that trajectories cannot leave S through the above edges of S', which completes the proof. Let F be the map from [@*-r, @*+r] into EE defined by F(,B)=Pg. The preceding lemmas imply that F is well defined. Lemma 24. Proof.

F is continuous.

Note that on Ef, o = W > O , on

E , W = - ~ E U ~ - ~ ( U )
3= -EU>O. This implies that trajectories which leave S through Ee must do so transversally. Together with Lemma 16 this proves the lemma. The following result completes the proof of the existence of a large pulselike solution.

Nonlocal Advection Efect on Reaction-Difusion Equations

535

Lemma 25. There exists ,BeE (B*-r, b*+v) such that F(/3,)€{( V , U , W ) 1 V=O, 1--6< U< U*(E),W=O}

.

This result implies that Y,(y;B,) crosses the U-axis.

Proof. Since [B*-r, p*+r] is a connected interval, and F is continuous, it follows that C=F([p*-r,P*+r]) is a connected curve lying EE. From Lemma 20, we know that F(P*+r) € E: and F(/3*-r) E E;. Therefore

{ P < P * + r ; F(P)EE;}

P(E)=SUP

is the one which we wish. This completes the proof of the lemma.

rn

3.2. The small pulse-like solution Setting e=O, (54) becomes

i

V'= CJ

U'=

w

W'=-j(U),

which is easy to analyse because V does not appear on the right hand side. To find a solution of (65) and ( 5 9 , we demonstrate that there exists a solution (U(X),W W ) of

which satisfies (67)

(CJ, W)(o)=(U,,0 )

and

( U , W ) ( + a ) = ( O 0, )

and

V*=lim W ) d Y -

for some U, > 0. We then set V(x)=

s:

U(y)dy

0

The phase plane corresponding to (66) is shown in Figure 5. The boldly drawn curve corresponds to the solution of (66)satisfying (67). We remark that V* is finite, since U approaches zero in the exponential order. This is the solution we wish to perturb to the case s>O. Before doing so we discuss further properties of solutions of (66), for which it is necessary to introduce the following notation. For v > O , let ( U ( x ;v), W ( y ;7)) be the solution of (66) which satisfies (U(O;v),W(O;v))=(v, 0). Let a=-f'(O), and choose 6>0 so that

536

M. MIMURA,D. TERMAN and T. TSUJ~KAWA W

\

Figure 5. Trajectories (U,W) of (66) and (67). Define u, el and epby u = { ( U , W ) I W 6 } , e , = { ( U , W ) I O< U<6,W=O}

and e e = { ( U ,W ) 1 U=O,--6<

W
.

The information we need about solutions of (66) is contained in the following.

Lemma 26. ( i ) There exists a unique v* > 0 such that lim ( U ( x ;v*), W ( x ;v*))=(O,0) 2-t

m

( i i ) Thereexistsr>Osuch t h a t f o r a n y ~ E ( ~ * - r , ~ * + r( U ) ,( x ; ? ) ,W ( x ; ? ) ) belongs to u for some x > 0. (iii) After enrering u, ( U ( x ;v*-r), W ( x ;v*-r)) leaves u through e , . (iv) After entering u, ( V ( x ;v*+r), W ( x ;T*+Y)) leaves u through e,. We do not give a poor of this lemma since it follows from straightforward phase plane analysis. The relevant trajectories are shown in Figure 5. The following notation is needed to discuss the three dimensional phase ; U,(X;7). WE(x;7)) space determined by (54). For 7>0, let Z E ( x ;~ ) = ( V * ( Xv), be the solution of (54) satisfying ZJO; v)=(O, 7,O). Define S,El, Epand E , by

Nonlocal Advection Effect on Reaction-Diffusion Equations

537

Since ( U ( x ;v), W ( x ;v)), u, e, and e, are, respectively, the projections onto the

(U,W ) plane of Zo(x;q), S,, El and E,, the following lemma is an immediate consequence of Lemma 26. r, Lemma 27. Let r be sufficiently small. Then, for any ~ € I o = [ ~ * - v*+r], Z,(x; q) belongs to Sfor some x>O. Moreover, after entering S for thefirst time, Z,,(x;v*-r) leaves S through El,and Z , ( x ; ~ * + r )leaves S through Ez. Now, S , El and E, are open sets. Therefore, by continuous dependence of solutions on a parameter, Lemma 27 remains true if Z, is replaced by 2, for small E . That is,

Lemma 28. Choose r as in Lemma 27. There exists E,, > 0 such that for any and 7 E Z,, then Z,(x; v ) belongs to S for some x >0. Moreover, after entering S for the first time, Z , ( x ; v*-r) leaves S through E l , and Z,(x; v*+r) leaves S through E2. a E [ 0, E,]

We assume throughout the remainder of this subsection that E € (0, aO) is fixed with lY

(69)

EO<--.

86

For € I,, and E E (0,

E,,),

let x,=inf{x>O lZ,(x;q)ES}

and y,=inf { x > x , 1 Z,(x; ?) sf S} . Define I , and Ze by 11={77€Zo l Z e ( Y 7 l ; q ) ~ ~ 1 l

and

I , = {v

zo

I ZAY, ; 77) c Ez}

*

Lemma 29. I , and Z, are nonempty. relatively open subsets of Zo. Proof. From Lemma 28, we know that v*-r GZ, and ;1*+rEZ,. W'=

-2E u2 -f( U )

2-

3 1

2&U--

U T - 2&S--

";Iu>o ,

On E,,

M. MIMURA, D. TERMAN and T. TSUJIKAWA

538

because of (68) and (69). Hence, whenever a trajectory leaves S through El, it must do so transversally, not tangentially. This, together with the fact that E, is open and the continuous dependence of a solution on initial data, implies that I , is a relatively open subset of I,,,on E,, U’=

w
Hence, any trajectory which crosses E, must do so transversally. this implies that I, is relatively open. An immediate consequence of Lemma 29 is

As before

Corollary 1. There exists 7. E I. such that Z,(x; ?,) enters S , but does not leave through either El or E,.

We shall prove that Z,(x;vE)is the desired solution. For convenience, we set Z,(x)=( Ve(x),Ue(x),W,(x))=Z,(x;q,), x,=xpeand y e = y p , . We must prove that lim Z,(x)=( V,, 0,O)

for some constant

V,>O

.

X-+m

The proof is broken up into a few steps.

Lemma 30. y , is infinite. That is, Z,(x) belongs to S f o r any x > x , . Proof. Suppose that y , is finite. Clearly Z,(x) cannot leave S through the V-axis since each point on the V-axis is a critical point of (54). By the assumption, Z,(y,) cannot belong to E, U E,. The only remaining possibility is that Z,(y,) belongs to E,, which we show, is impossible. Let n=(O, -1, 1) and

(70)

X = ( V ’ , U’,W ’ ) = ( U , w, -2E(VWS.U2)-f(U))

.

One finds that n is a vector normal to E8 pointing into S , while X is the vector field defined by the right hand side of (54). Recall that on E,, W
V>O

and

f(U)<-cU. 2

Therefore, o n E,,

because of (69). This implies that no trajectory can leave S through E3 and completes the proof of the lemma. w

Nonlocal Advection Effect on Reaction-DiJiusion Equations

539

Lemma 31. There exists a positive constant K such that V,(x)
for any x > x , .

Proof. Let

t s+=t

so= ( V , u, W ) € S v=-4 l

a

4

( V , u, W ) € S V > - W + K ( x J + l

1 1 1.

w+ K ( x J + -86a ,

a

86 a

and

t

s-= ( V , u, W ) € Sl V < -a4- W + K ( x e ) + -86 a

Note that So divides S into the two subsets S- and S+. Moreover, Z,(x,) belongs to S-, because - W,(x,)< 6 . We shall prove that Z,(x) belongs to Sfor any x > x , . This certainly proves the lemma. In fact, we shall show that no trajectory can leave S- through So. Let n = ( - l , 0 , 4 / a ) be the vector normal to Sopointing into S-, and X as in (70). Then, on So, n -X = -U - -

8~ U2-- 4 f (U ) a a

>-u--ou2+2u=u 8E a because of (69). Therefore, no trajectory may leave S- through So.

H

The following result completes the construction of the small pulse.

Lemma 32.

There is a positive constant V, such that

.

lim Z,(x)=(V,,0,O) 2-f-

Proof. Inside S , V'=U>O

,

U'=W
and W'=-2€VW-2EU2-f(U)>

(

-2E06+-

">

2

u>o ,

because of (68) and (69). Therefore, either (i) Z,(x) leaves S for some x > x , , (ii) V J x ) becomes unbounded inside S, or (iii) Z , ( x ) belongs to S for any x > x ,

540

M. MIMURA, D. TERMAN and T. TSUJIKAWA

and approaches a critical point as x++oo. From Lemmas 31 and 32, we know that (i) and (ii) are impossible. Since the only critical points of (54) are along the V-axis, this completes the proof. H 4.

Concluding Remarks

We have constructed two different types of pulse-like solutions by using singular perturbation techniques and phase space methods. With the choice of f ( u ) = u ( l - u ) ( u - a ) and k ( x ) = e ( 2 H ( x ) - l ) with sufficiently small E > O , we obtained the global picture of the pulse-like solutions with respect to the parameter a by integrating the analytical and numerical methods. This is shown in Figure 6B. The picture indicates that, though the branch in the absence of the aggregative effect ( E = O ) is unstable (Figure 6A), the new branch ( e > O ) proceeds to the right until it arrives a t a limit point and it turns back to the left with recovery of the stability. That is, while the smaller pulse is unstable, the larger one becomes stable. The smaller pulse acts as a “separator” or a “threshold“ of aggregation or extinction of the populations. Thus, we conclude that the existence of the stable pulse solution can be performed by the presence of the aggregative advection. It should be also noted that the behavior of branch of ( 5 ) shown in Figure 2 ( E is not necessarily small) is topologically similar to that of (9) shown in Figure 6B ( E is sufficiently small), through Y = E X . The study of nonsymmetric pulse-like solutions is also interesting, though we did not approach it here.

Figure 6A. Global solution branch of (9) with respect to a when E=O.

Figure 6B. Global solution branch of (9) with respect to a when p(s)=s and E is sufficiently small but not zero.

Nonlocal Advection Effect on Reaction-Dirusion Equations

541

Finally w e refer t o a reaction-diffusion equation exhibiting both lateral exhibition a n d excitability,

or by putting

EV=W,

where f ( u ) is t h e cubic function defined i n t h e preceding sections. Ermentrout e t al. [2] h a s recently s h o wn t h e existence of a t least tw o different types of pulse-like solutions w h e n O
This study is a future problem.

References W. Alt, Contraction patterns in a viscous polymer system, Proc. of “Modelling of patterns in space and time”, Lecture Notes in Biomath. 55 (eds. W. Jager, J. D. Murray), Springer-Verlag, (1984). I 2 1 G . B. Ermentrout, S. P. Hastings and W. C. Troy, Large amplitude stationary waves in an excitable lateral-inhibitory medium, SIAM J. Appl. Math., 44 (1984), 1133-1149. P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232. -, Boundary and interior layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521. -, Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biornath., 28, Springer-Verlag, (1979). P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equation to travelling wave solutions. Arch. Rational Mech. Anal., 65 (1977), 335-361. Y . Hosono and M. Mimura, Singular perturbation approach to traveling waves in competing and diffusing species models, J. Math. Kyoto Univ., 22 (1982), 435-461. [ 8 1 M. Mimura, Some convection-diffusion equations arising in population dynamics, Contemporary Math., 17 (1983), 343-351. 191 M. Mimura and K . Ohara, Standing wave solutions for a Fisher type equation with a nonlocal convection, Hiroshima Math. J., 16 (1986), 33-50.

542

M. MIMURA, D. TERMAN and T. TSUJIKAWA

[lo] A. Okubo, Diffusion and ecological problems: Mathematical Models Biomathematics, 10, Springer-Verlag, (1980).

Masayasu Mimura Tohru Tsujikawa Department of Mathematics Hiroshima University Hiroshima 730, Japan David Terman Department of Mathematics Michigan State University East Lansing, Michigan, 48824, U.S.A.

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 543-560 (1986)

On Small Data Scattering for Some Nonlinear Wave Equations By Kiyoshi MOCHIZUKI a n d Takahiro MOTAI Abstract.

We study nonlinear wave equations of the form dt2w-A w +mZw+f(w)=O

in Euclidean space. Here m 2 0 and f(w)represents power nonlinearities, the sine-Gordon nonlinearity and a cubic convolution nonlinearity. Under suitable restrictions on the nonlinearity, the scattering operator is proved to be defined on a dense domain of a neighborhood of 0 in the energy space. Key words: nonlinear wave equations, small data scattering, densely defined scattering operator, uniqueness of solutions

1. Introduction and Results

In this paper we study a small data scattering operator for the pair of equations (1.1)

at2W(t)+AW(t)+f(W(r))

=o

and (1.2)

at2w(t )

+Aw( t ) =0

i n ( x , t ) € R n x R . Here A = - A + m 2 w i t h A = ~ ~ = , a , 2 ( a , = a / a x j ) a n d m > _ 0and , f(w) represents some nonlinear perturbations including power nonlinearities st=,;j,lwIP~-~w with ~ , E Rand py> 1, the sine-Gordon nonlinearity sin w-w, and a cubic convolution nonlinearity ( V*lw12)w with real V = V ( x ) . Small data scattering theory for these problems has been developed in Strauss [ 5 ] , [6] (see also, Menzala-Strauss [2] and Pecher [4]). He discussed on the one hand conditions on f(w) for which the scattering operator can be defined on a whole neighborhood of 0 in the energy space Ze.Another problem discussed there is to find a wider class of perturbations for which the scattering operator can be defined on a large set. Mostly, the smallness of solutions of (1.2) is scaled in a certain Sobolev space H k , p . So the above large set will form a neighborhood of 0 in Z e n H k s p . Received June 18, 1985.

K . MOCHIZUKI and T. MOTAI

544

In this paper we restrict our concern to the latter densely defined scattering operator, and generalize some results of Strauss [ 5 ] , [ 6 ] . Our method is essentially the same as those developed in (51, [6]. Namely, the so-called Lp-Lq estimates for the free equation (1.2) will play a key role for our results. For the existence of the scattering operator, it is necessary to solve the Cauchy problem for (1.1)with - 00 initial time. Let w-(t) be a solution of (1.2), and let S ( t ) = k 1 l 2sin A 1 / 2 t . Then the integral version of this problem is given by

For k positive integer and l < p <

00,

the norm of Hk*P=Hk*p(Rn) is defined by

- - .,

a,) is a multi-index with norm where is the usual Lp-norm, a = ( a l , Ia]=al+ +a, and aa=ayl * . 3 s . The integral equation (1.3) will be solved in the space

-

.--

(1.5)

v= V k , 9, d ) = { u ( t ) € CAR; Hkzq);IIuIIv=SyP (1+ltI)'IIU(t)IIk,q< 00)

with suitably chosen k , q and d>O. Once the solution of (1.3) is shown to exist, we compare this w(t) with the free solution w-(t) in the energy space SYe with norm (1.6)

1

IIU(t)l l e =-J$IIA""(t)

1;

+ IIa,U(t)1I;Y

*

We denote by @ the following square in R2:

and by 9the part in

@ of

the closed quadrangle with vertices

Let d = d ( l / p , l/q) be a piecewise linear function of (l/p, l/q)€ 9 defined by (1.%

d = -I+---

n P

n 4

(in case m=O) ,

545

Small Data Scattering

Fig. 1. The case m=O ( n = 3 , p=4).

n---P -n+2+-+-

I=

4 n-2 P

Fig. 2. The case m>O ( n = 3 , p=4).

in P,P,P, n 4

in P,P,P,

in P,P,P,

n

in P,P,P,

where P,=(1/2+ l/(n+2), 1/2- l/(n+2)) and each P,PjP, means the part in CT of the closed triangle with vertices P,, P, and P,. For each p > l , the subdomain 9, of, 9is defined by

Cf. Fig. 1 and Fig. 2, where gP is shaded. We know from works of Strichartz [7] and Marshall-Straws-Wainger [ l ] that the free solution S(t)$ satisfies the following Lp-Lq estimates. Proposition 0. Let ( l / p , l / q )€ 9and let d = d ( l / p , l / q ) be defined by (1.7) (ix.,(1.7)0or (1.7)J. Then w e have

(1.9)

llS(t)$llq~Cltl-dll~llpfor any

R-W) ,

where C is a positive constant independent of t and $.

As regards the nonlinear term f(w)we require: Assumption. There exist a positive integer k, constants l < p l < and a belt domain g S c 9such that for any ( l / p , l / q )€ak,

. - -< p L

546

K. MOCHIZUKI and T. MOTAI

(1.12) Moreover,

(1.13)

2

c

l l f ( u ) - f ( d l l p ~c u=1 IIlu116y,'+

Il~ll2i'HlU- 41,

if two points (l/p, l/q), (l/p, 1 / ~E@* ) satisfy

(1.14) Under these conditions on f ( w ) we have the following

Theorem 1. ( i ) (Existence of the scattering operator) Suppose that .B'@n 9pl# 0 . For (Up, l/q) E.B'~ n eP, let d = d ( l / p , llq) and V= V(k,q, d) be defined by (1.7) and (1.5), respectively. Then there exists a 6 > 0 with the following properties: If w - ( t ) € Z e nV and Ilw-llv<S, then there exists a unique solution w ( t ) of(1.3) such that w ( t ) € X e nV, llw[lvs(4/3)11w-llv and (1.15)

~ ~ w ( t ) - w - ( t ) / ~ e - + Oas

t+-m

.

Furthermore, there exists a unique solution w + ( t )E 3Fen V of (1.2) such that (1.16)

Ilw(t)-w+(t)Il,-+O

as

t++m

.

The correspondence S : w - ( t ) - w + ( t ) defines the scattering operator. ( i i ) (Uniqueness of solutions) Let B be a convex domain in a k n g p , . Suppose that w - ( t ) € Z e nV(k,q, d) for each (l/p, l/q) € B and IIw-[IV(k.g,d) is bounded in B by a suBciently small constant. Then for any pair of points ( l / p , l/q), (l/fi,I/@) € B , the two solutions w(t) E V= V(k,q, d ) and G ( t )E P= V(k,4, d ) , where d=d(l/fi, l/@),of(1.3) coincide with each other. The proof of this theorem will be given in the next S 2. In $5 3-5 we give applications of this theorem to several concrete problems. The power nonlinearity f ( w ) = , I l ~ l p - ~isw considered in S 3. In this case we obtain the following results: Let r(n) be defined by

+

( n z 3n- 2

+ d ( n 2+3n-

2)2- 8n(n- 1)

if

m=O

if

m>O.

(1.17)

Assume that

n=1-4

Small Data Scattering

Then choosing k = l , pl=

547

.-.=pl=p and S k = ~ where p,l,

(1.19) we cam show that f(w) satisfies the above Assumption and @ p , l n 9 p # 0 . Moreover, if we assume

1<*

n=2kfl, 2kf2

(k22)

then choosing B'k=ep,X, we can have the same results. on p is given by

2(n- 1)


In [ 5 ] the condition

if m=O if

m>O.

Our condition (1.18) partly generalizes this, especially in case m=O and n= 2-4. Note that the number (1.17) for m=O is already uded in our previous paper [3], where is considered the case r(n)
n-1

(n=2-4).

(1.1) includes the sine-Gordon equation if we choose m = l and f(w)= sin w- w. This equation is considered in S 4. We put

i:

{

max 1,

(1.20)

[?I}

(21nf9)

3 (n=9) p l = 3 , p z = 4 , . . . , p i = Z+2, Z=max{2,k-l) I

gk= n =,+z,r u=1

where [a], a € R , means the biggest integer which does not exceed a. (Note that our results exclude the important case n=1.) Then it is shown that f(w) satisfies the Assumption and g x n.P8f0. Finally in S 5 we consider the cubic convolution interaction f(w)= (V*lw12)w. Suppose that V ( x ) is a real valued function belonging to LI(R"), where n 2 3 and

K . MOCHIZUKI and T. MOTAI

548

(1.21)

llz<

if m>O If we choose

(1.22)

I

k=l+[-1, n(n- 3) 2(n+ 1)

p=3

and

then it is shown that f(w) satisfies the Assumption and .G3kn.98#@. This problem is studied in [ 6 ] in the case m>O and our result is a new application to the case m=O. In connection with (1.18)1, we notice here the following: The requirement p 2 2 in (1.18)1 is never essential, and we can show the existence of a densely defined scattering operator for any p satisfying

which is already proved in Strauss [5] (Theorem 12) in the case m>O. verify this for m=O, we introduce the space

To

V={u(t)€ Ct(R;H”‘); lIuIIv=su~( l + l r l ) d I I U ( r ) I I H ~ . t P < tER

with l/q=(n+ l)/q-(n- 1)/2 and d=(n- 1)(1/2- l/q) (cf., e.g., Pecher [4] (Theorem 0. a)) or Mochizuki-Motai [3] (Proposition 2.3)). Let us consider solutions of (1.3) in the above space V. Then a n approximate energy method is applicable to obtain a suitable conservation property of energy. This and the Sobolev embedding H ~ q C , L P + make l us possible to follow the proof of Strauss [ 5 ] , and leadus to the desired conclusion.

S 2.

Proof of Theorem 1

( i ) First we shall show the unique existence of solutions of (1.3) for a fixed (l/p, l/q) € g k nPP,.We apply the contraction mapping principle in the space V= V(k,q, d ) . Put

J

--

Then by Proposition 0 and Assumption (l.lO),

Small Data Scattering

549

Here d< 1 by definition and we have assumed dp y>d p l > 1. Thus, J -m and it follows that

Similarly, it follows from Assumption (1.11) that

Now we choose 6, > 0 very small to satisfy 1

2 c 2 Sp-lIU=l

and put B(6,)={u€ V ; Ilullv<Sl}. (2.4)

Then we have from (2.2)

1

for u, u ~ B ( 6 ,,)

Il@u-@ull,
and for sufficiently small w-, say Ilw-Il.16, (2.5)

1 , 2

ll@ullv5 IIw-

we have from (2.3) 1

11"s- 7 llullv

61 .

These imply that @ determines a contraction mapping on B(6,). Hence, we see that there exists a unique fixed point w(t)E B(6,), which solves the integral equation (1.3) ~ I ( ~ (1.15) / ~follows ) ~ from ~ W Assumption - / ~ ~ . By (2.5) \ ~ W ~ ~ Moreover, (1.12). In fact, noting dp,>l, we have (2.6)

Ilw(t)-w-(~)llc
1'

-m

1'-

Ilf(w(4)llB
IIW(S)IIEYqds

550

K. MOCHIZUKI and T. MOTAI

Since w-(t) E Ze, this also shows w(t) E Re. Finally, we define (2.7)

w+(t )= w - ( t )-

1-

S(t - s)f( w(s))ds

.

--m

Then obviously w+(t ) E 2Fen V and solves (1.2). Moreover, (1.16) holds since we have (2.8)

w(t )- w+(t )=

1:

S(I - s)f(w(s))ds .

( i i ) First we shall show the assertion for (lip, l/q), (l/p, 1/4)E B satisfying (1.14) and (2.9) Note that the condition (2.9) implies the Sobolev embedding HkoGC,LQ. So, two solutions w(t) E V and +(t)E p of (1.3) can be compared in the space X = M t ) E CAR; L*);Ilullx=suP (l+ltl)doll~(f)llq< m} ,

where d,=min {d, d } . By (1.13) we have

Hence, if we choose w - ( t ) so small that

we conclude w = + in X . In general cases, we have only to note that any two points in B can be joined by a broken line contained also in B such that each pair of neighboring angular points satisfies (1.14) and (2.9). The repeated use of the above argument shows the assertion.

Small Data Scattering

55 1

S 3. Wave Equations with Power Nonlinearities In this section we consider (1.1) under the following conditions on f(w). These conditions are obviously fulfilled by power nonlinearities f(w) = C:=,l , I ~ I p ~ -with ~ w 2, € R. ( A l ) f ( u ) is a real valued Ck-function of u € R and f(O)=f(O)=

.-.=f'"(0)=0

( A 2 ) There exist constants k+l
For suitably chosen k and p,,

(f'"=ddkfidUk) .

- < p t such that

- - .,p l , we put I

(3.1)

*k=

n

ePp.k

9

Y=l

where SB'P,k is defined by (1.19). It then can be shown that a k n 9 , 1 + 0 and ( A l ) ,(A2)bring all the inequalities of the Assmption of S 1. Lemma 3.1. (a) If m=Oandp>(n+l)/(n-1), or i f m > O a n d p > ( n + 2 ) / n , then 9,, 0# . 9, is monotone increasing in p . Moreover, we have

jp.....-{(L,L); P 4 d = ~ } i f m=O 9,= IJPp= P 1 1 [9-{(-, d=O} -); i f m>O P 4

(3.2)

.

(b) Suppose that kO and n = l , we choose k = l . In this case, the upper boundary of aP,k is always the line l / q = 1/2.) Thus, we have for each p 1 < p l ,

Proof. All the assertions easily follow from the definitions (1.8) of PP and (1.19) of BP,*. Q.E.D.

.@'p,x

r(n) is given by (1.17) as the number p for which the lower boundary of Namely, solving the system of equations first meets with 9,.

_p _ _' _- 0 , 4 or

P

-I+---=-n P

n 4

l P

and

1-n 4 P

n+l (:P,Ps) (if m=O) 2

K . MOCHIZUKI and T. MOTAI

552

(if m>O), we obtain

(3.4)

(-,P1 -,91 -)P1 =

if

m=O

if

m>O.

Note that the upper boundary of .G’p,k goes down to { l / q = k / n } as p - ~ . Then we have the

Lemma 3.2. If n , k and ,o, satisfy one of the folIowing four conditions, then we have .G?P1.kn 9plf 8: (3.5)

k=max{l,

[+]I

, y ( n ) < p l < w and [:]+l
;

(3.7) (3.8)

n=7,9,

k=-

n-3 2

and

n- 1 ~


5n+ 3 nf3

Remark. In (3.5)-(3.8) is included the condition k + l < p l required in ( A 2 ) ,which is not necessarily used to show the above assertion.

Proof. First we require (3.5). In this case we can show that l / q in (3.4) is less than kin (if n=l and m>O, kln should be understood as 112). Hence, for any r(n)< p , < 00 the point ( l / p , l / q ) of (3.4) is contained in .58pl,x n PP1. Next we require (3.6). In this case, we can not have a common point which is contained in any a P 1 , k n 9 ’ p , .However, since the line l / q = k / n passes the pointP,=(1/2, 1/2-1/n)=(1/2, kin), each &3’pl,xnPPl is never empty. In the case where n=5-7 and k = l , or n=7, 9 and k=(n-3)/2, we see shrinks to P,=(1/2+ l / ( n + l ) , 1/2- l/(n+ 1 ) ) and becomes empty that @ p , x n PP for a finite ,o. The condition

that P, E g p is vrewritten k as

Small Data Scattering

553

n2- (2k- 3)n- 2k n2-(2kf l)n-2k

n+ 3 n- 1

-
'

and we can verify it for p=pI satisfying (3.7) or (3.8).

Q.E.D.

Lemma 3.3. Suppose that k and pI satisfy one of the four conditions of Lemma 3.2, and for each (l/pl, 11qJ € &3'p1,x n PPl, let p2, -,p1 satisfy

-.

pI
(3.9)

. - -< p l

Then we have (Upl, l/ql) E J k

n

Lemma 3.4. If n, k and pl, then we have g k n Pp,#0: (3.10)

k = m a x { 1,

(3.11)1

n=4, P1<

(3.11)k (3.12)

(3.13)

Q.E.D.

The assertion easily follows from (3.1) and (3.3).

Proof.

(3.11)1'

( : - k ) p l g L - kn.

and

[+]I

-

-,pLsatisfy one of the following six conditions,

, y(n)
k=-- n-2 - 1 , 2

7(4)
2Pl (if m=O) -pI2+2p1+ 1

n=4,

k=-

n-2 =1 and 2

n 2 6 , k=-

n=5-7,

n=7,9,


or

~

". pI<

n and - < p l < 2-

k=l

and 2 j p l <

k=-

n-3 2

and

3Pi2 -pI2+2p1+2

;

and

(if m>O) ;

- - .
- - .< p l < ( n - l ) p l + n ; < p L < n2+n-2

n- 1
2

[5]+l
< p L , 2
5$-J73
n-2 2

and

n2-3n-2

;

...
Proof. In cases where (3,10), (3.12) or (3.13) is satisfied, the assertion is already verified in the proof of Lemma 3.2. To show the assertion for the other cases, we use the above Lemma 3.3. As is easily seen, p1 < ( 5 + 2/%)/6 is the condition that pJq- llp < 0 on the point of intersection

(3.14)

K. MOCHIZUKI and T.MOTAI

554

(with n=4) of two lines d(l/p,l/q)=l/pl and l/q=l/np+(n-3)/2n(:PIP,). In this case, choosing ( l/pl, l/ql) c? gpl,l n ePI sufficiently close to the point of intersection

I(”-

(1’\=

(if m=O)

4(P1-1)’ 4Pi(Pi-1)

of two lines d ( l / p , l/q)=l/pl and p,/q-l/p=O, we see that (3.9) holds for pi satisfying (3.1l)I. Next, suppose that (3.11)1’ or (3.11)k is satisfied. Then we can choose (l/pl, l/qJ €epl,kfl 9pI very close to the point (3.14). Since pL<(n-l)p,+n, Q.E.D. this leads us to (3.9). Now we return to the nonlinear equation (1.1). Theorem 3.5. Suppose that f(w)satisfies ( A l ) and (A2), where k and pl, * * * , Let g k be defined by (3.1). Then f ( w ) satisfies the Assumption with this ak and all the assertions of Theorem 1 hold. pL are asgiven in the above Lemma 3.4.

Proof. Let l a l g k . Since we have

laaf(u)-a~f(u)llc

21

+C

+ . - - + a j(l = a

Ida(1)u *

* *

laa(1)u

.. . aa(~),-aa(l)v .. .

aa(j)ul

a(~l+...+a(jl=a

from (A2) and the Holder inequality it follows that lal,l

IIaaf(u)-aaf(v)IIps C

aa(’)ul

a(1)

2

lf(j)(u)l j=1

Z

f‘j)(u)-f”’(v)I

3=1

.Z {IIuII2iL.&p+

Py-j-I IIuII(pw-j)r*pI

3.”=1

x llu-ull(pv-j~+pa(11+...+a(j)=a C llaa(l)ullqp * .. lI~u(j)ulIrjp

where r! and rp are positive constants satisfying

-+-+ 1 1 r’

rl

... + - 1= I . r1

Moreover, since (l/p, l/q) E g kwe , can restrict them to satisfy

,

555

SmaN Data Scattering

follows from ( A l ) and (A2). By assumption

We then have l/q>1/2pv2l/q-k/n, and (1.12) follows also from the Sobolev embedding. Finally, we shall show (1.13). The argument used in the proof of (1.10) gives the inequality

-m II Ic , {i II II

IIJ(4

11

p

t c l )r ' p

Y=l

+ II II

2, )I:"-;:

7,

P}

where r', r are positive constants such that l/r+ 1/r'= 1 . also to satisfy (3.15)

-'

-_ 4 rP

and

IIu - II7 p

9

We can choose them

1 1 1 k -2 >--4 (p"--)r'P 4 n

*

Moreover, it follows from (1.14) that (3.16) I n fact, if l / p - l / j = l / q - l / B , this is the condition that (l/P, l/B)€.@pv,k. the other hand, if l/q= 114, we have

On

&-->()$Ll-ko, 1 4

P

4

P

n

which gives the condition that (l/p, l/q)€.GPpp,k. (3.15) and (3.16) imply the Sobolev embedding Hk,r

CL(PV-l)r'P

and

Hk.; G L ( P v - l ) r ' P

556

K. MOCHIZUKI and T. MOTAI

Q.E.D.

and we obtain (1.13).

S 4. The Sine-Gordon Equation In this section we consider the sine-Gordon equation dtw(t) - Aw(t)

(4.1)

+sin w(t)=0

in ( x , t ) € R"x R, where n 2 2 . We put

(4.2)

f(w)=sinw-w

and write (4.1)in the following form:

+

a;w(t)- Aw(t) w(t)+ f(w(t)) = O

(4.3)

.

Lemma 4.1. For any multi-index a we have

( Q r ( u ) = O if /a1< 3), where ap,a#(u)and ar(u) are smooth functions with bounded derivatives. Proof.

Note that

f(u)=sin u-u= -

\:?

~ ~ ( u ) = ( cu-l)a,u=os (1-0)

f(u) = -

s:

cos (0u)d0u3,

s:

Proof.

s:

,

C O S ( ~ U ) ~ ~ U ~ ~ cos , U -(Ou)dO~a,~aju

(1- 0) cos (0u)dOU ~ ~ : , , U

where a:,=a,a, and a;,,=a,ajak. equality give (4.4).

Lemma 4.2. gknp3+0.

(1-0) cos (8u)d0U2a,U,

Let k, p l ,

These and repeated differentiation of the last Q.E.D.

.. .,pL and B kbe as given in (1.20). Then we have

In case n=2, we have k=l, 1=2 and

Small Data Scattering

557

Thus, the point P,=(3/4, 114) is on the lower boundary {0=3/q-l/p}. In case n 2 3, we have

Obviously, PI=(1/2+l/(n+ l ) , 1/2- l/(n+ 1)) satisfies the first condition 0 1 314- l/p. The second condition is also satisfied since we have {n2-(2k+ l)n-2k}(l+ 2)
(cf., the proof of Lemma 3.2) for k and I given in (1.20).

Q.E.D.

Now, for equation (4.1) we have the

- -.

Theorem 4.3. Let k , p l , ,p1and G kbe as given in (1.20). Then the nonlinear term (4.2) satisfies the Assumption, and all assertions of Theorem 1 hold. Proof. It follows form Lemma 4.1 that laaf(u)l
C

l@u@‘du-B-p’ul+Cr

B+B’
C

Idr(au)a-r\ ,

la-1124

laaf(u)-aaf(41

-

c B+B’
211

pwB*uaa-p-P’ul+

{lu-211

laqau)”-‘l+

laBuaP’uaa’-F-B’u-aQvap’vaa-P-P’Vi}

lar(dU)a--T-dr(a21)a-rI}

,

where Cr=O if la1<3. Let Ial l + l a - - r l if la--r124. Then we have from (b) of Lemma 3.1.

With these inequalities, we can follow the same argument as in the proof of Theorem 3.5 to obtain

and

558

K. MOCHIZUKI and T.MOTAI

These show (1.11) and ( l . l O ) , respectively. Next note that 1 / 2 5 l/p51/2+k/n. Then

and it follows that

llf(u)llZ
l l f ( ~ ) - f ( ~ ) l l , ~ ~ ~ l l ul ll 4l ~7 +~ +l l ~ l l ~ ~ H ~, - ~ l l * where l/r=(q-p)/pq.

Then since a k ~ * 3 , k(1.14) ~G implies ’4,k,

l l l l ->->->--q-2r-3r-q

k n

l l l l ->->->---,

and

k n

B-2r-3r-g

Q.E.D.

and (1.13) is proved.

S 5.

Wave Equations with Cubic Convolution

In this section we consider the wave equation (1.1) with the following nonlinearity :

(5.1)

f(w)=(V*Iw12)w

9

where V= V ( x ) is a real function belonging to L’(Rn), z > 1 and n 2 3 , and means the convolution of V a n d [ w I 2 . Lemma 5.1. For z satisfying (1.21), let have

akbe defined by

(1.22).

*

Then w e

@,nP3+0.

Proof. The second inequality of (1.21) is obtained as the condition that 3/q- l/p > 1- l / z on the vertex 6n(n- 1)

6n ’

’ 6(n-

6n

of .9’8. Thus, if (n+1)/4<~<3n(n-1)/(6n-4) have @,.

n P8#0 .

1)

if

m=O

if

m>O

(if m=O) or <3n/4 (if m>O), we

Small Data Scattering

559

In case ln(n+ 1)/((4+2k)n+2k), which guarantees the required inequality. Hence PI @k n 93. Q.E.D. Theorem 5.2. Suppose that z satisfies (1.21). Let k , p and ak be defined by (1.22). Then the nonlinear term (5.1) satisfies the Assumption, and all the assertions of Theorem 1 hold.

Proof. Repeated differentiation of (5.1) gives d"f(w)=

c

CfIJ V*d&Yu)a"-P-'u

.

B+pia

By the Holder inequality and the Young inequality, we have

c I c c

Ija~f(w)Il,Ic

Btrla Birla

I/ v*:aBualull,,lla"-fI-rull~~p

I1 ~ l l * l l ~ ~ ~ ~ ~ ~ l l ~ l l ~ " - ~ - ~ ~ l l ~ ~ 9

where r , r' and t satisfy

Let I a l l k and (Up, l / q ) E e k . Then by assumption we can restrict them also to satisfy

Hence, the Sobolev embedding shows the inequality

lla"f(w)llpl CII ~ l l z l l 4 l ; , q

Y

which verifies (1.11). (1.10) can be shown by the same argument. Next note that

l l S ( W ) I l z l CII ~ l l n l l ~ l l ~ t l l ~ 1 1 2 v ~ and

where

560

K. MOCHIZWKI and T. MOTAI

T h e n w e can follow the s a m e line of proof of The ore m s 3.5 a n d 4.3 to obtain (1.12) a n d (1.13). Q.E.D. References [ 11 B. Marshall, W. A. Strauss and S. Wainger, LP-La estimates for the KleinGordon equation, J. Math. Pures Appl., 59 (1980), 417-440. [ 2 ] G.P. Menzala and W. A. Strauss, On a wave equation with a cubic convolution, J. Differential Equations, 43 (1983), 93-105. [ 3 1 K. Mochizuki and T. Motai, The scattering theory for the nonlinear wave equation with small data, J. Math. Kyoto Univ., 25 (1985), 703-715. [ 4 ] H.Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z.,185 (1984), 261-270. [ 5 ] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41

(1981), 110-133. Nonlinear scattering theory at low energy: sequel, J. Funct. Anal., 43 (1981), 281-294. [ 7 ] R.S. Strichartz, Convolution with kernels having singularities on a sphere, Trans. Amer. Math. SOC.,148 (1970), 460-471. [ 61 -,

Kiyoshi Mochizuki Department of Mathematics Shinshu University Matsumoto, Nagano 390, Japan Takahiro Motai Institute of Mathematics University of Tsukuba Sakura-mura, Ibaraki 305, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equationspp. 561-582 (1986)

The Near-field Finite Difference Approximation for Wave Propagation Problems in Infinite Media By Tomoyasu-Taguti NAKAGAWA Abstract. This paper is concerned with a numerical method for wave propagation problems where the computation of local nonlinear response to some exciting force is of main interest. The first part of the paper is devoted to a theoretical argument of the proposed method of finite differences for the solution in the near-field to the Cauchy problem of linear hyperbolic systems of first order. The method employs the coordinate transformation that maps Rninto a n n-cube. The numerical solution after the mappingback of coordinates to R" has the weighted L2-convergence property under certain smoothness and boundedness conditions in the continuous problem. The second part presents an application of this method to the time-dependent response of a soil-structure system subjected to seismic waves where the system has a contact nonlinearity between the soil and the structure. A computed result is given for a specific geometry and material constants. Key words: finite differences, hyperbolic problem, near-field approximation, soil-structure interaction, slip/separation nonlinearity, time domain analysis

1. Introduction

This paper is concerned with a numerical method for wave propagation problems where the computation of local nonlinear response to some exciting force is of main interest. A typical example of such problems is found in nonlinear interaction phenomena of a soil-structure system subject to seismic waves where the equation of elasticity is assumed to be linear. From the standpoint of practical calculation, the numerical modeling for radiation of scattered waves into infinity is a critical matter. If the nonlinearity does not exist at all, then the so-called frequency domain method can deal with these situations effectively, as found in the works of Lysmer-Kuhlemeyer [7], Lysmer-Drake [6], and Waas [16]. But if the nonlinearity is involved, as in the case mentioned above, the method in Received April 10, 1985.

T.-T. NAKAGAWA

562

the frequency domain can not be applied, so that the problem must be solved in the time domain. The methods proposed by Smith [13] as well as by Cundall and others [ 11 have been attempts to approximate the non-reflecting boundary condition on the artificial boundary in the finite modeling for time domain analysis. Engquist-Majda [2] has shown that prefectly absorbing boundary conditions on the artificial boundary for the general class of wave equations are nonlocal in both time and space, and has proposed “a hierarchy of highly absorbing local boundary conditions” to approximate the theoretical nonlocal one. Gustafsson-Kreiss [4] has discussed the validity of several local boundary conditions to be specified on the artificial boundary. The author has proposed in [8] a numerical method for the solution in the near field to linear hyperbolic systems of first order. The method is based on the use of the coordinate transform that maps R n into a n n-cube 9. The transformed system of equations is then solved inside 9 with the vanishing via a finite difference scheme for condition on the transformed infinity r=aQ shock phenomena in fluid dynamics. The combined use of coordinate transform with a scheme for shock computation yields the effect of a n absorbing boundary layer in the shrink mapping region. This numerical property calls for a mixed algorithm in the domain of practical application such that a n appropriate finite element scheme is used in computing the near-field portion having complex geomerty and/or material properties, while the proposed finite difference scheme is used in computing the far-field portion that may be assumed to have a simple structure on geometry as well as on physical properties. In the above context of research work, the present paper aims at: Firstly, to give a complete proof on the proposed finite difference scheme that has been provided with only a n outline of proof in [8]. Sections 2 and 3 deal with these theoretical arguments; and secondly, to present a numerical result on a specific application solved with the algorithm of a mixed finite element/ finite difference concept. The example is the dynamic behavior of a soilstructure interaction system under the condition of slipiseparation interface boundary of structure against its foundation. Section 4 deals with this example. 2. Problem Setting We consider the approximation of the following linear hyperbolic system of first order:

( ~ ( 0x ), = u , ( x )

,

X E R:

.

Here, the unknown variable u=u(t, x), the nonhomogeneous term f = f ( t , x),

The Near-field Einice Direrenee Approximation

563

and the initial data uo(x)are m-vectors; and the coefficients A , = A t ( x ) , i= 1,2, n , are m x m matrices. We assume that (2.1) is a symmetric hyperbolic system. Write the function space {LZ(R:)}"as L2(R:), or simply, L 2 , which is equipped with the norm II.II such that as.,

and assume that (2.1) is well-posed in E2. In what follows, we further assume ,n, is bounded and smooth, and is equal to a constant that each A , ( x ) , i= 1, matrix A,, for large 1x1, and that f ( t , x ) is also bounded and smooth with bounded higher derivatives. Let SZ be a n n-cube with side length 21, Z=l+ii (ii>O), set at the origin of the Euclidean n-space Rn,i.e.,

.- .

n

Q=JxJx

... X J

where J=(-Z,Z),

and let w c Q be a n n-cube with side length 2 such that n

w=IxZx

... xZ

where Z = ( - 1 , l ) c J .

, a n arbitrary function a(p)E C{(J)that satisfies Given ~ 2 . 4take

(2.2)

O
pEJ

€l

a ( v ) - ( Z - l ~ l ) ~ + ' in the neighborhood of

p=-tZ.

-

Here, the symbol denotes that the left-hand side is identical to the right-hand side except the existence of a scalar multiplier in the specified domain. The restriction p 2 4 is required for our theoretical argument. Define the mapping G G : y = ( y l ,* * * , Y ~ ) € Q ~ X = * *( X * ,~x ,~ ) € R E

Then, G maps Q into R; one-to-one, satisfying that G restricted on is is the identity mapping because of the second condition of (2.2). Through x = G y , (2.1) is written in terms of y as follows.

T.-T. NAKAGAWA

564

(2.4)

\v=u(r,y)=u(r, Cy) . We approximate (2 . 4 ) by finite differences as in the following. the i-th unit vector in R;, i.e., e,=(O,

Let e, be

. - - 1,, .-.,O). i-th component

Denote by h and k the space and time mesh widths, respectively, such that h=l/Nh and k=Rh where N , is a positive integer, and R is a n arbitrary fixed positive number. The mesh points ( t , , y , ) are set uniformly in the domain [0, TI x fi so that

[ t,=vk I

(2.6) where vc {O, 1,

-..,[ T l k ] } , and

,u=(,ul, - - - , E (is~ )the multi-index with each Denote by Q, Q, rh, and [0, TIh the sets of the above defined mesh points in 0,fi, r=an, and [0, TI, respectively. Let us call (rv, y , ) the generic mesh point. For simplicity of writing mesh-point valued functions, drop the suffices Y and ,u; then put in parentheses the arguments only when they have displacements from the generic mesh point ( t , y ) , or their appearance makes the situation clearer. For example, v,, v ,(y ,+h), v , ( t + k ) , and {vh(y,+h)-v,(y,-h)}/(2h), respectively, stand for uh(f,y , , * -,y n ) , V h ( t 9 Yi+h,Yz, * . . , Y J , Vh(t+k,Yir . . . , Y n ) , and { V h ( f , Y i , Yi+h,Yt+l, ...,YnIV h ( f r ~*l ,. - , Y , - ~ , Y , +.,-,. , ~ , ) } / ( 2 h )and ; v,(r) stands for V h(f,y lr. - . , Y , ) . Write as L , , , a ‘one-step’ difference operator for the approximation of p , € {0, + l , i 2 ,

.-.,i N , } .

...,

-

in which all the spatial variables y j , j f i, are considered as parameters. difference scheme for consideration is as follows.

The

The Near-feld Finite Difference Approximation

i

2 C' l...',=

565

1

bh=g

where

and

Scheme 2.

where

and si, h =

Iuh(yi

-

uh(yt-h/2)}/h

Remark. In (2.7),we restrict ourselves to those LI, which are in the form

T.-T. NAKAGAWA

566

of sum-of-products of Lllh’s. This restriction is essential in this paper due to our method of stability analysis. On the other hand, the choice of Li,, is rather conventional. Scheme 1, which is a modification of Friedrichs’ scheme in that e l i l . h U hreplaces {v,(y,+h)+v,(y,-h)}/2, and Scheme 2 which is the modified Lax-Wendroff scheme using half meshes are in fact two concrete examples of Li,h. The former is of theoretical interest in its simplest form whereas the latter deserves enough accuracy for practical use beside theoretical interest. There may be other possibilities of L i l h . Further, L , constructed from Lilh’s of mixed time steps k , such as Lh

L , h ( k / 2 ) L ,(k)Li,,(k/2) ,

with Li,, of (2.9) for n=2 [14],for example, is also within the scope of the present theory. As far as they are in the form of sum-of-products of Lt,,’s, the theoretical argument parallels the one to be dealt with in this paper, so we will not work out all the possibilities.

3. Error Estimation We shall begin with the stability argument about (2.7) by using the theory of pseudo-difference operators developed by Yamaguti-Nogi [ 171, Vaillancourt [15], Shintani-Tomoeda [ 111, among others. For technical reasons, we introduce the following extension to all the variables appearing in (2.7), regarding (2.7) as a n L*(R;)-valued difference equation in t. Let 8(7)be the function in 77 € R‘ defined as

Clearly &(T) is in Cp(R1). Then, introduce B,(y) defined as =4 Y J x .(Y)

9

Y R;

where r ( y ) is a C“-function such that O j r ( y ) j1 for y e R;, r ( y ) = 1 for y f a , and r(y)=O for large JyI. Define g , ( y ) , $ ( t , y ) , and Go(y)as follows.

In accordance with this extension, let

G(t, y )

be such that

The Near-field Finite Difference Approximation

567

where v ( t , y ) is the solution to (2.4). Obviously, this G ( t , y ) is formally the solution to the following system.

(3.1)

By the same extension procedure, (2.7) turns out to be

(3.2)

G,(t+k)=i,G,+kh,, { 8,(t)=G,, Y E R; ,

y€R; t=O

,

t € [ O , T]h ,

t+k€[O, TIh

,

in which

, the difference operator which is naturally defined from L , of (2.7) and .?,is by substituting Lii(y) and B , ( y ) for a , ( y ) and B i(y), i= 1, , 12, respectively, as well as i ( t ,y ) for g(r, y ) . Hereafter, we shall omit the symbol * from all quantities appearing in (3.1) and (3.2) for simplicity of notation. By this convention, we shall regard (2.4), and (2.7) together with (2.8) or (2.9), as those defined in R;. Before going into lemmas, we study the regularity of Bi(y). Since ai(y)is p times continuously differentiable with respect to yt (and infinitely continuously differentiable with respect to y j , j f i ) , Bi(y ) is p + 1 times continuously differentiable with respect to every y j , j = 1, -,n. This is shown by direct calculus such that for y E 8

-..

--

in which the highest derivative of a (y , ) with respect to y t appearing in the right-hand side is ( d /d y t)‘-l{ a (y t)-l} . Since A , ( x ) is a constant matrix for large 1x1, the derivatives of B,(y) in y E Q vanish in the neighborhood of r, so that the (extended) B,(y) is p + 1 times continuously differentiable in y E R;. Owing to [ l l ] , it is rather routine work to derive the stability condition of (3.2). Introduce the one-parameter family of operators P , associated with a n m x m matrix-valued function p(y, defined in R;x R; :

c)

P,w(y)=l.i.m.

c-11 elv.c

1

~ ( E - E ’ , hc’)w(c’)dc’d~

T.-T. NAKAGAWA

568

where c=(24=’2

,

n

Y*E=C Y,Et , 1=1

and * ( E ) and i(E,E’) are the Fourier transforms of w(y) and p(y,t’) with respect to y. I n order that P, is well defined as a family of bounded linear operators in L2,p(y, 5’) must be a n element of W, the set of m x m matrixvalued functions that satisfy certain conditions. For the definition of P, we adopt the one given in 3.1 of [ll]. Let p(A,) be the spectral radius of A , , i= 1, -,n, and let p be such that

-

9

,ij=max sup p(A,) . ISzSn z e R 2

Lemma 1.

L,,a of Scheme 1 satisfies thaf

+

l l L , a l l 5 1 O(h) if 15 l i p . Proof. The goal is to find the condition on 1 so that

(3.3)

IILt,awll5(1+O(h))llwll

for w € L 2

holds. We first note that by the symmetric hyperbolic assumption of (2.1), there exists a nonsingular matrix n,(y), n,(y)*n,(y)=s (3: identity matrix), by which B,(y) is diagonalized as b,(y)=n,(y)B,(y)n,(y)-l where b,(y) is a diagonal matrix. Since B,(y) is p+ 1 times continuously differentiable with respect to every yr, so are n,(y) and b,(y). Let lL(y,E ) be such that

4 ( ~E,) =3 - a h ) ( 1 - cos 5,)3+ ila,(y)B,(y)sin E, . This belongs to R by virtue of the assumed regularity of a,(y) and B,(y). Obviously it is the symbol corresponding to L,,a of Scheme 1. Put qi=,3IFI,. By matrix calculus, it holds that

(3.4)

-

11, = 2a, ( 1 -C[)( 1 -aJ,3

+

Ut2S[2(

3 -A2BTB,)

+i1(l-~~~(l-~~))a,s;(B~-~~) U ~ S [ -12bFbt)}n, ~ ( ~

=11:{2a,- (1 - CE)( 1 - a t ) 3 +

in which q = c o s E , and sc=sinE,. I n view of (3.4), (1, is made nonnegative definite if l s l / p . By noting that p 2 4 , Theorem 3.4 of [ l l ] is applicable to the above defined qt. Hence we have l l w l l - l l ~ ~ , h w l -O(h)llwll l~ This immediately implies (3.3).

for Vw€L2 .

Q.E.D.

The Near-field Finite Direrence Approximation

569

Lemma 2. Li,h of Scheme 2 satisfies that 11Li,h[l5 1+O(h) if 1 s l/p.

Proof. Introduce &(y,E ) such that

MY, E ) =3+2iRa,(y)Bt(y) sin Et/21S cos EJ2-t iRa,(y)B,(y)sin 5,/21 . Then the argument exactly parallels the proof of Lemma 1. In fact, we have 3-1f1, =4R2a,2s~4n~b,2{3 - R2ai2bi2}nr, where st=sin EJ2. Hence, if R 5 l / p , then S-If1, is nonnegative definite. Q.E.D.

Theorem 1. Schemes 1 and 2 are L2-stable in the sense of Lax-Richtmeyer i f R 5 l/p .

(3.5)

Proof. Under (3.5), it holds that

IILhII5 Z Ctl...tnII&l,hII n

* *

IILn,hIl=g ctl...tn{l+O(h)}n= l+O(h) . Q.E.D.

Let us derive the error estimate of vh. We begin by

Lemma 3. I f the solution u(t, x ) of the continuous problem (2.1) is in C.+l([O, TI x R:) where s=l for Scheme 1 and s=2 f o r Scheme 2 , then i f holds that

(3.6)

V(t+k)-LhV-kbh=hJ+’R(s)

--

at arbitrarily fixed point ( t .y ) with t € [0, T I , t+ k € [0, T I , and y € { ( y l , y,,); lyll s l - h , i= 1, 2, * -,n}. Here, R,,, = R,,, [v] is a linear combination of partial derivatives of v of up to s+lth order and of partial derivatives of g of up to sth order such that

-

a,

(3.7)

in which each derivative is evaluated at a certain point (t’, y’) satisfying t < t‘< t+kand Iyif-ytl
T.-T. NAKAGAWA

570

Proof. The proof is given by nothing but the standard procedure for finite differences, hence we will skip it. Lemma 4.

For any w ( x )E B'(R;), r s p + 1, it holds that

where W =w(Cy) and la1 = C L a , 5 r . Proof.

Due to a/ay,=(l/a(y,))a/ax,,we have

where P J a ] are polynomials of a(y,) and its derivatives of order up to a,-1. Noting that a(y6)is independent of y,, i f j , and is proportional to (I-[y,l)ptl Q.E.D. as lyil T I , (3.8) is established. Thus, we utilize a suitable weighting function V ( y ) to cancel the possible unboundedness of Rcsl due to a,(y)-at. Let us first introduce a scalar function $ ( y), $ E C;(R1),420,that satisfies

(3.9)

The parameter q will be determined later in accordance with the vanishing order p + l of a(y,) and the order of accuracy s of the scheme. Let a&,(y), y )defined such that a,(y), $,,(y), and ~ ; ~ (be

I

otherwise

(0

(3.10)

I

otherwise

,

10 '

if

-25yf5l-2h

if

-1+2h5y6 $ 1

The Near-field Finite DifferenceApproximation

571

Lemma 5. It holds thar

(3.11)

where C is some constant independent of h. Proof. For the first estimate, it is enough to study the behavior of ~ & ( y ) in the neighborhood of y,=I-2h. By the assumption of a(p) and +(p), we have there

-

a:i(y) (E+2W " H E + 2hIgt1- ( E + W ")/{h(E+h)qtl} (E=l-2h-y,) which is bounded for EZO. Hence ( ~ 2 + ~is( ybounded ) in D, and in R;. The uniformness in h is obvious. By the same reasoning, we have the second estimate, this time by checking the behavior in the neighborhood of y,= -1+2h. For the third estimate, we notice that the support of ~2+,(y)is the strip l - 2 h 5 y t 5 1 . In this strip, we have E:h(Y) -(Z-Y,)""{(l-y,-h)~"

- U--Y,)~+'}/h

-O(h=fq+'), -

which yields the desired estimate.

The last one is obtained similarly. Q.E.D.

Put V-q(y)=JJT=l+,(y), where (lt(y)=+(y,). Then we have Lemma 6. If A 5 l / p , then it holds that (3.12) where Mh and E , satisfy (3.13) Proof. For simplicity of notation, let us omit the suffix h from L,,,'s, a;,'s and E : ~ ' s .

-..

Case of Scheme 1. Since L,w is a linear combination of L,, L,,w with coefficients cil...tn satisfying C ctl...,,= 1 , it is enough to prove the case where L, consists of a single term L,, We prove this by induction for the L,,. special case (il, -,in)=(1,2, -,n):

-

--

(3.14) 41

+,,LI

- - *

L,w=LI

*.*

L,$,

- - *

+, w+kM ,...,+,

*-*+,,

w+kE, ..., w

T.-T. NAKAGAWA

572

with

llMl...n91* * gnWII=O(l)ll@1 * [IEI...nWII =O(h*+q+')llwII .

--

(knWII

- - .,

(The general case (il, in) is the matter of renaming the indices.) At first, we have, by using (3.10),

4tLiw=L,3~w+kMigiw+kE,w in which

- h) denote g,(~,+h)w(y,+ h) and g,(y,-h)w(y,--h), (Here [$,wl(yt+h)and [g,wl(~, respectively.) Therefore, Lemma 5 together with the boundedness of B,(y) yields that

Assume that

with

The Near-field Finite Diyerence Approximation

573

satisfied:

and

II~i+iWll=~(1)11+iwll . IIEt~II=O(hP+~'l)llw~~ By using the argument that parallels the proof for Scheme 1 , we obtain (3.14).

Q.E.D. Now we are ready to state the convergence theorem. Theorem 2. Assume that u ( t , x) to (2.1) is in BSt1([O,7'1 x R ; . Under the condition 15 lip, v,(t) to (2.7) satisfies the following weighted error estimate if qh s(p 1)- 1 :

+

(3.15)

max IIWqtu,-v)[lL~cn,(t)5 C - h * t f tO.TIh

where v = u ( t , Gy), and C is some constant independent of h f o r a fixed 1.

Proof. We partition R; into 5DA and 5DB such that W = { y I lyi151-2h, i = l , 2, QB=R;\BA ,

-.-,n)

T.-T. NAKAGAWA

574

and write W,{v,(t+k)-v(t+k)} in terms of W,{v,(t)-v(t)} as follows. (3.16) W&,(r+k) - v ( t + k ) }=(L,+kM,)K,{v,(r) - v(t)}+kE,{v,(t)- v(t)}+S, where S,=Si+Sf

with SA, and S: defined as

SL{;K,R(,,h"' S

q

t

Y€QA

7

y € W

'

y € W

K q I v h ( t + k ) - v ( r + k ) } - K q L h { v h ( t ) - v ( r ),} Y e5DB. The Si for y is derived from (2.7) and (3.6), together with (3.12), while Sf for y € D B is obtained by rewriting the trivial equality W,{v,(t+k)v(r+k)}=W,{v,(r+k)-v(r+k)}. The point of proof lies in the estimate of S, and in the treatment of k E , { v , ( t ) - v ( t ) } . We note first that v, is bounded in L2 if 15 l l p by Theorem 1. Since the supports of both v h and v are and obviously v is in L2(Q), it holds that

a,

ll~h(r)--(t)llL~(Q)S C

(3.17) uniformly in r € [0, TI,.

Therefore Sf is estimated as

IIS;IILz(Q) 5 2 c sup IW,(Y)I

~ ~ 5 n . n ~

uniformly in r € [0, TIh. For the estimate of S f , we apply Lemma 4 to (3.7). By choosing W , with q + l Z s ( p + l ) , we make the quantity Qmaxyfs.41-F,(y)R,,, I bounded uniformly in t € [0, TI, and in h. (The evaluatyn') for partial derivatives appearing in I?($) is generally ing point y'=(y,', different from the argument y of -W,(y), and could reach one mesh outside 5DA when y comes on the boundary of W. However, noting that y' is still n o less than one mesh away from the boundary of Q, we can conclude the uniform boundedness of Q.) Consequently we have IlSill =O(hstl), and thus

--

e ,

IIShllLZC0) = O(hs+')+O(h,+1)= O(hS+') uniformly in r E [0, TI,. The second term of the right-hand side of (3.16) is estimated as, with the use of Lemma 6 under (3.17),

After this absorption, Therefore this term can be absorbed into [(S,[[LztQ,. (3.16) yields that

515

The Near-field Finite Diflerence Approximation

The statement of Theorem 2 appears not well suited for application purposes because the quantities there are those defined in the transformed domain Q. Hence we shall rewrite Theorem 2 in terms of variables that are mapped back into RE. Let u, be the approximating solution v h with spatial variables y written in terms of x , that is, u,(t, X)=u,(t, G - l x ) . Then we have

Theorem 2'.

Under the same assumption of Theorem 2 , it holds that

(3.19) Here, CD is the weighting function O ( X ) = ~ : $=( ~x i ) in which $(x,) satisfies

($(x,)=l

for

lx,lil

Proof. The left-hand side of ( 3 . 1 5 ) is written as

where n

@ ( x ) = [ll $ N Y i ) z / a o l , = o - l z 2=1

-

Put ~ ( X , ) = [ ~ ( Y ~ ) ~ / ~ ] , ~ ,The ( ~ -fact I ~ that ~ ~ . GI, is the identity mapping and that a ( y , ) = + ( y , ) = l for Iyi151 implies the first relation of (3.20). Now we fix yi'=a, a < l , which is sufficiently close to I , and let xi'=@ be the transformed coordinate value through x = G y . Then xi where x i 2 @ is given in terms of Yt as x,=p+c

1:

(I-yt)-(p+1)dyt,

yi2a

with a positive constant c, so that we have z-yi= {pc-'(xt-/9)+(1-

a)-p}-l'p

,

x,2@

The case where yt'=a, a > -1, is sufficiently close to

.

-I is quite parallel to

T.-T. NAKAGAWA

5 76

the above. Hence we may write

I-

(3.21)

lyil

-WiI

+T)-'/~

where 7 is some positive constant. a ( y i ) - ( I - lyil)ptl, we have

-

as

Ixilbw

Putting (3.21) into

#(xi) ( I - 1 y , I ) q + I ( I - lyi1 ) (0 + l ) / Z = O( I x i I - ( z q + p + 3 ) / ( 2 p ) )

, $(y,).-(Z-llyil)qtl

as

Ixij +00

and

.

From the condition q z s ( p + l ) - 1, it follows that r = ( 2 q + p + 3 ) / ( 2 p ) > r o . Q.E.D. Imbed D into R ; , and regard uh as the function in ( t , x ) € [ O , TI xD. Then Theorem 2 immediately implies the following.

Corollary 1. Under the assumption of Theorem 2 , it holds that (3.22) Proof. The fact that u(t, x ) = u ( t , x ) and V U ( x ) = lover W C Q yields (3.22) from (3.15). 4.

Application-A Mixed Finite ElementiFinite Difference Scheme for the SoilStructure Interaction Problem

In structural analysis, the finite element method is extensively used because of the capability of modeling complex geometry and material properties. Thus, it is a practical idea to use a finite element scheme in computing the response of structure, while the method presented in the foregoing sections is used in computing the far-field portion of soil up to infinity. The following is a brief description of this concept [ 9 ] [12]. Take a two-dimensional model problem of Fig. 1. The entire domain is partitioned into two zones called Zones A and B. Zone A covers the structure and its near-by portion of soil, and Zone B covers the remaining portion of soil up to infinity, where the two zones overlap one mesh with the other. The overlapping area is called Interface Z A B . In Zone A , the response is computed with the finite element method, usually with the one using four-node quadrilateral isoparametric elements. The discretized equation is, in its general form, as follows.

where [ M I , [ C ] ,and [ K ] are the mass matrix, the damping matrix, and the stiffness matrix, respectively, { V }is the nodal variable vector, and {f}is the

The Near-field Finite Direrence Approximation

aA aB

577

‘A B I

I I

h

Zone A

<

-= ------+

> f-----+a,

Zone B I

I

-00

Fig. I. Two-dimensionalmodel problem of soil-structureinteraction external vector. This equation is integrated with the use of Newmark’s p scheme [lo]. In the entire algorithm, this computational part is called the near-field block. In Zone B , the response is computed with the method presented in this paper. For this purpose, the equation of elasticity is written as a n equivalent hyperbolic system of first order such that *,A,-+ av

at

ax

av A, az

where v is the unknown vector of five components, two of which are the horizontal and vertical velocity fields, and three of which are stress tensor fields; and A , and A , are 5 x 5 coefficient matrices whose elements are elasticity constants (Lame’s constants) and the inverse of the mass density. This computational block is called the far-field block. In I,,, the response in the two zones are related at each time step. This computational block is called the interface block. The task of this block is to supply the boundary values to the near-field block (i.e., those on boundary 8A) as well as to the far-field block (i.e., those on boundary 8s). Since the assumed unknown variables in the two zones are different, a numerical procedure of data conversion such as differentiation and line integration is required to provide adequate boundary data. Fig. 2 shows the flow diagram of the entire algorithm of the mixed finite element/finite difference approach. As a n application of this mixed scheme, we shall present a numerical result on the dynamic response of a soil-structure system where the structure is embedded partially in soil. Fig. 3 shows the goemetry of the structure and

578

T.-T. NAKAGAWA

>I

T ’ + T ~ + K]

I

(FAR-FIELD BLOCK) SCHEME I 1 NI ZONE B I

(INTERFACE BLOCK ( 9 ) ) NUMERICAL LINE INTEGRATION

OF

STRESS V A R I A B L E S TO O B T A I N THE EQUAIVALENT NODAL FORCE ON

aA

(NEAR-FIELD (3

-SCHEME

BLOCK)

N I

ZONE

A

(INTERFACE BLOCK ( A ) ) NUMERICAL

DIFFERENTIATION

DISPLACEMENT V A R I A B L E S O B T A I N THE STRESS ON

OF

TO

38

I N e N +

1; TN+

TI

I

Fig. 2. Flow diagram of mixed finite element/finite difference scheme

Fig. 3. Partially embedded structure with sliplseparation contact against foundation

The Near-feld Finite Diyerence Approximation

519

its finite element meshes. The contact surface of structure is not fixed to soil but may slipiseparate when the exciting force of incident seismic waves is applied. The slip/separation condition is expressed by joint element [3] as follows.

t

4

Fig. 4. Shear stress vs. relative displacement characteristic of the joint element

0.326

EM

Fig. 5 . Displacement field of structure at time 0.5 s Case (a): period T=0.25s Case (b): period T=0.50s.

5 80

T.-T. NAKAGAWA 81. 4

40.7

0.0 -4c. 7

-a!.

4

2. 7 E

.I. 2 % ^^

0.

oc

-1.39 -2.78 81. 4 40.1

c. 0 -40.7

-el. 4 2.78

1.39

0. 00 -1.39 -2.78

29E7.

I i

, I

I

1

1453.

c.

-1

-1493.

"1

-2987.

I

I

I

u

d l l I U U I U

v

2967.

I

1403. 0. -1493. -2987.

1

c

1

-

1

I I

1 ' I

I

2967. 1193. W

0. -1493. -2se1.

I

I

I

I /I

1493. 0. -1493.

-2987.

I

,k/

b l

l

_-I

I

I

I

l

I

l

I

I

l

I

I

I

2987.

I

I l

l \./I

l l

\ /i \/I V

I

!

1

I

\ L \/I

'

" I

I

l

v

\/i

" I

I

1

I/' " I "

\/I

/

kl v b

l

-

.&I l v

h

-.r

k

pc

Fig. 6. Time history at different nodes of structure (The case of period

T=0.50s.)

The Near-field Finite Diyerence Approximation

581

where r is the shear stress, cn is the normal stress, $ is the friction angle, k, and k, are appropriate coefficients, and ys, E,, uv, rv, and yo are the relative displacements in tangential and normal directions, the strengths of joint in normal and tangential directions, and the residual slip, respectively. The relationship of shear stress vs. relative displacement is shown schematically in Fig. 4. In order to numerically satisfy the above nonlinear relationship in the finite element modeling, a sub-iteration procedure is incorporated in the near-field block (Zone A ) for each cycle of time integration with the ,&scheme. In our specific computation, the material constants are: (structure) mass density p=0.12 tis2/m4,Lame’s constants R= 147000 tJm2 and G=625000 tf/m2; (soil) p=0.24 tfs2/m4,1=83300 tr/m2 and G= 125000tf/m2; (joint element) friction angle $=tan-’ 0.2, tensile strength=O, and shear strength= 10 tf/m2. The incident wave is given as the velocity field of a sinusoidal plane shear wave incoming vertically. Numerically, this wave is input as a generalized body force, that is, the inhomogeneous term f ( r , x) of (2.1), on a horizontal mesh line in Zone B at some depth from the ground. Let V, and T denote the amplitude and the oscillatory period of the incoming shear wave, respectively. Figure 5 shows snapshots of the displacement field in Zone A in the cases of V,= 1.0 m/s with (a) T=0.25 s and (b) T= 0.50s. Figure 6 shows the time history of response in case (b) at four nodes A , B , P , and Q (see Fig. 3 for the location) of the structure. The response slips into a steady-state oscillatory motion of period T after a short transient time interval. Apparently, the motion is not a simple harmonic one but contains components of higher frequencies, though the exciting force is a pure sinusoidal wave of single frequency. This is due to the nonlinear effect of slip/separation at the contact surface of structure.

5. Acknowledgment The author would like to thank Mr. Hiroo Shiojiri, Central Research Institute of Electric Power Industry, for general discussion, and especially for providing the numerical results presented in Section 4.

References [ 1 ] P. A. Cundall et al., Solution of infinite dynamic problems by finite modelling in

the time domain, Proc. 2nd Intern. Conf. Appl. Numer. Modelling, Madrid, Spain, 1978. [ 2 1 B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629. [ 3 ] R. E. Goodman and St. John, Analysis in jointed rocks, Chapter 11, “Finite Elements in Geomechanics” (ed. G. Gudehus), John Wiley, 1977. [ 4 ] B. Gustafsson and H.-0. Kreiss, Boundary conditions for time dependent problems with an artificial boundary, J. Comput. Physics, 30 (1979), 333.

582

T.-T. NAKAGAWA

[ 5 ] Z. Koshiba and H. Kumano-go, A family of pseudo-differential operators and a

[6]

[7]

[8]

[ 91

[lo] 1111

[12]

1131 [14] 1151 [16] 1171

stability theorem for the Friedrichs schemes, Proc. Japan Acad. Ser. A Math. Sci., 52 (1976), 000. J. Lysmer and L. A. Drake, A finite element method for seismology, Chapter 6, “Methods in Computational Physics, Vol. 1 l”, Academic Press, London, New York, 1972. J. Lysmer and R. Kuhlemeyer, Finite dynamic model for infinite media, J. Engrg. Mech. Division, ASCE, 95 (1969), No. EM 4, 859. T.-T. Nakagawa, Numerical solution in the near field to linear hyperbolic system with application to an elastic foundation problem, in “Computing Methods in Applied Sciences and Engineering V”, (eds.) R. Glowinski and J. L. Lions, North-Holland, Amsterdam, 1982. T.-T. Nakagawa and H. Shiojiri, A new method of time domain analysis for structure on a semi-infinite foundation, Proc. 8th World Conf. Earthquake Engrg, Vol. 111, Prentice-Hall, New Jersey, 1984, 761. N. M. Newmark, A method of computation for structural dynamics, Proc. ASCE, 85, No. EM3 (1959), 67. H. Shintani and K. Tomoeda, Stability of difference schemes for nonsymmetric linear hyperbolic systems with variable coefficients, Hiroshima Math. J., 7 (1977), 309. H. Shiojiri and T.-T. Nakagawa, A method for time-domain analysis of semiinfinite foundation-structure-water systems, in “Numerical Method in Geomechanics”, (eds.) T. Kawamoto and Y. Ichikawa, A. A. Balkema Publishing Co., Rotterdam, Boston 1985. W. D. Smith. A nonreflecting plane boundary for wave propagation problems, J. Cornput. Physics, 15 (1974), 492. G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506. R. Vaillancourt, On the stability of Friedrichs’ scheme and the modified LaxWendroff scheme, Math. Comp., 24 (1970), 767. G. Waas, Linear two-dimensional analysis of soil dynamics problems in semiinfinite layered media, Ph. D. Dissertation, Univ. California Berkeley, 1972. M. Yarnaguti and T. Nogi, An algebra of pseudo difference schemes and its application, Publ. Res. Inst. Math. Sci. Kyoto Univ., Ser. A, 3 (1967), 151.

Department of Applied Mathematics The Faculty of Science Konan University Higashinada Kobe 658, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 583-596 (1986)

Energy Decay for Nonlinear Wave Equations with Degenerate Dissipative Terms By Mitsuhiro NAKAO Abstract. Decay rates of solutions of the initial-boundary value problem for nonlinear wave equations; utt-uUzZ+u(x,ut)+p(x, u ) = f ( x , 1) u ( x , O)=uo(x) ,

ut(x, O)=ul(x)

on I x R + and ularx~+=O

are derived, where I is a bounded interval in R and u(x, v ) is a function such that

with r > - l , a(x)>O and l / u ( . ) E L p ( I )for some O < p < m . Key words: energy decay, nonlinear wave equation, degenerate dissipative term, energy method

0. Introduction and Result

In the present paper we shall investigate the decay property of solutions of nonlinear wave equations in one space dimension: (0.1)

u , ) + / ~ ( xu,) = f ( x , t )

utt-u,,+o(x,

on I x R +

with the initial-boundary conditions (0.2)

u(x, O)=u,(x)

,

uc(x,O ) = U , ( X )

and

u(x, t ) l a r x B + = O

where I is a bounded interval in R (the real line) and f, uo and u, are given data. When u ( x , v) and p(x, u ) satisfy certain dissipativity conditions, say, d x , ~ c ) ~ c 2 k o l ~ ko>O, t 1 ' + 2r>O , and @(x,u)u>O, it is already known that the energy

Received February 15, 1985.

5 84

M. NAKAO

for a solution u decays at the rate ( l + r ) - z / r (exponentially if r=O) as t - + q provided thatf(r) tends to 0 rapidly (cf. [ 5 ] . For more general or related results see Nakao [7, 81, Yamada [13] and Haraux [2].) Quite recently, in [9] we discussed the decay rate of solutions of linear wave equations (possibly higher dimensional) with u ( x , u,)=a(x)u, and P ( x , u)= 0 under the weaker assumption that a ( x ) 2 0 and l/a E L p ( Z ) for some 0 < p < 00. There we proved that E(u(t))5 C,( 1 t ) - ' ~with some constant C,depending on the initial data and k , where k denotes a certain index of the regularity of the solution u. Related results can be found in Russel [ l l ] where the assumption on a ( x ) is implicit and different from ours. See also Iwasaki [3] and Dafermos [l], where it was proved by dynamical method that lim,,,E(u(t))=O if ( u o , ul)€fZ1(Z) x L 2 ( Z )and a ( x ) > O on some open set in Z. The object of this paper is to extend the result of [9] to nonlinear equaUnfortions of the form (0.1) with u(x, u,) like u(x)Iu,jru,, a ( x ) > O , r > - l . tunately our technique is applicable only to equations of spatial dimension 1. For higher dimensional equations further devices and additional assumptions on u ( x , u,) and p ( x , u ) will be required. Here we state the precise hypotheses on u and 1.

+

A,. u ( x , v) is measurable in x E Z for each v E R, continuously differentiable in u € R-{O} for a.e. X E Z and satisfies the conditions: (0.3)

u ( x , ~ ) v ~ a ( x ) j v 1 ~with +~

r > -1 ,

where a ( x ) is a nonnegative measurable function on Z such that (0.4)

a E L"(Z)

and

a-l( .) E Lp(Z)

l/a)

where C( .) is a continuous function on (0, m). A,. !(x, u ) is measurable in x E Z for each u E R, continuously differential in u E R for a.e. x E Z and satisfies the conditions: ( i ) P ( x , u)u>k, $ , " / ( X , ?)d?>O for some k,>O, and ( i i ) for any M>O there exists C ( M )> O such that

A Nonlinear Wave Equation

5 85

Concerning the data (uo,ul, f )we assume A,.

(uo,u , ) € f i 1 n n H , xand ~ , f f W:;:(R+;L2(Z)).

Hereafter we denote by C, and C, various positive constants depending + ] ~ u l ~ ~ ~Also, l , by C we denote on ~ ~ ~ o l l ~llulllL2 l + and ~ ~ u o ~ ~ H ~respectively. generic positive constants. We set

Il~ll,=(\~

l ~ l ~ d x ) ~ ’O ~< , q<w,

and

llullm=esssup ZEI lu(x)l

for a measurable function u on I . For simplicity we write /I I] for 11 /I2. For a nonnegative integer k and l l q < 00 we denote by W:;:(R+;X ) the set of X-valued measurable functions u on R+ satisfying ~ , ( \ ~ ~ ! ~ u ( t ) ~ ~ ~ dfor t ) any ” * O ~

(with usual modification if q=

a). For

convenience we set

and

The following existence theorem is standard and the proof is omitted (cf. Lions & Strauss [4],Nakao [6] etc.).

Theorem 0. Under the hypotheses A,, A, and A, the problem (O.l), (0.2) has a unique solution u such that UE

w : ~ ( R +L2)n ; w : ~ ( R +~ ;, ) n L : , , ( ~ ~ ; ~ , n.i i , )

In what follows we are concerned with the decay property of the solutions in Theorem 0. Now. we shall state our results.

Theorem 1. Assume that r>O and p>2/r. (0.8)

Then, we have

E(u(t))lC0(1+t)-2/‘

provided that

so(?)<03

and 6 0 ( t ) ( r + z ) / ( ~ t 1 ) + 6 ( t ) ~ t 2 = ~ ( t - 1as- 2t-w. /~)

Theorem 2. Assume that - 1 < r I 2/p and B(x, u)r 0. (0.9)

provided that

E(u(t))lC,(l+t)-~o

Then,

M.NAKAO

5 86

where we set

if P 2 -W+ l)/r(r+3) if r
4p/(2+pr) -2(r+l)/r

Then, we have: (i) if

Theorem 3. Assume that r 2 0 and 2 / r > p 2 2 / ( r + 2 ) . p=2/(r+2), rhen (0.10)

.

E(u(t))lC,(log(l+t))-4p/(Z+pr)

provided that

and

sow( r + 2 ) / ( y i+t)(iog ( 1+ t))1+4/(7+~) as t+w ,

+d(r)(log (l+t))2/('ts)=o(1)

and (ii) i f 2 / r > p >2/(r+2), then

E(*(t))< C,(1 + t ) - 2

(0.11)

(P~fZP-2)/(P~+2)

provided that 6,( t ) ( r t 2 ) / ( ? + I ) ( l + t ) l t ' 2 - P 7 ) / Z p

+S(t)2=~((l+t)-4p/(2tpr))

as t-xc

and 6 , < w

.

Now, we make a further assumption on P ( x , u ) :

a

A,',

-@(x,

Iau

U)

I


a.e.

~ € 1 a, > O .

Theorem 4. Let r 2 0 and 2/r>p>2/(r+2). Under the additional assumption and

A,' we have: ( i ) g a > a , , 6,<

60(t)(rt2)/(rtl) + ~ ( ~ ) ( p r t 4 p t 2 ) / 2 p = O ( ~ - ( p r t 4 p t Z ) / ( p r t Z ))

as

f+w,

then E(u(t))Ic1(1+t)-4p/(p7+2)

(0.12)

and (ii) ifO
a,( then

t )( 7 t 2 )

/(1

t1,ts, +6( t ) ( p 7t 4 p t2) / 2 p =o(t - (1-8,)

(pr t 4 9 t 2 ) / ( p v t 2 ' )

as t+-

A Nonlinear Wave Equation

+

(0.13) for any

E(u( t ) )IC,(E ) ( 1 t ) E

587

+<

> 0 , where we set in the above a,- (2-pr)-I{2pr- (pr+2p-2)(pr+2)/2p} = (pr+2)/2p - 1

a, = ( p r+2p -2)/{2pr- ( 2-pr)a} O,,= (2 -pr)( 1 - aao-a,)/2p

and

.

The simplest example of a sufficiently smooth function a ( x ) that vanishes at xoEZ is a ( x ) = ( x - x , ) 2 . As this example shows, it is desirable that the hypothesis on a ( x ) includes the case 0 < p < 1/2. Unfortunately, however, the assumptions of Theorem 4 exclude such case if O
and

with some k,, k,>O, a , r 2 0 and O1(2-pr)/(r+2). Moreover suppose that 0 1

+co and

t+w

1.

6,<m ,

and a l < m ,

Derivation of an Energy Inequality and the Proof of Theorem 1.

Using the energy method we shall prove here the following energy inequality.

M. NAKAO

588

Proposition 1. Let - l < r < 2 / p and let u(x, t ) be the solution of (0.1)-(0.2) in Theorem 0 . Then we have

where we set

D(t)' + 2 = C{E(u(t ) )-E( u( t + l ) ) }+ C6,( t )

(? + 2 ) / ( 7

+l)

.

Proof. Multiplying the equation (0.1) by u, and using the condition (0.2) we have (cf. Strauss [12])

and

(1.3)

Now,

(1.4)

where we have used the assumption 2 - p r 2 0 . have

=A(t)2

Thus, by ( 1 . 3 ) and (1.4) we

.

From (1.5) there exist t l € [ t , r+1/4] and t z € [ t + 3 / 4 , t + l l such that llut(t,)III 2A(t).

5 89

A Nonlinear Wave Equation

Next, multiplying the equation (0.1) by u and integrating over we have for F(u)=llu,112+$I P ( x , p)dpdx:

$:

[ f t , f,] X

where we have used the assumption (0.5) and PoincarC's inequality. above inequality together with (1.5) implies immediately (1.6)

1:;

I

The

E(u(s))ds
+D(f)'+' sup

./E(uo)} .

t<S
Hence, there exists f * E [Il, t 2 ] such that E(u(t*))is bounded by the right-hand side of (1.6) (with larger C). Therefore, by the energy identity (cf. (1.2)) we have

I C{ sup dE(u(s))A( t ) + A ( f ) 2 +

s(t)2

t
sup d \ / E o ) } + D ( r ) r + Z

+o(t)Ttl

tiSlt+l

and hence (1.7)

sup E(u(s))I C { D ( f ) = l r + 2 ) ' ( p +sup l)

IIut(s)]I:-p7)'(p+1)

t<Sit+l

t<S
+S(p

+D(t)2(7t1) +D ( W 2 }.

Moreover, with the aid of the well-known inequality

Ilut(4 1 1 - l Cllu,(s) lll'zll~cz(~)1Y2 we can obtain easily the desired inequality (1.1).

Proof of Theorem 1. We assume 2 / p < r . C, will denote various constants depending on E(u(0)). In this case we have, instead of (1.4),

M. NAKAO

590

Using this we can derive, quite similarly to the proof of Proposition 1, the inequality

+

sup E(u(s))< C{D(t ) 2 + D(t)Z('+l) D(t)'+2+8(t)2} .

(1.9)

t_cSlt+l

It is easily seen from (1.9) that E(u(r))is bounded on R + if s,(t) and 6 ( t ) are

so (cf. [ 5 , 71). Therefore we obtain from (1.9) that

or

Now, applying the following lemma to this difference inequality the proof of Theorem 1 is completed.

Lemma 1 ([7]). Let $(t) be a nonnegative function on R t satisfying

with aTO, 0 1 1 and afunction g(t)>O. Then, $(t) has the following decay property: ( i ) i f a > O , 0 = 1 and Iimt-.- (logt)lt'/ag(t)=O, theiz$(t) O , 19< 1 and Iimt-- t(1--8)(1t1'U) g(t)=O, then $(t)< Co'(l+t)-(l-e)/U and (iii) i f a=O, 0 < 1 and g(t)O, where C,' denotes positive constants depending on C,, $(O) and other known constants. 2.

Proofs of Theorems 2 and 3

Starting from the inequality (1.1) in Proposition 1 we shall prove Theorems 2 and 3. First we note that the inequality (1.2) and the assumption on s,(t) imply

(2.1)

E(u(t))+lt O

1

36 0 ( i ) ( 7 f 2 ) ' ( 7 i 1 )

a ( x , u,)u,dxds<E(u(O))+C

I

Next, we consider the differentiated equation

i=l

591

A Nonlinear Wave Equation

where we set U=u, and @(x, u)=(a/au)B(x, u). When - 1< r < 0, d ( x , u,) may have a singularity at u,=O and the argument below is somewhat formal. However it can be made rigorous through appropriate approximate solutions (cf. [lo]). We utilize (2.2) to estimate ~ ~ u t z ( r ) ~ ~ . Proof of Theorem 2 . Assume that p(x, u)-0. from (2.2)

Since d ( x , u , ) 2 0 we have

which implies

Hence we have l ~ u t z ( r ) ~ ~03 ~ Cand 1 < consequently, by (1.1) and (2.1),

+

+

sup E(u(s))IC1{D(r)4p(r+2)’(4*+~~t2) D(j)2(1t1) D ( p + ~ 3 ( t ) ~ ) t_<S_ct+l

I Cl{D(t)v+6(t)2)

or

where we set p=min ( 4 p ( r + 2 ) / ( 4 p + p r + 2 ) , 2 ( r + l ) , r + 2 )

.

It is easily checked that 4p(r+2)/(4p+prf2) ’I= {2(r+ 1)

if ~ > - 2 ( r + l ) / r ( r + 3 ) if r
.

Now, applying Lemma 1 with 8=0, a = ( r + 2 ) / p - l (>O) to (2.3) we obtain (0.9). Proof of Theorem 3 . At this time we assume that r>O and 2 / r > p 2 2 / ( r + 2 ) (we set 2/r=03 if r=O). Recall that E(u(t))is bounded ( ( 2 . 1 ) ) . Then, by (2.2) we have

M. NAKAO

592

(2.4)

E(u,(r))IE(uc(O))+

s:

<E(ut(O))+C,

u)l Iu,I lu,,Idxds+

IP'(X,

s:

1:

IStu,tldxds

(l+llfcll)./Eo)ds

which together with the boundedness of

s:"

Ilf,(s)llds implies

E(u,(t))lC,(1+t)2.

(2.5)

Therefore, noting that 4 p ( r + 2 ) / ( 4 p + p r + 2 ) < r + 2 < 2 ( r + l ) , (1.1) the difference inequality for E ( u ( t ) ) : (2.6)

sup E(u(s))'+( 2

we can derive from

P7)l4p

t<Slt+l


+C ; i ( t ) ' 4 P t P l

+ 1)) + C6,( t )

(7t2)

}

+I)

f2) /2P

which yields, by Lemma 1, the estimates (0.10)and (0.11).

3.

Proof of Theorem 4

Here we assume again 2/r>p>2/(r+2), r 2 0 , and let us derive a sharper estimate than (0.11) under the more restrictive condition A,' on (a/au)/?(x,u). Due to the condition A,' we have, instead of (2.4),

1; 1

E(uc(t))<E(uc(O))+C IE(u,(O))

+c

st

lUla IUCI

Irrttldxds+]t

(E(u(s))"t a ) / 2

Ilfcll

IlUCtlldS

+ llS~ll~ll~c,lld~

and, using the estimate ( O . l l ) , (3.1)

+

E(u,(t))<E(ut(0)) Cl

with ro=(pr+2p-2)l(pr+2).

S:

+

{(I+s)-'o('cn) IlftW ll}ll4t(4lids

Since

$0" Ilf,(s)llds<

00,

we have from (3.1)

< C{(l+t)'-('+=)ro + l ) .

Combining (1.1) with (3.2) we can obtain, instead of (2.6), (3.3)

sup

E(u(s))'f(pr"2)/4p<

tSa
C'{l +(1+ t ) ' 2 - p "

('-oro--r0)'2P

1

A Nonlinear Wave Equation

593

and hence, by Lemma 1, E(u(t))I C1(1

(3.4)

+

t)-2'1

with

With ro replaced by r1 we can repeat the above argument, and consequently, by repeating this procedure indefinitely, we find (3.5)

E(u(f))
k = l , 2 , 3,

* - -

under the conditions (3.6)

{so(()( 7 t 2 ) / ( l t l ) ( 1 + ()@k +8( () (4P f-P7 + 2 ) /2P}( 1 +l ) " k - b O

as t+m where we set Bk=max {O, ( 2 - p r ) ( l - a ~ ~ - ~ - ~ ~ - ~ and ) / 2 pv}k = ( 1 - O k ) x ( 1 + 4 p / ( p r + 2 ) ) , and {T,};=~ are determined inductively by the relation

Since r1>ro, {r k }is nondecreasing and 1imk-- rk exists. In fact, we find that if a > a,- (2pzr+4-pzrr"-4p)/2p(2-pr), then r k = 2 p / ( p r + 2 ) for sufficiently large k and that if a l a , , then limk..- rk=ao/2-(pr+2p-2)/{2pr-(2-pr)a} (>O). Also we note that the conditions (3.16) are satisfied for all k under our assumptions on &,(() and s(()in Theorem 4 . The proof of Theorem 4 is now complete. 4.

Proof of Theorem 5

Here we assume A," and consider the case: 0 1 r 1 2 and 2 r / ( r 2 + 2 r + 4 ) < p . The proof of Theorem 5 is done in a parallel way to that of Theorem 4 . First we observe (cf. ( 2 . 4 ) , ( 3 . 1 ) ) that

(4.1)

Now, we see

M. NAKAO

594

and moreover

(4.3)

where we have used the assumptions {(r+2)&2}rT -p and u E L", and the fact:

From (4.1)-(4.3) and the boundedness of E(u(t))it follows that

Substituting (4.4) into (1.1) we obtain as is usual

(4.5)

sup E(u(s))'+ (2 t<8
< C,(1+t)ro{E(u ( t ) )-E(u(t+

+

1)) C60(f)(7+2)/(r + l )}

+C q t ) ( 4 P + P 7 + 2 ) / 2 P where we set po=r(2--pr)/2p(r+2)

(4.6)

< 1.

E ( u ( t ) ) I C , ( l + t ) - ' ~ ~ ' - ~ o ~. " ~ ' + ~ ~

Using (4.6) we have from (4.3) that

with

Thus Lemma 1 yields

A Nonlinear Wave Equation

595

a n d it follows f r om (4.2) that

E(u,(O)I C l ( 1+ t P

(4.7)

which is sharper t h a n ( 4 . 4 ) . By this a n d ( 1 . 1 ) w e obtain a sharper estimate than ( 4 . 6 ) . Repeated use of this procedure yields

E( u( t ) )< C,(1

(4.8)

+t )

-4p ( I - W ’ ( p r + Z )

under the conditions on 6,( t ) and 6( t ) :

where p,., T~ a r e determined inductively by

P,=(2-Pr)T,/4P

3

It is not difficult t o check that if r ( p r + 2 ) / 2 p ( r + 2 ) < a , the n pk=O for sufficiently large k a n d t h at if O < a < r ( p r + 2 ) / 2 p ( r + 2 ) , the n lim p k = b o = k-oo

2-pr 2p(r + 2 )

.r(pr+2)-2p(r+2)a p r + 2 - (2 - ~ r ) a

T h e proof of T h e o r em 5 is now finished.

References [ 1 ] C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in

[2] [3]

[4] [ 5]

Nonlinear Evolution Equations, ed. M. G. Crandall, Academic Press, New York, 1978. A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture Notes in Math. No. 841, Springer-Verlag, 1981. N. Iwasaki, Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains, Publ. Res. Inst. Math. Sci., Kyoto, Univ., 5 (1969), 193-218. J. L. Lions and W. A. Straws, Some non-linear evolution equations, Bull. SOC. Math. France, 93 (1965), 43-96. M. Nakao, Convergence of solutions of the wave equation with a nonlinear dis-

M. NAKAO

596

[6] [7] [8]

19] [lo] [ll] [12] [13]

sipative term to the steady state, Mem. Fac. Sci. Kyushu Univ., 30 (1976), 257265. M. Nakao, Decay of solutions of some nonlinear wave equations in one space dimension, Funkcial. Ekvac., 20 (1977), 223-236. -, A difference inequality and its applications to nonlinear evolution equations, J. Math. SOC.Japan, 30 (1978), 747-762. -, Asymptotic stability for some nonlinear evolution equations of second order with unbounded dissipative terms, J. Differential Equations, 30 (1978), 54-63. -, Energy decay for the wave equation with a degenerate dissipative term, Proc. Royal SOC.Edinburgh, lOOA (1985), 19-27. -, Periodic solutions and decay for some nonlinear wave equations with sublinear dissipative terms, to appear in Nonlinear Analysis, T.M.A. D. L. Russell, Decay rates for weakly damped systems in Hilbert space, J. Differential Equations, 19 (1975), 334-370. W. A. Straws, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. Y.Yamada, On the decay of solutions for some nonlinear evolution equations of second order, Nagoya J. Math., 73 (1979), 69-98.

Department of Mathematics College of General Education Kyushu University Fukuoka 810, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 597-605 (1986)

On the Stochastic Integral Equation of Fredholm Type By Shigeyoshi OGAWA Abstract. Given a pair ( Z ,{&I) of a random function Z ( f )( r 2 0 ) and an orthonormal basis {&I in L2(0,l), we are concerned with the stochastic integral equation of Fredholm type as follows, x(t)=f(t)+a

1:

L(t,s, w)x(s)ds+iS

1:

KO, s, o~)x(.++Z(s)

,

where the term J d # Z stands for the stochastic integral of noncausal type with respect to the pair ( Z , &I) and L, K and f are some random functions. In this article, we will show some results on the question of existence and uniqueness of solutions and apply them to the boundary value problems of stochastic differential equations containing (d/df)Z(t)as a coefficient. Furthermore, we will discuss the approximation of the solution of the integral equation by those of ordinary random differential equations. Key words: stochastic integral of noncausal type, Fredholm’s alternative theorem, boundary value problems of stochastic differential equations

S 1. Given a random function Z ( t , o)( O I t I 1, w € 52) defined on a probability space (52, .F P), and a n orthonormal basis {$%} in the Hilbert space L2(0,l), we are to study the stochastic integral equation of Fredholm type as follows: (1)

X(t)=S(t, w ) + a

1:

1:

L(t, s, o)X(s)ds+ K(t, s, ~ ) X ( s ) d + Z ,( s )

where f(t, o),L(t, s, w ) and K ( t , s, o) ( t , s € [0, 11) a r e some random functions and the last term, $ d+Z(s)stands for the stochastic integral of noncausal type with respect to the pair (2,{+n}) (see Definition 1 in the next paragraph). Our interest on this subject comes from the expectation that the integral equation of this type may give us a natural extension of the boundary value problem for stochastic differential equations: For example, let us consider the problem of analyzing the motion of a randomly perturbed harmonic oscillator with two sides fixed; Received May 2, 1985.

598

S . OGAWA

(x(o)=c,, x(I)=c, (el, c,;

constants) ,

where k ( w ) is a real random variable and h(r, w ) is a random function. Now, let K ( t , s, w ) ((1, s, w ) E [0, 1I2x9)be the Green function corresponding to the operator d2/dt2+k2(w),associated with the boundary condition cited in (2). Then following the usual method in classical calculus, we may rewrite the problem ( 2 ) into the stochastic integral equation:

(3)

K ( t , s, w)lh(s,o)ds+

K ( t , s, w)X(s)dZ(s).

The advantage of this interpretation becomes apparent when we take for Z ( t ,w ) such random function as the Brownian motion, whose sample functions are not differentiable. In such cases, the differential equation in ( 2 ) becomes nothing but a symbolic expression, while the integral equation (3) still maintains a concrete meaning, provided that the stochastic integral is understood in the sense of noncausal integration (cf., [l]). In the present article, we will show some results on the question of existence and uniqueness of S-solutions (a specified class of solutions). We will also refer to the relation between the integral equation of this type and the boundary value problem for the stochastic differential equation containing the quantity (d/dt)Zas a coefficient. Throughout the discussions, we understand by the random functions f ( t , W ) such measurable functions that satisfy the condition, f"!: i f ( t , w ) 1 2 d f < m ] = 1 and we denote by H the totality of such elements. When there is n o possibility of causing confusion, we will often omit the random parameter in the notations of random quantities.

S 2. We assume that the fundamental pair ( Z , {$n}) is chosen in such a way that Z(O)=O and that for each n the integral Z($,)= 4,(s)dZ(s) has a definite meaning. This posed, we introduce the sequence of random functions { Z i ( t ) } defined by the formula; Z $ ( t ) = C L Z($,) 5," $,(s)ds.

s:

Definition 1. ( i ) We say that a random function f(t, w ) is integrable , with respect to the pair ( Z , I$,}), provided that the on [a, b] ( ~ 1 011) limndms:f(t, w ) d Z $ ( t ) exists in probability. The limit is denoted as s:f(f,6J)ddYr). ( i i ) Moreover, if for each t the function is integrable on [0, t ] and the sequence $ i f ( s ,w)dZt(s)converges in probability to the function f(s, w)d+Z(s) in the L2-sense as n+m, then we say that f ( t , w ) is strongly integrable and denote by S the totality of such random functions.

s,"

Definition 2. A random function X ( t , w ) is called the S-solution of the equation ( 1 ) if it satisfies the equation with probability one and belongs to the class S.

Stochastic Integral Equation of Fredholm Type

599

Here are some notations and terminologies that are of frequent use in the discussions ; (a) For a random function f ( t , w ) and a random kernel H(r, s, w ) , we set, f ( t ,w ) = f ( u , o)d+Z(u)and R(t,s, w ) = H(u, s, w)d+Z(u). (b) For a kernel H ( t , s, w ) such that, P[$: IH(t, s, w)I2drds<m ] = l , we use the same symbol H to denote the linear operator on H defined by; ( H x ) ( t ) = H ( t , s, o ) x ( s ) d s ( x ( - )E H). (c) A random kernel H ( t , s, W ) and the corresponding linear operator H are said to be of S-class if ( H x ) ( . )E S for any x( .) E H, and the totality of such linear operators is denoted by L ( S ) . As stated in paragraph 1, we are to study in this article the question of existence and uniqueness of S-solutions of the equation (1). For this aim, we will pose the following two assumptions on the regularities of the pair ( Z , {q$,}) and the random quantities L , K and f.

$:

$: $:

$:

( H , 1 ) The sequence { Z t (-)} converges in probability to Z ( as n+m and lim Z ? ( l ) = Z ( l )also in probability.

a)

in the L2-sense

n-m

( H , 2 ) ( i ) The functions f ( r ) and K ( t , 1 ) belong to the class S. ( i i ) The kernels K , (=(a/as)K(r,s, 0)) and L are sufficiently regular in the sense that they belong to the class L ( S ) and that the following equalities of Fubini type; ( L x ) - (t ) = ( L x ) ( t ) ,( K , x ) - ( t )= ( X , x ) ( t )hold for any x ( - ) € H . Moreover, E(1, a ) E H. a,

Remark 1.

If we take the real Brownian motion for Z ( . ) and the system then the assumption ( H , 1) is satisfied. In this case, we set; Z f ( t ) = C l r i s Z(ezrrLk.) n ezrrrk8ds. We say that a kernel H(r,s) is piecewise differentiable in t if it is represented in the form; H ( t , s)= Cfzi llr,(s),l.~+l(S))(t)Hl(t, s), where {rt(s): i=O, 1, . . . , p } aresomesmoothfunctionssuch thatO=rO(s)
of functions {ezzznc: n E Z } for the basis

{$%},

$:

$:

Proposition 1. I f a random kernel H ( t , s, w ) is such that almost all sample functions are piecewise di'erentiable in t , then the kernel satisfies the condition ( H , 2 ) , namely; H E L ( S ) and ( H x ) - ( r ) = ( f f x ) (for t ) any x(-) E H . Proof.

By integration by parts, we obtain,

S. OGAWA

600

Hence, by virtue of the assumption ( H , l), we obtain; s-lim n-a,

s:

7i+1(8)At

(Hx)(u)dZ$(u)= r((a)At

i=O

Remark 2. It is worthwhile to notice that the derivative K J t , s, o)of the Green function K(r, s, w ) of the usual Strum-Liouville problem, corresponding to the random operator LP(w)=(d/dt)(p(t,o)d/dt)+q(t,o) ( p ( t ,o) is a smooth random function), is piecewise differentiable in t for almost all o. Therefore, all results obtained under the assumption ( H , 2 ) can apply to the case of such boundary value problems.

S 3.

Results We begin with the case a=O in (l), that is;

(4)

X(t)=f(t)+((KX))(t)

s:

where ( ( K X ) ) ( t ) = K(t, s, w)X(s)d+Z(s)

Proposition 2. Besides the assumptions ( H , 1) and ( H , 2 ) , we suppose that P [ z (1, 1)# 11= 1. Then, there is a one-to-one correspondence between the S-solution x ( . ) of the equation (4) and the H-solution y ( . ) of the following random integral equation;

where

(ii)

and

Proof. Let x ( - ) be a n S-solution of (4),then by integration by parts we find that;

(7)

x ( t ) =f(t)+K(t, l ) - W ) - ( K A W

.

Taking the stochastic integration on both sides of (7), we obtain the following

Stochastic Integral Equation of Fredholm Type

60 1

relation by virtue of the assumption ( H , 2 ) .

from this we get the relation, %(l)=(l/(k(l, 1)- l)){(k$)(l)-f(l)}. Substituting this into (S), we see that the y(t)=%(t)satisfies the equation (5). It is obvious from (7) that the correspondence x( -)+y( - ) (=%( -)) is one-to-one. Conversely, given a n H-solution y(.) of ( 3 , we put (9)

Then by ( H , 2 ) we see that x(-) E S a n d that; n(t)=(%zrf)-(t)+(oy)(t)=(Mf)(t)+ ( G y)(t )=y(t ),hence by (9) we get the relation

Following the same argument given above, we see from (10) that x( - ) satisfies the equation (7) and so the equation (4) since x(.)€S. That the correspondence y( .)+x( a ) (given by (9)) is one-to-one follows immediately from the fact, % ( . ) = y ( . ) . The random equation (5) being a family of ordinary integral equations, parametrized by w , we can apply to this the classical theory of integral equations. I n the present case the kernel G(t, s, w ) is of Hilbert-Schmidt type with probability one by virtue of the assumptions ( H , 1) and ( H , 2). Therefore, by the well-known alternative theorem, we see that the necessary and sufficient condition for the equation (5) to have a unique solution is that the homogeneous equation, y(t)=(Gy)(t)does not have a nontrivial solution. Thus, taking Proposition 2 into account, we have obtained the

Theorem. The stochastic integral equation (4)has one and only one S-solution, ifl the homogeneous equation, x ( t ) = ( ( K x ) ) ( t )does not have a nontrivial Ssolution. Now let us study the general case, (1). We notice that the random operator G(w), determined by the kernel C ( t , s, o),being compact for almost all w , the set S J w ; G ) of spectrums of G is at most countable. Therefore the operator ( a l - G ) ( w ) ( I is the identity operator) is invertible for every real constant a , except at most countable cases, with probability one. For example, let &(a)}be the set of absolute values of all elements in S J w ; G ) , arranged in decreasing order and put Z(G)=UZ=l { a € R: P[lal = l , ( w ) ] >O}. Then the set B(G) is at most countable and we see that for every real a Z(G), the operator ( a I - G ) ( o ) is invertible with probability one. Keeping this in mind, we introduce the last assumption on the kernel K ( t , s, w ) ; ( H , 3)

f"k(1,l ) f l ] = l

and

165 S,(w; G ) (P-a.s.)

.

S . OGAWA

602

Proposition 3. Under the assumptions ( H , l ) , ( H , 2) and ( H , 3), the stochastic integral equation (1) has a unique S-solution for every real a with at most countable exceptions. Proof. The assumption ( H , 3) implies that a n S-solution x(.) of ( l ) , if it exists, also satisfies the following random integral equation;

where

I:

( i i ) ( B x ) ( t ) = B ( t , s, w)x(s)ds

and

B ( t , s, w ) = U t , s, o ) + K ( t , l ) { ( z - G ) - 1 ( 2 w - , -{Ks(Z-G)-'('2XL(*, S , w ) ) - } ( t ).

3,

w))-}(l)

Since the kernel B ( t , s , w ) is of Hilbert-Schmidt type for almost all w , the operator B(w) is compact for almost all w . Therefore, as we have seen above, for almost every real constant a ( f 0) (with at most countable number of exceptions) the operator (Z-aB)(w) is invertible with probability one and in such case, the equation (11) has a unique solution which must belong to the class S . Conversely, we can easily verify that every solution of (11) also solves the equation (1).

As a n application of the results, we consider a simple example as follows; Example. Consider the equation,

where W(t,w ) is the real Brownian motion, a ( t ) is a continuous function and K ( t , s, w ) is the Green function corresponding to the differential operator Z ( w ) with appropriate boundary conditions, such as; x(O)=constant and x( 1 ) = 0 . In this situation, the equation (13) can be viewed as a n alternative expression of the following Sturm-Liouville problem (see the discussions in the next paragraph) :

(13)'

d l

[ - P ( o ) - x Z ( t ) x ( t ) = h ( t ,w ) ,

x(O)=const. and x(l)=O ,

where (d/dt)Z(t)=aa(t)+p(d/dt)W(r)and h(r) is such that, f ( t ) = ( K h ) ( t ) . The equation (13) corresponds to a special case of the boundary velue problem ( 2 ) where, c,=O and (d/dt)Z(t)=(d/dt)W(r)+a(t).In this case the things become extremely simple, for we have; E ( 1 , 1 ) = 0 , g ( t , s, w ) = -K,(t, s, w ) and (?If)(t)=

603

Stochastic Integral Equation of Fredholm Type

f(t), therefore the assumption ( H , 3) is satisfied for almost all p. Thus, by Proposition 2, we find that for almost all pairs of constants ( a , p ) the equa-

tion (13) has a unique S-solution.

S 4.

A Comment on the Approximation of the Equation (1)

In this paragraph, we are to examine the legitimacy of the stochastic equation (1) as being a n alternative expression for a formal stochastic boundary value problem such as (2). Of course the answer must depend on the meaning that we give to those formal expressions. One natural way for this may be to understand them as being a limit case of approximative sequence of ordinary equations, for example;

(x(O)=c,

x(l)=c2 .

,

We notice that for the problem (2)’ there corresponds a n integral equation of the following form,

+

x ( d =f(4 a ( W ( t ) + ((W),(d

( 1 )’

((W),(t)=

1

\:

K ( t , s, w)x(s)dZ?(s).

Since Z, has a differentiable sample function with probability one, the relation between the problem (2)’ and the integral equation (1)’ is clear. Hence for the legitimacy of (1) as being a n alternative expression for the stochastic boundary value problems, it is desirable that the solution of (1) becomes in some sense a limit of the sequence { x ~of } solutions of the equations (1)’. Fortunately this demand is satisfied in the framework of our theory as we shall see in what follows. For simplicity of discussion, we replace the assumptions ( H , 1) and ( H , 2) by slightly more restrictive ones;

( H , 1)’

s-limZi(.)=Z(.) n-m

( H , 2)’

(P-as.)

and

l i m Z t ( l ) = Z ( l ) (P-a.s.) n-m

Almost all sample functions of L ( f ,s,w ) , K J t , s, w ) are piecewise differentiable in t and f ( t , w ) is such that f(s, w)dZ$(s)=f(., w ) (P-a.s.)

.

n-m

Remark 3. ( i ) The assumption ( H , 1)’ is satisfied in the case that the Brownian motion is chosen for the process Z .

S . OGAWA

604

( i i ) In order to denote the integral with respect to Zg, we use the notation, (Z,H)(r, s, o)= H(u, s, o)dZ?(u). We notice that if a kernel H ( t , s) is piecewise differentiable in t , then for any x( € H we have, s-limn+w{Z,(Hx)). (.)=(l?jc)(.) (P-a.s.) by virtue of the assumption ( H , 1)'.

$:

a )

Let N,(w) be a n N-valued r.v. determined by the formula,

then we have, P[(Z,K)(l, l ) f l , Vn>N,(o)]=l since (ZnK)(l,l)=k(l,1) (by ( H , 1)'). On the other hand, the operator (I-G)(w) is invertible with probability one by virtue of the assumption ( H , 3). Thus, taking Remark 3 (ii) into account, we see the existence of a n N-valued r.v. N 2 ( o ) such that; P[1 g S,(w; GJ, Vn>N,(o)]= 1, G , is the random operator corresponding to the kernel (Z,,g)(r, s, 0).(For example, it may suffice to put;

where [ ] A [means ] the operator norm of A as a n operator from L2 into itself.) Therefore, following a n argument similar to that in the proof of Proposition 3, we confirm that for a sufficiently large n ( > max ( N , , N , ) ) the equation (1)' can be rewritten into the following form; x ( t ) = ( C e , f ) ( t ) + a ( B , x ) ( r ) (n>max W1,Nz)) ,

(15) where

).K(f, (16) ( i 1 ( ~ ~ f ) ( r ) = f ( r ) + { ( Z - G ~ ) - ' ( z ~ ( ~ f ) ) } ( l1) - {K*(Z-G=)-l(z~(~f))}(r) 9

1: d=W,

( i i ) (B,x)(t)= B J t , s,

Bn(t,s, o)x(s)ds

s, w ) + K ( t , U U - G J 1 ( Z n ( W - , s, 4 ) ) W -{K,(Z-GG,)-'(Z,(21L(., s, o)))}(t) f

It is important to notice that the kernel B , is of Hilbert-Schmidt type for almost all o and that, limn-IB,-Blzdsdf=O (P-a.s.). Now let a , be such that (a,)-l g Z(B),then the operator (I-a,B) is invertible with probability one and for this value of a the equation (1) has a unique S-solution which we denote by x"(.). This given, we set

$:$:

1:

N,(o)=max i n ; l a o l { \ ~ l ~ , - ~ l ~ s 1/2 d tl~(z-aoB)-l/l>l} }

then as in the case of the operator G,, we see that;

Stochastic Integral Equation of Fredholrn Type

P[(a,)-' E SJw ;B A

Vn

605

> N,(o)l= 1 ,

hence for a sufficiently large n ( > m a x ( N l , N , , N 8 ) ) the equation (1)' can be solved uniquely for each o and we denote the solution by x,. If we put, X J t , ~ ) = l ~ ~ , ~ ~ , ( ~o) ) -(N,=max x,(t, ( N l ,N , , N , ) ) then it is not difficult to see that, s-limndSX,(*)=x"(.) P-as. Thus we have obtained the following: Proposition 4. Under the assumptions ( H , l)', ( H , 2)' and ( H , 3), the following statement holds; For every constant a , such that ( a o ) - Isf ,Z(B), the modified equation (15)'

has a unique solution X,, which has the following properties; ( i ) s-limfi+mX,(.)=x"(.) (P-a.s.) ( ii ) For a sufficiently large n the function X , solves the equation (1)'

References [ 1 ] Ogawa, S., Quelques propri6tts de I'inttgrale stochastique du type noncausal, Japan J. Appl. Math., 1 (1984),405-416.

Kyoto Institute of Technology Matsugasaki, Sakyoku Kyoto 606, Japan

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Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 607630 (1986)

An Approach to Constrained Equations and Strange Attractors By H i r o e OKA and Hiroshi KOKUBU Abstract. Because of the very thin nature in their shapes of several strange attractors of ordinary differential equations, it is likely that they are expressed in terms of constrained equations. In order to clarify such a situation, the notion of generalized vector fields is introduced, and their normal forms as well as unfoldings are discussed. Constrained systems are characterized in the class of generalized vector fields. An application is proposed for the case of the Lorenz attractor by a computer simulation. Key words: constrained system, strange attractor, normal form, generalized vector field, infinitesimal deformation, unfolding

Introduction Several strange attractors i n three-dimensional systems of ordinary differential equations, such as t h e Lo r en z attractor (Fig. l), the Rossler attractor,

40

i

20 -

10 t . . . . , , . . . , -10 0

.

.

.

r

10

.

.

. i

20

x-

Figure 1. The Lorenz attractor. (reprinted from [S]) Received July 6, 1985.

608

H. OKAand H. KOKUBU

etc., have nearly two-dimensional very thin natures in their shape. Taking notice of this character, some authors regarded them as the ones constrained on adequate surfaces and tried to find out some properties of these strange attractors from this point of view ([Rl], [Tl], [Loz]). In order to make such studies systematically, we extend our object to a class of ordinary differential equations involving the one not solved by its derivatives. Such a differential equation is called a n implicit differential equation or a generalized vector field. In this paper, we begin with the local study of implicit differential equations around a point in the phase space and, especially, we prepare a theory of normal forms and versal unfoldings for them. Our next purpose is to characterize constrained differential equations in the space of implicit differential equations. We give a definition of the constrained system in § 4 and discuss the singular perturbation of constrained systems employing their normal forms and unfoldings. Finally we propose a n example of singularly perturbed equations which exhibits a Lorenz-like strange attractor by numerical experiments. The connection between this system and the Lorenz equation can be explained in terms of degenerate singularities of a (generalized) vector field. This paper is a survey article for our recent works ([OKl], [OK2], [O]). The organization is as follows: S 1. Historical survey and motivations § 2. Definition of generalized vector fields S 3. Normal forms and versal unfoldings for generalized vector fields S 4. Constrained systems and their perturbations S 5. Constrained Lorenz-like attractor § 1.

Historical Survey and Motivations

Investigations of strange attractors by constrained systems seem to have been begun by Takens and Rossler independently. In this section we obscurel y mean a system constrained on a surface by ‘constrained system’ and later we give a definition of it. Takens [Tl] gave a n example of a constrained system as shown in Fig. 2. Though he only illustrated a picture, his constrained system seems to model the Lorenz’ strange attractor [Lor]. In fact, he alluded to the resemblance of the graph of its first return map to that of the Lorenz attractor. Rossler [ R l ] also considered such systems on a double fold type surface (Fig. 3), and in [R2], he proposed four typical constrained systems as prototypes of continuous chaos. One of these systems called ‘the spiral type chaos’ is shown in Fig. 3 (a). Among these prototypes, a constrained system modelling the Lorenz attractor is contained, which is essentially the same as the Takens’ model (Fig. 2), though thz latter is constrained on the cusp surface

Constrained Equations and Strange Attractors

609

, Figure 2. Takens’ constrained model [Tl].

Figure 3. (a) The spiral type chaos proposed by Rossler [R2]. (b) A strange attractor observed in the equation (1.1).

(the Whitney’s pleat). Moreover, Rossler [R2] gave a n explicit form of differential equation as follows:

1

x=-y+ax-bz

p=x+1.1

E i = (1- z”(x+ z ) - EZ

(.=$)

which realizes a spiral type strange attractor by a numerical experiment when ~=0.03,a=0.1 and b= 1.

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H. OKAand H. KOKUBU

We generalize this equation into the following form :

where E is a small parameter (or a small multi-parameter). When E=O, the equation reduces to,

t

X=fo(x, Y ) 0= g o b , Y ) .

For the surface defined by the equation g o ( x , y ) = O , we use the term characteristic surface. The characteristic surface of the equation (1.1) consists of three planes z = i1 and z= - x , and the observed attractor is along the stable part of this surface. (Fig. 3 (b)) As for the constrained model of the Lorenz attractor, neither model indicated above reflects the symmetry of the original Lorenz equation, which is invariant under the transformation, ( x ,Y , 4

-

( - x , -Y,

4.

Recently, Lozi [Loz] proposed another constrained model respecting this symmetry. His model, shown in Fig. 4, has also the cusp type constraint surface.

Figure 4. Lozi’s constrained model of the Lorenz attractor [Loz]. Since constrained models for the Lorenz attractor (or other strange attractors appearing in O.D.E. systems) can be considered as a sort of the idealization of the original attractor, one may expect to use such constrained models for the study of strange attractors. All these constrained models mentioned above, however, illustrate only naive pictures and do not have any analytic expression such as differential equations. The only exception is the Rossler’s equation (1.1). This equation

Constrained Equations and Strange Attractors

611

seems suggestive, for it is a n exmaple which can be considered as a constrained model of the spiral type chaos with analytic expression. Though Rossler does not clarify the method of its derivation, and though the Rossler's equation has too special a form to generalize, this example shows that the equation of the form (1.2) may play the role of a constrained model with analytic expression. Our interest is to find out some general method to obtain a n equation of the form (1.2) from a given strange attractor. We note that, as long as the small parameter E is non zero, the equation (1.2) is a system of ordinary differential equations solved by their derivatives, but when E becomes zero, some of the variables are n o longer independent and the equation is not, in general, within the class of ordinary differential equations solved explicitly by the derivatives. Therefore we enlarge the class of differential equations containing both equations of the form (1.2) (especially, the equation with E =0) and autonomous ordinary differential equations solved by the derivatives. In this framework, we try to investigate a general correspondence between constrained models and strange attractors. We consider the differential equations of the following type: (1.3)

A(x)2= V ( X )

x E R"

where A is a matrix-valued function of X, and z, is a vector-valued function. We call it a n implicit differential equation or a generalized vector field on R". The class of implicit differential equations contains both equations of the form (1.2) and ordinary differnetial equations solved by the derivatives. Moreover, by a coordinate change x = $ ( y ) , the equation (1.3) is tranformed to, A ( $ ( Y ) ) .W v ) 3 =V ( $ ( Y ) )

.

Therefore, this class is invariant under the action of coordinate transformations. In the subsequent sections, we concentrate on the local classifications of implicit differential equations and their perturbations. At the end of this section, we remark on the recent work of Benoit [B]. For the equations on R S with the small parameter multiplying one of the derivatives, he gave a definition of trajectories of reduced system with E = O , and obtained a condition for the convergence of trajectories of a singularly perturbed system to those of the reduced one as E tends to 0. Moreover he investigated some topological and analytical properties for trajectories of such equations. In his work, he mainly employed methods of the non-standard analysis. It seems for us that his results may give a useful tool for our study: once we have obtained a constrained model of the form (1.2) of some strange attractor, we could analyse such equations using Benoit's results.

S 2.

Definition of Generalized Vector Fields In this section we give a general coordinate-free description of generalized

612

H . OKAand H . KOKUBU

vector fields on a manifold. Throughout this paper, M is an n-dimensional C"-manifold and T M denotes the tangent bundle of M with the bundle projection IT. By X ( M ) , we denote the set of all vector fields on M , that is, the smooth sections of the vector bundle T M . A bundle homomorphism A of the vector bundle T M is the C"-mapping from T M to itself whose restriction on each fiber T,M ( x E M ) is the linear endomorphism A ( x ) of T , M . The set of all bundle homomorphisms of TM is denoted by H O M ( T M ) . Definition 2.1. A generalized vector field on M is the pair ( A , v ) of a bundle homomorphism A of T M and a vector field v on M . Therefore the set of all generalized vector fields on M , which is denoted by @ X ( M ) ,is nothing but the product of HOM ( T M ) and X(M). On a n arbitrarily chosen local chart, we can identify a generalized vector field ( A , v ) with the following local expression:

(2.1)

4'5)* i= v(E)

which is a n implicit differential equation on Rn introduced in S 1. Thus the generalized vector field on M can be considered as a globalization of the implicit differential equation on R". From this fact we define the solution of a generalized vector field as follows: Definition 2.2. Let y be a smooth map from a (possibly, infinite) interval Z to M . We say the map y is a solution of the generalized vector field ( A , v) if y satisfies,

for all t € Z . The above expression makes sense, for dy(t)/dt is a tangent vector of M at y ( t ) . Note that the existence and the uniqueness for solutions of generalized vector fields are not guaranteed in general. We give two examples illustrating each case. Example 2 . 3 . x x = - 1 By putting z = x 2 , this equation reduces to z= -2, so we can easily obtain the explicit solution: z =x2 =x,2 - 21

where x , is the initial condition. From this expression, it is clear that all solutions arrive at the origin, x=O, in finite time, and cannot be extended beyond this time. Especially there is no solution starting the origin at t = O . Fig. 5 (a) exhibits such a configuration of solutions.

Constrained Equations and Strange Attractors

613

(a) (b) Figure 5. Configuration spaces of solutions. (b) x i = - x (a) xx=-1 Example 2.4. x k = - x We can also solve this equation explicitly and obtain the configuration of solutions as in Fig. 5 (b). In this case, for each initial condition, there exists a solution which can be extended at infinity. But this equation does not have the uniqueness for solutions at x=O as indicated in Fig. 5 (b).

In the above two examples, the existence and the uniqueness are violated only at x=O. This holds in general; for any generalized vector field ( A , v), the existence and the uniqueness of solutions break only where A is degenerate. Next we consider the notion of equivalence and transformation for generalized vector fields. Let ( A , v) be a generalized vector field on a manifold M . As stated above, we identify the generalized vector field with the differential equation (2.1). Thus it is natural to consider that the transformed equation of (2.1) by a coordinate change C=$(E) is equivalent to (2.1) itself. Moreover multiplying the non-singular matrix valued function P(E) to both sides of (2.1) does not change the equation essentially. Strictly speaking, P is the bundle isomorphism of T M , that is, a n invertible bundle homomorphism. (The set of all bundle isomorphisms on M is denoted by ISOM ( T M ) . ) Consequently we arrive at the following definition of equivalence:

Definition 2.5. Let ( A , V ) and (A’, v’) be generalized vector fields on M . We say these generalized vector fields are equivalent if there exist a bundle isomorphism P of T M and a diffeomorphism $ of M such that,

-

( A’, v’) = (P* T$ * A * ( T$)-’, P T# * u*

holds. The pair (P, (p) is called a fransforrnation of the generalized vector field. In the sequel, we focus our attention on the study of the local structure of generalized vector fields around a n arbitrarily chosen point x, in M . Expanding a generalized vector field into the Taylor series at xo, we change it, up to order k, by transformations of the generalized vector field as simply as possible. Such simplified generalized vector field is called a k-th order normal form for generalized vector field. T o obtain all k-th order normal forms for generalized

614

H. OKAand H. KOKLJBLJ

vector fields can be interpreted as the local classification of them up to order k .

S 3.

Normal Forms and Versa1 Unfoldings for Generalized Vector Fields

This section is devoted to proposing a general framework of the normal form theory for generalized vector fields and to give several examples. For the results obtained in this section, we refer to [OK2]. First we briefly review the method to obtain normal forms for (ordinary) vector fields. The normal form theory for vector fields has been developed by PoincarB, Siegel, Sternberg, Arnold [A2], Takens [T2], Ushiki [U], and others, We mainly follow Ushiki’s method. Let v be a vector field on M . Our interest is to know how the vector field v changes by a given diffeomorphism q5. For this purpose, taking a one-parameter family q5t of diffeomorphisms connecting the identity at t=O and $ a t t = l , we investigate the way of deformation &v of v in terms of a differential equation on the space Z ( M ) of vector fields. Recall that every vector field Y o n M generates a diffeomorphism as the time-one-mapping of the flow defined by Y, that is, q5=exp Y . We call Y the infinitesimal generator of q5. In this situation we consider the following one-parameter family of diffeomorphisms, @=exp r Y , and deform a given vector field v by @. Then a formula in differential geometry gives,

where [ , ] denotes the Lie bracket for vector fields. The left hand side of (3.1) is called the infinitesimal deformation of v by Y. Since {V}forms a oneparameter group of diffeomorphisms, (3.1) defines a differential equation on Z ( M ) , that is, (3.2) which describes the way of deformation of v. Integrating this equation Conversely, in from t = O to t= 1, we obtain the transformed vector field &v. order to simplify some terms of v, we have only to find appropriate infinitesimal generators and to solve the equation (3.2). This is the idea of the normal form theory for vector fields. Especially the Jordan normal form theory for

Constrained Equations and Strange Atrractors

615

matrices is nothing but the first order normal form theory for vector fields. For details and the practical method of computation, see Ushiki [U]. The normal form theory for generalized vector fields is essentially the same but slightly more complicated. As the first stage, we consider the theory of the leading order. Let ( A , v) be a generalized vector field on M and let x , be a point in M . We consider the following two cases: ( i 1 v(xo)+O, ( i i ) v(xo)=O. 3-1. When v(x,)#O, by fixing a n arbitrary local chart, we can consider the pair, ( A , ; ~,)‘(A(Xo), v(x0))

of a matrix and a vector where A ( x , ) is the matrix obtained by the restriction of A on the tangent space of M at x , . Such a pair ( A , ; v,) is identified with a n n x (n+ 1)-matrix and we call it a n extended matrix. The equivalence for generalized vector fields induces a n equivalence for extended matrices. Definition 3.1. Extended matrices ( A , ; v,) and (A,’; v,’) are equivalent if there exist non-singular matrices P, Q € GL(n,R) of order n such that, (A,’; v,’)=P.(A,; v,).Q

where Q denotes the non-singular matrix,

of order n+ 1. The classification of extended matrices with respect to the equivalence is very simple: we obtain the following theorem by a n easy linear algebra argument. Theorem 3.2.

Every extended matrix ( A , , v,) € M ( n , n+ 1; R) (where v,#O)

is equivalent to one of the following two forms:

where r is the rank of A,,

ek=Yl,O, - . . , O ) E R k

( k = n - r or r )

H. OKAand H. KOKUBU

616

and Z, is the unit matrix of order r. 3-2. We consider the case v(xo)=O: the point x, is called a n equilibrium point of the generalized vector field ( A , v). In this case, the linearization of ( A , v) at x, is naturally defined as,

Mxo), W

X O ) )

where Dv(x,) is the Jacobian matrix of v at xo. This yields the following definition.

Definition 3.3. An (n-dimensional) linear generalized vector field is a pair of two real square matrices of order n. It is convenient to identify a linear singular vector field ( A , B ) with the expression AA+B by introducing a parameter 1. We call such a n expression a matrix pencil of order n. ?en (n, R) denotes the set of all matrix pencils of order n. Clearly, Pen (n, R ) = M ( n , R) x M(n, R) where M(n, R) is the set of all real matrices. We define a n equivalence relation among matrix pencils similarly to the previous sub-section 3-1.

Definition 3.4. We say two matrix pencils AA+B and A’A+B’ are equivalent if and only if there exist non-singular matrices P and Q of order n such that the relation,

A’A+B’=P* (AA+B)* Q-l , holds as a polynomial of A. We divide matrix pencils into the following types:

Definition 3.5. ( i ) A matrix pencil AA+B is non-singular if A is nonsingular, that is, det A f O . Otherwise we say A1+B is singular. ( i i ) A singular matrix pencil AR+B is non-degenerate if det ( A J + B ) f O as a polynomial of 1. Otherwise we say AA+B is degenerate. Note that a non-singular matrix pencil A + B is equivalent to, Z,J+

Q(A-’B)Q-l

by choosing P=QA-’ in the above definition 3.4, where I,, is the unit matrix of order n. Moreover, by a suitable choice of Q , it reduces to,

where J is the Jordan normal form of A-lB.

Therefore the equivalence

Constrained Equations and Strange Attractors

617

classes of non-singular matrix pencils have one-to-one correspondence with the Jordan normal forms of matrices. So the classification of matrix pencils is a generalization of the Jordan normal form theory for matrices. The classification of matrix pencils in the above sense was completely done by Weierstrass and Kronecker with the help of elementary divisors. We only state the result in a rough manner. For the proof and details, see Gantmacher [GI. Theorem 3.6 ([GI). Every matrix pencil is equivalent to a block-diagonal matrix pencil, each block of which has one of the following forms: ( i ) Z,A+J,(c) where J J c ) is a Jordan cell of order m of Jordan normal form with the eigenvalue c. W e denote ImA+Jm(c)by the symbol c".

m

We assign it the symbol pm.

The symbol is

(iv) The transpose of (v)

-

(O)]g

E

~

.

E~

.

The symbol is 7".

The symbol is (9,h ) .

h

W e call this block-diagonal matrix pencil the Weierstrass-Kronecker normal form (abbrev. W-K normal f o r m ) of a matrix pencil. Moreover the W-K normal form is uniquely determined up to the order of disposition of blocks. In order to represent a W-K normal form, we use the symbols indicated above; e.g. the symbol (0, 1).7,P-O signifies the matrix pencil,

We can easily count up all equivalence classes of matrix pencils of fixed order n by combining these symbols. As a n example we show, in Table 6, the classification of matrix pencils of order 2 except for non-singular ones.

H. OKAand H. KOKUBU

618

Table 6. Classification of the singular matrix pencils of order 2. Degenerate

Non-degenerate Norma' form

Symbol

Codim.

Normal form

Symbol

Codim.

2

2 4

3-3. In the above two sub-sections, we considered the normal form theory for generalized vector fields of the leading order both at a n equilibrium point and at a non-equilibrium point. Here we proceed to the higher order normal form problem, for which we adopt the same strategy as that for (ordinary) vector fields: we take a one-parameter group of transformations for a generalized vector field generated by a n infinitesimal generator, and compute the infinitesimal deformation. Let ( A , u ) be a generalized vector field on M , that is, A E HOM(TM) and u E Z ( M ) . Let (P, $) be a transformation of the generalized vector field, where P EISOM(TM), the set of all bundle isomorphisms of T M and $ E Diff ( M ) , the set of all diffeomorphisms of M . (P,$)#(A,u ) denotes the transformed generalized vector field of ( A , u ) by (P,$), that is,

(3.3)

(P,$ ) $ ( A ,u)=(PoT$oAoT$-l, PoT$ouo$-')

.

This means that the product group of ISOM(TM) and Diff(M) acts on HOM(TM) x Z ( M ) ,where (3.3) induces the group structure as follows:

(P.$ ) - ( Q +)=(PoT$oQoT$-', , $04), and yields the semi-direct product group ISOM(TM) >a Diff ( M ) . Thus the following proposition is obtained:

Proposition 3.7. The semi-direct product group ISOM( T M )>a Diff ( M ) acts on 8 X ( M ) in the manner as ( 3 . 3 ) . We define the exponential mapping from @)3E(M)to ISOM(TM)>aDiff ( M ) . To begin with, we prepare several notations. Let ISOM(TM, T M ) be the set

Constrained Equations and Strange Attractors

619

of all pairs ( F , f) where F is a smooth invertible bundle mapping of TM and f is a diffeomorphism of M satisfying, mF=fon,

(n is the bundle projection of T M )

.

ISOM(TM, T M ) forms a group under composition, and there is a group isomorphism, u:

ISOM(TM)>aDiff( M )+ ISOM(TM, T M ) ,

defined by,

(P,4)

+

(POT474 ) .

On the other hand, we can identify any bundle homomorphism R E HOM(TM) with a vector field on TM in terms of the following local coordinate representation : (x, E ) 3 (x,0,E, R(x1.E) .

(Here the bundle projection of T ( T M ) is Tn.) We define the mapping, K:

HOM(TM) + 5 ( T M )

by this identification and denote the image K ( R )of R by YE 5 ( M ) , T Y is a vector field on T M , that is, TnoTY=T(no Y)=T(id,)=Id,,

K,.

Since, for any

,

we can consider the sum K,+TY. For any t c R , the exponential mapping of this vector field on TM defines a n element of ISOM(TM, T M ) , which is denoted by exp t(r,+TY). Using above notations, we define the exponential mapping for generalized vector fields. Definition 3.8.

The exponential mapping of ( R , Y) E @ 5 ( M )is defined by, exp t ( R , Y)=u-loexp t ( r , + T Y ) ,

for sufficiently small t . The infinitesimal deformation of the generalized vector field ( A , U) is, thus, given by,

d ( exp t ( ~ Y, W , dt

u) .

t=D

To compute the infinitesimal deformation, we identify the bundle homom orphism A with a (1, 1)-type tensor field through the natural vector bundle isomorphism,

620

H. OKAand H. KOKUBU

Horn ( T M )N T*M@ TM

,

where Hom ( T M ) is the vector bundle over M whose fiber at x E M is the vector space Horn ( T , M , T , M ) , the set of all linear maps from T,M into itself. The next lemma is a fundamental result in differential geometry.

Lemma 3.9.

where @=exp t Y and - P Y A denotes the Lie derivative of the tensor field A with respect to the vector field Y . From this lemma and a n easy calculation, the infinitesimal deformation is obtained as follows:

Theorem 3.10. exp t(R, Y ) # ( A v, ) = ( R o A - P Y A , Rev-[ Y , v]) . With a local coordinate representation,

a

where R = C Ri,-@dxj,

axi

A=

a C Aij--@dxj, ax,

Y=

C

Yi-

d

8%

and

v=C vt-. d

ax,

Using this theorem, we can calculate normal forms for generalized vector fields after Ushiki’s method to obtain those for vector fields. Since computations for individual cases are complicated, and various types of degeneracy occur for non-linear terms, we cannot exhaust all possible normal forms with the specified leading terms. We give below several examples, each of which has some type of the W-K normal form as its leading terms. These examples deal with most nondegenerate cases in their non-linear terms, called generic normal forms.

Proposition 3.11.

where a is a constant.

The generic normal form of order 2 for

,uzis given

by,

Constrained Equations and Strange Attractors

Proposition 3.12. symmetry,

62 1

The generic normal form of order 3 for p 2 - 0 with the

( x ,Y , z )

-

(-x, -y, z)

is given by, (az+ O ( ? ) ) x = y i x z + p x 3 j=X

-txZ+bz2+qz3

Z=

where O(2) denotes terms of order 2 and a, b, p , q are constants.

Proposition 3.13. The generic normal forms of order 3 for (0, 1).$.0 with the same symmetry as that of Prop. 3.12 is given by, (az+ 0 ( 2 ) ) x = y i x z + p x 3 j=bxz+qx3

(3.4)

z = ~ x ~ ~ z ~. + s z ~ 3-4. The final subject in this section is the perturbation of generalized vector fields. For any given generalized vector field ( A , u), we consider a smooth family ( A 2 ,vl) parametrized by 1~ R*. We call the germ of the family ( A 2 ,vl)at 0 E RXa k-parameter unfolding of ( A , v) if ( A o ,v o ) = ( A ,v). Our interest is to obtain a n unfolding of ( A , u ) which contains all possible types of perturbations of ( A , v), which is called a versal unfolding. Definition 3.14. A n unfolding ( A 2 ,v2) (1 G Rk)of ( A , v) is called versal if, for any unfolding (Ap’,up’) ( p E RL)of ( A , v), there exist a mapping 1=B(p) and a family of transformations ( P 2 ,$ 2 ) parametrized by 1such that the followings hold:

‘W)=O ( 4 ’ 7

Up’)

(Po,$o)=(IdTM, id,) , and , for any p E R’ $scp, )#(&p) vs(p)1

= (Peep, 9

9

9

.

In other words, any unfolding of ( A , u ) is derived from the versal unfolding of ( A , u).

Of course, the versal unfolding for a given ( A , u ) is not unique. We say a versal unfolding of ( A , v) is miniversal if the number of parameters is minimal among all versal unfoldings of ( A . u ) and we call the number the codimension of ( A , u). The notion of versal unfolding for matrices was first introduced by Arnold [ A l l . We can construct a miniversal unfolding of a matrix A in the following way. The set of all matrices conjugate to A is called the orbit of A , which is a submanifold of M(n, R). A versal unfolding of A is given as the germ of a family transversal to the orbit at A . We introduce a natural inner product into M ( n , R) given by,

622

H. OKAand H. KOKUBU

=tr (AsCB).

Then a miniversal unfolding is obtained by taking one complementary (say, for example, perpendicular with respect to the inner product defined above) to the tangent space of the orbit at A . All such considerations work for extended matrices (3-l), matrix pencils (3-2) and even for generalized vector fields (3-3) as they are. Introducing appropriate inner products, we can show the following: Theorem 3.15. A miniversal unfolding of an extended matrix ( A o ;v,) is given by ( A o ;v , ) + ( G ; u ) where G and u satisfy, A , . t G + v o ~ t u = O , and c G . A , = O .

Corollary 3.16. Miniversal unfoldings of normal forms of extended matrices given in Theorem 3.2 are,

0 0 . 0 0 1,' eT where A=(Atj) and

Theorem 3.17.

Y=(Y*)

sre unfolding parameters.

A miniversal unfolding of a matrix pencil AA+ B is given by,

where G and H satisfy, G . t A + H - t B = O , and t A . G + t B - H = O .

Corollary 3.18. Miniversal unfoldings for ,urnand ( 0 , ~)-Y,I"-' are given by the following ones, respectively:

Constrained Equations and Strange Attractors

623

where a , and ,E, are parameters. Especially the codimension of these matrix pencils is m and m+ 1, respectively.

From the corollary 3.18, we can easily obtain the bifurcation diagrams of matrix pencils. As a n example, we show the bifurcation aspects of (0, l).vl. The codimension of the matrix pencil (0, 1)-$ is 3. Hence all possible perturbations of (0, l).yl are contained in a 3-dimensional space indicated in Fig. 7.

Figure 7. Bifurcation diagram of versal families for the matrix pencil (0, I).$.

We can obtain versal unfoldings for generalized vector fields in a similar fashion. Here we do not go into details but give only one example. We consider the unfoldings ( A 2 ,u2) of the normal form (3.4) given in Proposition 3.13 satisfying the following conditions: ( 1 ) v,(O)=O for any 1, ( 2 ) ( A 2 ,v,) has the symmetry of Proposition 3.13, ( 3 ) A , [resp. v,] is a t most of order 2 [resp. 31.

Proposition 3.19. for (3.4) is given by,

In the above category of unfoldings, the versal unfolding

i

(E

(3.5) where

E,

a , /3 and

+az+ O(2))i=ax +y kx z + p x S 9 =,EX +bxz+ qxS

r are parameters.

i =y z i xz iz2+

SZS

H. OKAand H. KOKUBU

624

S 4.

Constrained Systems and Their Perturbations Let us recall the equation (1.2) of

S 1,

(4.1)

Within the framework of generalized vector fields, we shall completely characterize such equations including the case of E = O . A bundle homomorphism A of TM is called of constant rank if, for any x E M , the rank of A(x) is independent of x. In this case, we say A is of corank r if the rank of A equals n - r where n is the dimension of M and r is a non-negative integer.

Definition 4.1. A constrained system of corank r on M is the pair ( A , v) of a bundle homomorphism A of T M of corank r and a vector field v on M . A constrained system on M is a constrained system of corank r for a n integer r . We denote the set of all constrained systems [resp. of corank r ] on M by B E ( M ) [resp.&X('j(M)]. The set & f ( M ) is a subset of @ f ( M ) . It is easy to see that, for any constrained system ( A , u ) of corank r and any transformation (P,$), the transformed system (P, $ ) # ( A ,v) is again a constrained system of corank r . Thus the group ISOM(TM)>aDiff ( M ) acts on the space Q5(rj(M) (and, as the result, & f ( M ) ) . Especially, applying the map, GE'O'(M)+ f ( M ) ,

( A , v)

-

A-'v

,

we can show that the category of & f ( O ) ( M and ) the action of ISOM(TM)>a Diff ( M ) to it is equivalent to the category of the space Z ( M ) and the action of Diff ( M ) to it by the coordinate transformation. In other words, the constrained system is a natural extension of the vector field. In this framework, the equation (4.1) can be identified with a family ( A s ,u,) of constrained systems parametrized by E satisfying ( A o ,u,) G QX("(M), that is, a n unfolding of a constrained system of corank r . This fact leads us to the following idea: a singularly perturbed family of vector fields is nothing but a n unfolding of a constrained system. At the end of this section, we analyse several examples of perturbation problems from this point of view. We can also define the characteristic surface for our constrained systems.

Definition 4.2. characteristic surface

Let ( A , u ) be a constrained system of corank r on M . The 9of ( A , u ) is given by, 9= {xE M I u(x) E Im A(x)} ,

where ImA(x) is a linear subspace of T,M consisting of all images of linear endomorphism A(x) of T,M.

A standard transversality argument shows the next proposition:

Constrained Equations and Strange Attractors

625

Proposition 4.3. For any generic ( A , v) in G X ( 7 ) ( M ) ,the characteristic surface of ( A , v) is a smooth submanifold of M of codimension r. The normal form problem for constrained systems is solved by reducing it to that for generalized vector fields. Let ( A , u ) be a constrained system of corank r. Similarly to S 3, we put,

('40;vo)=(A(x,), U ( X 0 ) )

7

for a n arbitrarily chosen point x, in M , and we call ( A o ;v,) the leading part of ( A , v) a t x,. Since ( A , u ) is o f corank r , A , is a linear map of rank n-r. If u,fO, the leading part of ( A , v) is classified as in Theorem 3.2. For simplicity we give results only for the two dimensional case.

Theorem 4.4. Suppose that ( A , v) is a constrained system on a two dimensional manifold M and that its leading part (A,; u,) at x, 6 M is equivalent to

(:

:;3.

Then, the infinite order normal form of ( A , v) is given by

(4.2) In other words, every finite order parts of ( A , u ) at xo except for the leading part is eliminated by suitable transformations. This result corresponds to a formal version of the rectification theorem for vector fields. For the case (ii) of Theorem 3.2, there exist various kinds of normal forms according to the degeneracy of higher order terms. We give below a partial result.

Theorem 4.5. Let ( A , v) be a two-dimensional constrained system of corank 1 whose leading part ( A o ,v,) is equivalent to

(:

;;9 .

Then the first order normal form of ( A , v) is one of the following:

H. OKAand H. KOKUBU

626

Moreover the case (i) is also the infinite order normal form. For the case (ii), its generic second order normal form is given by, iy+ax2

(4.4)

where a is a constant. Its characteristic surface is the parabola given by y= *ax2. In contrast with the case of generalized vector fields, we cannot obtain, a t present, versal unfoldings for constrained systems in a general manner. The main difficulty is that the space & X ( M )is neither a vector space nor a manifold. We give below a n example of versal unfolding of a constrained system, which is based on a cumbersome computation for this special case.

Proposition 4.6.

The versal unoldifng of (4.2) is given by,

3 (3

(;I

(4.5)

9

where E is an unfolding parameter. This versal unfolding can be expressed in terms of differential equations as follows:

t

(4.5)

ER= 1

9=0.

As well as the above system, we consider the following, which are unfoldings of (4.3) and (4.4), respectively:

(4.7)

t t

€X=a-rX j=1

+

ER=a +y+ /3x ax2

Y=l-rx

where E , a and /3 are parameters. Each of them describes a typical local orbit structure of the equation of the form (4.1). For instance, the phase portrait of the well-known Van der Pol equation, Ejt-(l-x2).R+X=O

,

is given in Figure 8, in which the local orbit structures around the points A , B and C correspond to the phase portraits of the equations (4.9, (4.6) and (4.7), respectively.

Constrained Equations and Strange Attractors

627

Ty

&-

I

Figure 8. An illustration of the phase portrait of the Van der Pol equation for small positive E .

S 5.

Constrained Lorenz-Like Attractor

The last section of this paper is a n application of our theory of normal forms and their unfoldings for constrained systems. As we have mentioned in S 1, the equations [Lor], (5.1)

i=-ux+uy,

P=rx-y-xz,

i=-bz+xy,

have a nearly two-dimensional strange attractor, called the Lorenz attractor (Fig. 1) for some values of parameters u, r and b. We have proposed, in [OKl], the following differential equation: E X =y

+a x z -

xs

3 =A x + B y + pxz

(5.2)

i =cZ+w ,

as a constrained model of the Lorenz attractor with analytic expression. In fact, the equations (5.2) exhibit a Lorenz-like strange attractor in a computer simulation; when parameter values are, E:

0.03, A : 0.7, B: 0.7, C: -1.0, a: 0 . 7 , p: - 1 . 0 , 7: 1.0,

628

H. OKAand H. KOKUBU

Figure 9. A strange attractor observed in the equation (5.2). These two figures exhibit the same attractor from different directions.

Figure 10. The graph of the Lorenz plot of the attractor of Fig. 9. the strange attractor indicated in Fig. 9 is observed. We have also taken the Lorenz plot, that is, the successive local maxima in the z-coordinates of the numerical data. The graph of the Lorenz plot is shown in Fig. 10. See [OK11 for more precise data of numerical experiments. We have derived the equation (5.2) somewhat heuristically in [OKl]. Here we shall give a n explanation for the reason why the equation (5.2) and the Lorenz equation seem to have similar attractors in a computer simulation.

Constrained Equations and Strange Attractors

629

The equation (5.2) can be transformed into, E X =BEX+ Y + a X Z -

[

x3

P =A X + ( p - B a ) X Z + i=cz+y x 2 ,

BXS

by the linear change of variables, X=x,

Y=~-BEx,

Z=z.

We can show that this system is a n unfolding of the generic third order normal form of the constrained system with symmetry whose leading part is (0, 1).$.0. On the other hand, Ushiki and we [UOK] have shown that the Lorenz equation can be regarded as a subfamily of the following system of ordinary differential equations,

1

x=y 9 =A’x+ B ’ y i x z + a’yz+px3 2 = C’Zl x2+ p’z2+sz3 .

+qx2y+ ryz2

This is the versa1 unfolding of the generic third order normal form [U] of the ordinary vector field whose linear part is conjugate to the Jordan normal form,

iH B 9. The linear vector field of the above type corresponds to the matrix pencil 02-0,that is,

k : $+k : :I. 1 0 0

0 1 0

As is illustrated in Fig. 7, and easily checked by simple calculation, the matrix pencil (0, 1 ) - $ - 0 is contained in the closure of the orbit of the matrix pencil 02.0. In other words, the linear singular vector field of type (0, l ) . v l - O is a (singular) limit of the linear vector field of type 02.0, which gives a relationship between the equation (5.2) and the Lorenz equation. We have shown in [UOK] that some degenerate singularities of vector fields can be considered as ‘organizing centers’ of strange attractors: we have proved that a strange attractor which is qualitatively the same as the Lorenz attractor is observed in an arbitrarily small perturbation of a degenerate singularity of the vector field of type 02.0. It seems, therefore, for us that the above mentioned relation of the equation (5.2) and the Lorenz equation

H. OKAand H. KOKUBU

630

m a y provide a clue t o finding o u t a general correspondence between strange attractors i n systems of ordinary differential equations a n d their constrained models.

Acknowledgement. W e express our sincere gratitude t o Professors M a s a y a Yamaguti a n d Shigehiro Ushiki for their valuable advice a n d encouragement. References [All

V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys,

[A21

26 (1971), 29-43. __ , Lectures on bifurcation in versa1 families, Russian Math. Surveys, 27

(1972), 54-123. [B] [GI [Lor] [Loz] [O] [OK11

E. Benoit, Thesis, UniversitC de Nice, 1984.

F. R. Gantmacher, The Theory of Matrices, Vol. 2, New York, (1959). E. N. Lorenz, Deterministic nonperiodic flow, J. Atom. Sci., 20 (1963), 130-141. R. Lozi, Thesis, Universith de Nice, 1983. H. Oka, in preparation. H. Oka and H. Kokubu, Constrained Lorenz-like attractors, Japan J. Appl. Math., 2 (1985), 495-500. in preparation. [OK21 -, [Rl] 0. E. Rossler, Chaotic behavior in simple reaction systems, Z. Naturforsch., 31a (1976), 259-264. , Continuous chaos, New York Acad. Sci., 316 (1976), 376-392. [R21 [S] C. Sparrow, The Lorenz Equation: Bifurcations, Chaos and Strange Attractors, Appl. Math. Sci., Vol. 41, Springer-Verlag, 1982. [Tl] F. Takens, Implicit differential equation: some open problems, Lecture Notes in Math., 535, Springer-Verlag, (1976), 237-253. [T21 , Singularities of vector fields, Publ. Math. I.H.E.S., 43 (1973), 47-100. S. Ushiki, Normal forms for singularities of vector fields, Japan J. Appl. Math., [U] 1 (1984), 1-37. [UOK] S. Ushiki, H. Oka and H. Kokubu, Existence d’attracteurs &ranges d a m le dtploiement d’une singularitt dtgCnCrte d’un champ de vecteurs invariant par translation, C. R. Acad. Sci. Paris, 298, SCr. I, (1984), 39-42. ~

~

Department of Mathematics Kyoto University Kyoto 606, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equationspp. 631-44 (1986)

On the Existence of Progressive Waves in the Flow of Perfect Fluid around a Circle By H i s a s h i OKAMOTO~ and M a y u m i S H ~ J I Abstract. We consider a free boundary problem for incompressible perfect fluid with surface tension. The problem to be considered is as follows: A perfect fluid is circulating around a circle r (see Fig. 1). The outward curve r is a free boundary to be sought. We assume that the flow, which is confined between r and 7, is irrotational. On the free boundary, surface tension works and makes the free boundary circular. On the other hand, the centrifugal force caused by the circulation of the flow makes the fluid go outward. Hence the balance of these two kinds of forces determines the geometrical properties of the free boundary. We show that there exist progressive waves, which are periodic motions of the fluid and are the exact solutions corresponding to the solitary waves.

Fig. 1. Key words:

S 1.

free boundary, bifurcation, progressive wave, surface tension

Introduction

W e consider a nonstationary flow of perfect fluid with a free boundary a r o u n d a circle. T h e problem t o b e considered here can be regarded a s a model f o r a flow ar o u n d a celestial body. W e consider a plane through an equator of a celestial body a n d a two-dimensional flow i n this plane. W e assume that f o r a fixed time t t h e flow region is enclosed by tw o closed Jorda n curves and r(t). is an equator of a celestial body a n d ~ ( t is) a free

r

r

Received April 2, 1985. Revised July 20, 1985. + . . . Partially supported by the FGjukai.

H. OKAMOTO and M. SHOJI

632

boundary which is outside r. These two curves enclose a doubly connected domain, which is denoted by Qr(,,. Then the fluid lies in Q,(,, (see Fig. 1). For simplicity we assume that the inward curve r is the unit circle in the plane. We also assume that r ( t ) is represented as r ( t ) = { ( r , 0); r = r ( t , 0 ) } in terms of a function r = y ( t , 0 ) and the polar coordinates ( r , 0). Then the problem is formulated as follows. Problem. Find a time-dependent closed Jordan curve r ( t ) ,a stream function V = V(t,r , 0 ) and the pressure P(t, r , 0 ) satisfying the conditions (1.1-8)

-0

in

QT,,

,

--

Here u, g and w o are prescribed positive constants. A is the Laplace operator with respect to ( r , 0 ) . Krc,) is the curvature of r ( t ) , the sign of which is chosen to be positive if it is convex. For the physical meaning of this problem and derivation of these equations, see [7] or [8]. We only note that V is a stream function for the flow, i.e., the velocity vector is given by (aV/ay, -aV/ax) and that (1.4, 5) is the Euler equation written in terms of V. Our purpose in this paper is to show the existence of progressive wave solutions, i.e., solutions of the following form 7=7(0-~ct)

,

V= F(r, O-cr)

,

where c is a constant. When c=O, this solution reduces to a stationary solution, which is studied in [7]. We can easily find a simple stationary solution given by

Progressive Waves in the Flow of Perfect Fluid

633

where r , > 1 and a > O are constants (see S 2 below). This is a radially symmetric solution and we call it a trivial solution. We will prove that there exist progressive wave solutions in a neighborhood of the trivial solution. The propagation speed c is determined by the wave number and the magnitude of circulation. The important consequence of this result is that the trivial solution is not asymptotically stable, since the progressive wave solution is a time periodic solution. This fact constitutes a striking contrast to the case of viscous fluid (see Beale 111). Indeed, in 111, Beale considers viscous incompressible fluid flow above a plane-like bottom with finite depth (a model of flow in the ocean). He proves that the rest state (motionless fluid with a horizontal plane as a free surface) is asymptotically stable by virtue of the viscous and capillary forces. Furthermore the rate of convergence to the motionless state is proved to be O(t-''2) (see, Beale and Nishida [13]). We also show that if the circulation of the flow is zero, there is no stationary solution other than one in which the free boundary is a circle. This fact is worthy of notice because even if the circulation is zero there are progressive wave solutions which are not circles. Finally we remark that we show the existence of the progressivz waves by the bifurcation theory. In utilizing the theory, we take the propagation speed as a bifurcation parameter. This fact provides for distinction between our problem and the problem for flows in a n infinite domain over the straight line or plane (i.e., the ocean problem). Indeed, the wave number ranges over all the real numbers in the case of the ocean problem. On the contrary, in our problem, the wave number can take only the integer multiples of 2n. This makes the spectrum discrete, whence we can use the bifurcation theory. Our problem differs from the dissipative system of evolution equations like reaction-diffusion equations defined on R. In some dissipative system, the existence of the progressive wave solution is known but its propagation speed is uniquely determined by the system. Therefore such progressive waves do not fall into the framework of the bifurcation theory (see 14, 61). This paper is composed of four sections. In section 2 we reformulate the problem and give a precise version of the theorem. Section 3 is devoted to the proof of the theorem concerning the existence of progressive waves. In section 4 we show the uniqueness theorem for the stationary flow when the fluid does not move but only the surface tension works on the free surface. Finally we give a derivation of the formula used in S 3 concerning variation of domain. Acknowledgment. The authors wish to express their hearty thanks to Prof. H. Fujii who kindly read the original version of the manuscript and gave us much useful comment and encouragement.

634

S 2.

H. OKAMOTO and M. S H ~ J I

Existence of Progressive Wave

We begin with the fact that there is a trivial stationary solution in which the free boundary is a circle. The stationary problem is to find a closed Jordan curve and a function V such that the conditions (2.1-5) below are satisfied: (2.1)

AV=O

in

Q,,

(2.2)

V=O

on

r,

(2.3)

V=a

on

r,

(2.4)

1

2 /VV12-x+cKr=constant r

,

on

r

Here Qr is a doubly connected domain bounded by and 7. Note that in the stationary problem the condition (1.3) reads that V is constant on r. So we denote the constant by a. The constant a represents the magnitude of the circulation. The case of a=O corresponds to the case where the circulation vanishes. If a=O, then V=O by virtue of (2.1-3). Hence, in this case, the fluid does not move anywhere. The conditions (1.4-6) reduce to (2.4), which is known as the Bernoulli equality. We now define r , > l by l r ( r o 2 - l ) = u 0 . Let To be a circle of radius ro with the origin as its center. We put V,(= V,(r))= (a/log r,,) log r for 1< r < r,. Then {rn,V,} is a stationary solution, i.e., satisfies (2.1-5). Our aim is to prove the existence of progressive wave solutions. We exclusively consider the solutions near the trivial solution. Hence what we really do is to show the existence of functions U E C3+"(S1)and V € C3+"(Dr) such that r is given by y ( t , O)=r,+-u(O-ct). Here the symbol C3+"implies the Holder space. These functions are governed by (1.1-8). If we introduce a new variable $=O-ct and if we note that a/& is replaced by -c(a/&b), then these governing equations are, in the present case, expressed as follows: r-

(2.8) (2.9)

:r

(r-Vr: )+-=O :T

in

a,,

Progressive Waves in the Flow of Perfect Fluid

635

(2.10)

P=aKr

(2.11)

on

r,

(2.12)

The equation (2.10) is equivalent to saying that cr(aV/ar)+(l/2)IVV I2+P--g/r does not depend on q5. On the other hand, by (2.6), the equation (2.9) is rewritten as follows: c-

:(

T)+-

a 1 -lVVlz+P-ar(2

r-

r

>-

-0

in Qr

.

Hence we see that ~ r ( a V / a r ) + ( l / 2 ) 1 V V ~ ~ + P -does g / r not depend on r . fore the condition (2.9-11) is reduced to the equation below: (2.13)

av

1

cr-+-lVV12+aK,-x=constant dr 2 r

on

There-

r.

Next we consider the equation (2.8), which is rewritten as

Hence we have V(7(q5),$)= -(~/2)r($)~+constant. Now we can reformulate the problem. We first define symbols:

ru: a

closed Jordan curve represented by (r,+u($), $)

Qu={(r,$)€R2; l
K,: the curvature of

05$<27r},

ru .

Problem. Find a function ing the following conditions:

K€

C S t a ( Sand ) a function V E C s + a ( ~satisfyu)

(2.14)

(2.17)

.

in Q,,

cr-+av dr

1 IVV12--+aK,=constant J! 2 r

On

ru

H. OKAMOTO and M. SHOJI

636

Note that the problem above reduces to the sataionary problem (2.1-5) when the constant c vanishes. We solve this problem by a method very similar to that in [7]. Namely we regard this as a bifurcation problem. In [7, 91 the first author proved that nontrivial solutions to (2.1-5) bifurcate from the trivial solution. In this case we viewed the parameter a as the bifurcation parameter. As a consequence we see that there are a n infinite number of nontrivial stationary solutions. To show the nontrivial solutions to (2.14-18), we fix a and we take the propagation speed c as a bifurcation parameter. We now give a framework by functional analysis. We define a mapping F which is defined near the origin of R x C3'"(9) x R and takes its value in PtU(S1) x R. To define F, we first solve (2.14-16) which is a Dirichlet problem with respect to V for a given u E C3+"(S1), and we denote the solution by V,. Namely we define V, by

We now define F as follows:

Here the constant E, is defined so that F(c;0, O)=(O, 0 ) . Namely we put

Using a pull-back (To,O)+(r,+u(O), 0), we regard F,(c; u, E ) as a n element of Then it is clear that Clta(S1).

{r,,

V,} is a solution to (2.14-18)

t ) .

F(c; u, E ) = ( O , 0 ) .

Of course (0,O)corresponds to the trivial solution. We show the existence of nontrivial zero-points of the mapping F, i.e., we prove the following

Theorem. For n € N, define c, by (2.19)

c =-

a ro2log ro

nrO3R,

a2 nR,rO4(logro)2

where R,=(ron+r,-n)/(ron-ro~n). Let n € N be fixed. Suppose c, € R and that c,#c, for all m which is not equal to n . Then c, is a bifurcation point of F .

Progressive Waves in the Flow ofPerfect Fluid

637

Remark 1. Note that the parameter a 2 0 is fixed. It is important to observe that in the case of a = O there is no stationary solution other than the trivial solution but a progressive wave solution does exist even in this case. For the first statement, see S 4. Remark 2. The assumption that c,#c, for m f n implies simpleness of the multiplicity in some sense. But for appropriate values of parameters u, g, wo, it happens that c,=c, for n f m . Even if this is the case, c, is still a bifurcation point. But the behavior of the bifurcating branches becomes much more complicated. The situation is the same as that in [lo], see also Fujii, Mimura and Nishiura [3]. So we only consider the bifurcation under the simpleness assumption.

S 3.

Proof of Theorem

This section is devoted to the proof of Theorem. We first show that F is a C1-mapping and derive the concrete expression of the FrBchet derivative of F at ( c ; 0,O). The proof is similar to that in [7]. In fact the differentiability of F is proved in the same way as in [7]. Hence we omit it. We put b = c r o + a / ( r o log y o ) . Then the derivative is represented as follows. DuFdc; 070) DwFAc; 090)

(3.1)

(3.2) (3.3) (3.4)

D,F1(c;0, O ) W = -

caw ro log y o

aw

D,F1(c;0,0)2= -2

D ; F , ( ~o;, O ) w = r ,

DtFAc; 090)

j

( A € R)

28

W(O)~O

(WE

,

C S + ~ ( S, ~ ) )

0

DtF,(c; 0,0)2=0

(JER) t

where U is a solution of the Dirichlet problem below:

'AU=O

(3.5) U=-bw

in l < r < r o , on r , on r = r o .

These formulas are derived in the Appendix. Admitting these formulas, we calculate the critical points of D,,cF, i.e., we determine the conditions under which D,,eF(c; 0,O) fails to be a n isomorphism from C3+"(S1) x R onto Clta(S1) x R. For this purpose we look for ( w , 2 ) E CSt0(S1) x R such that D,,aF(c; 0, O)(w,2)=

H. OKAMOTO and M. SHOJI

638

(0,O).

To express this equation concretely we expand w in the Fourier series: ca

w=

C s,

n=l

m

sin (no)+ 2 c, cos (no) . n=o

Then the solution of (3.5) is expressed as follows.

Hence we have

Therefore we have D,F,(c; 0, O)w=c,x "something"

On the other hand, we easily obtain D,F,(c; 0, 0)=2rrroc, .

By these equalities we see that there is a (w,A ) # ( O , 0) satisfying D,,tF(c; 0,O) x(w,A)=(O, 0) if and only if c satisfies (2.14) for some n = 1 , 2 , In fact (w,R)=(cos no, 0) and (w,A)=(sin no, 0) are eigenvectors for c = c , . By the method used in [7] we can show that D,,tF(c; 0,O) is a n isomorphism if and only if c sl {cn}. To show that cn is a bifurcation point, we use Theorem 1.7 of Crandall and Rabinowitz [2]. This theorem ensures bifurcation from simple eigenvalue. So we introduce the following function space: X m t a -- {U

€

Cmta(S') ; ~(6')f4 2 ~ O)} -

( m € N, 0 < a < 1)

.

Then we can easily show that F is a mapping from R x X3+"x R into X I t a x R. Indeed U E X3+" if and only if the curve ru. is symmetric with respect to the x-axis. If this symmetry holds, then the solution of (3.5) is symmetric with respect to the x-axis. Hence we see that F : R x X S t ax R + X 1 + a x R. Henceforth we consider the mapping F restricted to R x X3+"x R. We now verify the conditions in [2]. By the assumption for c , the kernel of D,,cF(c,; 0,O) is spanned only by (cos no, 0). Then verification of the conditions in [2] is easy except that D,D,,eF(c,; 0, O)(cos no, 0) @ Range D,,tF(c,; 0,O) (nondegeneracy condition). We can show this by the formula (3.1-4). Differentiating these formulas by c, we see that D , D , , c F ( ~ ,0,; O)(cosno, O)=constantx

Progressive Waves in the Flow of Perfect Fluid

639

(cos nu, 0). On the other hand, the range of D,,;F(c,; 0,O)is orthogonal (in the L2-sense) to (cos no, 0). Hence the condition above (the nondegeneracy condition) is verified. The proof is now completed. Q.E.D.

Remark 3.1. Numerical computation shows that the bifurcation occurs subcritically (see Fig. 2). This, however, does not imply that the bifurcating solutions are unstable. In fact, the nonstationary problem (1.1-8) is not of the form ut=F(u), where subscript means the differentiation, but we are dealing with the evolution equation of the following form: (3.6)

utt=@(u, ut,

Fig. 2.

Ute,

ue?use, useel

.

A schematic bifurcation diagram. Every branch occurs subcritically.

(The reduction of (1.1-8) to (3.6) is found in [14].) Analyzing (3.6) in a way which is standard in the bifurcation theory, we can see that both the trivial solution and the bifurcating solution are marginally stable in the sense of the linearized stability. The stability analysis of the nonlinear equation (3.6) is very hard. It requires more analysis to determine the stability in the nonlinear sense.

Remark 3.2. As for the stability of the progressive wave solutions or the stationary solutions, nothing has been rigorously obtained so far. But the geometrical properties of stationary solutions are studied in [ l l , 121. They show that the figures of the stationary solutions are not so simple. On the other hand, it is very difficult to simulate the nonstationary problem for a long range of time. Our computation (see Fig. 3) shows that the progressive waves exist for appropriate time. But a desicive answer for the stability is not yet known.

640

H. OKAMOTO and M. SHOJI

I=

n 0

27t

0

271

Fig. 3-a. A solitary wave which travels to the right. Difference ~ ( tB)-ro , is plotted from r = O to r=90.

Fig. 3-b. A solitary wave which travels to the left.

Fig. 3-c. A one-peak solitary wave and a two-peak solitary wave. In Fig. 3 the parameter a is fixed to be 0.1.

S 4.

Uniqueness of Stationary Solution

Here we prove a uniqueness theorem for the stationary problem. goal is to show the following

Theorem 2. Suppose that a=O and that g>O. (2.1-5) other than y = ~ , , ,

Our

Then there is no solution to

V=O.

Remark. Here we do not assume that y lies near r0. We only assume that y is a C*-curve. The assumption that g > O is indispensable. In fact, if

Progressive Waves in the Flow of Perfect Fluid

641

g < 0 , existence of nontrivial solutions can be proved by the bifurcation theory.

Proof of Theorem 2. Let {r, V } be a solution. Then, by (2.1-3), we see that V vanishes identically in 9,. Therefore oKr=g+E r

(4.1)

on

r,

where 6 is a constant. We have only to show that r is a circle. We first prove the case where o < g . Let A (resp. B ) be a point on which has the largest (resp. smallest) distance from the origin. Then we have

9 - = a -~ 9, ( and ~ )O A Z O B . OA

OB

By the definition of A and B, it holds that K , ( A ) Z 1/OA and that K , ( B ) S l / O B . Therefore we obtain a/OA 5 aK,(A)=g/OA-gglOB+aK,(B) Z g / O A - g / O B + a/OB. By this inequality and the assumption a < g , we obtain O A S O B , which implies that r is a circle. We now prove the case where a z g . We assume for simplicity that the curve r is represented by the equation r = r ( O ) . Then by (2.5) we have

It is known that total (oriented) curvature of a closed Jordan curve is equal to 2n, i.e.,

\r

Kr(r(O)'+ ~'(B)2)1'2dB= 2~ .

As for the proof, see, e.g., Hopf [ 5 ] . By (4.1) and this fact (4.3)

2na=g

S:

we have

( r ( ~ ) 2 + r ' ( e ) z ) " z / r ( ~ ) d ~,+ E L ,

where L, is the length of the curve it on [ 0 , 2 n ) . Then we obtain

r.

We multiply (4.1) by

(4.4)

The second equality is proved by the formula

r2 and

integrate

642

H. OKAMOTO and M. SHOJI

Eliminating E from (4.2-4) we obtain

By the isoperimetric inequality the left-hand side is nonpositive and it vanishes if and only if 7 is a circle. Then, by o Z g > O , we have

On -

the other

:5

r(B)dBIL7 and that This is i.e., 7 is a circle, which completes the proof. Q.E.D.

hand, it obviously holds that

:$ (y(B)2++r’(B)z)1/z/r(B)dB~ - 2 ~ . Hence the equality must hold.

possible if and only if f = O ,

Appendix. Here we show how the formulas (3.1-4) are derived. The procedure here is the same as that in [7]. In fact we borrow a lemma which is proved in [7]. We first prepare notation. Let f be a C1-mapping from Cmt1+ftU(S1) into Cm+l+a(S1). Here m>O a n d j 2 0 are integers and O
(A.1) (‘4.2)

AW,=O

W,=O

on

in

r,

Q,,

W,=f(u)

on

r,.

Of course the second equation of (A.2) implies that W,(r,+u(B), O ) = f ( u ) ( S ) . In terms of W , we define mappings T and T I by

where a/&, is a differentiation along the outward normal to following lemma holds:

ru.

Then the

Lemma. T and TI are C’mappings from some neighborhood of the origin in c m t 1 t j t a (Sl)into Cm+a(S1).If u belongs to C1for some I > m + 2 + j + a , then the derivative of T is represented by (A.4)

D,T( u)v = auu

I

rl

+I(W,)v -

(r,+u)v’-u’v afo {(ro+u )+~( K ’ ) ~ } ~ / dB ~

Here V is the solution of

(A.5)

AV=O

in Q,,

Progressive Waves in rhe Flow of Perfect Fluid

643

(A.6) Here D , f is the Frichet derivative.

The function 2’( W,) is given by

The derivative of T , is represented by

If we a d m i t this lemma, t h e proof of (3.1-4) is easy. In fact we put f ( u ) = - ( ~ / 2 ) ( r , + u ) ~ + a + ( c / 2 ) r , ~ . Then t h e l e m m a shows that (‘4.7)

D,T(O)w= D,T,(O)w

dU azv, w , =-+dr ar2

w h e r e U is given by (3.5). Observing that iVVl=aVu/i3v, w e obtain (3.1) by (A.7). O t h e r formulas are easy to see. Q.E.D.

References J. T. Beale, Large-time regularity of viscous surface wave, Arch. Rational Mech. Anal., 84 (1984), 307-352. M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalue, J. Funct. Anal., 8 (1971), 321-340. H. Fujii, M. Mimura and Y.Nishiura, A picture of the global bifurcation diagram in ecological interaction and diffusion systems, Phisica D, 5 (1982), 1-42. R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982). 343364. H. Hopf, Differentical Geometry in the Large, Lecture Notes in Math., No. 1000, Springer, Berlin, 1983. H. Hosono and M. Mimura, Singular perturbation approach to travelling waves in competing and diffusing species models, J. Math. Kyoto Univ., 22 (1982), 435-461. t 7 1 H. Okamoto, Bifurcation phenomena in a free boundary problem for a circulating Bow with surface tension, Math. Methods Appl. Sci., 6 (1984) 215-233. On a nonstationary free boundary problem for perfect fluid with surface 1 8 1 -, tension (to appear in J. Math. SOC.Japan). [ 9 1 -, Stationary free boundary problems for circular flows with or without surface tension, in Proc. U.S.-Japan Seminar on Nonlinear PDE in Appl. Sci., eds. H. Fujita, P. D. Lax and G. Strang, Lecture Notes in Numer. Appl. Anal., Kinokuniya/North-Holland, Tokyo/Amsterdam, 1983, 233-251. -, On the 4-dimensional O(2)-equivariant bifurcation equation arising in a stationary free boundary problem for a perfect fluid (preprint).

644

H. OKAMOTO and M. SHOJI

[ll] H. Fujita, H. Okamoto and M. Shaji, A numerical approach to a free boundary problem of a circulating perfect fluid, Japan J. Appl. Math., 2 (1985), 197-210. 1121 M. Shaji, An application of the charge simulation method to a free boundary problem, to appear in J. Fac. Sci. Univ. Tokyo. [13] J. T. Beale and T. Nishida, Large-time behavior of viscous surface waves, to appear in Lecture Notes in Numer. Appl. Anal. Vol. 8, North Holland-Kinokuniya. [14] H. Okamoto and M. Shbji, Dynamical system arising in nonstationary motion of a free boundary of a perfect fluid, Kbkytiroku of RIMS No. 559, 19-41. Department of Mathematics Faculty of Science University of Tokyo Tokyo 113, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 645-662 (1986)

On Laminar Boundary Layers with Suction By Taira SHIROTA Abstract. The basic mathematical questions of the two-dimensional, incompressible and laminar boundary layer theory are considered, where prescribed pressure gradients are not always negative. From certain a priori estimates, the existence of separation points of solutions to Prandtl equations with suction and Prandtl’s approximation to Navier-Stokes equations along Fife’s consideration are obtained by means of continuations of Oleinik’s local solutions. In this regard it continues the work of Matsui and myself. Key words: laminar boundary layer, separation points, Prandtl equations, suction

1. Introduction Let x , y be coordinates parallel and normal to the wall and u, v the corresponding components of velocity. Then the Prandtl equations of motion considered here are

u -au + v - = -au -+v-, dp (1.1)

ax

8%

dx

ay

ayz

au av -+-=o

aY

in

aY

[0,Z)X [0,a)

with boundary conditions

(1.2)

u=O

and

v=vo(x)

on y=O

and

(1.3)

u(x,y)+U(x)

as y+oo

uniformly in x on any compact subset of [0, I ) . Here ( d p / d x ) ( x ) = p , ( x )is the prescribed, non-negative pressure gradient, v , ( x ) the prescribed suction and U ( x ) the main-stream velocity related by

(1.4) Received April 1, 1985.

2 p ( x ) + U2(x)=constant .

T. SHIROTA

646

Since the governing equations are parabolic if x is taken as the evolution variable and u20, it is also necessary to specify a n initial profile for u at some station of x : (1.5)

u(x,y)=uo(y)

for x=O

and

CJ(O)>O

.

From a n engineering point of view ([lo], [l l ]), the station of zero skin friction of solutions to the problem is calculated approximately or numerically, and the value of the suction velocity, which is sufficient to prevent separation, is predicted approximately for flows with the downstream stagnation point. But from the degenerate parabolicity of (1.1) in y=O together with boundary conditions (1.3) and (1.4), even the existence of the exact, classical solution of the boundary layer equations up to and including the point of zero skin friction has not yet been obtained. Thus, from a mathematical point of view the behaviors of the exact solution and its derivatives at such a point cannot be used directly in discussion as in a n engineering approach. On the other hand, for the Prandtl approximation to Navier-Stokes equations Fife has defined a certain laminar class of the solutions of Navier-Stokes equations and succeeded in obtaining a n approximation theory in the case where the pressure gradients are strictly negative ([l] and its references). But it is very difficult to obtain a proper description of the high Reynolds number, steady, laminar Navier-Stokes’ flows, even if only the flow field upstream of the separation is considered. The purpose of this paper is to provide a mathematical basis for the theory of Prandtl equations mentioned above. In Section 4 we shall show the existence of the separation point, assuming behaviors of the main stream and the suction velocity near the rear stagnation point, whose definition is given in Section 2. In Section 5 we shall try to develop a connection between the preseparation solutions to Navier-Stokes equations and the Prandtl equations along the Fife’s considerations. Our results will give no ultimate solution to the problem, but they are hoped to be useful for further mathematical investigation of the problem ([2]).

2. Notations and Definitions For the sake of simplicity we are concerned with flows past a flat section of boundary. Then the stationary Navier-Stokes equations are uU,+UU?l=y(C1,,+Uyy)-Px

(2.1)

C1Z),+UUy=Y(Z),,+21,,)--Py

U,+Uy=O

, , in

[0, I] x [0,

.

a)

(The subscripts in (2.1)denote the partial differentiation with respect to the corresponding variable.)

On Laminar Boundary Layers with Suction

647

The Prandtl equations (1.1) are formally derived from (2.1) by the following manner: By the transform x=x,

and

qeu-1’2y

72x9 v ) = ~ - ” z 4 x Y, ) ,

a(x,v)=u(x, Y ) 9 P(x, v)=p(x, Y ) 9

we have ii,,-liii,-cii,-p,=

(2.1)’

-j, = -lJ(d,,

-Mi,,

,

+uscx+ui)c,

fue,)

,

ii,+s,=o . Neglecting the terms of the right-hand sides in the above, we obtain the dimensionless Prandtl equations ii,,

- vii, = p,( x )

-Eli,

u,+v,

,

=0

where p , ( x ) = p , ( x , 0). Then returning to the original coordinates ( x , y ) and setting also

4%Y)=u1’2v(x,77) ,

,

u ( x , y ) = U ( x , ‘7) PAX) = P , ( x )

9

we obtain (1.1). Now the substitution of the independent variables in (1.1) of the form

(2.3)

x=x

9

$=#(x,y)

3

where

reduces formally (1.1), (1.2), (1.3) and (1.5) to Mises’ form: putting w ( x , #)= u2(x,y ) , we have

with the boundary conditions (2.5)

w ( x , Y ) =o

on y = O ,

(2.6)

w(x,#)-U2(x)

as $-roo

T. SHIROTA

648

uniformly in x € [0, I ] ,

(2.7)

407

=.10(9)

9

where

Next in order to indicate sets of data or solutions to the above equations, for an interval [0, I ] let Bo([O,I] x [O,m)) be the Banach space of uniformly bounded continuous functions in [O,I ] x [0, m) with supremum norm. For any a ( O < a < 2 / 3 ) and any y,>O let C"([O,I ] x [ y o , m)) be the Banach space of Holder continuous functions, i.e.,

IM{IXl--Xz1"2+ IY, -YzlY for ( x i , Y , ) E 10, I1 x [yo7 a) ( i = 1 , 2 ) and Iu(x1, Y I ) -u(xz, Yz)l

M=M(yo, u )

.

nyo>o

Let C"([O,I ] x (0, m)) be C"([O,I ] x [ y o , m)). Furthermore let B"([O,I ] x (0, 00)) be Bo([O,I ] x [0,m))n C"([O,I ] x (0, m)). We also define spaces Bo([O,a)), C"((0,m)) and B z t U ( ( 0m)) , by the same way. Then we shall take a function u € B z t u ( ( 0 ,a)) as an initial datum in (1.5) if uy(0)>O,

u(O)=O,

(2.8)

u-U(0) VU,,(Y)

as y+m -P,(O)

w($)

-u o ( 0 ) u , ( ~ =)0 ( y 2 )

as Y->O ,

Let us denote the set consisting of

€ I % a if and only if

" , "+ w(O)=O

(2.9)

,

and

which is a strong compatibility condition. such data by I z t a = Z Z t a ( ~U, ( 0 ) )and

by Pi". Obviously

for y 2 0

u,(y)>O

, 4 Gw$h$h B"((0, m))

,

w+(O)>O,

9

ws($)2O

49)+ U2(0) lJ4WwILIL -2p,(O) - V O ( 0 ) O=0(9) ~

for $ 2 0 , a s $-+m and as 9 - 0 .

Let F& "([O, I ] ) be the space consisting of functions w ( x , $) such that

On Laminar Boundary Layers with Suction 0

, 0, , 04 , v'Gw,+€B"([O,I1 X ( 0 , m)) for (x, $1 € I0,ZI x (0, I k$l-B , w ( x , 9)> 0

649

9

Iw,I

q d x , 0) > 0

for

XE

00)

,

[O,I]

where kdepends only on wand ,8€ (0, 1/2). Let F2,'"([0,Z))=nosl~
Then if w ( x , 4)€ F v a ( [ O ,I ) ) is a solution to (2.4)-(2.7) for some W , , ( $ ) € Z ~ ~in (2.7), putting

we have the classical solution (u, v) to (1.1)-(1.5), provided U ( x ) > O for x E [0, I ] ([6]). Here we say that (u, v) is a classical solution to (1.1)-(1.5) if

Let the space consisting of such functions described above be represented by F2([0,ZJ) and put

Now we define the separation points of solutions to (1.1)-(1.5) (2.4)-(2.7) respectively.

and to

Definition 2.1. A point (s, 0) is a separation point of a solution (u, v) to (1.1)-(1.5) if ( u , v ) E F Z t a ( [ 0s),) and

(2.11)

lim inf

K&X,

y)=O

,

x<s, (x,~)-+(s,O)

or equivalently a point (s, 0) is a separation point of a solution (2.7) if w E F2,'"([0, s)) and

(2.12)

lim inf

w+(x, + ) = O

w

to ( 2 . 4 ) ~

.

x
We denote this separation point by s(uo) and s(wJ respectively. fact, we obtain that

Then in

T. SHIROTA

650

S ( U 0 ) =s(wo)

for u, corresponding to w , by (2.10). Finally we define the laminar, pre-separation class CP of solutions to NavierStokes equations.

Definition 2.2. The class of solutions ( u y ,nu,p”) to Navier-Stokes equations in a domain D”=[O, I,] x [0, 21 is said to be laminar and pre-separation if the following conditions are satisfied on D’, ( i ) uv=O for y=O and O
l4u:,+u;JI

,

14u:,+u;J

I M

and

l~(uL+u;~)AI M ,

(iv) for some positive constants a, b (=b,) a - m i n {l, ~ - ” ~ y } < u ” (yx),< b - m i n 11, I J - ~ ’ ~, ~ } where I , is dependent on Y and m>I>I,>infvl,>O. For the interpretation of this definition see Fife [ l ] and its references and Section 4. Here and hereafter all functions described above are assumed to be continuous.

3.

Lemmas on Local Solutions

Here we shall consider the local solution o ( x , 9)to (2.4)-(2.7) with initial data wo($)E rye constructed by Oleinik [ 6 ] . First we remark that the solution 0 ( x , ~ ) ~ F ~ ~ “ ( 0for , 1 some , ] ) Z,>O is monotone non-decreasing in $ 2 0 for all x € [0, Z,], provided wo(9)€ IFa ([4]).

Lemma 3.1. Let w ( x , $ ) be the locaI solution constructed by Oleinik with initial data € I F : “ . Then w ( x , 9)E F2,’“([0,I,]) for some lo, and for some $o > O and k>O

Corollary 3.2. Under the same assumption in Lemma 3.1, the section o ( x , - ) belongs to ITafor any x E [0, Z,], i.e., the section w ( x , .) satisfies the strong compatibility condition (2.9) replacing 0 by x . The above assertions are very useful to continue the local solution obtained in [6] and is first announced in [4]for a special case, but the proof in our case is accomplished by the same way as in [4] and hence omitted here. In the following, in particular in Section 5 we need to obtain uniform

On Laminar Boundary Layers with Suction

65 1

estimates of solutions with respect to the data. T o obtain such estimates, we set k , and k , be mino,,,lo W(x) and max,,,,,, j2p,(x)I for some positive 1,<<1.

Lemma 3.3. Let W , be the set of I:" such that if w , € W,, inf w 0 + W>a

for

9 E [O, $,I

where a and $, are some positive fixed numbers. Then for some positive l,
Lemma 3.4. If a solution w(x, $) €F2,fu([0,I,]) a,(+) E I y asatisfies: for some positive b

with the initial datum

then for some positive constant 1, depending only on b. k , , k, (l2
Proof. Let f be a smooth, monotone nondecreasing function such that for some positive A , << 1

(f )$

f ( $ ) = ~ l , $ ~ ' ~ +- b

2f

($)

if

$
if

$>-4 3

and

f($)sb

for any $ 2 0 .

Let $ ( x , $)=f(+)(l+e-"") for a constant a>>1. Then as in Lemma 2 in [ 6 ]and from the proof of Theorem 3 in [ 6 ] ,comparing w with $, we obtain our assertion for 1,<<1. Here we remark that l / z w + + is bounded from above, but that

%q$*+= O(+-"".

Furthermore let k , be the maximum of the absolute values of 2p,, u, and their first order derivatives over [0, I,]. Let W , be the subset of Zyusuch that wo(+) E W , if the absolute values

Iw&$) I , Iwo+($) I , I l/woOo++($) over [0,

m).

I I M < 00

Then we have the following

Lemma 3.5. Let w o ( + ) € W,. F2,+"([0,I,)) with initial data coo($)

Assume that for the solution w(x,+ ) €

652

T. SHIROTA W(X,

Then f o r any positive

E

9)2 4

if (x,$1 E [O,2,) x [O, $01

<< 1

op(x, $) 2 ( 0 - c )

Here

$1

*

(x,4)E [O, 1 0 ) x [O,$,)

if

depends only on M , a ,

E,

k , , k,,

$0

*

but is independent of a. itsev.

Proof. Let f be the same function as in the proof of Lemma 3.4. Then for a sufficiently small A , 2 0 we see that

Then by comparison theorems as in the proof of Lemma 4-8 in [6], using the constants mentioned above we can estimate values of Iop(x,$)I, -o,(x, 4) and - d w ( x , $ ) . o + + ( x , $) from above over [0, I,) x [0, m) successively. Therefore from the relations

and o + ( x ,0) 20 ,

we obtain our assertion. Corollary 3.6. For a solution tion 2.1 can be replaced by

F2,fa([0,s ) ) the condition (2.12) in Defini-

lim inf

(3.2)

z<s,

o(x, $)/$=O

.

(Z,Y)'(S,O)

Proof. If the condition (3.2) is violated for some a E F v a ( [ O ,s)), we see that for some Go and a>O

Then from Lemma 3.5, it implies that

for e<<1 and some obvious. 4.

$1,

which contradicts (2.12). The converse assertion is

The Existence of the Separation Point

Suppose that the main stream U(x)€ Co([O,I ] ) n C"([O,I ) ) , U(x) > O for x E

On Laminar Boundary Layers with Suction

65 3

[O, I ) , U(I)=O, U,(I)= --oo and that p , ( x ) 1 0 as x - t l . Then we have

Theorem 4.1. For any constant suction vo< 0 and for any uo€ ZZta, there , to ( 1 . 1 ) - ( 1 . 5 ) , exists the separation point (s,0 ) of the solution (u, v)CEF 2 ( [ 0 s)) where s < I . T o prove the assertion of Theorem 4.1, we may take ing u, b y (2.7) such that

w, €

ZYacorrespond-

where k , is a constant dependent on a,,. For some constant k 2 k , let H(x,$)=2(2k$-(k$)”/”)

s:

p,(r)dr

.

Then by the definition it implies that

W O ,$1 2 @,($) ,

H ( x , 0) =0

3

for

XE

[0, I ]

.

Hence if

as in Theorem 3 of [6] we see that

where we assume that w € F g a ( [ O ,I ) ) . Taking k as a function of x, we shall examine under what condition on k (4.2) is valid. Let us calculate L ( H ) : L ( H )= ~ 2 / 2 - ( k $ ) ~ (‘ k~ $. ) ’ / z U 3 ( ~ )

+ 2 ( 2 k $ - ( k $ ) 3 / 2 ) p , . ( ~ ) + -( u o ) 2 - 3 (k$)’/’) k U 2 ( x ) ( 2

T. SHIROTA

654

3 1.~42 - - e“2U3(~)k2+2(2E - E”/”)p,(x) L ( H )= -4

where OLE< 1. Therefore (4.2) is valid for (x, 9)E [0, I ) x [0, l / k ]if and only if

(4.4)

(-vO)(2-

for ( x , E ) E [O,I ) x 10, 11. Let k ( x ) = g / U ( x ) where g is a positive constant. Then since

we need only to verify when

is valid for (x, E ) E [0,I ) x [0, 11.

Lemma 4.2. The inequality (4.3) is valid f o r k and x if ( - u 0 ) < ( 3 / 8 ) d J ( x ) k f o r the constant k , or if the k is of the f o r m : k = g / U ( x ) f o r any but fixed constant g , provided

(4.6) Proof. From (4.4) we see easily the validity in the first case. Considering (4.5) as a quadratic inequality of g , we obtain that the discriminant is negative if the second case of Lemma 3.1 is valid. F o r , from (4.5) the discriminant < O if and only if for any ( E [0, 11

uo2 3d2-E””(2-26+(l/2)E3’2) -< Y

(2-(3/2)E1’2)2

Since

we obtain the second assertion.

655

On Laminar Boundary Layers with Suction

Here we remark that we may restrict our consideration in the proof to the neighborhood of 1. Therefore we may asume that V ( x ) ,p , ( x ) are monotone nonincreasing and (4.6) is valid for any x € [0, I ) . Now we assume that (4.3) is satisfied. Then by the definition of H ( x , 9) we see that for k ’ 2 2 k and for ( x , 9) c [0, I ) x [0, k l - l ]

(4.7) Let G ( x , 9)be the function defined over x € [m. s ] for some m and s such that

’j’

~ ( x+ ), = 2 k ‘ + { ( l - r + )

p,(t)dt+r@

J(J)

where

+

I(x - m) m(s -x ) s-m

fb)=

,

O<<m<s
and

r>_k’(x).

Then by the definition and (4.3) it implies that

f(m)=m,

f(s)=Z,

f(x)>_x

and

for x € [ m , s ] . Next assume that

Hereafter we shall try to find the condition from which (4.9) is valid. To obtain the condition, assuming that k‘(x) and r ( = r ( x ) ) are monotone nondecreasing, we shall calculate L(G):

\””

L(G)1 y ( r - ~ k ’ V ~ ( x ) ) ~ / * 4 k ’ r p , ( t ) d f 2.

656

T. SHIROTA

< 4ur1/ek’3‘2U(x)p,(x) (l-s)(x - m ) s-m

Therefore to find (4.9) it is enough to obtain the inequality

+

2 ~ U ( , ) k ’ ~ / ~ , ~ ~ ~ (I -m s ))( xk’r-I(I- m )

+(

+

u,)k’( ( I - m)(s- x ) 2(I - s)( x -m ) ) I(s-m) for x c [ m , s ] . -

Setting r=4k‘, we see that the above inequality is satisfied if

(4.10)

+

4u V(x )klZ(x) (I -x) (s - m ) ( - uO)3k’(x)(I -x)(I - m) 1 <(s-m)--(I-m) 4

for x e [ m , s ]

Here if s=m, (4.10) is not valid, but if s=I, then (4.10) is reduced to

3 4 ~ U ( ~ ) k ’ ~ ( ~ ) ( I - ~ ) + ( - u ~ ) 3 k ’ ( x ) ( I - x ) O we see that (4.10) is valid for some s if

3 4

~ u ~ ~ ~ + ( - ~ , ) ~ ~ ~ ( ( I - x ) / C I ( X ) )for ~ / ~ x< c- [m,I ]

i

Since U( x) /( l- x) > >for l x c [m, I ] , provided m>O, there exists g,>1 in (4.1) we put if g. U(x)-’
k ( x )=k , k(x)=g.U(x)-’

where g = 8 ( - ~ ~ ) ( 3 u ) - ~Then . (4.3) is valid for such a k ( x ) . Furthermore we see that 1 2

g - U(x)-I<- g1( U ( x) ( Z -x))-1~~

for any x2ml>>0. Thus if we take m, ( < I ) such that ko

On Laminar Boundary Layers with Suction

657

nzvm,, then there exists s c (m,, 1) for which (4.8) a n d (4.10) a r e valid for m= in,. Therefore we see that even if U E FGu([O,s)) n Co([O,s])

1’

w ( x , +)i2k’+{(l-r+)

P,(t)dt+r+

l(.c)

for (x, 9) € [m,, s]X [ 0 , r-’].

Hence it implies that and

w&,O)=O

O<s
if w ( 0 , +) satisfies (4.1). T h e assertion of Theorem 4.1 follows from Lemma 3.3, Corollary 3.2 and the above fact, as in 141. 1) T h e conditions U ( x ) a n d p ( x ) mentioned in Theorem 4.1

Remark 4.3. a r e satisfied for

2

U ( x )= (1- x)’ -; 0 < 6 < -

(

2 ) When U ( x ) = ( I - x ) ’ we obtain as in the above that i f

u,?iu<3.2-’/2 then the assertion of Theorem 4.1 is also valid for U ( x ) = ( Z - x ) - ’ . But in the result obtained by the approximation method, u,,, a r e regarded as larger numbers than 3.2-‘% to obtain the same conclusion (1101). It is very interesting to compare our method with the approximation one.

5. The Prandtl Approximation First we give a simple example of the laminar, preseparation class rl, of solutions to Navier-Stokes equations as a n interpretation: Example 5.1.

p+L 4

u1/2

For positive constants (I, b such that b>a>O we set ax ,

b-a b

uY(y ) =-

1-exp ( - ( 1 / 2 ) ~ - ’ / ~ h y )a -I- -Y 1-exp(-v-1/2b) 2h

for y E [ 0 , 2 ] . T h e n {(u”,u”, p ” ) } is such a class. In fact they satisfy al5o the Prandtl equations for 10, I ] x 10, 2) with a n arbitrary I , the value of the suction / ~ b the pressure gradient (1i4)u’/’a > 0 for a n y v . being ( 1 / 2 ) ~ ’ and Though there is n o real exapmle of O={(u”, u”,p’)} with all u > O in either the numerical o r engineering point of view, the hypothesis of the existence of

T.SHIROTA

65 8

such a class 0 seems plausible since we restrict our attention to the preseparation flow field. (See L. G. Leal; J. Fluid Mech. (1973), vol. 59, part 3, pp. 513-535 and W. R. Briley; J. Fluid Mech. (1971), vol. 47, part 4, pp. 713736, together with [l] and its references.) Let CJ be a class of solutions to Navier-Stokes equations defined in Definition 2.2. Then from Definition 2.2 for any Y we may take p i , 5; E C’+~([O, Z,]) and the initial data i i ; ( ~€) P C Q with respect to the solution of (2.2) as foIlows: For a positive constant B

(5.1)

~B:(X, O)-pY:(x)l a rnin {I, 7 )

(5.2)

if x f [0, l,] , if O < ~ < v - l ’ ,~ if O < x < l , .

if O < ~ I Y - ~ / ~ ,

Ipz(x)l , lp:Jx)I
provided (5.2)’

7) f B”((0,v - ” ~ ] ) ,

nv(O,7) is monotonous, z‘i;,(O, Ip(x:, 7)l , IPi2(x,7) I Ib,

for ( x , 7 )E [O,L1 x [O, ~

- l / .~ l

Finally, putting (5.3)

( U ” ) Z ( X )=(ii~(C0))2-22BY(X)+2P”(O)

,

we may take the exterior flow W ( x ) as follows: (5.4)

0 < b,-l I(UV),(x)I b,

if x f [O, Z,]

provided that M > b, and b, is a sufficiently large number. Then assuming B > 2( 1+b,) we mention the following Theorem 5.2. For the positive constants B , M , a , b,, b,, b, letflow (u”,v’,pu)€ Qandsatisfy (5.2)’. Then i f v g v , for a certainv,=v,(B, M , a, b J , thereexists the Prandtlflow (a*, V’, p’) satisfying (5.1), ( 5 . 2 ) and (5.3) up to x = l , such that for Q constant C=C(B, M , a , bi) (5.5)

IUY(X, Y ) - i i ” ( X , y)l I C P

I.

on [O, l,] x [O, P

Proof. By virtue of the streaming functions @ ( x , 7) and P ( x , 7) of (2.1)’ and (2.2), we reduce them to Mises’ form: for o ” ( x , $)=(i2’)z(x, 7 ) and for S ( X , s;)=(u”)’(x, 7)

(5.6)

d/wYw+J;

-ijU(X, 0 ) o i-2B:(x, 0 )=v’/2&

,

On Laminar Boundary Layers with Suction

659

(5.6)’

where

Pf; = -2Yri:,+2(p:(x, =2uf; +2Y’/2

p-1a:, ,

K

7j)-p:(x, 0 ) )

fi”=(X,p’)dv’ ,

f; =p;=Y(Zj~?+Yi):t)-Yya”~-YZj”Zj; Let W ( x , $ ) = w ” ( x , + ) - z ’ ( x , we obtain (5.7)

$1.

.

Subtracting (5.6) (with $=&’=$”) from (5.6)’

L, w= l/isw,* - w,-3; w,-c ” ( x ,9)W=u“2fU(x, 9 )

where

(5.8)

where

Now we assume that for the problems ( 2 . 4 ) - ( 2 . 7 ) with wo(+)=z;(+), u,(x)= V;(x),p , ( x ) = p : ( x ) there exists the solution w u ( x ,$) E F y a ( [ O ,m,])such that for some positive constant C; independent of Y C ; g ( + ) l z ” ( x 9, ) 1 b 2 ,

O
Then from Definition 2.2, (5.7) and above inequalities it implies the following

T. SHIROTA

660

Lemma 5.3 ([l]). For u 5 M - l and x < m , IW”(X, $ ) - W ” ( X ,

$)I

,


where C depends on M , B , a , bi and C,l, but independent of m,. Next in order to obtain the solution G” mentioned above up to I,, we put (5.9)

A,,)=sup

i

I ; for any u’
G’”(x, $) exists in F v a ( [ O ,lAlv]) and satisfies

1

~ ” ‘ ( x$), 2- C,g(Q) on [O, I A I,] 6

Then from (5.8) and (3.1) it follows that for some ( i O > O and for any u I 1 if $<$o

G;(O, $ ) > ( C , - E )

.

Therefore from (5.2), (5.4) and by Lemma 3.3 we see that for some I , and for any v < 1 there exists the solution Gy(x,9)with the initial datum G; such that for lo<<1 and for any U I1 GYX,

0)€ F % a ( [ O , l o ] ).

Then from (5.2), (5.4), (5.8) and by Lemma 3.4 we obtain that for any u < l

1 6

wy(x,$)>- C,-g($)

The last inequality means that

>O.

A (1) < SUP 1,

x
if

.

Assume that

( U I (C,(2C)

-lY)

.

Let 1 be a number such that l < A , , , and A ( , , - l < < l . Then from (5.9) and Lemma 5.3 we have that for u < M - l and for x < l , ~ I IW”(X,

$)-G”(X,

$)I


where C depends in particular on (1/6)C5. Here we remark that C>>max { M , C 3 } . Therefore from the inequality C , g ( $ ) < ~ ~ 9) ( x for , X E [0, I,] we see that for x i l , ~ l GYX,

9)2 C,g($)-Y l ’ T $ 2

1 2

- C5g($)

if C 3 ( C ) - 1 2 ~ 1 /since 2, C 3 2 C , . Accordingly by Lemma 3.5, Lemma 3.3 and Lemma 3.4 it implies that for Y < ( C , ( ~ C ) -the ~ ) ~solution z j Y ( x 4) , can be con-

On Laminar Boundary Layers with Suction

661

tinued such that

for some 6 > 0 which is independent of I,.

Thus we have that A , , ) <

A-1)2)’

Applying once more the same consideration as above we may conclude that for any Y < ( C ~ ( ~ C ) there - ~ ) ~ exists

Then from the above inequality and Lemma 5.3, it follows finally that for v<(C3(2C)-1)2 IW”(X,

Q) - W ” ( X , $1 I IYl’ZCQ

for x g l , and C, in particular, depending on (1/6)C5. Thus returning to z?(x, v), ii”(x,7) from ~ ” ( xqU), , ~ ” ( x$’) , by using streaming functions @’, $!’ respectively we obtain that lC”(X,

q)--u”(x,

v)1 IClJ1’2

on

[O,/,I x [O, 11 ,

provided Y < ( C ~ ( ~ C ) - ~(For ) ~ . details in the last process see also [l].)

Remark 5.4. 1) The constant vO=vO(B,M , a, b,) in Theorem 5.2 tends to zero rapidly, if a tends to zero. 2) Signs of uo(x)don’t take a part in Theorem 5.2, but the non-positiveness of v, plays a n important role in obtaining the class @ in Definition 2.2. 3) To obtain ii;(7)€Z2+“we use a cut off function x(v/dY),the monotonicity of ii”(O,v), and the second inclusion in (5.2)’ which is derived from interior regularities of the solutions of Navier-Stokes equations defined over

References P. C. Fife, Considerations regarding the mathematical basis for Prandtl’s boundary layer theory, Arch. Rational Mech. Anal., 28 (1968), 184-216. -, Corrigendum, Considerations regarding the mathematical basis for Prandtl’s boundary layer theory, Arch. Rational Mech. Anal., 46 (1972), 389-393. [ 2 ] J. Glimm, Singularities in fluid dynamics, Mathematical Problems in Theoretical Physics, eds. R. Schrader, R. Seiler and D. A. Uhlenbrock, Springer-Verlag, 1981, 86-97. [ 3 1 A. M. Win, A. S. Kalashnikov and 0. A. Oleinik. Linear equations of the second

662

T. SHIROTA

order of parabolic type, Russian Math. Survey, 17-3 (1962), 3-146. [ 4 1 S. Matsui and T. Shirota, On separation points of solutions to Prandtl boundary layer problems, Hokkaido Math. J., 13 (1984), 92-108. 151 S. Matsui and T. Shirota, On Prandtl Boundary Layer Problem, Lecture Notes in Numerical and Applied Analysis, to appear. [ 6 1 O.A. Oleinik, On a system of equations in boundary layer theory, U.S.S.R. Comp. Math. Phys., 3 (1963), 650-673. Mathematical problems of boundary layer theory, Uspehi Mat. Nauk, 23, 171 -, NO. 3 (1968), 3-65. Weak solutions in the Sobolev sense for a system of boundary layer equa[ 8 1 -, tions, Amer. Math. SOC.Transl. (2), 105 (1976), 247-264. 191 J. Serrin, On the mathematical basis for Prandtls’ boundary layer theory: An example, Arch. Rational Mech. Anal., 28 (1968), 217-225. [lo1 S. N. Brown and K. Stewartson, Laminar separation, Ann. Rev. Fluid Mech. 1 (1969), 45-72. 1111 J. C. Williams, 111, Incompressible boundary-layer separation, Ann. Rev. Fluid Mech., 9 (1977), 143-144. Department of Mathematics Hokkaido University Sapporo, 060 Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 663-673 (1996)

Initial Value Problem for Kac’s Model of the Boltzmann Equation By Y a s u s h i SHIZUTA and H i d e k o NISHIYAMA Abstract. The initial value problem for Kac’s one-dimensional model of the Boltzmann equation is investigated. We perform the Fourier transformation to the linearized equation and study the associated eigenvalue problem depending on a parameter f . The C”-dependence on of eigenvalues and eigenfunctions are established. The proof makes use of the fact that the complete set of eigenfunctions of the linearized collision operator consists of Hermite functions. The global solutions for the Cauchy problem are constructed by standard arguments. Key words: Kac’s model, Boltzmann equation, global solution, linearized equation, decay estimates

S 1.

Introduction

W e study Kac’s model of the Boltzmann equation. T h e unknown F is a function of the space variable x E R, the velocity variable v € R and the time t>O. T h e rate of change of F with time is described by the following nonlinear integro-differential equation :

z=dF at

-~--+Q(F, ax

F).

T h e explicit form of the collision integral Q ( F ,F ) is given by

where

v‘=v cos 0-v, sin 0 ,

v,’=v sin O+vl cos 0 .

W e assume that Z(0)>0 is a n integrable function o n [-n,n]. Note that the weight Z(0) is related to the rate a t which the binary collision (v,v,)+(v’, vl’) takes place. W e assume furthermore that Z(B)=Z( -0). This guarantees the microscopic reversibility. In other words, the chance of a collision that takes Received April 8, 1985.

Y.SHIZUTA and H. NISHIYAMA

664

velocities (v,v,) into (v', vl') equals the chance of one with the reversed effect. Our aim is to find the solution to (1.1) under the prescribed initial condition F(0, x, v)=Fo(x,v) .

(1.2)

It is known that the Maxwell distribution is the only stationary solution for (1.1). Hence we introduce a new unknown function f by setting F=g+g1l2f, where g ( v ) = ( 2 ~ ) - exp ' / ~ ( - v 2 / 2 ) . We obtain

x=

(1.3)

at

af

- v--+Lf+r(f, dX

f ),

First we study the linearized equation of (1.3) by using the semigroup theory. This enables us to get the decay estimates for solutions to the linearized equation by means of suitable norms. Then the global solution of the initial value problem (1.3), (1.4) is constructed by a standard method. Here the abovementioned decay estimates play a crucial role. In studying the linearized equation, we consider the eigenvalue problem (L-i&)$=R$, depending on a parameter E E R .

We prove the asymptotic expansions

in a neighborhood of E = O . This agrees with the result of Ellis-Pinsky [ l ] for the linearization of the Boltzmann equation with cut-off hard potentials. Our proof of this fact follows their lines but more lengthy computation is needed in showing the C"-dependence of eigenvalues and eigenfunctions on 5 . This is explained by the fact that Kac's caricature is a one-dimensional model of the Maxwellian gas.

S 2.

Linearized Equation In this section we consider the linearized equation of (1.3), namely,

Kac's Model of the Boltzmann Equation

665

L is usually called the collision operator and is defined formally in the preceding section. We set B = - v ( d / a x ) + L . Let L2(R,) be the Hilbert space of square-integrable functions on R . We denote the inner-product and the - ) and 1.1, respectively. It is known that L is a bounded symnorm by metric operator in L2(R,) with the complete set of eigenfunctions h,(u), n= 0, 1, formed by the Hermite functions. Recall that ( 0 ,

...,

h,( v) = exp (- u2/4)H,(u ) =( - 1)" exp ( ~ ~ / 4 ) ( a /exp a x )( -~ u2/2) , and that

\

m

for j f k ,

h,(u)h,(u)dv=O -m

=j!(2i~)l/~ for j = k

.

The eigenvalue of L corresponding to the eigenfunction h,(u) is given explicitly by 1,=0,

(2.2)

\'

A,=

(sin" ~ + c o s *0- I ) I ( B ) ~ B

for n> 1 .

--E

See Kac [3] or Griinbaum [2] for the proof of these facts.

Let

.=\= Z(O)d0. -77

Then we have the expression L = - v + K , where K is a compact symmetric operator. It is easily seen from (2.2) that I,=Rz=O and that 1,<0 for nfO, 2. Hence L is negative semi-definite. The nullspace of L is spanned by h,(v) and Mu). By performing the Fourier transformation with respect to the variable x , we get from (2.1)

df=(-iEu+L)f,

(2.3)

at

where

f = f ( ~u ),= (2n)-'/2 We set g(E)=-iEu+L.

\

m

e-izy(x, v)dx. -m

9(6(E)), the domain of k(f),is taken

to be maximal.

666

Y.SHIZUTA and H. N I S H I Y A M A

Then 3(E)is a maximal dissipative operator for any E E R. Hence h(()is the generator of a contraction semigroup on L2(R,). We may regard B(E) as a perturbed operator of A^(E)= -iCu-v. Since the perturbation is a compact operator, we can study the spectrum of j ( E ) by using a generalization of Weyl's theorem. See Kato [4]. In particular, if EfO, the essential spectrum of j ( t )is the set of complex numbers R such that ReA=-v. We see also that all the eigenvalues of &(E) lie to the left of the imaginary axis, if (#O. When 5-0, any eigenvalue to the right of ReR=-pp/2 converges to zero. Here p=minn+o,2 (-An). Set 2ni for E sufficiently small, where C is a small circle centered at the origin and is positively oriented. By the second resolvent equation

-A^( 6)) +( R - A^( E))-fK(R-B(E ) )

(1- B(E ) ) - 1 =(1

-1

-1

,

it is shown that P(E) is a continuous function of E in the sense of operator norm. Consequently, dim P(f)L2(R,)=2 for E small enough. We obtain also a bound of the resolvent R(E, 1 ) = ( R - & E ) ) - 1

.

The precise statement of this fact will be given in the next section. Now we turn to the eigenvalue problem

&E)f=V.

(2.4)

We shall prove the local existence and differentiability of two eigenvalue branches satisfying 17-0 when E-0.

Theorem 2.1.

There exists a neighborhood U ojE=O and junctions

-

I,:

u- c

4,:

U

L2(R,)

(j=O, 2) (j=O, 2)

, ,

such that ( i ) A,(O)=Ofor j = O , 2. ( i i ) ( R j ( E ) , Cj(E)) satisfies (2.4) for U and j=O, 2. (iii) (aA,/aE)(O)=O for j=O, 2. Let 1,(E)=E2Cj(E). Then &(E) is a C"junction on U for j=O, 2. Furthermore, C,(O)
Kac's Model of the Boltzmann Equation

667

Proof. We set p*=min {1, p}. Recall that p=min,,,,, ( - i n ) . We denote by P the orthogonal projection onto the nullspace of L, namely, Pe,=e,

=O

for n = O , 2 , for n#O,2

Here en=e,(v) is the eigenfunction of L , obtained by normalizing h,,=h,(u). We rewrite (2.4) as

f=(P-L+ivE+R)-'Pf,

(2.5) assuming that R e R > -p*/2.

We substitute Pf=Coeo+C,e,

into (2.5) and then form the inner-product of (2.5) with eo,e,. Co=Co(G(E, 4eo, eo)+Cz(G(E, 9 e 2 ,e,)

(2.6)

We obtain

,

C,=C,(G(E, 4 e 0 , e,)+C,(G(E, Alez, ed'

Here

We set

A nontrivial solution exists if and only if

W E , R)=det (DjAE,4 ) = 0

(2.8)

.

We substitute i = e z C into (2.8) and compute each entry as follows.

The last term equals (G(E,E2C)ive,-G(0, O)ive,, e,)

= -E(G(E, E'C)(iv+EC)G(O, O)ive,, e,)

.

Y.SHIZUTA and H. NISHIYAMA

668

Here we used v e , = l / ~ e , - , + l / n f e , + , .

It follows that

Djk(E,E2C)= -E2C(G(E, E2C)ej,e,)

.

+Ez(G(5,S'C)(iv+EC)G(O,O)iue,, e,)

Let us set DjAE, E") =E"jAE,

(2.9)

6) .

Then the equation (2.8) is equivalent to the equation

W E , 1:)= det (MJE, 5 ) )= O .

(2.10)

When E=O, this reduces to the equation (2.11)

(tl-+- 3c+-=o

3

M ( 0 , C)=C'-3

.

2123

The two roots are

We apply the real implicit function theorem It is easily seen that z,
2j(E)=E2Cj(E) ,

i=O, 2

.

Then these functions satisfy (2.8). Now we wish to prove that the E-dependence is actually C". To this end we show that each entry M,,(E, z ) is a C"-function of (E, z ) . We set T(E,1:)=G(E, EzC) for convenience. Lemma 2.2.

T(E,c)e, is a C"-function of (5, C) for n=O, 1,2,

*

- -.

Proof. First we observe that T(E,5) is a strongly continuous function of

( E , 5). If E f O , T(E,C) is a C"-function of two variables in the operator norm. We give the explicit form of the partial derivatives of T ( f ,5). Set Fl =Fl(5,5)= -iv- 251: , F$=F3(E, C)=-2C F,=F,(E,C)=--2 .

9

Fz=F,(E, 5)= - 2E , F,=F,(E, 11)=-E2 7

Kac’s Model of the Boltzmann Equation

669

Then (2.13) Here M is a n integer depending on m, n. N ( s ) is a n integer depending on m,n, s, where l < s < M . {a,(s),a&), .} is an ordered N(s)-tuple of integers depending on rn, n, s and each a,(s) equals one of the numbers 1,2, 3,4, 5. The product of operators appearing in the summation on the right side of (2.13) should be formed according to the order 1,2, -,N ( s ) . The expression (2.13) is proved by induction with respect to m and n. Next we note that

..

--

converges to a limit as E-0, C+z, where z E C . essentially the recurrence formula (2.14)

w,= z/Te,-,

+ z/n+e,+

In proving this, we use

.

The limit may be written formally as

to which a precise meaning is given by using the relation (2.14). This completes the proof of Lemma 2.2 and hence the proof of the assertion (iii) of Theorem 2.1. We need to check the construction of the eigenfunctions #o(C), $z(E) corresponding to the eigenvalues J o ( ( ) , Rz(C), respectively. We solve the homogeneous linear equations (2.7) where R is now replaced by I j ( E ) , j = O , 2. Note that R o ( E ) f J , ( E ) for CfO. We get a nontrivial solution C,,,,(t), C2,j(E), j=O, 2. Then we substitute p f = C Oj(E)eo+ , C2,j(C)e2

into (2.5) and obtain the eigenfunctions f,(E), j = O , 2. The E-dependence is clearly C”. We normalize these eigenfunctions as follows. Set

Y. SHIZUTA and H. NISHIYAMA

670

we have

( $ J ~ ($Ej )( -, 5 ) ) = l

Now l e t j f k and l e t j , k=O, 2. I j ( 5 ) ( $j ( 5 )

9

$k(

for i = O , 2

.

Using B*(C)=B(-E), we obtain - 5 ) )=I k - El($ j ( E ) ,

$ A -E ) )

*

But, I , ( - Q = I k ( ( ) . To see this, we note that b(5,I ) = D ( - E , 1). Therefore, if 1 is a root with parameter 5, then 1 is a root with parameter - E . The only It follows that possibility is lk(-E)=x(3. Vj(E)

-I A E ) } ( $ j ( E L

$A -E ) ) =o

.

Since I j ( 5 ) # I k ( [ ) . We conclude that ($j(5),$*(-El) =O

for E small enough, if j f k .

Thus the proof of Theorem 2.1 is completed.

Remark. We note that b(5,I)=D(E, 1). Consequently, if I is a root with parameter 5, then 1 is also a root with parameter 5. It follows that I o ( E ) , I,(E) are real functions.

S 3.

Decay Estimates First we give the following estimates for the resolvent R(E, 1).

Proposition 3.1. 6 > 0 such that

(i)

For each a>O small enough, there exists a constant

IINE, IIl<Ml holds for any (5, I ) satisfying IEl<6, 111 >a, Re 12 - p / 2 . Here M I is a constant not depending on E, I . ( i i ) For each 6 > 0 , there exists a constant j3>0 such that

II W E , 4ll I M, holds for any ( E , I ) satisfying l E l 2 6 , R e , ? - j 3 . pendent of 5, 1.

Here M , is a constant inde-

Proof. See, for example, [6] for the proof. Note that the corresponding fact is well-known for the Boltzmann equation with cut-off hard potentials. We use now a result from the semigroup theory. The semigroup etB^(E)

Kac's Model of the Boltzrnann Equation

671

is represented as the inverse Laplace transform of the resolvent R(E, A). Although the integral is not absolutely convergent in general, we may shift the path to the left by virtue of Proposition 3.1. Let 6 be sufficiently small. Then, for IEl<6 and f E S ( B ( E ) ) , we have the expression

For IEI 2 6 , f € B(g(E)),we have (3.2) Note that a, ,8 are the constants given in Proposition 3.1. The integrals on the right sides of (3.1), (3.2) are bounded in norm by Cle-utlfl and C,e-Ptllfl, respectively. Here C,, C, are constants not depending on E, t . The second term on the right side of (3.1) is estimated by using Theorem 2.1. Let f € L Z ( R , ;L2(R,)) and let m

f(E, 0 ) = ( 2 n ) - ~ ' ~ \ ciZff(x,v)dx . --m

Then the mapping f-Ff is a n isometric isomorphism from L2(R,;L2(R,)) onto We define the semigroup etB by L Z ( R ~L2(R,)). ; n

(etBf)(E,. ) = e t i ( ~ ) f ^-1 ( ~., We may also define etBin Sobolev spaces. Definition 3.2. ( i ) We define Hc to be the space of square-integrable functions of the variable v taking values in the Sobolev space H c ( R , ) . The norm in H , is given by

J-w

J-w

( i i ) Let L2(R,; L'(R,)) be the space of square-integrable functions of the variable u with values in L1(R,). We define ELto be H,nL2(R,; L1(Rz)). The norm in EL is given by the sum of the norm in Hc and the norm in L2(R,; L1(R,)), and is denoted by [I 11 EL. The following decay estimates are obtained in Kac's model.

Theorem 3.3. ( i ) Let 1 2 0 and let f E EL. Then

IIe t B f l l c C,llfllEc(l l +t1-l" . Here the constant C, does not depend on t , f.

Y.SHIZUTA and H. NISHIYAMA

612

( i i ) Let 1 2 0 and let f € E l . Let, furthermore,

irn

$ j ( v ) f ( x v)dv=O ,

(a.e.

XE

R)

,

--m

for j=O, 2.

Then lleCBfllrlCzIlfIIEL(1+t)-5'4*

Here the constant C, does not depend on t , f .

S 4.

Existence of Global Solutions

In this section we give a n outline of the construction of the global solution for the initial value problem (1.3), (1.4). First we note that the quadratic operator f , f)is a bounded operator. The proof is given in Grunbaum [2]. We have

r(

I W , 911 I 2 v l . f I lgl for f , g E L2(R,). This is easily generalized to the following proposition.

Proposition 4.1. Let 12 1 and let f,g € H , . Then ( i 1 l l w - 7 g ) 1 1 1 ~ c ~ l l f lIlSlll. ll ( i i ) Ilr(f, g ) l l L z ( R u : L 1 ( R z )i) c 2 l l f l l l /\g111. Here C,, C, are constants and C , depends on 1. We construct the solution to (1.3), (1.4) by iteration. we rewrite the equation as follows. (4.1)

For this purpose,

s:

f(t)=etBfn+ e ( c - s ) Bff ( s ) ,f(s))ds.

Set f,=ecBfo,

f , = e"fo+

1'

e ( c - S ) B r ( f n - l f,-,(S))dS (s),

7

for n 2 l . We define X to be the space of H,-valued continuous functions of the variable t 2 0 . For f E X , we set

lllflll =sup t>O

+

-

( 1 t)1'411f(t)lll

Then the sequence of functions defined by (4.2) converges to a limit in X as n+m, if f , E E , and furthermore (1 f [IEL is sufficiently small. This can be proved by repeating the arguments by Nishida-Imai [ 5 ] . The limit function

Kac's Model of the Boltzmann Equation

673

is t h e desired solution. To show the strong differentiability of the solution with respect t o t w e introduce auxiliary function spaces. W e define G, to be orm the space of functions such t h at ( ~ + ~ Z ~ ) - ~ T~h eC nH ~ in - ~G,. is given by

T h e final result is the following theorem.

Theorem 4.2. Let 12 1 . Then there exists a constant E > 0 depending on I such that, i f f , E El and l \ f o l l E c < E , the solution f f o r the initial value problem (1.3), (1.4) exists a n d f C C o ( [ O ,w ) ; H , ) nC1([O,0 0 ) ; G,). Moreover we have the decay estimate

where C is a constant depending on 1.

Remark. T h e local existence theorem for (1.3), (1.4) can be proved by th e st a n d a r d method. References [ 1 ] R. Ellis and M. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl., 54 (1975), 125-156. [ 2 ] F. Grunbaum, Linearization for the Boltzmann equation, Trans. Amer. Math. Soc., 165 (1972), 425-449. [ 3 J M. Kac, Foundations of kinetic theory, Proc. Third Berkeley Sympos. on Math. Statist. and Probab., 1954/55, Vol. 3. Univ. California Press, Berkeley and Los Angeles, 1956, 171-197. [ 4 1 T. Kato, Perturbation Theory for Linear Operators, Springer Verlag, New York, 1966. [ 5 1 T. Nishida and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation, Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976/77), 229-239. [ 6 ] H. Nishiyama, Convergence of approximate solutions for Kac's model o f the Boltzmann equation, Hiroshima Math. J., 15 (1985), 1-25.

Department of Mathematics Nara Women's University Nara 630, Japan

This Page Intentionally Left Blank

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 675-684 (1986)

The Initial Value Problem for the Equations of the Motion of Compressible Viscous Fluid with Some Slip Boundary Condition By Atusi TANI Abstract. We consider an initial value problem for a flow of a compressible viscous fluid with some slip boundary condition in a domain Q c R 3 . We assume that Q is a bounded or unbounded domain whose boundary r belongs to the class CZtu, a E (0, 1). Our aim is to prove the unique existence (locally in time) of a classical solution of the problem in Holder spaces. Key words: compressible viscous fluid motion, slip boundary condition

S 1.

Introduction and Main Theorem

Compressible viscous isotropic Newtonian fluid motion is described by the system of differential equations:

(xDp--pv.u,

[p, density; u, velocity; 0, absolute temperature; f , outer force; p=p(p, O ) , pressure; S=S(p, B ) , entropy; p=p(p, O ) , coefficient of viscosity; p’=p’(p, O), second coefficient of viscosity; K = K ( P , O), coefficient of heat conductivity; D= D ( u ) , deformation tensor whose elements are given by D j k = (1/2)(V,u,+V,~,), j , k = l , 2, 3; D/Dt=a/at+(u.V); V=(V,, V p ,V 3 ) , V,=a/ax,, j = 1 , 2, 3; D : D= D,,D,,; I,, identity matrix of degree n]. Here and in what follows we use the well-known notations of vector analysis and the summation convention. And we should refer to [3, 41 for notations not stated here explicitly. We have already considered various initial-boundary value problems for

Received May 2, 1985,

A. TANI

676

(1.1) ([3, 4, 51). The aim of the present paper is to establish the unique solvability, local in time, of the following new initial-boundary value problem for (l.l), which has been described in [6]: (1.2) (1.3)

( p , v, 0 ) I,=o=(po,

v.n=O

vo, Oo)(x)

9

(some kind of slip condition),

Pn.T=O

and one of the classical boundary conditions for the temperature

0=0,

(1.4)

or

rVO-n=re(O-O,)+g

[n=(n,, n,, n J , unit inner normal vector; r , unit tangential vector]. So far as the present author knows, there is no investigation of exact, time-dependent flow with slip condition for both compressible and incompressible viscous fluid. It is to be noted that the argument in [6] is somewhat incorrect, so that we give in this paper a rigorously correct and complete proof of it. Our main theorem is as follows: Theorem.

Suppose that

( i ) Q is a bounded or unbounded domain in RS with the boundary ing to W 3 + na~ , (0, 1);

r belong-

( i i ) ( p o , v o , ~ ,G'~+~(O)X )~ ~ + ~ ( 8~ )' x~ ( O 8<)-p o,~ p o ( x ) S , i j 0<8,< o, oo(x)Sgo(po.po,@, and Go are all constants); (iii) 0, E cgz+a,i+a/z =,t (r,ErX [O, TI), r e ,o,, g € w:fp,""/"(r,), r,>O ( T > O ) ; (iv) f E 'G::;/z(eT-i2x[O, TI); m ) x (0, > 0, 2pf( v ) (,% p', K , P, S)(p, 0) E V'+"((o, P , r , So ( aS/a@) 3 ~ ~ 2 0 : (vi) the compatibility conditions between the system of (1.1) and the initialboundary conditions (1.2)-( 1.4), which we omit to write down because of clearness, are valid. Then there exists a unique solution ( p , v, 0 ) of (1.1)-(1.4), which belongs to [B'+"(&J)n { p " Z p ( x , t ) >O)l

x w:f:.l+a/y(QTf)x [ ~~?:f~~~+~/~((e,~) n { O * Z O ( X , t)>o}] (p* and 0" are positive constants) f o r some T' E (0, T I .

Remark. The initial-boundary value problem for (l.l),(1.2), the other slip condition (cf. [l]) (1.5)

v.n=O,

v-r=KPn-r,

K>O

,

and the general singular boundary condition for the temperature (1.6)

r,(O-O,)-(l-r,)VO-n=g

,

Osr,Sl

Slip Boundary Problem for a General Fluid Flow

671

will be discussed in the forthcoming paper [7]. (1.3) is a singular case of (1.5) in the sense that (1.3) corresponds to the case K = m in (1.5). In the present paper we use the characteristic transformation h':,$,,: 0, onto QTby virtue of (1.3)l and is given by the formula

(x, t)w(xo(x, t), to=& which is one to one from

x=xo+

(1.7)

5:"

C(X0,

r)dr ,

O(Xo,

t0)=17:,~,,v(x, t ) ,

for only the first equation in ( l . l ) , differing from the previous ones 13, 4, 51. Then the equation ( 1 . l ) I is uniquely solved by p ( x , t)=n::;"P(xo,

(1.8)

to)

p(x0,to)=po(xo)e w [

,

- ~ i o v r ~ ( Txwo] , ,

where 17Ev;t0 is the inverse mapping of n:;It,,, V:= C V , V=(V,, V,, V 3 ) , V,= a/axo,,( j = l , 2, 3), 9=(g,,)=(ax/i3x0)-l. Hence the problem (1.1)-(1.4) can be reduced to the following initial-boundary value problem with respect to w= q ~e ),- w o , wo=t(uo,oo): D

t , w; V ) w + Q ( x ,

(1.9) I

1

or

in

QT

w/,=o=o

on 9 ,

B ( x , t , w ; V ) W = $ ( X ,t , w )

on

n,V,f6,,n,V,-2n,n,n,V, Bik=d

r, w)

O

m,V,

( j = 1 , 2 , k= 1 , 2 , 3 ) ,

(j=k=4) ,

O

,

rT,

D

0

0

A. TANI

678

with p and (v,O)replaced by (1.6) and w+w, respectively. Therefore it suffices to solve the initial-boundary value problem (1.9). And we only give the proof of the main theorem concerning the boundary condition (1.4)l, since we can prove such quite to the other one (1.4)2.

S 2.

Linearized Problem In this section we consider the following linearized problem of (1.9):

l"j = *( x,

I,

w ; V)$+Q(x, t , w )

in

QT

,

on 9 , on r T .

*d=0

b ( x , t , w ; V)@=$(x,t , w ) Here w is a given function belonging to

9T {w -

<&Z+u,l+a/? z,t

(QT);

wIc=o=O, l \ w l I ~ ~
}9

where M I is a n arbitrary positive number and M , is a positive number determined later. 2.1.

Parabolicity and complementing condition

Lemma 1. The system of differential equations (2.1) is uniformly parabolic in the sense of Petrowsky with modulo of parabolicity 6 i f we take T in such a way that M~T< SO

9

0 <~

3

{ ( =M I +

Iluollg')T)/{l- ( M I +

II vn11:))TI < ~o

9

M ~ T <1

where Mnis a positive root of the equation 1 -3x-6x2-6x3=0. Proof. Since det [ M ( x .t, w ; iC)-JZ41=(J+

IE12/al)2(J+IE12/adJ+ IE12/a4)

( a ,=a, =PIP, a, =p/(++p'), and (1.7) and (1.8) imply the estimates

I? ? \

119

it is sufficient to take F in such a way that

a, = p W d d

Slip Boundary Problem for a General Fluid Flow

Here for

u E { K , p , S8},we

679

use the notation

a = m a x o(p, 0) SP,O

,

g = m i n ~ ( p19) , 9P.O

%,e=[po exp [-3(l+3C,)M8Tl, Po exp [3(1+3C,)M8TlI X [@o-M1T,@o+M,T] . Q.E.D.

Now we proceed to check the complementing condition in the case of

Q=R:, r={xS=O}, which is crucial in our investigation. Lemma 2.

any

There exists a positive constant 6’ smaller than 6 such that for

t’=(el,5,) E R2 and any v € C1 satisfying

(2.3)

Re v

z -6’I5’lZ ,

[{’I4+

I Y ~ ~ > ,O

the row vectors of matrix B ( x , t, w; i f ) g ( x ,t , w ; i5, v ) ( ( x , t ) E rT, fixed) are linearly independent modulo M = n : = , (E3--5:(?)), where M ~ tX , w; , ie, v ) is an adjugate matrix of M ( x , t , w ;ic)-u14, and E;(‘)’s are the roots in f 3 of det [ M ( x , t , w; it)- vZ4] =0 with positive imaginary parts. Proof. The roots

tT(r)(5’,v ) of det [ M ( x , t , w; iE)-v14]=0 are given by

In the present case, the boundary operator B ( x , t, w; i f ) i s

0

O1

If we denote by C:=, a ( s ) f i -the l remainder term when we divide B(x, t , w ; i f ) . M ( x , t , w; i f ,v ) by M , then a ( 4 ) = ( a j i ) ) 1 s jis , kgiven s, by the following formulae: h

A. TAN]

680

(2.4)

a::)=

If v = O , then we define a(4)as the limit value of the where u,=p/(p+p’). above formulae as v+O. After considerably lengthy calculations, we obtain (2.5)

det a ( 4 )

= p Z ( ~ , E ~ a , 3 ~ 4 3 ) - l ~ ~ ~ 1 ~ ~ ~ ~ 3 ~ ( ~ ~ ~ l ~ + ~ ~ ~ 3 ~ ) 3 (

x (E; (1) -E; x (EJ ( 4 ) -E,

(4))2((E:

(3)

(l))z(,e;

(4)

-E; -E,

(1))(6; ( 3 )

(3))

-E;

(4))

.

Since B,>O ( r = l , 2, 3,4) follows from Lemma 1 and the condition (2.3), it is obvious that det > 0. Hence it is sufficient to take 6’ as a n arbitrary positive constant smaller than 6, but for later purposes we take 6’ in such a way that

a’=-

1 min {us,u4,6) 4

.

Q.E.D.

Poisson kernel and Green matrix of R3,. Poisson kernel HI and Green matrix H , of R3, are defined in the same manner as those in [3, 4, 51: 2.2.

x B b , t , w ; V)Z,(Y,-E, ro’-r0; x,

,

I;w)IYo,3=od~o’

where r+ is a contour enclosing all El(‘)( r = 1 , 2 , 3 , 4 ) , a 4 = ( a i j * r )is) the inverse matrix of a(’) and 2, is the fundamental solution of the system of equations

aw

--=.P/(x,

ar

t, w;V J W

Slip Boundary Problem for a General Fluid Flow

681

By Lemma 2.5 in [ 5 ] , we have the following estimates of a4 for any q)€{Re q 2 -p,IIm 41, 177’1 5p8(lE’14+lqlz)/1’4,15’14+lq1z>O) (for B, and &, see Lemma 2.5 [ 5 ] ) , (C’=E’+iv’,

(otherwise) . Tracing the proof of Lemma 3.14 in [3], we obtain

Lemma 3.

(otherwise) ,

I Dr7D,’H0I(y,r ; E, r,,)5 C 3 ( ~ - ~ 0 ) - ( 2+ r3 )t/ 2’ exp s ’ ~ - ~ l ~ - E l ~ ./ ~ ~ - ~ ~ ~ l 2.3. Solution of (2.1) Similarly to [ 3 . 4 ] , we can obtain the following lemmas concerning the regularizer R of the problem.

A. TANI

682

where

1

’Rp$ = -

1: SKI dr,

Hick”)(a-jj’,t-ro)c(k”)(jj’)$(jj’,ro)dy’ .

where C, ( 2 1) and C, increase monotonically in T and M , . C+O as T+O, and N = N ( T ,M , , M , ) increase monotonically in T , M I and M,. Returning to the problem (2.1), we need to evaluate .@ and q5. seen that E W3+, implies the estimates

r

$411F,fa)

11$i,

9

1 \ 6 9

$311?;a)

sc8

.

From (1.7), (1.8) and (2.2) it follows that (l-ta)

+

IIPllQ,

5 CAT7 Ml) CIdT, MJM2

1 1 ;9 : 11

5 C,(T, Ml)+ ClO(T9 MAM2

7

hence 9

It is easily

Slip Boundary Problem f o r a General Fluid Flow

683

S 3. Nonlinear Problem (1.9) We construct the sequence {w,(x, t ) } of the successive approximate solutions as follows :

t

w,(x, t)-O€G, , w,(x, t ) is defined as a solution

* of (2.1) assuming W = W , , - ~ E ~ ~ .

Then the result in § 2 implies that w,(x, t ) uniquely exists and belongs to G T ,n=O, 1,2, -. Applying the estimates in $ 2 to the equation concerning W " - W ~ - ~we , obtain

.

(3.1) where C,,-O as T-0. Therefore the sequence {w,(x, t ) } converges to some function w(x, t ) uniformly if we choose T'E (0, TI so as to satisfy C,,(T', M,, M , ) < 1. The uniqueness of the solution of (1.9) is proved by the fact that the difference of two solutions supposed to exist satisfy the inequality analogous to (3.1). The positivity and boundedness of p and 6 are obvious from our construction method. Thus the proof of our main theorem is completed.

A. TANI

684

References [ 1 ] J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik, 8, Springer, 1959. [2]

B. A. COJIOHHWKOB, 0 KpaeBbIX

a a ~ a q a xAJIR JIWHeMHLIX

[3]

[4] [5] [6] [7]

o6qero

naph6onusec~uxCWCTeM

MaT., 83 (1965), 3-162. A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. RIMS, Kyoto Univ., 13 (1977), 193-253. -, On the free boundary value problem for compressible viscous fluid motion, J. Math. Kyoto Univ., 21 (1981), 839-859. -, Two-phase free boundary problem for compressible viscous fluid motion, Ibid., 24 (1984), 243-267. -, The initial value problem for the equation of motion of general fluid with some slip boundary conditions, Keio Math. Semi. Rep., 8 (1983), 77-83. -, The initial value problem for the equations of the motion of compressible viscous fluid with general slip boundary condition, in preparation. AW@@epeHqWaJIbHbIXYpaBHeHWZi

Department of Mathematics Keio University Yokohama 223, Japan

BWJIB, TpYAbI

Index A

a priori estimate, 40, 100, 118 absolute Maxwellian, 40 activator-inhibitor, 158 ADE (Alternating Direction Edition), 296 ADENA-I, 299 ADENA-11, 303 ADENA computer, 299 AD1 method, 294 adiabatic exponent, 460 adjoint, 61, 78 advection, 507 aggregating phenomena of population, 385 Airy, 439 alternative theorem, 601 approximate solutions, 591 artificial viscosity, 393 astrophysical contexts, 461 asymptotic behavior, 58, 111, 112, 387, 403, 481 asymptotic expansion, 371 asymptotically stable, 82, 83 attractive, 248 attractive basin, 248 augmented Lagrangian method, 45 1 B

Banach scale, 90, 91 barotropic gas, 98 barycentric domain, 347 bifurcation analysis, 145 bifurcation point, 636 block Jacobi method, 294 blowing-up problem, 419 Boltzmann equation, 38, 663 boundary condition, 41, 645 boundary layer, 90

boundary operator, 41 boundary value problem for stochastic differential equation, 597 Brownian motion, 599 B(u, u), 420 B(u, u), 420 E(u, u), 420 B(u,u), 420 buffer memory unit, 299 Burgers equation, 11 1

C canonical equation, 27 canonical one form, 26 Cantor’s ternary set, 259 Cauchy problem, 40, 57 Cauchy-Kowalewski theorem, 40, 50 cavity flow problem, 454 cell Peclet number, 323 central difference scheme, 227, 269 C,,-group, 48 chaos, 222, 227, 239 Chapman-Enskog expansion, 43, 94 characteristic equation, 46 characteristic (integral) curve, 75 circumcentric domain, 349 class I, 168 199 class 11, 169 200 classical solution, 50, 70, 92 codimension, 621 collision cross section, 38 collision integral, 663 collision (summational) invariants, 39 compact, 601 competition system, 240 compressible Euler equation, 30, 44, 93 compressible Navier-Stokes equation, 44, 94

686

compressible viscous flow, 481 compressible viscous fluid, 675 compressible viscous and heatconductive fluid, 481 condensation of singularities, 26 1 conservation law, 39, 40 constructed by Oleinik, 650 constrained Lorenz-like attractor, 627 constrained system, 608, 624, 629 contraction mapping, 549 contraction mapping principle, 49, 381, 548 convergence of cutoff approximation, 56 critical eigenvalues, 193 critical line, 2 15 crosswind diffusion, 332, 339, 353 C,-semigroup, 57 cubic convolution nonlinearity, 543 curvature, 632 cutoff assumption, 44 cyclic reduction method, 288 D

D”,174 D: ( E , a), 168 de Rham’s functional equations, 269 decay estimate, 64, 72, 670 decay property, 583, 590 decay rate, 112, 584 decomposition of a stationary solution, 407 degenerate singular point, 241 densely defined scattering operator, 544 difference inequality, 590, 592 diffusion-induced instability, 147, 159 diffuse reflection, 41, 81 dimensionless Prandtl equations, 647 discrete maximum principle, 326 dissipativity conditions, 583 doubly asymptotic, 229

Index

E

effective hyperbolicity, 13 effectively hyperbolic, 1 1, 15, 17 eigennilpotent, 60 eigenprojection, 60 eigenvalue, 58 energy, 583 energy estimate, 119 energy identity, 589 energy inequality, 587 energy method, 113, 587 energy space, 543, 544 essential spectrum, 58 Eulerian coordinate, 1 16 Euler equation, 460, 632 Euler’s finite difference scheme, 222 evolution of a star, 460 existence theorem, 585 explicit scheme, 282 exterior stationary problem, 48 1,482 F

fast Poisson solver, 296 Fatou set, 248 Fife-Mimura class, 199 finite difference scheme, 123 finite propagation speed of exponential decay, 1 15 fixed boundary problem, 98 fixed point theorem, 465 Flower theorem, 248 fluid dynamical limit, 42 fold-up principle, 174 formal normal form, 241 Fourier series, 71 Fourier transform, 60, 64 fractal, 259 fractal object, 247 free boundary, 63 I free boundary problem, 98, 116 Friedrichs’ scheme, 566 frozen branch, 2 1 1 fundamental equation, 231 fundamental existence theorem for

Index

687

ordinary differential equations, 467 fundamental matrix, 14 fundamental solution, 481, 482, 484

hyperbolic conservation laws, 112 hyperbolic snap-back point, 240 hyperbolic-parabolic type, 112

G

I

Galerkin method, 452 rn,172 Gauss curvature, 22 generalized vector field, 61 1, 612, 613, 616, 619, 623 Gevrey class, 7, 52, 53 ghost solution, 225 Gierer-Meinhardt model, 166 globally in time, 99, 122 global shadow assumptions, 162 global singular assumptions, 163 global solution, 57, 58,69,71,74,81, 89, 98, 672 global solutions near zero, 72 Grad’s angular cutoff, 40 gravitation, 122 Green’s formula, 77, 78 Green matrix, 680 Gronwall inequality, 106

ILLIAC-IV, 291 implicit differential equation, 61 1 implicit function theorem, 668 implicit scheme, 283 incompressible flow, 48 1, 482 incomplete HV-decomposition, 314 incompressible or nearly incompressible media, 445 index, 584 infinitesimal deformation, 614, 619 initial boundary value problem, 42, 74, 80, 98 initial-boundary conditions, 583 initial layer, 90, 91, 94, 371 initial value problem, 108 integro-differential equation, 663 interior transition layers, 200 invariant manifold, 230, 233 inverse power law potential, 38, 52 iteration scheme, 445 J

Hamiltonian, 25 hard ball gas, 38, 40 hard potential, 62 Hausdorff dimension, 274 Henon’s mapping, 225 Herman ring, 249 hidden symmetry, 173 high Reynolds number, 646 Hilbert expansion, 43, 89 Holder quotient, 52 Holder space, 98 holomorphic dynamical system, 233, 247 homoclinic point, 229 Hopf bifurcations, I37 horse-shoe, 229 “hump” effect, 159

J(t), 427 Julia set, 247, 264

K Kac’s model, 663 k-jet, 241 Koch’s curve, 259, 271

L lacunary series, 262 Lagrangian mass coordinate, 116 laminar and pre-separation, 650 laminar, preseparation class, 657 Lane-Emden equation, 478

688

Lane-Emden function, 478 Laplace transform, 60 Lax-Wendroff scheme, 566 le developpement de Friedrichs, 432 le nombre d’Ursell, 443 le theorkme abstrait non-lineaire de Cauchy-Kowalevski, 442 Lebesgue’s singular functions, 270 I’equation de KadomtsevPetviashvili, 431 I’equation de Korteweg-de Vries, 43 1 I’equation deux-dimensionnelle de Boussinesq, 432, 437 les ondes longues d’ampleur h i e de surface de l‘eau, 431 Levi condition, 4, 15 Levi-Lax condition, 4 limiting absorption principle, 77, 84 line method, 118 linearized Boltzmann operator, 62, 81, 83 linear Cauchy problem, 46 linearized equation, 664 linear hyperbolic system of first order, 562 linear initial boundary value problem, 77 linearized system, 48 1 linear Vlasov equation, 374 linear wave equations, 584 Li-Yorke, 222 Lp-Lq estimates, 544, 545 local bifurcation, 169 local bifurcation theory, 167 local Maxwellian, 40, 93 local singular-shadow assumptions, 161 local solution, 44,48, 52, 650 local stability assumptions, 164 local unstable manifold, 228 locally in time, 97 Lorenz attractor, 246, 608, 627 loss of smoothness, 52, 54 L2(R,”)-valued difference equation, 566 L2-stable, 569

Index

M Mach angle, 485 Mach cone, 481, 483 Mach number, 483, 484 main-stream, 645 Mandelbrot set, 247, 251 marginally stable, 639 matrix pencil, 616, 622, 629 maximal covariance, 173 maximum principle, 115, 326, 421 Maxwell distribution, 664 Maxwell equation, 376 Maxwellian, 39 Maxwellian gas, 664 memory bank, 297 memory conflict, 298 MIMD, 284 miniversal, 62 1 Mises’ form, 647, 658 mixed finite element/finite difference scheme, 576 mixed finite element method, 445 Monge-Ampkre equation, 22, 25 monotone matrix, 335 monotone scheme, 391 monotonicity, 77, 326 moving boundary condition, 108 multiple coexistence, 165 multiplicity, 58, 60

N Nash-Moser implicit function theorem, 19, 32 nested disection ordering, 309 Neumann layered class, 200 Neumann slit, 200 neutral, 248 Newton’s method, 247 Newtonian potential, 460 non characteristic Cauchy problem, 13 noncritical eigenvalues, 192 noncutoff potential, 40, 52 nonlinear effect of slip/separation,

Index

58 1 nonlinear hyperbolic conservation laws, 43 nonlinear systems, 19 non-linear wave equation, 25, 543, 583 non-negative, 358 non-negative solution, 357, 358 nonnegativity, 57, 80, 326 nonnegativity of solution, 50 non-standard solution, 42 normal, 248 normal form, 241, 243 normal forms for generalized vector field, 613, 620 normal forms for (ordinary) vector field, 614 normal form problem for constrained systems, 625 number density of gas particles, 38 numerical examinations, 4 12

0 odd-even reduction method, 288 one-dimensionalization, 292 one-dimensional shock profile, 82 orthonormal basis, 597 Oseen’s hydrodynamical potentials, 483 overshoot, 332, 353

P parabolic basin, 249 parallel computation, 279 parallel simulation, 282 parallelization by dimension, 289 parameter dependent problems, 445 Parseval’s equality, 60, 66 partial upwind, 333, 340 Peclet number, 321 penalty approach, 446 perfect absorption, 41 perfect factorization, 3

689

perfect fluid, 631 period doublings, 138 periodic boundary condition, 70 periodic point, 249 Petrov-Galerkin approximation, 342 Petrowsky-parabolic, 428 phase space methods, 5 19 @(u, v), 426 piston problem, 108 Poincare’s inequality, 589 point Jacobi method, 291 point SOR method, 292 Poisson equation, 290, 379, 460 Poisson kernel, 680 Polya’s space-filling curve, 271 polytropic gas, 98 population model, 239 positive solution, 358 positive trace, 14 positive-type matrices, 335 power nonlinearities, 543 Prandtl approximation, 657 predation-mediated coexistence, 129 predatory-prey models of LotkaVolterra type, 129 pressure, 445 pressure gradient, 645 prey-predator, 159, 165 prey-predator system, 240 principal symbol, 18, 20 probability density, 43 probability density of gas particles, 38 progressive wave, 631 propagation speed, 112, 633 (pseudo-) differential operator, 40 pseudodifferential operators, 3, 54, 91 pseudoeigenvalue, 72 pulse, 507

Q quadratic maps, 250 quadrature point, 345

690

R random integral equation, 600 random operator, 601 rational map, 250 reaction-diffusion equation, 246, 507 recovery line, 2 15 recovery of stability, 168 reduced set, 201 reduced solutions, 202 reductive perturbation method, 111 regularity, 70, 584 regularizer, 681 regularizing operator, 3 repulsive, 248 residue calculus, 495 resolvent, 58 resolvent set, 58 resolvent spectrum, 58 reverse reflection, 41 Riemann function, 262 Riemann invariant, 113 Riemann problem, 116 Riesz’ representation theorem, 77,78 Ritz-Galerkin finite element approximation, 323 Rossler’s attractor, 246 R-S scheme, 302 R(a), 201

S S. Russell, 439 S scheme, 302 saddle connection curve, 230, 233 scale of Banach space, 44, 54 scattering operator, 543, 546 Schauder base, 270 scrambled set, 224, 240 secondary bifurcation line, 213 Seelig’s model, 166 selective reduced integration, 453 self-gravitation, 460 self-similarity, 271 semigroup generator, 59

Index

semilinear equation, 357 separation point, 649, 653 shadow method, 168 shadow system, 185 shallow water waves, 443 shock layer, 90 shock wave solution, 116 Siege1 disk, 249 SIMD, 284 sine-Gordon nonlinearity, 543 singularity, 591 singular-shadow edge, 169, 208 singular integrals, 492 singular integral operators, 2 singular integral operators of Calderon-Zygmunt type, 371, 379 singular limit point, 207 singular perturbation technique, 134, 140 singular perturbation method, 168, 51 1 singular wall, 207 slip boundary condition, 675 slip/separation condition, 579 small data scattering, 543 smoothing operator, 49, 55, 68 snap-back repeller, 240 Sobolev embedding, 548, 550 Sobolev space, 98 Sobolev’s theorem on smoothness of composed functions, 469 soft potential, 71 soil-structure interaction, 576 solitary wave, 631 solutions, 507, 583 sound speed, 484 spatially homogeneous case, 42 spectrum, 601 specular reflection, 41, 70 splitting-up method, 31 1 S-solution, 598 stability of the discrete model, 389 stable manifold, 228 stable region, 249 stationary flow, 82, 481

Index

stationary Navier-Stokes equations, 646 stationary patterns, 397 stationary problem, 82 stationary solution, 40, 82, 83 stochastic integral equation of Fredholm type, 597 stochastic integral of noncausal type, 597 Stokes equation, 445 strange attractor, 241, 246 stream function, 632 strictly hyperbolic, 15 strong compatibility condition, 648 strong hyperbolicity, 13 strong solution, 78 strongly hyperbolic, 11, 12, 15 strongly well posed, 20 structure matrix, 274 subprincipal symbol, 14 subsonic, 481 substitution operator, 266 successive approximations, 380, 38 I suction, 645, 653 super attractive, 248 supersonic, 48 1, 483 surface tension, 631 s(x), 426 symmetric hyperbolic, 563 symmetric hyperbolic system, 460 symmetry-breaking bifurcations, 159 symmetry-breaking destabilization, 168

T Takagi function, 261 test function, 78 the <-dependence is actually C", 668 The (-dependence is clearly C",669 topological transversality, 230 total (oriented) curvature, 641 trace operator, 41, 76 trace theorem, 74, 76 translation of a stationary solution,

69 1

40 1 transonic, 481, 483 triangular finite elements, 454 truncated versal family, 245 turing assumptions, 160 U

ultimate symmetry-breaking stabilization, 169 ultimate symmetry breaking bifurcations, 2 13 unfolding, 62 1 unique existence, 2 1 unique extension, 21 uniqueness of solutions, 546 unstable manifold, 228 upwind element, 340 upwind-type finite element method, 319

V vacuum, 98 Van der Pol equation, 626 variation of constants formula, I13 vector pipeline, 284 vector processor, 297 velocity, 445 versal family, 245 versal unfolding, 621, 626 viscous compressible and heatconductive fluid, 97 Vlasov-Maxwell equation, 369 Vlasov-Poisson equation, 370, 379 Vlasov-Poisson limits, 371 vortex, 457

W wake regon, 483, 485 water waves on the beach, 127 weak solution, 42, 77, 78 Weierstrass function, 259, 261, 262 263, 264

692

Weierstrass-Kronecker normal form, 617 weight function, 351 weighted error estimate, 573 weighting function i,b (y), 570

Index

well posed, 13 Z

zero skin friction, 646