Elliptic and Parabolic Problems: Rolduc and Gaeta 2001 : Proceedings of the 4th European Conference Rolduc, Netherlands 18-22 June 2001 : Gaeta, Italy 24-28 September 2001

n

- / Vu*V(fidx+ Ja

/ (h2{u*) + f(-,u*)) (pdx + (b,ip). Jn

(4.7)

Prom (H2) and the boundedness of (un) in Y it follows that (wi, n ) is bounded in Y, and thus there is a subsequence (w-ij) which is weakly convergent in Y. Since X is dense in Y from (4.7) we deduce that each weakly convergent subsequence of (wi |n ) must have the same weak limit, and thus the entire sequence (»!,„) must be weakly convergent, i.e., uii in —*• to*. Prom wi]n(a;) 6 dji(un(x)) and the convergence properties of (wi, n ) and (u„) we easily get W[(x) e dji(u*(x)). Setting w2 •= h2{u*) e dj2{u*) we see that the limit u* satisfies [ (Vu*V

[ {wt + f(-,u*))
(4.8)

where wl 6 djk{u*), k = 1, 2, and thus u* is a solution of the HVI (1.3) according to Definition 2.1. Next we are going to show that u* is in fact the greatest solution. To this end let u be any solution of the HVI (1.3), then u e [u, u], i.e., u < u0 = u, and satisfies -Au + w1 = w2 + F{u)+b

in X*,

(4.9)

where uik e djk(u), k = 1,2. Since w2 < h2(u) < h2(uo), from (4.9) it follows that it is a lower solution of (4.2) defining the iterate ui: and trivially u0 = u is an upper

38 solution. Thus there exist solutions of (4.2) within [u, u0]. However, by definition ux is the greatest solution of (4.2) less than UQ, which shows u < u\. By induction one can show that u < un for all n and thus u < u*, i.e., u* is greatest solution of the original HVI (1.3). The proof for the existence of a least solution u* can be done in a similar way replacing /i 2 by h2 in (4.3) and starting with «o := M, where w„+i is the least solution within [u n ,u]. (b) Compactness of the solution set S By (a) the solution set S of (1.3) is nonempty. Let (vn) C <S be any sequence of solutions, i.e., (»n,iBi]n,iB2,n) £ X x Y x Y such that -Avn

+ whn = w2,„ + F{vn) + b in X*,

(4.10)

where uifcjn e djk{v„). Let c > 0 be a generic constant. Since ||i>n||y < c for all n, it follows that ||wfc,n||y < c f ° r a n n a n d k = 1,2, which by using (4.10) implies ||«n||x < c. Thus there exist subsequences (u,), (u>k,j), k = 1,2, satisfying Vj —*• v in X, and due to the compact embedding X C Y, Vj —> v in y , and w^j - 1 » t in Y with Wk(x) 6 9jfc(v(a:)). Replacing in (4.10) n by j we may pass to the limit as j —> oo, which shows that the limit v is a solution of the HVI (1.3), i.e., v G «S. Finally, the strong convergence of Vj -> JJ i n X follows from (4.10) replacing n by j and taking as special test function (p = Vj — v which yields hi ~ v\\x = I l V ( ^ ~v)\2dx

= - [ VvV(vj - v) dx

+ / (w2tj-u>i,j + f(-,Vj)){vj-v)dx+(b,Vj-v).

(4.11)

The weak convergence of (VJ) in X and its strong convergence in Y along with the boundedness of (wf.,j), k = 1, 2, in Y imply that the right-hand side of (4.11) tends to zero, which yields Vj —> v (strongly) in X, and thus the compactness of S. D

References 1. S. Carl and S. Heikkila, Nonlinear Differential Equations in Ordered Spaces, Chapman & Hall/CRC, Boca Raton, 2000. 2. K. C. Chang, Variational methods for non-differentiable junctionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. 3. F.H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. 4. F. Dem'yanov, G.E. Stavroulakis, L.N. Polyakova and P.D. Panagiotopoulos, Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics, Kluwer Academic Publishers, Dordrecht, 1996. 5. Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995.

39 6. P.D. Panagiotopoulos, Hemivariational Inequalities and Applications in Mechanics and Engineering, Springer-Verlag, New York, 1993. 7. E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. Ill, Springer Verlag, Berlin, 1985.

n an operator equation for Stokes boundary value problems T. D. Chandra Department of Mathematics, State University of Malang, Malang, Indonesia J. de Graaf Department of Applied Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands Abstract The Stokes boundary value problems (SBVP) can be found in many areas of engineering, for example, in fluid mechanics when the Reynolds number is small. This study shows a method of solving the non homogeneous SBVP Av - Vp = - / ,

e

CI,

v-v =-h,

x e

n,

v(x)

x

= a.(x),

x

(*)

E 9fi,

by translating it into an operator equation at the boundary dd of the domain fi. First, we show that the general solution of the system, without boundary condition, i.e.

f Av - Vp = - / , x e n, \ v-u = -h, x e ii, is of the form

v =aH+Vf + V{M (V- an) + M (V- Vf + h)+ 7 w v} , p =-h-V-aK-V-Vf, with an is a harmonic extension of a which denotes a tangent vector field at the boundary dU, 7 is an arbitrary but fixed function, and Af(g), Vf, 7 ^ are respectively defined by AAf(g) =-g, x S U,

^Af(g) =0,

x e on,

A(£>/) = - / , x e n, Vf =0, x e dn,

40

41 and

f MHM =o, x e n, I | ; 7 w =7, a: 6 da Our operator equation relates the (unknown) tangent vector field a to the given boundary condition a. As an application, we present full explicit solutions of SBVP (*) for some simple domains such as the interior of a disk and of a ball.

1

Introduction

In recent years, a number of studies have been made of finding a method to solve the Stokes equations / nAv - Vp = 0 , {la b) \ V-v = 0, ' which usually emerges in studying the steady flow of an incompressible, viscous fluid at low Reynolds number. For example, Sheng and Zhong [1] and Padmavathi, et all [2] mention some solutions such as the Naghdi-Hsu solution, the Papkovich-Neuber solution, and the Boussinesq-Galerkin solution. Inspired on the above ideas, we develop a general theory of solving the non homogeneous Stokes boundary value problems (SBVP) (see [3, p. 31]) fj,Av -Vp V-v

= -/, = -h,

x x

G e

Q, ft,

(2a,b,c)

!

v(x) = a.(x), x € dQ., Briefly, we solve the nonhomogeneous Dirichlet and Neumann problem that leads to an operator equation at the boundary dQ of Q with the unknown is a tangent vector field a at the boundary dQ. We obtain the solution of SBVP (2 a-c) can be 'parametrized' by au, the harmonic extension of a to the interior of Q. Finally, we give some applications of the method to find the solutions of SBVP (2 a-c) for some simple domains.

2

General Theory

On a fixed open domain Q C R " with piecewise smooth boundary dQ, we consider the system of Stokes equations

' "Av ~JP = -{'

(3a,6)

V-w = —h. Here / : O. —• R n , h : Q —> 1R are given functions which depend on the position vector x, which we usually suppress. In the sequel, we put /j, = 1. For some given boundary condition v = a at dQ, we want to solve v and p from (3a,6). From [2, 3], we know that the pair (v,p) is a solution of (3 a, 6) if we put

(v

= * + W,

(4

b)

42 with (4>, ip) any solution of the system A = - /

(5a'6^

A^ + V-tf, =-h.

It will turn out that the boundary condition to the system (3 a, b) is related to the boundary condition of (5a,b) by means of an operator equation. For later convenience, we now split the solution {4>,4>) of (5a,b) into several pieces.

2.1

A Dirichlet problem

Consider the Dirichlet problem Ac/> =-f,

x

4> =<*, where a : dSl —> IRn 4> = Vf + ocn, with

6

SI,

.


{6a b)

x e on,

'

is an arbitrary but fixed tangent vector field at dSl. Next, split •A(vf)

= - / , * € =0, x e

Vf

si, 90,

{7ab) l a 0J

' '

and AaH aw

= 0, = a,

x x

e SI, e dSl.

, g a ft. '

Note that 2?/ and aH can be obtained, at least in principle, by means of Green's function [4, pp. 86-87].

2.2

A Neumann problem

Consider the Neumann problem Aip = -V-<j>-h

= -V-an-

V-Vf-h,x

e Q,

{9a b)

f* =7,* e 9 a,

'

on where 7 : dSl —¥ IR is an arbitrary but fixed function. For this Neumann problem, we impose the compatibility condition J jda = - I(V-an Jan Jn

+ V-Vf

+ h) dr = - [(V-Vf Jn

+ h) dr = 0,

(10)

since / V - a « d r = / aHnda = / a n d a = 0. Jn Jan Jan Now split f/>=JV(V-a M )-l-A/'(V-X>/-l-/i)+7*Ar, with AjV(V- an)

lM(V-an) on

(11)

= - V - an, x e SI,

= 0,xedSl,

{Ua b)

'

43 AAf(V-Vf

+ h) =

-V-Vf-h,xeQ,

and kin*

= 0, x

9

e Q, r- an

( 14a > 6 ) v

^ 7 M =7, x e aa.

' '

Note that, because of Aan = 0, one has A(-ix-a„) = -V-a„,

(15)

where a; is again the vector position. The solution for the non homogeneous Neumann problem (12a,6) can now be written Af(V-an)

= -±x-aH+ip(a\

(16)

where ip^°^ is the solution of the homogeneous Neumann boundary value problem f AV>(a) = 0 ,

2.3

x

e

Q,

A parameterized class of solutions

For any tangent vector field a. : dQ 9f2 —> IR" and any function fi 7 : dQ —• 1R, the pair (v,p) with f v =aH+Vf + VM(V-an) { p =4-V.a»-V.D/,

+ VM(V-Vf)

+ VJV(h) + V 7 ^ ,

,..

.,

(18a 6)

'

is a solution to the system (3o,6). The problem we have to solve now is : How are a and 7 related to a prescribed vector field a at dQ ? It will be explained that for a the solution of an operator equation at the boundary has to be taken. This operator equation involves all the data : a, f, h.

2.4

The operator equation at the boundary dCl.

In this section, we derive an operator equation which relates a : dQ —>• IRn to the prescribed data (a, / , h). From (4a), we obtain the requirement v = a = <j> + Vi>, at dQ,

(19)

or a + VAf(S7-an)

= a - VlnM - VM(V-Vf)

- VAf(h), at dQ.

(20)

If we take at each x g dQ, the inner product of identity (20) and the normal vector n, we get 7 = an. (21)

44 Therefore, we obtain the operator equation for an unknown tangent vector field a at the boundary a + VN {V-a-H) = a-V{a-n)nM

-VN

{V-T>f) -VN{h),

at dfl,

(22)

where the right hand side is a tangent vector field on dfl that can, in principle, be calculated from a , / , and h. If we can solve a from the operator equation (22), the solution of the non homogeneous SBVP

{

Av - Vp = - / , V-v

x e

= -h,

x

ft,

6

fi,

(23a,6,c)

D(:E) = o(x), a; e dfl, is given by (18 a,b) or equivalently by v = an + X>/ + V { - | x - a „ + V ( a ) } + VAf(V-Vf) + V.V(/i) + V ( a - n ) ^ , p =-h-V-an-V-Vf. (24a,6) Next, as an application, we present full explicit solutions of homogeneous SBVP for some simple domains such as the interior of a disk and of a ball.

3

Applications

3.1

The interior of a unit disk

Here, we consider the homogeneous SBVP with the domain is a unit disk, i.e. ' Aw - Vp = 0, x £ D = {(a;, y)\x2 + y2 < 1}, V-v =0,x£D, „(*)

= a(x) = ± { a

n

(

- £ $ ) e-

+

K ( $ $ > ) e « } ,„

e

S A

(25a,6,c) with &o = 0 to satisfy the compatibility condition. According to the theory, we have to solve the boundary operator equation (22) a +VjV(V-a M ) = o - V ( a - n ) „ v ,

(26)

since / = 0, and h = 0. Therefore, at first, we study the action of the operator a i—• a + VJV(V-a„),

(27)

on the 'basic' tangent vector fields : cos(6) J t„=

<

" S i n S ),n = 0, -sin(fl) i cos(0) | e

iNg

'

(28) v

cos(0) ' n 6

. '

'

45

Using the notation z = x + iy and z = x — iy, we obtain the harmonic extension of tn

Ki)z("+1)+K~il)z<""1),neN' \{\)Z

+

\{ I

2\ 1 )*M~1)

+

Un =

°'

(29)

)2(H+1).-ne:

l[ I

that leads to 2 tn,

VJV(V-t^)| r = i =

?T. xz .

(30)

0,71 = 0, 2

n

'

71 t -

We conclude the left hand side of the boundary operator equation (26) is a mapping a^a

+ VAf(V-an),

(31)

that maps (" tn i—> \tn, n e Z, n ^ 0, (32) in

tn,n = 0.

Therefore, given the boundary condition a, we can solve the boundary operator equation cos(#) (26) 6) for an. IFor example, given the 'basic' boundary condition a = I • )a\ \ em6,n G we obtain a-V(a.n)w Hence, a , = 2(a - V {a-n) 3.1.1

=U u)

=

1

.)z~»-l(

1

) z ^

(33)

_n+l

Solution

Since / = 0 and ft = 0, we obtain from (24 o, 6) the solution of the homogeneous SBVP is n = Q„+VA/'(V'a 7 1 ) + V ( a - n ) m ,

p=-V-an.

Below, we present a table of solutions for various 'basic' boundary conditions

(34)

46 boundary condition (a)

velocity (w)

-2i(n + l).zn

co8 e

V

pressure (p)

() J +

!l(-i')*""'< 1 -* J >

( -sin(fl) \ e - i n * n e N

2i(n + l)z"

K0-KD« (COS^)e^,neN V

sin

0 - 2 ( n + l)z n

(0) /

+ s i

( i )(!)'-p->

/ COS(0) \

{ sin(0) )

- 2 ( n + l)z n

ine

e

'"

6 P <

'Any' solution (v,p) of the homogeneous SBVP in the interior of a unit disk is a linear superposition of these elementary solutions. Note that for every fixed \z\ < 1 the value of the basic solutions decreases exponentially if \n\ —> oo. This implies that any linear superposition of basic solution with coefficients {cn}, not increasing faster than polynomially, lead to a solution of the homogeneous SBVP on the open disk D. If, by oo

way of example, one assumes that \ J \ncn\ < oo then both the velocity v and the pressure n=l

field p extend continuously to the closed disk D.

3.2

The interior of a unit ball

In this section, we consider the homogeneous SBVP with the domain is a unit ball, i. e. Aw-Vp V-u v(x)

=0, =0, = a(x),

x e B = {(x,y,z)\x2 + y2 + z2 < 1}, x e B, x 6 dB = {{x,y, z)\x2 + y2 + z2 = 1},

with a : dB —• M 3 is the prescribed velocity field at the boundary.

(35a,6,c)

47 3.2.1

Some notations and preliminaries

In this section, we introduce some notations that will be used later. The closure of B is denoted by B = B U dB. For a tangent vector field a, one has for all x 6 dB : ax = 0. We will use fat greek lower case symbols for tangent vector fields. Extensive use will be made of the usual vector algebra and vector calculus in IR3 applying the usual nabla notations. V / = grad/, V-u = divu, V x v = rot v. (36) Also the Euler operator £ = x • V = x^ + y-g- + z-^, and its resolvents (£ + s)~ , s > 0, given by (£ + sy1 f(x) = f X-'HXx) d\, (37) Jo play an important role. For vector fields v the operators £v and (£ + s) _ 1 v are defined componentwise. We mention some commutator properties for £ and its resolvents. They follow from straightforward computation : V£ = V + 5V,

A£ = 2A + £A,

V ({£ + s)" 1 /) (x) = (£ + s + ly1 V / ( x ) .

(38)

Next, we discuss some lemmas. Lemma 1 If a continuous vector function

IR3 satisfies

A = 0 ,

B,

a; G £ , i 6 9B

then

by applying 'rotation' on both sides V x 0 = V x ( « Vy?) = (V-Vtp)x + (V - (x -V)V = -{£ + 2)Vtp. (39) So a candidate for Vip is given by -{£ + 2) _1 (V x )) = V x ( V x (£ + 1)"V) = V (V- ((£ + l)"1*/))) - A(f + l)" 1 ^. = V(£ + 2)" 1 (V-<£)-(£ + 3 ) " ^ = 0. So the vector field -(£ + 2) - 1 (V x
48 3. we conclude that u = Va, with ACT = 0. However also xu oo

= £a = 0. Expand a

oo

in spherical harmonics a = Y^On. Then £a = VJnff n . We conclude that an = 0 if n=0

n=0

n = 1, 2, • • •. Therefore cr must be constant and hence u = 0. Lemma 2 If a continuous vector function 4> • B —)• M3 satisfies A<j>{x) = 0, x € B, 0-x = 0, x e 3 S , i/ien can be written (j> = 4>a + *"i£/i <£0 =

a; x V<£,

(40)

*i = V x - ^ x + KN'-iJv^ + i r ^ x .

(41)

with both if : B —> M orad x ; -B -*• IR harmonic. If we require ip(0) = x(0) = 0, both ip and x are unique. Moreover x satisfies [—(£ + l) 2 + | + \{£ + I)" 1 ] X = V-<£ = V - 0 : . Proof First, construct x from V-0, i.e. x = [-(£ + I) 2 + | + K f + I ) " 1 ] " 1 V " <£> a n d define (^ according to (41). Put (f>0 = qb - <j>v Observe that V-0 O = 0. Next, we obtain x-0 = 0. Further, since A(j>1 = 0, also Acj)0 = 0. Hence, according to the preceding lemma, we obtain q>0 = x x Vip. Finally, note that untill now ip and x a r e fixed up to an additive constant. By taking
= 0 , x e B, = a;i5(a;) + ex, x € 9B, c € IR.

T/ierj to can 6e written w = V£~lfd + ex. Note that c = ^ / 8 f l tu(x) -a; du. Proof Straightforward computation yield Aw = A(V£ _1 i9 + ex) = 0. Next, by taking the inner product w(x)-x at the boundary, one finds x • w = x • V£ _ 1 $ + x • ex = &(x) 4- c, which is equal to x • w = x • xd + x • ex = i?(x) + c. Finally, take the inner product w (x) • x at the boundary, calculate the integral over the boundary, and use the mean value theorem for harmonic functions, to obtain / JBB

wxda

=

x-xi!)(x) da+. JdB

exx

da = 47n9(0) + ATTC,

JdB

-I

c = — / w • x da. 47T , 4?r J In the preceding lemmas, we gave threedE different recipes for constructing harmonic vector fields from harmonic functions. For later convenience we introduce a notation for these recipes.

49 Definition 4 Let hj[ IR, j = 0,1,2, denote the harmonic vector fields defined by h0[ip]

x i-> x x V(/j(x),

hilx]

x M. V X - (£*)x + \{\*\2 ~ 1)V [(£ + \yl£} x M- Vf "1i9.

h2m

x,

Remember that IR3. Let an : B —> M 3 be the harmonic extension of a. Then there exist unique harmonic functions (p,x,$ '• B ^>- TR, with ip(0) = x(0) = &(Q) = 0 such that aH = h0[(p] + h^x] + hzltf] + ex, with c = j ^ JdB(ax)

da. Note that if d = 0 and c = 0, then a is a tangent vector field.

Proof The proof is divided into several steps. 1. Step 1. Split the vector field a into a tangential and a normal component as follows : a = at + an + en, with c = ^ JgB(a• n) da, o„ = {a-n-c)n and at = a — an- en. Note that at is tangential. 2. Step 2. Define $ as the solution of the Dirichlet problem Ai?(x) = 0 , x € B,

•d(x) =an

— c,

xedB.

3. Step 3. Note that for any x e dB, the vector field a - V£ _1 t?(x) — ex is tangential. 4. Step 4. The harmonic extension x i-> aH(x) — 'V£~1'd(x)—cx satisfies the conditions of lemma 2. This means it can be uniquely written as /io[
In our case, V-« = 0 implies

Solution

In a similar way as in the section unit disk, we have to solve the boundary equation (22) (42) and at first, we study the operator a i - ^ a + VJV(V-a„).

(43)

Based on theorem (5), we know that any tangent vector field at the boundary a can be splitted into half] + hi[x]- Therefore,

50 1. if a = h0[x] = x x V
(44)

or at the boundary (|a;| = 1), h0[ip] i—s- /i 0 [p].

(45)

2. if a = /ii[x]||s|=i = Vx - x£x, we get a + VAA(V-a„) = /»i[x]||»|=i + hi [~X + \£{£ + s) _ 1 x] l|.|=i = /.i[|£(£ + i)-1x]lw=i.

(46)

or at the boundary (|ai| = 1), hxM _ > / » ! [ ! £ ( £ + 1 ) " 1 * ] .

(47)

Hence, given the boundary condition a, we can solve the boundary operator equation (42) for aH. For example, given a = fix, with Afi = 0, i?(0) = 0, we have {a-n)nM

= £~xfi,

a-V(a-n)nM

=

fix-V£-lfi

= h1[-£-1fi]\lxl=1.

(48)

2

Therefore, according to (46), an = hi [-2(£ + \)£~ fi]Below, we present the solution of the boundary operator equation (42) for various boundary condition a. boundary condition (a)

Oi-H

x x Vip

x x Vtp

Vx - x£x

hi [2(£ +

\)£-lx]

hx [-2(£ + \)£~*fi]

fix

Next, we make some special choices for the right hand side a of the boundary operator equation. These special choices involve harmonic polynomials of degree m (Qm) on which the Euler Operator £ acts as a simple multiplication (£Qm = mQm). Using solutions (24o,6), v = a7i + VAf{V-an) + VjnN, p = -V-aM, (49) we obtain a table of solutions as following boundary condition (a)

velocity (v)

pressure (p)

x x VQ m

x x VQ m

0

VQ m - mQmx

VQ m - mQmx + \{m + 3)(|ai|2 - l)VQ m

(2m + 3 ) ( m + l ) Q m

QmX - (*j£) {\xf - l)VQ m

(2m+3)(m+l) p

WmX

51 'Any' solution (v,p) of the homogeneous SBVP in a ball is a linear superposition of the elementary solutions. Remarks similar to those after the table in section 3.1.1. apply here as well. For neat (functional) analytic considerations (also for the general case) we refer t o our forthcoming paper (see [5]).

References [1] X.X. SHENG and W.M. ZHONG, General Complete Solutions of the Equations of Spatial and AxiSymmetric Stokes Flow. Quaterly Journal of Mechanics and Applied Mathematics 44 (1991), p.537-548. [2] B.S. PADMAVATHI,G.P. R A J A SEKHAR, and T . AMARANATH, A Note on Complete General

Solutions of the Stokes Equations. Quaterly Journal of Mechanics and Applied Mathematics 51 (1998), p.383 - 388. [3] R. TEMAM, Navier-Stokes Equations. Amsterdam: Elsevier Science Publishers B.V. (1984) 526pp. [4] I. STAKGOLD, Green's Functions and Boundary Value Problem. Toronto: John Wiley &: Sons, Inc. (1998) 689pp. [5] J. DE GRAAF, D. CHANDRA, and R. DuiTS, On A Boundary Operator Equation For Stokes Flow, (to appear).

Local stability under changes of boundary conditions at a far away location M. Chipot, A. Rougirel Institut fur Mathematik, Universitat Zurich, Winterthurerstr. 190, CH-8057 Zurich, Switzerland Abstract We study the asymptotic behaviour of the solution to lineax and nonlinear parabolic problems in cylindrical domains becoming unbounded in one or several directions. In particular we show the local stability of the solutions under changes of boundary conditions at a far away location. This generalizes a previous work in which the data depended only of the cross section of the domain.

1

Introduction

In many physical situations the mathematical analysis starts after noticing that the cylindrical domain where the phenomenon is taking place being large in one direction, one can consider only what happens in a cross section reducing the difficulty by one dimension. We have justified rigorously this reasoning for elliptic and parabolic problems in [3], [4], [5], [6], We refer to [2] where others differential equations are involved, for a systematic treatment of this question. Let us give a simple example illustrating our results for evolution problems. Suppose that n, = ( - £ , £ ) x ( - l , l ) (1.1) where £ is a number that we will let go to +oo. Let U( be the weak solution to the parabolic problem dtut - Aue = f(t, x2) in (0, T) x Qe, ut{t, x) = 0 on (0, T) x dtte, (1.2) ut(0, x) = u0(x2) in tie, (we denote by x = (£1,2:2) the points in R2, dSli denotes the boundary of Clt). Our data / = f(t,x2), u0 = uo(s 2 ) are depending only on the section of fi^— i.e. are independent of Zj. So it is reasonable to suppose that in a neighbourhood of x\ — 0, ui is more or less

52

53 independent of x\. This should be especially true for £ large. Then, if it is the case, in the neighbourhood of 0, u^ ~ «„, where Uoc is the solution to: dtux - aP.tioo = f(t, x2) in (0,T) x (-1,1), uoo(f,a;) = 0 on (0,T) x { - l , l } , Uoo(0, x) = ^0(2:2) in (-1,1).

(1.3)

More precisely, we have proved (see [4] Theorem 2.1 or [2] Theorem 9.2) that, for any given £0 > 0, ue -> Woo in L°°(0,T;L2{Qeo)) n L2(0,T;H\Ctlo)), with a speed larger than any power of l/£. In this paper, we would like to show in addition that the independence of the source term and the initial condition with respect to £ and x\ is, in some sense, not essential. Let us explain this idea reformulating the above problems in the following manner. Let u\ be the solution to dtu) - AuJ = ft(t, x) in (0, T) x Qt, u\{t,x) = 0 on (0,T) x dne, u\(0,x) = uot(x) in Q(,

(1.4)

and uf be the solution to , f dtu\ - !\u\ = ft{t, x) in (0, T) x Qt, u2{t,x) = Q on (0,T)x {-£,£) x {-1,1}, dnu2e{t,x) = 0 on (0,T) x {-£,£}. x (-1,1), ^ u\(0,x) = Mo,(;r) in n^.

(1.5)

Then we will prove, under certain assumptions, that u\ - u? -> 0

in L°°(0,r;L 2 (^ 0 )) nL 2 (0,r;if x (n £ o )).

Note that now the source term and the initial condition depend on £ and x\. This setting generalizes the previous one since, if, in (1.5), fi{t,x) = f(t,x2), u0l(x) = 110(0:2), it is clear that uj = u ^ where u^ is the solution to (1.3). Moreover this local stability result is intuitively clear since the problems whose solutions are u\ and u2 differ only on the vertical parts of <9fi which are rejected at infinity. We will study more general problems than (1.4), (1.5). In the next section we will consider the case of a general parabolic linear problem. In section 3 we will consider a quasilinear case.

2

T h e linear case

Let us consider a bounded open set of R" given by ne=(-£,£)P

xu)

(2.1)

54 where w is a bounded open subset of R™ p , 1 < p < n. For x = (x\, ...,xp,xp+i, R" we will set Xi = (xi, •••,xp), X2 = (xp+i, ...,xn).

...,a;n) €

We denote by du>0 a measurable subset of du and we will assume altough it is not essential that |dw0| > 0. (2.2) Then we set

r 0 = (-e,ey xduo,

r1 = r0ud(-£,£)p

xu>

H^(Qi;Ti) = {ve H1^)

; v = 0 on I\}

i = 1,2.

(2.3)

and We equip these spaces with the norm (recall (2.2))

I M 2 i n i = { j [ \Vv(x)\dx}1/2.

(2.4)

Let Vf, Vf be two closed subspaces of HQ(QI; F0) equipped with the norm (2.4) such that H^Qf, r \ ) C Ve\ V? C Hl(Sl,; T0).

(2.5) p

Note that V} and Vf contain functions not vanishing on (—£,t) x f l w \ <9w0. If fteL^TiH^Slt-.To)'),

(2.6)

where ^ ( f i ^ ; T 0 )' denotes the dual of Hg(Qe; F 0 ), then for a.e. t e (0,T), the restriction of ft(t) to V? defines a continuous linear form on VI, i = 1,2. Thus we get a functional that we still label ft satisfying feeL2(0,T;Vi'). (2.7) We will need in the following to control the growth of fe at infinity that is why we impose the condition lAlLWiffiffW) ^ c ( r + !) W > 0 (2-8) where c and a are positive constants. Remark that (2.7)-(2.8) imply that it holds that I/*ILW;V?')
W > 0 , V t = l,2.

(2.9)

Moreover, as already explain, we allow also a dependence of the initial condition u0t with respect to £ and X\ but we must add the assumption K k n ( < c ( £ Q + l)

W>0.

(2.10)

We consider then coefficients eL°°((0,T) x RP x ui)

aij = aij(t,x)

Vi,j = 1, ...,n.

(2.11)

To simplify the notation we set A = A(t,x) = (aiJ)iJ=l

n

(2.12)

55 and we suppose that for some A, A > 0 it holds that |A(t,a;)| A|£|2 a.e. te (0,T), a.e. i 6 f x w, V£ e R".

(2.13) (2.14)

(|£| denotes the euclidean norm of f, | | the norm of matrices subordinated to the euclidean norm of K.".) Let us set = {ve L2(0,T;Vi)

ffHCri^.V?') 2

I «* £ L 2 (0,T;V7')} •

(2.15)

2

For /f G L (0, T; .Hp (fi<; r 0 )'), u0l 6 L (f2^) there exists a unique u\ solution to (

uieH\0,T;Vi,Vi>), ft(u\,v)2

+J

A(x,t)Vu\Vvdx (fe,v)

I u\{0,-)=u0t{-)

= mV'(0,T),VveVi,

2

mL {Qe).

(We refer the reader to [7], [1] for a proof. X>'(0,T) denotes the usual space of distributions on (0,T), (•, -)2 the usual L2(f^)-scalar product.) Remark that if du>0 = dus then H^QfJi) = H£(£li). Hence, if we choose p = 1, V? = H^{Qt), Ve2 = ^ ( f ^ T o ) then (2.5) holds and for i = 1 (resp. i = 2), (2.16) is the weak formulation of (1.4) (resp.(1.5)). Then we would like to show Theorem 2.1 Under the above assumptions, in particular if (2.5), (2.8) and (2.10) hold then, for any IQ > 0, any r > 0, there exists a constant C independent of I such that Q \u\ - U?|L°°(0,T;L2(J1 < 0 )) + \u\ ~ U?Ua(o,T;ffi(n<0)) ^ Jr'

(2'17)

To prove this theorem we will need several lemmas. Let us first show Lemma 2.1 (Poincare Inequality) Let v be a function of HQ(£II;TO). a constant C — C(w) independent of £ and v such that \v\2,nt
Then there exists

(2-18)

(V^ 2 = {dx +1, •••jC'x„).) See [3] for a proof. We can now estimate u\. We have indeed Lemma 2.2 Leti G {1, 2} andu\ be the solution to (2.16). Then there exists C = C(\,p) a constant independent of £ such that

M2L»(„,T;1»(nt)) + [P4\\lntdt

^ C(fa + !)•

(2-19)

56 Proof of Lemma 2.2. We take v = u\ in (2.16). We obtain for a.e. t £ (0,T) 1 d.

•il^ + /

AVu\Vu\dx

= < l / J ( * ) | . | | V u j l | w

J SI/

where | |» denote the dual norm in VI'. Using (2.14) and the Young inequality we derive

5 | I « J I U + A | | V « J I | ^ < ^ I / , I ; + ^|IV«JI|^

Integrating on (0, i) we obtain for a.e. t \v-m\lne

+ *Jo ||V*4l|2,„,d*< luoM^ + lflMl^dt-

(2.21)

With (2.9), (2.10) we obtain min(l, A) | | u i ( t ) | ^ + ^ I I V ^ H U * } ^

C

^

+

1

(2.22)

and the result follows.

• Next we show Lemma 2.3 Let u\ be the solutions to (2.16), i = 1,2. Let k be an integer k > 1. Then there exists a constant C = C{\, A, k,n) independent of I such that \u\ ~ u2e\l°o(o,T;L2(nl/2k+1)) + J

2

||V(MJ -

u\)\ on

dt


(2-23)

Proof of lemma 2.3. In Rp we consider a smooth function p = p{X{), 0 < p < 1 such that ( 2 - 24)

P=l

on (~\'l)

'

p= 0

outside (-1,1)",

\VXlP\ < C,

(2.25) (2.26)

where C is some positive constant. Then, for a.e. t, for any t\ € [0,£\, it holds that x = (X 1 ; X 2 ) H . (u\ -u2e)(t)p2{f)

e H^QcT,).

(2.27)

57 According to (2.5), this function is a suitable test-function for the problem (2.16) for i = 1,2. Taking the difference of the equations of u\ and uj with v equal to the above test-function, we derive

(jM

~ U^' ^ ~~ U^p2)

According to [4], {u\ - uj)p L2(0,T))

+

J

AV e

~ U?)V{(U< " u2e)p2}dx = °-

^

€ #H0,T;#01(^,r1),.ff01(ft6r1)')

^(uj-u?)p,«) = (-(ui-u?),t;p)

(2-28)

and it holds t h a t (in

Vue^cn/.ri).

(2.29)

Thus (2.28) becomes l_d_

2~dt

\{u\

- u\){t)p\lni

+ J AV{u\ Jn, in,

- u2)V{{u\

- u2)p2}dx

= 0

(2.30)

in L 1 ( 0 , T ) . Performing the derivation in space in the above integral, we get

2d*

(u\ - uj)(t)p\lni

•

- u?)V(u] - u2)p2dx

+ f AV(u\ .in, Jn,

2 - / AV{u\ 1 In,Jn,

= - u2)pdx.

- u\)Vp{u\

(2.31)

Using (2.13), (2.26) and the the Young inequality ab < ^ o 2 + eb2, this last quantity bounded by 2AC — / \V{u\ - uj)\\u] - uj\pdx 1 Jn, in,

< —-5- / \V(uj - uj)\2dx Jn,, «" i1 Jn h

+e

\u\ -

uj\2p2dx

Jn,, Jn tl

and, with the Poincare inequality of Lemma 2.1, also by ^ f "1

/ \V{u\-u2)\2dx Jnh

\VX2{u\-u2l)\2p2dx

+ C'ef . Jntl

where C" is a constant independent of £— recall t h a t p is independent of X^. Going back to (2.31) and using the ellipticity condition (2.14), we obtain choosing e = A/2C" ±-\{u\ - u2)(t)p\lnt

+ X f |V(uJ - uj)\2p2dx Jn,

at

< J / |V(uJ <-i Jnh

Integrating this inequality between (0, t), it holds that, since p = 1 on

H - «?)(*)ll,n,l/a + 1

X

f[

Jo Jn,l/2

u2)\2dx. fi^/2,

l V K - «/)l2(fa ^ Si Jof IJn l V ^ " ^ l ^ ' L

h

k

The inequality (2.23) follows easily choosing l\ = £/2 . Lemma 2.3.

This completes the proof of

•

58 Proof of Theorem 2.1. From (2.23) we derive \u\ - u2Al°°(o,T;mn,/2k+1))+ J

=J <-

||V(uJ - i

{l«« " u2Al~{0,T.,LHnt/2k))+Jo ||V(ui - t * ? ) ! ! ^ * ]

Iterating this inequality we obtain— using (2.23) at the last step

\\V(ul-uMlnlf2k+1dt

Now, by Lemma 2.2,

jrTiiv(«j-«?)ii*inidt<2jfTiiv«iii^dt+2jrTiivu?ii^dt < C ( ^ 2 a + l).

Then we derive from (2.32) that it holds that fT 2 l«J " u e\l~(o,T;mnl/2k+l)) + ^ ||VuJ - u2t\\2n^h+dt 2

<

l2a + 1 C-^—^

where C = C(X, A, k,p, n). We select then k such that 2(/c + 1) — 2a > 2r and we choose £>1 large enough such that £/2* +1 > £0. We obtain + !"£ \ut ~- '"/ ut\LlL2/2 (0,r im )) < 0tT.H ^lL~(0,T;L2(!!, 0 )) "I" ; if»(«/„)) -

C -^r^

(2.33)

where C = C(A, A,p,n). Note that, since uj — u | is vanishing on a lateral part of f^0, the L2-norm of the gradient is equivalent to the i71(Q£0)-norm— see also Lemma 2.1. This completes the proof of Theorem 2.1.

• 3

The quasilinear case

In this section we consider again

fi, = (-«,
(3.1)

We set F0 = {-£,£)" xdu, l

Hl(ne; r 0 ) = { u e H {Ue) ; v = 0 on T 0 }

(3.2) (3.3)

59 and we consider Vl, V2 two closed subspaces of Hg(Q,f, T0) equipped with the norm (2.4) such that H^Qe) c Ve\ Vl C Hl{Q,t- r„). (3.4) Let us denote by atj = aij(t,x,u), such that

i,j = l,...,n Caratheodory's functions i.e. functions

(t, x) i-> aitj(t, x, u) is measurable VM G R, u H-> aitj(t,x,u) is continuous a.e. (t,x) G (0,T) xW x ui.

(3.5) (3-6)

We write the matrix of the coefficients under the form

(ah3)=A = A(t,x,u)=^2u))

™

where Ai is a p x n matrix and A2 is a (n — p) x n matrix. We assume that A satisfies the following usual hypothesis, ^••?>A|f|2

\A(t,x,u)\

V £ G R " , V U G R , a.e. {t,x) G (0,T) xW


x u,

V a e R , a.e. (i,a;) G (0,T) x l ' x u ,

(3.8)

(3.9)

for A, A > 0. Furthermore we suppose that \/u,v G R, a.e. (t,x) G (0,T) x f x u the following holds |A(t,s,u)-i4(t,a;,i;)| < C w ( | u - ? ; | ) , (3.10) where w is a nondecreasing positive function on R + such that

L

ds

o+ ^ 2 ( s )

(3.11)

Under the assumptions (3.5), (3.6), (3.8), (3.9) one can show that, for i = 1,2, there exists u\ solution to f

uleH'(0,T;Vl,Vl'), (u't,v)2+ • ,I ^ - v A(t,x,u' i(-i,-,* - . - . - e«)Vu'lVvdx --f-~ Jn, (ft,v) mV'(0,T),Vv€Vi,

{ u\(0,-)=u0l(-)

(312)

mL2{ne),

for uQ[ G L2(Qe) and fe G L 2 (0,T; ^(fi^To)')- For an existence proof we refer for instance to [1]. We will also assume that (2.8), (2.10) hold and that for some A' > 0 it holds that for every i = l,...,p, j,k — l,...,n \dxtaid(t,x,u)\
a.e. (t,x) G (0,T) x W x u, \/u e R.

(3.13)

60 Theorem 3.1 For any £0 > 0, r > 0 there exists a constant C independent of £ such that Q

\ui - •"?|i,(o,T)xn(o < jT-

(3.14)

In particular u\ — u\ —> 0 in L 1 ((0,T) x Q^,). Before to give the proof of Theorem 3.1 let us introduce some lemmas. Lemma 3.1 Let i e {1,2} and u\ be a solution to (3.12). Then there exists C = C(X,p) a constant independent of £ such that lu
A{t,x,u\)Vu\Vu\dx

=

l).

(3.15)

{ft,u\).

Using (3.8) it follows that

\jt\4\lsit + A\Vu\\\22nt < |//|.||V«j|| 2in/ .

(3.16)

The rest of the proof is then absolutely identical to the proof of Lemma 2.2. We leave the details to the reader. This completes the proof of the Lemma.

• We turn now to the Proof of Theorem 3.1. Since w is a positive function, by (3.9) for any e > 0 there exists a S = <5e < e such that

r-£r = l-

(3-17)

We define then F€ by '0 fz ds / -^j-^ sc w2J(s) Js, (s 1

if z<Se H6€
(3.18)

if e < z

It is clear that Fe defined this way is a piecewise (^-function such that F't the derivative of Fe, belongs to L°°(R). It follows that if we denote by p the function introduced in (2.24)-(2.26), for lx £ (0,£) it holds that FeP2 = Ft(u\ - ul)p2 ( £ ) G Hl$lt).

(3.19)

61 Taking v = Fep2 as test-function in (3.12) and performing the difference of the equations, we get - uj), {u\ - uj)p2} + /

-{u\ (d

A(t,x,ule)X>ulV{Fep2}dx/ A{t,x,u2)Vu}V{Fep2}dx

= 0. (3.20)

Jilt liii

Let us set

He(z)=

['Fc(t)d£.

Jo

For smooth functions it holds that d / He(u\ - u\)p2dx = ( ^ ( « J - uj)p, Fc(u\ - u2t)pj. dt ./nt By density this holds also for Up 5 11/) <1S above. For simplicity we will denote resp. Aij(t,x,uj) by Aij(uj) resp. Aitj(uj). Then, from (3.20) we derive

-I

Aij(t,x,u\)

uj)p2dx+

H€(u\ -

i fit

/ A(u))VuJVp 2 • Fcdx - / A{uf)Vu2tVp2 Jilt

• Fedx =

Jfic

- J A(u\)VulVFe-p2dx+ [ A(u])Vu2VFf_- p2dx. Jn / n,t JJnntt Since p 2 is independent of Xi we obtain, setting Vjr, = (dxi E--=jtf

Ht(u\ -

(3.21)

u2)p2dx+

/ Ai{u})VulVXlP2-FedxJiii

/ AxiuDVufoxtp2

• Fedx =

Jiii

- I A{u\)V{u\-u\)WF(-p2dxin,

I (A{u\) - A ( u | ) ) V ^ V F e • p2dx. Jilt

(3.22)

Using (3.8), (3.10) we derive easily E<-\

[ \V{u} -

u\)\2F'tp2dx

Jiii

+C f

J [6
uj{u\-u2t)\Vu2\F[\^(u\-u2)\p2dx.

(3.23)

{[8 < u\ - u2 < e] denotes the set {x G Qt \ S < (u\ - u2)(x) < e} where F[ = ^ •) Therefore we have E < -A f \V{u\ - u2t)\2F^p2dx + C [ | V « ? | v ^ | V ( u J - u2)\p2dx. Jnt

Jiii

(3.24)

62 Using in the last integral the Young inequality

a ffl2+ 62

^

£'

we get, since in the above integral we in fact integrate on [5 < u\ — u2 < e]. E<-\

[ |V(i4 - u2)\2F'ep2dx + ^ [ \V(ul - u2)\2F'ep2dx Jn, 2 JJnn Jn, c 2 C f \V4\2P2dx ZX

J[S

\Vu]\2p2dx.

(3.25)

\Vu2t\2p2dxdt.

(3.26)

2-^ J[6
Going back to (3.22) and integrating in t we obtain / He(u\-u2t)p2dx+ Jn,

I I A(i4)VuJV X l p 2 -Fedxdt Jo Jn, / / AxiuDVufoxiP2-F(dxdt Jo Jn, lo Jn, C2

<

f*

If

2A J0 J\s 0 in the above inequality and noting hat Ht{z)^z+,

Fe^XR+,

where ( ) + denotes the positive part of a function and x«+ the characteristic function of K+ we obtain / {u\-uf)+p2dx+ Jn,

I I A1(ule)Vu1eVXlp2dxdt Jo J{u)-u?>o] / / A1{u2l)Vu2VXlp2dxdt< 2 u\-u2>0] Jo J\u\-u >0]

0 (3.27)

for a.e. t E (0,T). We set then a,ij(t,x,z) = / aij(t,x,s)ds. Jo For u e H1^),

(3.28)

it holds that for a.e. t e (0,T), di:j{t,x,u)

€ H\tt)

(3.29)

and for every k — 1,..., n ru

dXka,ij(t,x,u)

= a,ij(t,x,u)dXku+

dXkaij(t,x,s)ds. Jo

(3.30)

63 With this, if we denote by E' the difference of the two last integrals in (3.27) we have E' = / Jo

Y2

u\dXip2dxdt-

aij(ul)dx 1

i=i,..., •A«, -«?>°]

;PP .7 = 1,...,71

zJ / aij(u2e)dxu2edXip2dxdt. / i=i,...,P- W-"?>o]

/

Jo

J=l

(3.31)

n

(For simplicity we drop the dependence in t, x in the coefficients.) Thus, by (3.30) we obtain J'= / Jo

J2

{ dXja^{u\)

i^..,PJH-n]>o)

{

dXjaiij(s)ds}dXip2dxdt-

- I

Jo

J

j=l,...,n

/

V]

/

\dXjaij(uj)

dxa,ij(s)ds>dXip2dxdt.

-

j = l,...,n

Hence,

J = l,...,n

—/

Jul

= / Jo

Yl i=i,...,P j = l,...,Tl

dXja,ij(s)dsdXip2 > dxdt

'

J

I \ d^AhAu\ + w) - Kii^dxiP2 Jn

'

>•

—

dXjaij(s)dsdXip2

/

} dxdt

(3.33)

where we have set to = (uj — u2)+. Integrating by parts we obtain E>

= / Jo

{ ~(«ij( u £ +w) - Hj(u2e))dliXip2

Y,

i=i,.../ f l < I

dXja,ij(s)dsdXip2 \ dxdt.

- / 7

"?

J

j = l,...,n

(3.34) Indeed, since u\ — uj = 0 on (—£,£)p x eta (see (3.4)) and 9Xip2 = 0 on d(—l,l)p x w, it holds that (aitj(u2e + w) - aitj(u2))dXip2 = 0 on d£V Using the definition of aitj, we rewrite E' under the form E'=

/ -

Y" j=l,...,n

M

/

aij(s)ds92..,..p2 + /

cVa^^dsc^p2 Uzdi.

64 Since the a^- and dXka,ij are bounded and

K/(f)|,Kp2(f)|
(3.35)

where C is a constant independent of £i, we derive for some constant C = C(p,A,A') since p = 0 outside (—H.i,l\f : \E'\
(3.36)

Recalling the definition of E' and going back to (3.27) we obtain for some C independent of£i / {u\ - u})+p2dx <j-

(u\- u2e)+dxdt

/

(3.37)

thus by (2.24) / {u\ — uj)+dx < — / / (wj — Jntl/2 *i Jo Jfi(l Integrating in i it follows that /

/

(uj - ufj+dxdt < — \

j

uj)+dxdt.

[u\- u2)+dxdt.

Since clearly — u\, —uj are solution of similar equations we arrive to / / \u\ - u\\dxdt < — / \u\ - u2e\dxdt. Jo Jntl/2 *-i Jo Jntl

(3.38)

Setting £i = £/2k we get / / \u\ -u\\dxdt Jo Jnl/2k+i

<— / *• Jo

/ \u\ - u\\dxdt. Jnt/2k

(3.39)

Iterating this formula we obtain rT

r

n

i-T

/ / Wi - u2t\dxdt < — / / \u\ - u\\dxdt l Jo Jnt/2k+i Jo Jat

- ¥* {[ L Wt ~uj]2dxdt)'2

Ti/2| |i/2

^

(by the Cauchy-Schwarz inequality). Using now Lemma 3.1 we derive easily that f j \u\ - u\\dxdt < ^Tl'M Jo Jnt/2k+1 *•

-

u\|1=.(o,T;W(n/))r1/2|n/|1/2

4(<2° + 1)1,V/2

65 for some constant C independent of L Choosing then k in such a way that k+l-a-p/2 r and I such that £/2k+1 > l0 we obtain /

f

Jo Jnlo

\u\-u\\dxdt
>

(3.40)

£

with C independent of L This completes the proof of the theorem.

• Acknowledgments. The work of the authors has been supported by the Swiss National Science Foundation under the contract #20-58856.99. We are very grateful to this institution for this help.

References [1] M. Chipot: Elements of nonlinear analysis, Birkhauser, (2000). [2] M. Chipot: £ goes to plus infinity, Birkhauser, (2002). [3] M. Chipot, A. Rougirel: On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, to appear. [4] M. Chipot, A. Rougirel: On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), 3, p. 319-338. [5] M. Chipot, A. Rougirel: Remarks on the asymptotic behaviour of the solution to parabolic problems in domains becoming unbounded, Nonlinear Analysis, Vol. 47, Issue 1 (2001) p. 3-12. [6] M. Chipot, A. Rougirel: Sur le comportement asymptotique de la solution de problmes elliptiques dans des domaines cylindriques tendant vers I'infini, C. R. Acad Sci. Paris, t. 331, Srie I, p. 435-400, 2000. [7] R. Dautray, J.L. Lions: Mathematical analysis and numerical methods for science and technology, Springer-Verlag, (1988).

On some variational problems with an infinite number of wells A. Elfanni Universitat des Saarlandes Fachbereich 6.1- Mathematik Postfach 15 11 50 D-66041 Saarbriicken, Germany Email: [email protected]

Abstract We give a necessary and sufficient condition for nonexistence of minimizers for some variational problems allowed to have an infinite number of potential wells and subject to some special linear boundary conditions. Namely the gradient of the boundary condition is related by a convex relation to some wells spanning a hyperplane. We also prove that the minimizing sequences generate a unique Young measure which is supported by the wells related to the boundary data. The variational problems we consider in this note are certainly special cases of the complete work of Priesecke [F.] but the condition we present here does not involve the computation of the convex envelopes.

1

Introduction

Let fi denote a bounded domain of R " (N > 2) with Lipschitz boundary T. ip : KN —> R be a continuous function and W{tp) the set of wells of tp i.e. W{
I
Let

(1.1)

We assume t h a t

0 Vw
(1.2)

If H denotes the following hyperplane of R w H = {w£KN/wfj,

= 0}

(1.3)

(w • n denotes the scalar product of w and fi) we assume t h a t there exist w\,...,

wN N elements of W(^>) such t h a t w{ eHVi

= l,...,N

66

(1.4)

67 0G

ri^CoK),^)

where Co(tui)^=1 is the convex hull of the uVs, riH(Co(«;i)-I1) tively to the topology of H.We denote by W„1,0°(fi) the set W^°°(Q) = {ve

(1.5)

denotes its interior rela-

W1-00^) I v(x) = 0 on T}.

Then we consider the following minimization problem inf

/


/

v(Vv(x))

(1.6)

It is well known that inf

dx = |0| tp*'{0).

(|Q| is the Lebesgue measure of Q and tp** is the convex envelope of

Ar = Span(uij - WI)J=2,...,JV.

(1-7)

where Span(a;)i denotes the vectorial space spanned by the vectors
N

t=l

i=l

By (1.5)

We adopt the following notations H+ = {weIlN N

/ w-n>0},

H. = {weR /w-li<0},

TJ+ = {w<EKN N

Tl- = {w£~R

W(

)* = % ) \ K

i= N

J

W/J,>0}

I

w-n<0}

l,...,N}

The case where W(ip) spans a proper subspace of R was studied by M. Chipot and C. Collins (see for example [C], [C.C.]). In this paper, we are concerned by the case where W(
68

2

A necessary and sufficient condition for nonexistence of minimizers.

In this section we give a necessary and sufficient condition for nonexistence of minimizers. The problem we study in this note is certainly a special case of the interesting work of G. Friesecke (see [F.]) but our condition does not involve the computation of the convex envelope of ip (see the four-well antiplane shear problem studied below.) Theorem 1. We assume that 0^%),

(2.1)

then the infimum in (1.6) is not attained if and only ifW{
tp(Vu(x))dx = 0.

(2.2)

Hence by (1.1), (1.2) one has VM(I)

e W{ip) C ~H+ a.e. in Q,.

Therefore Vu(i) • n > 0 a.e. in Q. Using the following variant of Poincare's inequality / \u(x)\dx
Jn in

\Vu{x) • n\dx

(2.3)

(2.4)

Jn

btains where C is a constant one obtains [ \u{x)\dx
( Vu{x)dx-ti.

in

Jn

(2.5)

Integrating by part one gets

/ Vu(x)dx = 0. in

(2.6)

Jn Combining (2.5)-(2.6) one deduces that

u = 0 in fi. Since 0 £ W() one has by (1.2) ip(0) > 0 so that (2.2) is impossible. Now assume that there exist u>_i e W($) n H+ and w0 G W($) n H_. Since 0 G riH{Co(wi))i=h

N

there exist k wells (k > N — 1) among the Wi's that we can assume to that Span(wj)i = _i, 0 ,...,j: =

RN

0 e Int(Co(u)i)i=_li0>...,fc) where Int(A) denotes the interior of A. Therefore there exist B-i,B0, that k

k

t=-l

i=-l

Then we consider the following function it

(a;) = f\ Wi • x + 1, x e RN where f\ denotes the infimum of functions. It is clear that V«(i) = Wi a.e. in R " (i =

—l,0,...,k).

Due to (2.8) one has k

f\ Wi-x<0,

VieR".

We denote by 5 the following open subset of KN k

S = {x e RN I u(x) = /\wi-x

+

l>0}.

We claim that 5 is bounded. Indeed one has k

f\wi-x

+ l>0

V x€ S.

i=-l

Then k

Y Wi • {-x) < 1 V x e S where V denotes the supremum of numbers. Thus

\x\ \ /

Wi

• t £ < 1 V x e S.

Hence k

\x\

inf S6S»-' .

\ v/ Wi • y < 1 V x e S

70 where SN ' = { j € TLN / \y\ = 1}. Since SN that

1

is compact there exists y* € SN

inf V Wi • V = V yes"-' i=-l i=-\

1

such

Wfy*

From (2.8) one has k

\J We claim that

wt-y*>0.

k

V Wi • y* > 0. Indeed assume that

(2.10)

k

V

Wi

• y* = 0.

Then Vi = - l , 0 , . . . , f c u;i-y* < 0, but from (2.8) one has k

J2 frv>i • y* = o, Pi e (o, l). Then Wi- y" =0 Vi = 1,.. .,fc and (2.7) implies that y* = 0 which contradicts the fact that y* 6 S^" 1 . Thus (2.10) holds and

N^-r-^

Vies. w

V i = - i i ' 2/

Therefore 5 is a bounded open set. Taking 0 = 5, one has u € Wo1'00(ft). Since the ujj's belong to W($) and u verifies (2.9) we deduce that it is a minimizer of the minimization problem (1.6) with Q, = S. Then one can construct a minimizer on Q by covering it with scaled copies of 5 using Vitali's covering lemma.

In the absence of minimizers we turn to the study of minimizing sequences. We will follow the ideas in [C.C.] taking now into account of the data of our problem. Namely we will use the fact that a Lipschitz function vanishing at the boundary and verifying (2.3) is identically equal to 0. Then we have Theorem 2. Assume that W{
71 Let (u„)„ be a minimizing sequence of (1.6) such that \\un{x)\\, ||Vw n (z)||
a.e. in U

(2.11)

for some constant C independent of n (\\-\\ denotes a norm in R or KN), then one has un —y 0 uniformly in fi

Proof. From (2.11), by a compactness argument there exist u G WQ'°°(Q) quence unk such that u„, —• u uniformly in Q Vunk ->• Vu in L°°{Q)N - * weak

and a subse-

(2.12)

when k —¥ oo. Now, the bounded sequence of gradients generates a Young measure on R " (see [P.]) in the sense that there is a probability measure vx on RN and a subsequence of Vunk -still labeled Vu„k- such that for any continuous function F on ~RN one has F{Vunt)

-± f

F{X)dvx(X) in L°°(Q) - * weak.

(2.13)

JR." /R"

Considering first (2.13) 3) with F = tp and since (un)n is a minimizing minimizin sequence one gets f 1 • 0 = f [ Jn

ip(X) dvx{\)dx

JSIJR"

It follows that

L


/R"

One deduces that

Supply C W(p) C B+, for a.e. i £ f !

(2.14)

where Supply denotes the support of vx. Considering then for F in (2.13) the function F(X) = A • n Vunk (x) • M -*• /

^ • A* dvx(\)

in L°°(n) - * weak.

12) one deduces that Combining this with (2.12) Vu(i) • M = /

A • fi dvx{\)

a.e. in Q,

(2-15)

but by (2.14) one has /

X-n dvx(X) > 0 a.e. in Q.

(2.16)

JR"

Combining (2.15), (2.16) and (2.4) one deduces that u = 0. Since the sequence has 0 as unique limit point the whole sequence converges towards 0. One has finally obtained u„ —• 0 uniformly in Q

72 Vu„ ->• 0 in L°°(Q)N - * weak

(2.17)

This completes the proof of theorem 2. • In the following theorem we express the behaviour of the minimizing sequences in terms of Young measure. As we will see, they have a precise pattern governed only by the N wells Wi and the boundary condition. More precisely we have Theorem 3. We assume that WW

C H+.

If (u„) is a minimizing sequence of problem (1.6) satisfying (2.11) then un —> 0 uniformly in Q.

(2-18)

Moreover the sequence of gradients VM„ generates a unique homogeneous Young measure on R w given by

vx = ^2a>S»>i

(2.19)

i=l

where SWi denotes the Dirac mass at the point Wi and the cti 's are the constants appearing in (1.8). Proof. The assertion (2.18) is a simple consequence of theorem 2. Let (VX)X€Q be the Young measure associated to Vun verifying (2.13). Combining (2.13) and (2.17) one obtains /

A • fj, dvx{\) = 0 a.e. in

fl.

(2.20)

Due to (2.20) and the fact that Suppi^ C W(
(2-21)

^ = X^<w i=l

Since vx is a probability one has N

$^ft = l,

ft>0.

(2.22)

i=l

Let B be a ball included in Q. From (2.13) applied to F(X) = \ and (2.17) we obtain /

/

JBJKN

A dvx{\)dx = 0.

73 By (2.21) this reads

0

= Y, I ^x)dx

(2.23)

ax • W{.

,=IJB

Hence by (1.8), (2.22) and (2.23) one has - - / pi(x)dx = a{ Vi = 1,

,N.

Since this is true for any ball B, it comes out that j5i(x) = oti(x) a.e. in Q, Vi = 1 , . . . , iV and thus (2.21) reads N

vx = 2_,ai&wi a.e. in fi. This shows that vx is necessarily the homogeneous Young measure in (2.19) R e m a r k 1. If W{ip)* n H is not empty, then the bounded minimizing sequences of (1.6) converge toward 0 (cf Theorem 2) but they may not generate the same homogeneous Young measure. Indeed when we have wells satisfying (1.8) one can construct a minimizing sequence of the minimization problem (1.6) satisfying (2.11) and whose sequence of gradients generates a Young measure given by (2.19) (see [C.]). Let us illustrate our results by the following example Example (the four-well antiplane shear problem). Take TV = 2 and $(x, y) = (x2 — l) 2 + (y2 — l) 2 . Let us consider the square of vertices the four wells «Ji = (—1, —1), w2 = (1, —1), w3 = (1,1) and u>4 = (-1,1). Let a be a point of one of the segments ]u>i,Wi[, i = 2,3,4 •

y

w3 1

\

1

.1 I 1

/ \

\\

\

/

s

l 1

/•

/

1 m

1

!

• \

x

1

/ \

Wl

/ 1 1

1 1

/

\

1

1

/

/

1

w2

The four-well antiplane shear problem.

74 or ]w2, uii[, i = 3,4 or ]w3,u>4[ and consider the minimization problem (1.6) with the boundary condition u(x) = a • x on T. If a belongs to the diagonals ]wi, »3[ or ]u>2,u>4[ one can immediately conclude that the problem in hand attains its infimum. Indeed for example if a e]wi, w3[ then the two other wells w2 and w\ are respectively in the two halfspaces separeted by the straight line passing by the wells w\ and w2. Now if a belongs to one of the sides of the square for instance a €]wi, w2[ then the other wells are all in one of the half-spaces determined by the straight line passing by the wells w1 and w2. One can conclude that the minimization problem does not admit minimizers. So we have studied this problem, considering special boundary conditions, without computing the convex envelope of $(z, y) = (x2 - l) 2 + (y2 - l) 2 . Notice that if a belongs to the interior of the square of vertices W\, w2, u>3 and UI4, one can also conclude that the problem (1.6) admit minimizers using the second part of the proof of theorem 1. Now if a e {i«i, w2, W3, w4}, one can easily see that the Lipschitz function x —> a • x is a minimizer. References [C] M. Chipot : Numerical analysis of oscillations in nonconvex problems. Numerische Mathematik, 59, (1991), p. 747-767. [C.C.] M. Chipot and C. Collins : Numerical approximation in variational problems with potential wells. SIAM J. of Numerical Analysis, 29, 4, (1993), p. 473-487. [D.] B. Dacorogna : Direct Methods in the calculus of variations, Springer-Verlag, Berlin, (1989). [F.] G. Friesecke : A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc. Royal Soc. Edinburgh 124 A (1994), p. 437-471. [P.] P. Pedregal : Parametrized measures and variational principles, Birkhauser, 1997.

Boltzmann Equation for Quantum Particles and Fokker Planck approximation Miguel Escobedo Departamento de Matematicas Universidad del Pais Vasco Apartado 644 Bilbao 48080. Spain

Abstract We consider in these short notes a simple case of homogeneous Boltzmann equation for quantum particles and its Fokker Planck approximation. Namely the Boltzmann Compton and Kompaneets equations. We solve the Cauchy problem associated to these two equations and prove global well posedness for a large set of initial data. We describe the long time behaviour of the solutions and prove in particular that the solutions of the Boltzmann Compton equation with suficiently large initial data undergo Bose condensation in infinite time. We finally show that for some initial data, the Kompaneets equation is a good global approximation of the Boltzmann equation for all time t > 0.

1. Boltzmann equations for quantum particles Consider a homogeneous dilute gas of two species of interacting q u a n t u m particles. Let f(p, t) and F(p, t) be the densities of particles of each species t h a t at time t have a m o m e n t u m p e l 3 . Following [UU], [GLW] and others, t h e evolution of / and F may be described, under t h e hypothesis of molecular chaos, by the following system: dF

= Qltl(F,F)

+ Q1,2(F,f)

F(0,.)

= Fin

! T h e collision terms Qiti(F,F) and Q2,2(f,f) t h e same species a n d are given by

Q iii (*,*)= fff

stand for collisions between particles of

WM[*'*',(l + T*)(l + T*,)-***(l + '-*')(l + '-*'.)]dp.dp'dp'.-

J J Js?

75

76 The collision terms Qi,2(F,f) and Q2,i(f,F) the two different species, they are given by

stand for collisions between particles of

W1,a(F,f',(l+rF)(l+Kf,)-Ff,(l+TF')(l+Kfl))dp,di/di/„

Q i , 2 ( F , / ) = fff

with a similar expression for Q2,i(f,F). In both cases, Wij(p,pt,p',p't) are given measures and r and re may take the values —1,0 or 1 depending on the quantum properties of the particles. If F (respectively / ) is the density function of particles which obey the Fermi statistics (in short which are fermions), then r = — 1 (respectively K = — 1). If on the other hand F (respectively / ) is the density function of particles which obey the Bose statistics (in short which are bosons), then r = 1 (respectively K = 1). Finally, if F (respectively / ) is the density function of non quantum particles then T = 0 (respectively re = 0). We consider in these notes the interesting example of the Boltzmann-Compton equation which describes the photon- electron scattering. We denote by F the density function of the electrons (which are fermions) and / that of the photons (which are bosons). This makes that r = — 1 and re = 1. We assume that the electrons and photons have low energy and low density from where, after some physical considerations we may (i) suppose that the electrons are at non relativistic and classical equilibrium F(p) = (ii) neglect the photon-photon interaction and Bremstrahlung effects. Since the electrons are assumed to be at equilibrium, the first equation in the corresponding system (S) drops. Moreover, by (i), the main contribution in the right hand side of the second equation of (S) comes from the Q2,i term, corresponding to photonelectron interactions. Then, the system reduces to the following equation for the photon density: %(jp,t) = f

S{p,p',m)[e-Mf(p',t)(l

+ f(p,t))-e-^y(p,t)a

+

f(p',t))}dp',

where m is the mass of the electron (the photons have no mass), and S(p,p',m) is a function explicitly known (cf. for instance [EMV2]) which depends on the type of the interaction photon- electron. If we assume, for the sake of simplicity, that f(p,t) = f(\p\,t) = f(k,t) with k = \p\, after some straightforward calculations, using in particular specific properties of the function S, one obtains the Boltzmann-Compton equation

(1)

k2^

= j^°{f

(1 + f)e~k - f(l + f')e-k')k2k'2b(k,k',m)dk'

= Qm(f,f) A similar homogeneous Boltzmann equation arises if we consider bosons-fermions interaction, with fermions at equilibrium. This is also a short range interaction and is considered in [LY1] and [LY2] by numerical computations.

77 The system describing the interaction of non quantum particles, corresponds to the case r = K = 0. Moreover let us consider for our purpose, an homogeneous gas composed of heavy particles, with density function F, and light particles with density function / , under the physical assumptions: (iii) low concentration of heavy particles: so that we may neglect the collisions of heavyheavy particles. (iv) in the heavy-light collisions, the momentum of the heavy particle undergoes a relatively small change. We may then suppose that the heavy particles are at equilibrium, so that the first equation drops, and the main contribution in the right hand side of the second equation comes from the Q2,i term, corresponding to heavy-lights particles interactions. Therefore, the density of light particles satisfies the classical homogeneous Boltzmann equation (2)

%(p,t)

= J 3{w(p +

q,q)f(p',t)-w{p,q)f{p,t)}dq.

where w(j>, q) is a given function, which depends on the equilibrium state of the light particles and the kind of interaction heavy-light particles. The behaviour of the solutions to these homogeneous Boltzmann equations may be very different for quantum and non quantum particles. In particular, in the case of quantum particles Bose-Einstein condensation is possible asymptotically in time as it is proved in [EM2] and we briefly explain below (see Section 5). 2. Fokker P l a n c k a p p r o x i m a t i o n It is well known that Boltzmann equations for gases of classical particles may be approximated by Fokker Planck equations (cf. [LP] §21). The interest of this approximation is that, as we shall see below, the resulting equation is a partial differential equation which is simpler than the original Boltzmann equation. But on the other hand this approximation may happen to be only valid for p in a subdomain of R3 Consider for example the classical Boltzmann equation (2). By the condition (iv) above, the measure w(p,q) decreases rapidly as q increases. Therefore, the main contribution in the integral term comes from the region where q is small. The idea is then to perform the following formal Taylor's expansion: w(p + q,q)f(p + q, t) ~ w(p, q)f(p, t) + q • Vp(w(p, q)f(p, t))

+ 5 5>*& { W ( P ' 9 ) / ( P '' ) ) ' from where we obtain the following Fokker Planck equation:

78 with Ai = /

qiw(p,q)dq,

Bitj = -

qiqjw(p,q)dq

This, and more general situations for classical particles, have been extensively studied in the physical and mathematical litterature. Among the recent works where this question is considered from a mathematical point of vue, we may quote in particular: [D], [DL], [AB] and [VI]. The general mathematical framework of these studies is to consider a family of "regular" cross sections wE which concentrate as e —> 0 to the region where q = 0 and show that, under suitable conditions on w and we, the solutions fe of i

dt

(P>t) = / {weip + q,q)fE{p +

q,t)-we(p,q)fe{j>,t)}dq,

converge to the solution of the Fokker Planck equation, as e —• 0.

We understand very well at this stage that, for classical particles, as soon as one of the two species is at equilibrium, the remaining Boltzmann equation is linear, and so is the approximating Fokker Planck equation. If on the other hand, the particles which are not at equilibrium are supposed to have a quantum nature, the equation remains nonlinear, and so is the resulting Fokker Planck approximation. Let us show this with the Boltzmann Compton equation.

3. Fokker Planck approximation of (1): t h e Kompaneets equation The Fokker Planck approximation of (1) which is known as Kompaneets equation [K] reads:

(3)

| =

fc

-l^K

+ /2 + /»-

k

>^>°-

The formal deduction of this equation from the original Boltzmann Compton equation is as follows. One first considers that the main contribution in the collision integral />oo

/ {f'(l + f)e-k-f(l Jo comes from the region \k' — k\ «

+

f')e-k')k2k'H(k,k',m)dk',

k so that:

/(*') ~ f(k) + (fc' - k)f{k)

+ i(*' -

e~k' ~ e-k - (k' - k)e~k + -(fc' - fc)5 from where

k)2f"(k)

79 (

r°°

/ (/' (1 + / ) e"fc - / (1 + / ' ) e~k') k2 k'2 b(k, k')dk' Jo

. ~ A(k)f(k) with

+ (A(k) + B(k))f'(k) i

r°°

A(k) = / ((fc' -k)-Uk' Jo 2 B(k) = =- / 2 Jo

+ B(k)f"(k)

+ 2B(k)ff

- k)2)b{k, k',

(it' - k)2b(k, k',

+

A(k)f.

m)e-kk'2k2dk'

m)e~kk'2k2dk'.

So that, if we define a(fc) = exp / -gf-^rda Jo B{a) and 7 (*)

= B(fc)exp(-/' JO

^4da) #((7)

we have

* ^ ~ 7 ( f c ) ^ [ a ( f c ) ( | + / + /')]. On the other hand, the law of conservation of the total number of particles in the process of scattering requires that the equation should have the form: 9/ dt

-V- J

where J is the photon flux in momentum space. Because of the isotropy of the photon distribution function, the divergence of the flux has the form:

Therefore we get that j(k) is a constant. Finally, using the explicitly known function b(k, k', m) (cf. for instance [EMV]) it is easy to show that as k/m - > 0 w e have, B(k) ~ fe4. This gives 2df

-ci-[k\ldJ-x

dt

dk

dk

+f+

f% as im- 0 .

for some positive constant C. It is important to notice that in the classical Boltzmann equation under condition (iv), the collisions for which p — p' is small are much more important than collisions with large momentum changes. On the other hand, since Compton scattering is a short range

80 interaction, the change in the momentum p of the colliding photon is not small. But the change in its energy \p\ is small, (due to the exponential decay of the term S(p,p')e~l p 'l when | |p| — |p' 11 is large). Therefore, the main contribution in the integral comes now from the region ||p| — |p'|| small. Therefore, in order to obtain the Kompaneets equation, the Taylor's expansion is performed in terms of |p| instead of p. This introduces a boundary at the point \p\ = 0. This boundary arises from the very nature of the photon electron interaction and plays an important role in all the following (cf. in particular Remark 3 below). Finally, let us emphasize that, in the Kompaneets equation the nonlinearity comes from the terms (1 + / ) i.e. quantum effects. Remark 1. If we do not assume that the particles of one of the two species (the light particles in the first example or the electrons in the second) are at equilibrium, the approximation may still be performed but it gives rise to a more complicated integro differential equation. In the simplest case, we may consider just one specie of interacting classical particles. The system reduces then to one single equation

QiAfJ)=

[ I f w[f(J/.>W,t)-f(p.,t)f(p,t)]dptdpfdj/l,

JR3 JRS JR3

For some particular types of collisions, the measure W is still concentrated in the region where p — p' is small. This allows to perform again formally a Taylor's expansion in the integral collision. A simple example for classical particles undergoing grazing collisions with Coulomb potential and with radially symetric distribution is the following:

a/ dt dt k(x,x')

o

W£{/<X,fc(a;.a;')[|f(i.0/(i'.*)-/(a:.*)£(i'.0]«&'}I =

mm{x3/2,x'3/2}

which can also be written as:

/•oo

A(f,x) B(f,x)

= a(x)

Jo

k(x,x')f(x')dx',

= ^ M f m i n ^ / y i / J } / ^ , ^ ^ Jo c{x) =

M^l

81 (c.f. [V2] and, in particular the ocurrence or not of finite time blow up ). Notice that Kompaneets equation could be written as:

| = ^S + (4fc

+fe2+ 2fe V)g + 4 fe (/ +

f).

The study of Boltzmann Compton and Kompaneets equations, has deserved a large amount of interest in physical litterature from the late 50's to the mid 70's: [Dr], [ZL], [C], [CCS], [CL]. They have been considered either by formal asymptotics or by numerical and computational methods. The Kompaneets equation is currently used to describe the distorsions in the cosmic background radiation (cf. [P] and [S]).

4. S o m e properties of B o l t z m a n n C o m p t o n and K o m p a n e e t s equations The Boltzmann Compton and the Kompaneets equation present some joint features and some differences that we briefly describe below. First of all, if we consider the entropy

r

5(/)=y°°S(/(fe),fc)/c2dfc,

t s(x,fc)= (1 + x) ln(l + x) — x hix — kx then, along the trajectories of the two equations we have:

|W))>0. On the other hand, a very important property of the Boltzmann Compton equation (1) is that the total number of particles is constant, i.e.:

«>

i[

k2 f(k,t)dk

= 0.

If we want that property to be true also for the solutions to the Kompaneets equation, which is a second order P.D.E in the domain k > 0, we need a flux condition at fc = 0 and asfc—> +oo. Indeed, integrating by parts at least formally gives: d_ |°° /(fc, dt j 0

t)k2dk

=

J " ± { k ^ + f' + f)}dk.

JQ

We have then to impose the flux condition: ki(% + f2 + f)-¥0 as fc->-0 and fe -> oo, ofe to get (4). The Kompaneets equation has been considered with condition (5) in [CL] from a numerical and formal point of vue. (5)

82 Finally, the functions

are steady states of the two equations. Notice nevertheless that for every // > 0 : /

f„(k)k2dk<

Jo

f0(k)k2dk

/ Jo OO

This shows that there are no such steady states for a total mass greater than N0. We have then to consider also the distributions Fa such that k2Fa{k)

k2 = -jrK-_- + a80, e —1

a>0

which are also stationary solutions of the kinetic Boltzmann Compton equation. Notice that the ocurrence of Dirac measure is for the function k2 f(k) which is the density function in the radial variable k since J f(p)dp = u>„ j f(k)k2dk. It was already observed by Bose and Einstein (cf. [B], [El], [E2]) that for systems of bosons in thermal equilibrium a careful analysis of the statistical physics of the problem leads to enlarge the class of steady distributions and to include also the solutions containing a Dirac mass at the origin.

5. The Cauchy Problem and asymptotic behaviour of solutions for (1) and (2) We expose in this section our main results, which show first that the Cauchy problem for the two equations (1) and (2) are globally well posed. We then consider the asymptotic behaviour of the solutions as t —> +00. We prove that the solutions of the Boltzmann Compton equation undergo Bose condensation in infinite time if the density of particles is larger that N0. Finally we show that for some initial data, the Kompaneets equation is a good approximation of the Boltzmann Compton equation in the whole space k € (0, +00) and for all t > 0. Let us define k2

f°°

and consider the set: poo

£0 = {heL1(0,oo),h>0,

/ (1 + k)h(k)dk < 00}. Jo

Theorem 1. [EM2] Iffin £ £0 there exists a unique global solution f 6 C([0, 00), £0) to the Boltzmann Compton equation, such that /

Jo

f(k,t)k2dk=

fin(k)k2dk

Jo

=: m

Vt > 0.

83 Moreover rod

(i) ifm = Na : lim /

\f(k, t) - fjk)\k2dk

=0

(ii)ifm > N0 : k2f{-, t) -+ k2f0 + {N0 - m)5 and lim f \f(k,t)-f0(k)\k2dk = 0,\/k0>0. '-*00 Jk>k0 Remark 2. Theorem 1 shows that the Cauchy problem for Boltzmann Compton is stable: we have existence of global solutions which converge asymptotically to the equilibrium states. T h e o r e m 2 . [EHV] Assume that fin is a bounded and continuous function over (0, ex)), such that lim k4fin(k) = 0. Then there exists a unique global classical solution k—*oo

to the Kompaneets equation (2) such that f(k, 0) = fin(k) such that, for some constant C: /(*,*)<§ 6

<>

<

df limfe4(g fc->oo

Vfce(0,l),Vi>0, + / +

/

2)=0.

Ok

Moreover, there are fin, bounded and continuous over (0,+oo), such that for some T* > 0 the unique solutions satisfies: lim fc4(|( + / + / 2 ) = 0 k-tO

lim k4(%

fc-s-o

Vt€(0,r*)

OK

ok

+ f + f2) > 0

\ft>T*.

For some f^, steady state of the Kompaneets equation: f(k, t) —^f/j,(k)

uniformly on compacts sets of R +

Remark 3. The Cauchy problem for Kompaneets equation is well posed with the conditions imposed in (6) only. In particular, no boundary condition is needed at k = 0. This is due to the degeneracy of the elliptic operator in Kompaneets equation at k = 0. On the other hand, the Cauchy problem for the Kompaneets equation with condition (5), which ensures the conservation law (4) to hold, is unstable for a large class of initial data. This is not so surprising since, as it was said above, the Kompaneets equation is an approximation of the Boltzmann Compton but only in the region k/m —> 0. Therefore, in principle, the solutions of the Kompaneets equation describe the photon density only in that region, and there is no particular reason for these solutions to satisfy the global

84 condition (4). It is actually shown in [EHV] that the solutions constructed in the second part of Theorem 2 are driven by the following Burgers-type equation:

They behave like smoothed shock waves moving to the left as t increases. More precisely, there exists some positive constants C\, C2 and C3, depending on the initial data fin such that: -+

if

0 if

z>C2 0
uniformly on compact subsets of (C2,oo) and (0, C2) \imx2f(k,T*)

= C3 > 0 .

fc->0

This makes that the boundary condition (3) at the origin, is violated at the origin in finite time. This in turn, implies that condition (4) is not fulfilled anymore. Theorem 1 and Theorem 2 show that, for some initial data, the Kompaneets equation can not be a good approximation of Boltzmann Compton globally in the whole region k e (0, +00) and for all time t > 0. The next result shows, on the other hand, that this may be true for some initial data. Theorem 3. [EM2]

Assume that b(k, k') = ek'2

ek''2

and take a e £>(R) even, suppa C [—2,2], a > 0 on [—1,1] with J0°° a{z) dz = 1, / z2a(z) dz = 2. Define

b£(k,k') = b(k,k')

a

k

^ -

\

ae[z)

=

Ia(£).

For 0 < fin < /o, let fe e C([0,00), L 1 (0,00)) be the solution to the Boltzmann equation (1) corresponding to the cross-section be and the initial datum fin which is given by Theorem 1. Then, for all T > 0, lim ||fe 2 / E - fc2/*:||c([o,T],£2(o,oo)) = °.

fx is the unique solution to the Cauchy problem , 2 dfic

!

k

d ,

4

,

2

dfK \ 1

-df= dk^ (fK + f * + -8k)}> /*(*,<>) = /«.(*) for k>0.

85 Remark 4. Since fK < f0, the function fK is the unique solution of the Kompaneets equation with initial data fin satisfying fK(k,t)<^

Vke(0,l),Vf>0,

given by Theorem 2. In that case, we also have that

and therefore JJtW —> fu

uniformly on compacts of [0, oo)

t—¥00

with JVM = /

fink2dk.

Remark 5. Theorem 3 is not true for all initial data such that J fink2dk < N0. Notice indeed that, by Theorem 2 we know some of these data, for which the solution /if is such that for some T* > 0:

\imk4(% + f + f2)>0,

fc-»o

OK

Vt>T*

and J

/-OO

-J

fK(k,t)k2dk<0

and /if (*) —^ /i/

uniformly on compact subsets of (0, oo)

for some v such that /•OO

W„ < / Jo Since on the other hand, for every e > 0:

fin{k)k2dk.

lim | | / e ( i ) - / M | | i = 0 t—*oo

with p, < v we may not have / e —>• JK for < large. Remark 6. The long time behaviour of the "singular solutions" (i.e. such that M{n > N0) to the Boltzmann Compton and Kompaneets equation can be precisely described. The strong differences in their behaviour is then easily observed. For Kompaneets equation it is proved in [EHV] that for a large set of integrable initial data, with Mi„ > N0, the solution satisfies C(i) fK(k,t) - ^ a s t -> oo, k ->• 0,

86 where C(i) is a bounded function of t. On the other hand, it is shown in [EMV] that the solution of the Boltzmann Compton, with the same initial data behaves like: /(M)~

k

i2 + —e~mkt,

1 _

as

i - > o o , 0 < kt < 1.

In the second case one can see that 1.2

fc2/(£> t) ~ - T — - + kt2e-mkt,

as

t -s- oo, 0 < fci < 1

and the formation of the Dirac mass clearly appears in the second term of the right hand side.

References [AB] A.A. Arsenev, O.E. Buryac, Math. USSR Sbornik 1991. [B] S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Z. Phys. 26 (1924), 178181. [CC] S. Chapman, T.G. Cowling,The Mathematical Theory of Non-Uniform Cambridge University Press 1970. (Third ed.)

Gases,

[CL] R.E. Caflisch, C D . Levermore, Equilibrium for radiation in a homogeneous plasma, Phys. Fluids 29, 748-752 (1986) [CCS] G. Chapline, G. Cooper and S. Slutz: Condensation of photons in a hot plasma; Phys. Rev. A, 9, 1273-1277 (1974). [C] Cooper, G. Compton Fokker-Planck equation for hot plasmas, Phys. Rev. D, 3, 2412-2416 (1974). [El] A. Einstein, Quantentheorie des einatomingen idealen Gases, Stiz. Presussische Akademie der Wissenschaften Phys-math. Klasse, Sitzungsberichte 23, (1925), 3-14. [E2] A. Einstein, Zur Quantentheorie des idealen Gases, Stiz. Presussische Akademie der Wissenschaften, Phys-math. Klasse, Sitzungsberichte 23, (1925), 18-25. [DL] P. Degond, B. Lucquin-Desreux, Math. Models and Meth. in Appl. Sci. 1992. [D] L. Desvillettes, Trans. Th. and Stat. Phys. 1990. [Dr] Dreicer, H. (1964) Kinetic Theory of an Electron-Photon Gas, Phys. Fluids 7, 735-753. [EMI] M. Escobedo, S. Mischler: Equation de Boltzmann quantique homogene: existence et comportement asymptotique, C. R. Acad. Sci. Paris 329 Serie I (1999) 593-598.

87 [EM2] M. Escobedo, S. Mischler: On a quatum Boltzmann equation for a gas of photons, J. Math. Pures Appl. 80, 5 (2001), pp. 471-515. [EHV] M. Escobedo, M.A. Herero, J. J.L. Velazquez, A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma Trans. Amer. Math. Soc. , 350 (1998) 3837-3901. [EMV1] M. Escobedo, S. Mischler, J.J.L. Velazquez, Asymptotic description of Dirac mass formation in kinetic equations for quantum particles, preprint E.N.S., Paris 2001. [EMV2] M. Escobedo, S. Mischler, M. A. Valle, Homogeneous Boltzmann equations for quntum relativistic particles, preprint E.N.S., Paris 2001. [GLW] S. R. Groot, W. A. van Leeuwen, Ch. G. van Weert, Relativistic kinetic Theory, North Holland Publishing Company, 1980. [K] A.S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4 (1957) 730-737. [LY1] E. Levich, V. Yakhot: Time evolution of a Bose system passing through the critical point. Phys. Rev. B 15 (1977), 243-251. [LY2] E. Levich, V. Yakhot: Time development of coherent and superfluid properties in a course of a A-transition. J. Phys. A: Math. Gen., 11 (1978), 2237-2254. [LP] E. M. Lifschitz, L. P. Pitaevskii: Landau and Lifschitz, Course of Theoretical Physics Volume 10, Pergamon Press 1981. [P] P.J.E. Peebles: Principles of Physical Cosmology, Princeton University Press, (1993). [ST1] D. V. Semikov, I.I. Tkachev: Kinetics of Bose Condensation, Phys. Rev. 74 (1995), 3093-3097.

Lett,

[ST2] D. V. Semikov, I.I. Tkachev: Condensation of bosons in the kinetic regime, Phys. Rev. D, 55 (1997), 489-502. [S] A. Stebbins: The CMBR Spectrum, report of the Fermi National Accelerator Laboratory, Fermilab-conf-97/134. [UU] E.A. Uehling, G. E. Uhlenbeck: Transport Phenomena in Einstein-Bose and FermiDirac Gases. I Physical Review 43 (1933) 552-561. [VI] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rat. Mech. Anal., 148 (1998), 273-307. [V2] C. Villani, These d'habilitation, E.N.S. Paris, 2000. [ZL] Ya. B. Zel'dovich, E. V. Levich: Bose Condensation and Shock Waves in Photon Spectra, Sov. Phys. JETP 28 (1969), 1287-1290.

Boundedness of global solutions of nonlinear parabolic equations Marek Fila* Institute of Applied Mathematics Comenius University 842 48 Bratislava, Slovakia Email : [email protected]

Abstract We consider superlinear parabolic problems for which blow-up in finite time occurs for some initial data but other initial data may yield global solutions. We review results concerning the question whether or not global solutions are necessarily bounded.

1

Introduction

In some classes of nonlinear parabolic problems, solutions may cease to exist in finite time by becoming unbounded. Such a phenomenon is called blow-up in finite time and it has been a subject of intensive mathematical studies in connection with various fields of science such as plasma physics, combustion theory and population dynamics, for instance. In this survey we consider problems for which blow-up in finite time occurs for some initial data but other initial data may yield global solutions. We discuss results which say that global solutions are necessarily bounded or, in other words, blow-up in infinite time does not occur. The problems that we review include semilinear (Section 2) and quasilinear (Section 3) equations with Dirichlet boundary conditions, the Cauchy (Section 4) and Stefan (Section 5) problems for semilinear equations, the Cauchy problem for a semilinear equation in similarity variables (Section 6), problems with nonlinear Neumann (Section 7) and nonlinear dynamical (Section 8) boundary conditions, and also systems (Section 9). The last two sections are devoted to a brief description of the methods (Section 10) and applications (Section 11). "The author was partially supported by VEGA grant 1/7677/20.

88

89

2 2.1

Semilinear equations with the Dirichlet boundary condition Power nonlinearity

Consider the problem ut = Au+ |u| p_1 u x e D, t > 0, u = 0 x e 5D, £ > 0, u(x,0) = u0(x) i£D, where p > 1, u0 e LC°(D) and D is a bounded domain in RN. For this problem it is well known that there exist choices of u 0 for which the corresponding solutions tend to zero as t —> oo and other choices for which the solutions blow up in finite time. If we are interested in other types of behavior we may consider the possibility that the solution u(-,t; u0) exists globally but is not uniformly bounded. The study of the question whether the last possibility may occur was initiated by Ni, Sacks and Tavantzis [28]. The results of [28] can be expressed roughly in the following way. If p < (N + 2)/N, D is convex, u 0 > 0 and u is a global (classical) solution then u(x,t) < C(uo,D,p),

xeD,t>0,

where the constant C(uo,D,p) > 0 depends on the shape of u0 near dD. If p > (N + 2)/(N — 2), JV > 2, then there are global unbounded L1-solutions. The very interesting question whether these unbounded global weak solutions are actually classical was left open for a long time. Recently, Galaktionov and Vazquez [17] proved that for p = (N + 2)/(JV — 2) this is true provided D is a ball and w0 is radially symmetric. On the other hand, if p > (N + 2)/(N-2) andp < l + 6/(JV-10) for N > 10 then any unbounded global L1-solution blows up in finite time in the L°°-norm provided D is a ball and u0 is radially symmetric and decreasing in r. An improvement of the result from [28] on the boundedness of global solutions was given by Cazenave and Lions [2]. They removed the assumptions on convexity of D and nonnegativity of UQ and showed that global solutions are uniformly bounded if p < (N + 2)/(iV — 2) (without giving any explicit dependence of the bound on the data). An a priori bound of the form \u(x,t)\
xeD,t>0

(1)

for any global solution u was established in [2] for p(3N — 4) < 3N + 8. Later, Giga ([19]) derived an a priori bound of the form u{x,t) < C(||uo||i~p),£>,p),

xeD,

t >0

for any nonnegative global solution u when (N — 2)p < N + 2. The results from [2] and [19] were improved by Quittner ( [31]) who proved that (1) holds for the whole subcritical range (N — 2)p < N + 2.

90 The first universal bound (independent of initial data) has been established recently in [16]. More precisely, if (N - l)p < N + 1, r > 0 and u is a global nonnegative solution then u{x,t)r. (2) Even more recently, Quittner has derived a universal bound of this kind in [32] for (N - 2)p < N + 2 and N < 3.

2.2

Exponential nonlinearity

Consider next the problem: ut = Au + Aeu i € B i ( 0 ) , < > 0 , u = 0 x 6 dfli(O), i > 0 , u(x,0) = uo(x) i £ Bi(0), where Bi(0) = {x e R N : |x| < 1}, u0 is a radial continuous function on B1(0) vanishing on dBi(0) and A is a positive parameter. For JV = 1,2 it was shown in [5] that every global classical solution is uniformly bounded even for general domains and initial data. On the other hand, for N > 10 and A = 2(iV-2), global unbounded classical solutions do exist (cf. [22]). For 3 < N < 9 it was known that global unbounded L1-solutions exist for certain A > 0 (see [23]), which makes the problem of existence of unbounded global classical solutions even more interesting. It was conjectured (see [23] and [42]) that such solutions might exist for some dimensions 3 < N < 9 and A > 0. A result from [12] rules out this conjecture, as far as radial solutions are concerned. Namely, the main aim of [12] was to prove that every global classical solution is uniformly bounded if 3 < N < 9 and A > 0.

2.3

Chipot-Weissler equation

In this subsection we discuss the problem ut = Au - \Wu\q + Xu" xeD,t>0, u = 0 x € 3D, t > 0, u(x,0) = u0(x) > 0 ieB. Here p, q > 1 and A > 0. Under the assumptions 3<-7T, p+ 1

A>0

or

q= ^ r , p+ 1

A > 2"(p+ l)(p - l)-*" 1 ,

and (N — 2)p < N + 2it was shown in [6] that global increasing solutions are bounded if D is a bounded domain in RJV. If q > p and A is small enough then all solutions are global, bounded and decay exponentially to zero provided D C RN is a domain (possibly unbounded) where the Poincare inequality is valid (see [40]). Conversely, if D is such that the Poincare inequality is not valid then there always exist unbounded solutions (cf. [40]). These results of

91 [40] were slightly improved and given a more complete form in [38]. In particular, the restriction on smallness of A was removed for global existence. Studies of global solutions of similar problems with more general nonlinearities / (u, Vu) were performed in [24] and [35].

2.4

Nonlocal nonlinearity

The following nonlocal problem is considered in [7]: ut

u du u(x,Q)

Au +

g(JDF(u)dx)f(u) x e D, t > 0,

0

xeT-L, t > 0,

0 u0(x)

x e r 2 , t > o, x el>,

where D is a bounded domain in RN, 3D = T1L)T2, I \ DF 2 = 0, U

f e C\R),

/ > 0,

J f(u) du < oo,

F(u) = J f(v) dv,

— oo

g : (0, oo) —> (0, oo)

—oo

is a locally Lipschitz function.

The assumptions on / and g imposed in [7] guarantee blow-up in finite time if UQ is large. Sufficient conditions for boundedness of global solutions are given in [7] and their sharpness is discussed there. A particular case of the main result from [7] was proved independently in [1] where the method from [5] was also used. A priori bounds for global solutions of nonlocal equations of this type and also of the form ut = Au + f(x,u)-

—J

f(x,u)dx

were established in [33].

2.5

Equation with a localized reaction t e r m

In [36], the following problem was studied: ut = &u + up(x0(t),t) u = 0 u(x,0) = u0(x) > 0

xeD,t>0, x G dD, t > 0, ieB.

where p > 1, D is a bounded domain in RN and x0 : [0, oo) —> D is a Holder continuous function. The results in [36] say that if either i 0 = 0 6 D or J\f = 1 then every global solution is uniformly bounded. The former result is in contrast with the fact that for the problem discussed in Subsection 2.1, boundedness of global solutions holds only for p which is Sobolev subcritical ((TV - 2)p < TV + 2).

92

2.6

Periodic forcing t e r m

Consider the problem ih = Au + m(t)f(u) u = 0 u(x,0) = u0(x)

x £ D, t > 0, x e dD, t > 0, x e D,

where D is a bounded domain in Ft", / is a superlinear function and m is positive and periodic. An a priori bound for global solutions of this problem was derived in [34].

2.7

Abstract equation

In [9] we studied the initial value problem for evolution equations of the form d

^+Au(t)=F(u(t)),

in a Banach space where A is the infinitesimal generator of an analytic semi-group and F is a nonlinear function such that the initial value problem possesses non-global solutions for some initial data. (See [9] for details of the setting which are omitted here). We gave sufficient conditions (on A, F and the underlying spaces) to ensure that global solutions are necessarily bounded in a suitable norm. We also gave several examples of problems to which the abstract result applies. They include some nonlocal equations and equations of order higher than two.

3

Quasilinear equations with the Dirichlet boundary condition

Several works have been devoted to the study of boundedness of global solutions of the degenerate problem ut = A(|u| m - 1 u) + /(w) xeD, u = 0 xedD, u(x,0) = u0(x) x € D,

t>0, t >0,

where m > 0. It was shown in [28] that for m > 1, D convex, uo > 0 and f(u) = up, all global solutions are bounded if m < p < (N + 2)/N, while global unbounded solutions exist iip/m > (N + 2)/(N — 2),N> 2. The results from [17] that were mentioned above also apply to this problem if f(u) = up, D is a ball and u0 is radial (cf. [17] for details). In [5], boundedness of global solutions is proved for general D and UQ and nonlinearities / that include / s , ,„-i /(«) = H P »:

r ^ p>max{l,m},

f iV-21 m > max JO, - ^ — - j ,

p N +2 —<^ 3 ^ -

This result was improved in [25] in several ways. Instead of |u| m _ 1 u a large class of continuous increasing functions $(u) was taken there. Non-power growth of $ was allowed

93 for N < 2 as well as more general / ' s . For N = 1, Suzuki ([41]) proved boundedness of global solutions for m > 1 for some nonlinearities / that are not allowed in [5] and [25]. Arguments and results from [5] were generalized to parabolic p-Laplace equations in [26]. For any global solution, an a priori bound of the form Hx,t)\

< C(\\u0\\L«,(D),D,p,rn)

has been recently established in [39] for f(u) = |u| Km
, p_1

x € D, i > 0 w under the assumption

10m+ 2 <m + —— , if JV > 1,

and a universal bound of the form u(x,t)
xeD,t>r>0

has been derived there for global nonnegative solutions provided N +2 1 <m 1. N

4

Cauchy problem for a semilinear equation

Consider the problem ut u(x,0)

= Au - ii\Vu\i + vP i e R " , ( > 0 , = u 0 (a;)>0 xeRN.

Herep, q > 1 and /u > 0. If fi = 0 and (N-2)p < N + 2 then all global solutions decay uniformly to zero (cf. [37]). (An earlier study of global solutions of this problem with fi = 0 was performed in [21] where sign-changing solutions with fast decay (as x —> oo) were considered). If /x > 0 then there exist global solutions such that (cf. [40]) : lim u(x,t) = oo

5

for all x e R w .

Stefan problem for a semilinear equation

Consider now the following reaction-diffusion problem with free boundary: ut u(x, 0) u(s(t),t) s'(t)

= = = =

uxx + v? xe (0, s(i)), t > 0, u0(x) > 0 x e (0, s 0 ), s(0) = s 0 > 0, ux{0,t) = 0 t > 0 , - U a ; (s(t),t) t > 0 ,

where p > 1. This problem can be viewed as a simple model of a chemically reactive and heat-diffusive liquid surrounded by ice. Here u > 0 represents the temperature of the liquid phase, and the ice is assumed to be at temperature 0. Global existence, stability and blow-up for this problem were studied in [18], for previous studies see references therein and in [15]. In [15] we showed that if u is a global solution then u < C ( s 0 , ||uo||ci([0,so]).

94

6

Cauchy problem for a semilinear equation in similarity variables

Consider a solution of the problem 11, = A« + « p u(x,0) = u0{x)>0

x e RN, xeRN,

0
which blows up at t = T. For a E R " one can rescale u using backward similarity variables as follows

s = -log(r-t),

j/ = - ^ L ,

w{y,s) =

(T-t)^u{x,t).

Then w satisfies y • Vw + wp

w. = Am

y

2 w(y,-logT)

= T^u0(a

-w y € R " , s > — logT, p-1

y

(3) yeRN.

+ VTy)

Under some assumptions on u0 and p, it has been shown in [27] that for 6 > 0 there exists C = C(p, N, 6) > 0 such that for any global solution of (3), it holds w
for s> -\ogT + S.

In the original variables this can be equivalently formulated as a universal bound (independent of «o) for the blow-up rate.

7 7.1

Nonlinear boundary conditions Heat equation

The following problem is studied in [4]: ut

= Au

—

= f(u)

u(x,0)

x G D, t > 0, x£dD,t>0,

= UQ(X) X&D.

Here I? is a bounded domain in R and v is the outer normal to dD. The result from [4] as applied to the particular case with f(u) = \u\p~1u says that global solutions are bounded if (N — 2)p < N. This means that in this case there are no global positive solutions since there is no positive steady state and the trivial steady state is unstable from above. The main result of [4] and several elements of its proof were generalized in [24].

95

7.2

Porous medium equation

In [25], Lieberman studied the problem A$(u) x e D, t > 0, dv u(x,0)

/(«)

xedD,

«o(^)

x £ D.

t > 0,

Here D is a bounded domain in R N and $ is a continuous increasing function. The results from [25] say that global solutions are uniformly bounded if the growth of / is super linear but subcritical (which may be faster than polynomial if N = 1,2). For the precise meaning of "superlinear" and "subcritical" see [25].

7.3

Porous medium equation with absorption

The large time behavior of solutions of the problem ut

= (um)xx - aup \x\ < 1, t > 0 , |x| = 1, t> 0,

u(x,0)

= UQ(X) > 0

W<1,

was discussed in [8]. Here a > 0 and p,q> m > 1. Among other things it was shown in [8] that if p < 2q — m or if p = 2q — m and ma < q then blow-up in infinite time does not occur for any initial data while blow-up in finite time occurs for some initial data. On the other hand, if p = 2q — m and ma = q then all solutions are global and unbounded. Corresponding results for m = 1 can be found in [3].

8 8.1

Dynamical boundary conditions Laplace equation

Consider next the problem du

Au du

u(x,0)

0

x e D, t> 0,

h(u)

x&dD,

u0(a;) > 0

t > 0,

xedD,

where A is the Laplacian in x, D is a bounded domain in R N , N > 2, with smooth boundary, u is the outer normal to dD, h e C1 and u0 belongs to some appropriate function space. We shall consider superlinear nonlinearities h for which blow-up in finite time occurs if initial data UQ are "large". As an example we can take h{u) = |w| p_1 u — au,

p > 1, a > 0.

It is shown in [13] that all global solutions are bounded provided the growth of h is subcritical, i.e. (N — 2)p < N if h is as above. But under this assumption only an

96 a priori bound for l i m s u p ^ ^ ||w(-,t)||C(n) i s obtained. To derive an a priori bound for su Poo llu('i*)llc(ft) a stronger growth restriction on h is needed in [14]. As an application of the latter result we proved existence of sign-changing solutions of the corresponding stationary problem.

8.2

Parabolic equation

The paper [14] is devoted to the study of the problem tH = Au + \\u\p-lu du

du

, q,

x£D,t>0,

i

„„

-sr + w- = nM u at dv u(x,0) = uo{x)>0

xedD,

t > 0,

xedD,

where Q is a bounded domain in R w , v is the outer normal on dQ, p, q > 1, A, fj, £ {—1,0,1}, max{A, fj,} = 1 and \p+/iq > 0. Under our assumptions on p, q, X and n, blowup in finite time occurs for some initial data. We proved in [14] that if (JV — 2)p < N + 2 and (N — 2)q < N then all global solutions are uniformly bounded. To obtain an a priori bound for global solutions in terms of suitable norms of M0 a stronger condition on p and q is needed. Namely, if N--^)p
+-

and

(N - - ) q < N + -,

then sup||u(-,£;u0)||z,~(n) < C x(|KIU~(0), IKII/fi(n)) > oo ' provided u is a global solution. As an application of this a priori bound we proved in [14] the existence of sign changing solutions of the problem Au - lu^u

= 0

^ dv

=

x £ D,

Wo-^

xedD,

if / 5\ 1 Kp
9

Weakly coupled system

Consider the system = vt = u = V = u(x,0) = v(x,0) = «t

Aw + ?)p Av + uq 0 0 u0(x) > 0 v0(x)>Q

x e D, t > 0, x e D, t > 0, xedD, t > 0. xedD, t>0,

x e ID, iefl,

97 where p, q > 1 and D is a bounded domain in RN. It was shown in [16] that if max(p,g) < 1 + 2/(N + 1) and r > 0 then there is a constant C(D,p,q,r) > 0 such that for all global solutions it holds u(x,t) +v(x,t)

< C(D,p,q,r),

x e D, t > T.

An a priori bound (which depends on initial data) for global solutions has been established in [35] under the assumption max(p,g) + l TV + 1 pq - 1 " 2 and under the weaker assumption max(p, q) + 1 pq-1

TV ~^

N

if D = R .

10

Methods

10.1

Energy methods

The basic idea of the proof of the boundedness of global solutions in [4]-[9], [11], [12], [22]-[24] is similar. We illustrate it here for the simplest problem considered in Subsection 2.1. We show by contradiction that the W/1,2(D)-norm of any global solution must be bounded uniformly in t, and then we use known results to conclude that the C(Z))-norm is also uniformly bounded. Suppose limsup ||u(-,t)||v^i,2(£)) = oo. t—*oo 2

A uniform L (D)-estimate (or the concavity method) enables us to rule out the possibility that hm||u(-,t) 11^1,2(0) = oo. In this part of the argument, the superlinear growth of the nonlinearity (as u —> oo) plays a crucial role. On the other hand, if limsup ||u(-,i)||vKi.2(D) < oo, then the o;-limit set of u contains for every number B large enough a stationary solution w with ||w||W^2(D) = B. (The assumption that the growth of the nonlinearity (as u —* oo) is subcritical is needed here.) This leads to a contradiction since there is a constant K (depending on UQ) such that ||u||iyi,2(£)) < K for every stationary solution v in the oi-limit set of u.

98 Both parts of the argument rely on the existence of a suitable Lyapunov functional. In general, this method does not yield any a priori estimate of ||u(-, i)||iv1.2(D) in terms of some norms of UQ. Other energy and interpolation arguments were used in [2]. The method from [2] was further developped in [31, 33] by bootstrapping and employing maximal regularity results.

10.2

Scaling arguments

The variational structure together with scaling invariance are the main tools in [19]. If we rescale a solution u of the equation ut = A H + up, as v(y, s) = \r-lu{x

+ Xy,t + As),

then v satisfies the same equation. This fact, combined with the existence of a Lyapunov functional, yield a contradiction argument to prove the a priori bound in [19]. A different contradiction argument, relying on scaling, energy and Hardy's inequality, gives the universal bound in [32]. In order to handle problem (3), this approach was extended in [27]. For problems without variational structure, a scaling argument was used in [35] in order to prove a priori estimates for global solutions.

10.3

Zero number techniques

The method of our proof of boundedness of global classical solutions in [12] is based on the intersection-comparison with a particular self-similar solution. It was inspired by the proof of Theorem 15.1 in [17] where a similar result was shown for a different problem. However, unlike the case in [17], the nonincrease of the number of zeros of the difference of two solutions, by itself, is not sufficient. We also employ the more subtle "zero number diminishing property" which says that the number of zeros drops when a multiple zero occurs. The nonincrease of the zero number also plays a crucial role in [41].

10.4

Smoothing effect in weighted Lebesgue spaces

To prove (2.2) in [16], we start from the classical argument of Kaplan which yields that for any r > 0 there is T\ £ (0, r/2) such that Ju"(T1,x)
(4)

where ipi > 0 is the first eigenfunction of —A in HQ(SV), normalized by / n ipi — 1. Since Ci<5 < ipi < C2<5, <5(a;) = dist(x,dD), the bound (4) suggests that it is rather natural to

99 consider the problem in L% = Lq(D, S(x) dx) spaces. With this motivation, we developped in [16] both linear and local nonlinear theories in these spaces. Our main result in the linear theory is an optimal Lj - Lrs estimate for the heat semigroup. In view of this estimate for the heat semigroup, the local existence and uniqueness theory in L\ is similar to the usual L9-theory in dimension JV+1. In particular, we proved local existence and uniqueness for solutions in L\ for q > qc = |(JV + l)(p — 1) and local nonexistence for q < qc. We also established a nonlinear version of the L\ — Lre estimate whose particular case q = p, r = oo, plays a crucial role in deriving the universal bound (2) from (4). A similar idea is used in [27] but weighted Lebesgue spaces are replaced by convolution Lebesgue spaces there.

11 11.1

Applications Blow-up

The first application concerns blow-up in finite time. It requires some knowledge of the corresponding stationary problem. For example, if one can show that there are no steady states then all solutions of the parabolic problem blow up in finite time (cf. [1, 5]). A slightly more sophisticated application of this type says that a solution blows up in finite time if it starts above a maximal steady state provided this steady state is unstable from above, cf. [3, 4, 6, 8], for instance.

11.2

Existence of stationary solutions

The second application goes in the opposite direction. Some knowledge of the parabolic problem is used to draw conclusions on the corresponding stationary problem. We describe roughly a simple illustration of this idea. Assume a stable steady state is known to exist. If Mo belongs to the boundary of its domain of attraction and the solution u starting from u0 is global then there must exist another (unstable) steady state which is in the w-limit set of u. For dynamical proofs of existence of solutions of elliptic problems we refer to [13, 14, 29, 30, 34].

11.3

Existence of periodic solutions

The a priori bound discussed in Subsection 2.6 was used in [34] to prove the existence of solutions which are periodic in t.

11.4

.^-connections

The result from [12] mentioned in Subsection 2.2 plays a fundamental role in the study of connections between equilibria by solutions which blow up but continue to exist in a weak sense (see [10, 11]).

100

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101 M. Fila, P h . Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces and Free Boundaries 3 (2001), 337-344. M. Fila, P h . Souplet, F. Weissler, Linear and nonlinear heat equations in L\ spaces and universal bounds for global solutions, Math. Ann. 3 2 0 (2001), 87-113. V. Galaktionov, J.L. Vazquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. P u r e Applied Math. 50 (1997), 1-67. H. Ghidouche, P h . Souplet, D. Tarzia, Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary, Proc. A.M.S. 1 2 9 (2001), 781-792. Y. Giga, A bound for global solutions of semilinear heat equations,Comm. Math. Phys. 103 (1986), 415-421. S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. P u r e Appl. Math. 16 (1963), 327-330. O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincare, Anal. Non Lineaire 4 (1987), 423-452. A. A. Lacey, D. Tzanetis, Global existence and convergence to a singular steady state for a semilinear heat equation,Proc. Roy. Soc. Edinburgh Sect. A 1 0 5 (1987), 289305. A.A. Lacey, D.E. Tzanetis, Global, unbounded solutions to a parabolic equation,J. Diff. Equations 101 (1993), 80-102. G.M. Lieberman, Study of global solutions of parabolic equations via a priori estimates. P a r t I: equations with principal elliptic part equal to the Laplacian,Math. Methods Appl. Sci. 16 (1993), 457-474. G.M. Lieberman, Study of global solutions of parabolic equations via a priori estimates II: porous medium equations,Comm. Appl. Nonlin. Anal. 1 (1994), 93-115. G.M. Lieberman, Study of global solutions of parabolic equations via a priori estimates III. Equations of p-Laplacian type, in Singularities and Differential Equations, Banach Center Publications, Vol. 33, pp. 199-221, P W N , Warszawa (1996). J. Matos and P h . Souplet, Universal blow-up estimates and decay rates for a semilinear heat equation,preprint. W.M. Ni, P.E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations,J. Diff. Equations 5 4 (1984), 97-120. P. Quittner, Boundedness of trajectories of parabolic equations and stationary solutions via dynamical methods,Diff. Int. Equations 7 (1994), 1547-1556.

102 [30] P. Quittner, Signed solutions for a semilinear elliptic problem, Diff. Int. Equations 11 (1998), 551-559. [31] P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem,Acta Math. Univ. Comen. 68 (1999), 195-203. [32] P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem,Math. Ann. 320 (2001), 299-305. [33] P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems,Houston J. Math., to appear. [34] P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems,NoDEA, to appear. [35] P. Quittner, Ph. Souplet, A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discr. Cont. Dyn. Systems, to appear. [36] P. Rouchon, Boundedness of global solutions of nonlinear diffusion equation with localized reaction term, preprint. [37] Ph. Souplet, Sur l'asymptotique des solutions globales pour une equation de la chaleur semi-lineaire dans des domaines non bornes, C. R. Acad. Sci. Paris, Serie I 323 (1996), 877-882. [38] Ph. Souplet, Geometry of unbounded domains, Poincare inequalities and stability in semilinear parabolic equations, Comm. PDE 24 (1999), 951-973. [39] Ph. Souplet, A priori and universal estimates for global solutions of superlinear degenerate parabolic equations, Annali Mat. Pura Appl., to appear. [40] Ph. Souplet, F.B. Weissler, Poincare's inequality and global solutions of a nonlinear parabolic equation,Ann. Inst. H. Poincare, Anal. Non Lineaire 16 (1999), 335-371. [41] R. Suzuki, Boundedness of global solutions of one dimensional quasilinear parabolic equations,,!. Math. Soc. Japan 50 (1998), 119-138. [42] D. Tzanetis,Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation,Proc. Edinb. Math. Soc. 39 (1996), 81-96.

Transmission problems for degenerate parabolic equations Takeshi Fukao Department of Mathematics Graduate School of Science and Technology Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan Email : [email protected] Abstract In this paper, we consider a transmission problem for nonlinear two degenerate parabolic PDEs, in such a setting that a fixed bounded domain in R 3 is separated by a moving interface into two subdomains. The degenerate parabolic PDEs are formulated on each of these subdomains and moreover the so-called transmission condition is imposed on the moving interface. The purpose of this paper is to prove the existence and the uniqueness of a solution.

1

Introduction

We consider the following initial-boundary value problem for degenerate parabolic equations in a cylindrical domain Q := (0, T ) x Q, where 0 < T < oo and fl is a bounded domain in R 3 with smooth boundary T. Moreover, we suppose t h a t 0 is separated by a time dependent smooth interface Tm(t) into two subdomains fii(£) and Ch{t), t h a t is : Q, : = n-i.(t) U n2(t) U Tm(t) for each t € ( 0 , T ) : «,-Aft(a)+v-Vu = / ft(«)=ft(«),

^

r

in Q; := U t e ( o r ) { i } x f2i(t),

+ ^ i ^ = °

i = l,2,

on E m := U ( 6 ( 0 , T ) W x r m ( t ) , (1)

^

^

+ noA(«) = p

o n S , := U e ( o , r ) W * W W

\ I W * ) } , » = 1,2,

u(0) = UQ on Q. Here ut := du/dt, A u := E L I ^ V ' ^ ? ) and V u := {du/dxudu/dx2,du/dx3). We denote by u+ := u+(t,x) t h e 3-dimensional unit vector normal to T m (i) at x e T m ( t ) , pointing to fi2(i), by v~ : = i/ _ (£,:r) t h e vector — v+ and by v := ^(a;) t h e 3-dimensional unit vector normal to V at x € T. no is a positive constant, / and p are given functions prescribed on Q and E, respectively. u0 is the initial d a t u m on Q. fa and fa are functions defined on R and which satisfy t h e following conditions:

103

104 • Pi • K —• K is a Lipschitz continuous and non-decreasing function for i = 1,2. • A(0) = 0 and \k(r)\ > c f t |rf - c^, where /3;(r) :=

JQr

Vr e R and i = 1,2,

f3i(s)ds, i = 1,2 and c ftl c^ are positive constants.

In the above problem, (1)2 describes a relation between Pi(u) and feW) on the moving interface T m (i). It is called a transmission condition. In our geometric setting, the following two cases are allowed, the first one is that <9fi;(*) = F u r m ( i ) and dQ,j{t) = Tm(t) for (i,j) = (I, 2) or (2,1), and the second one is that a part of <9f2;(i) is included in T for i = 1,2. For a typical example of the latter case, see [7]. In this paper, we assume that the following conditions are satisfied: ( A l ) For each t 6 [0,T], there exists a C2-diffeomorphism © := ©(£, •) on fi, such that 6 £ C2(Q) and Q(t,Qi(t))=Qi{0),

i = l,2,

O(i,r m (t)) = r m (0),

Vt€[0,T].

This assumption (Al) says that the motion of Tm(t) is prescribed and smooth in time and the C2-regularity of 6 is actually used in the existence and uniqueness proofs of solutions (Lemma 3.3, Lemma 4.2). (A2) We assume that the convection vector field v belongs to C 1 (Q) 3 and that the following properties are satisfied: div (v) = 0 in Q, +

v • v = vxm

(2)

on E m ,

(3)

v i / = 0 on£:=(0,T)xr,

(4)

where u Sm := v-Em(t,x) is the normal speed of Tm(t) at (t,x) € S m . We define the time-dependent function (3 : Q x R —* R P(t,x,r)

:= Pi(r)xQ1(t) + P2{r)xn2(t),

where x
dP(t x,u)

+

+ v - Vu

= f

onE;

=

«(0) =

in Q,

UQ

(5)

in O.

For the solvability of (5), it seems that the time-dependent subdifferential operator theory is useful (cf. [1, 4, 7, 8, 9, 10, 16]). In sections 3 and 4, we prove the existence and uniqueness of a solution.

105

2

Definition of t h e weak solution and t h e main result

We begin with the definition of a weak solution of (5) and then formulate our main result. To this end we introduce some function spaces. We denote by V the Sobolev space iJ 1 (Q) with norm j • |y given by 1/2

\u\v := {|Vu|£ 2(n) + n 0 | u | | 2 ( r ) |

Vu € V,

which is an equivalent norm to the standard one of H1 (f2). The dual space of V is denoted by V* and the duality pairing between V* and V is denoted by {-,-)v.y. The duality mapping F : V —> V* is defined by (Fu 1 ,u 2 )v«,v := / Vui • \7u2dx + n0 / uiu2aT Jn Jv

Vuj,u2 € V.

Moreover, we define an inner product (•, -)y in V* by (u\,u*2)v •= « , F~lv*2)v.y

\ful,u*2 6 V*.

Then V* is a Hilbert space with the following dense and compact imbeddings: V ^ L2(Q) ^

V*.

Now we give the definition of a weak solution of (5). Definition 2.1 u : [0, T] -* L2{0.) is called a weak solution of (5) ifue Wh2(0, T; V*) n L°°(0,T;L 2 (f2)), /3(u) := f3(t,x,u) G L2(0,T;V), and u satisfies (5) in the variational sense : - JQ ur]tdxdt + JQ Vp(u) • Vvdxdt 4 n0 J E f}{u)rjdTdt — JQ u(v • Vrfidxdt = JQ fndxdt + / s prjdTdt + Jn uori(0)dx V77 e W,

' ^

where W := {77 € Hl(Q) | rj(T) = 0 on fi}. Our main result is stated as follows. Theorem 2.2 Assume that (Al) and (A2) hold. Let f € L2{Q), p e Cl{Y,), u0 e L2{Q) and f3(u0) € if 1 (f7). Then there exists one and only one weak solution u of (5). The existence proof for (5) will be given in section 3, using the abstract theory of time-dependent subdifferential operators. The uniqueness proof will be given in section 4.

106

3

Existence proof

Let us prove the existence of a weak solution of (5), applying the existence result in [16] (or [8]) for a nonlinear evolution equations including time-dependent subdifferential operators and a perturbation of the form: f ut(t) + d(t>\u{i)) + g(t,u(t)) B f*(t) \ w(0) = u0 in V,

in V* for a.e. t £ [0,T],

(?)

where the functions (pf, g(t, •) and f*(t) for each t £ [0, T] are defined as follows: • cf/ : L2(Q) —> (—oo, oo] is defined by ftiz) '•=

/ / \

JJ fii(t)

01(z(x))dx+ f

iizeL2(Q),

j32(z(x))dx

JJ n f 2(t)

if z e V*\L 2 (fi).

Clearly 0* is a proper, lower semi-continuous and convex function on V. • g(t, •) : L2(tt) -> V* is defined by (ff(t,z),ri)vy •=-

I z(x)(v{t) • Vn)(x)dx Jn

L2(Q).

V77 &V,VZ£

• /* £ L2(0, T; V*) is defined by (f(t),ri}v*,v

:= { f(t,x)V(x)dx+ Jn

f p(t,x)n(x)oT Jr

Vr,eV.

It is easy to see that (5) is equivalent to (7). Therefore, it is enough to solve (7) with the above-defined functions $, g(t, •) and /*(*). The characterization of dcj/ is given by the following lemma. L e m m a 3.1 Let z £ D(90') and j3(z) := /3(t,x,z(x)). Then z* £ d<jt{z) in V* if and only if (F-1z*,w)^(fi) = {P{z),w)^(p) V»6l2(fi), (8) where (V)L 2 (S)) denotes the inner product in L2(Q). statement that j3(z) £ V and {z*,T))v.y

Moreover, this is equivalent to the

= / V/3{z) • VVdx + n0 J P(z)ndT Jn Jr

Vr? £ V.

(9)

The next lemma is concerned with the demi-closedness for g(t, •). Lemma 3.2 The operator g(t, •) : D(g(t, •)) = L2{Q.) -*V*,te 2

[0,T], satisfies :

\g(t,z)\ v. < C0<j>\z) + Ci Vt £ [0,T], Vz £ L 2 (Q), (10) where C$ and C\ are positive constants, and if {4>tn(zn)} is bounded and zn —> z in V* and tn —> t, when n —> oo for {zn} c L2(Q) and {tn} C [0, T], then g(tn,zn) —+ g(t,z) weakly in V*. Furthermore, the set Lr := (J0 0.

107 The proofs of these lemmas are quite standard, so we omit them. In the following lemma, we check the time-dependence of ft and the conditions required in order to apply the abstract result in [16] (or [8]) to (7). Lemma 3.3 There exists a positive constant C2 with the following property : for every s, t € [0,T], s
+ \ft{z)\ll2),

t

\4> {zi)-ft{z)\
(11)

+ \ft{z)\).

(12)

Proof. For every z G D(ft) := L2(Q), we take zx(x):=z(Q-1(s,e{t,x)))

Viefi.

Then, by the geometric assumption (Al), zx e L2(fl) = D{ft) and the mapping &t,s '•= Q(s, -)~ L ° ©(*,-) i s a C2-diffeomorphism on fi for every s, t G [0,T] such that ©M (£k{t)) = &i{s), i = 1,2. Moreover there exist two positive constants C 3 and C4 such that | d e t ( j e t , ( z ) ) - 1 |
4

Uniqueness proof

In this section, we prove the uniqueness of a weak solution of (5) obtained in the previous section. Let u\ and u2 be two weak solutions. Then taking the difference of their weak variational forms (6) we have - JQ(ui - u2)rjtdxdt + JQ(V/9(wi) - V/?(u2)) • Vndxdt - jQ(ui - u 2 )(v • Vn)dxdt +nQ / E ( / ? K ) - P{u2))r,dYdt = 0 \fr,eW. Moreover, we define the function b : Q —> R as b{t,x):=i

( /?M*,s))-/?Mts)) Ul(t,x)-u2(t,x) [ 0

jf

^ (t } A ' l r ' if «i(t,a;) = « 2 (i,:r), (i

U

which is non-negative and bounded on Q. With this function b, Green's formula implies - f (u1-u2){nt Q

+ bAV+v-Vn}dxdt+

f {PiuJ-Piui)) E

l^+n0ri\ "

dTdt = 0

\fneW. (13)

108

Now we take smooth and positive approximations bE of b such that b
a.e. on Q, e
. .

where M0 is a positive constant. In order to prove the uniqueness, we consider the following backward problem for the auxiliary linear parabolic equation for any given ieD(Q): t]Et, + bEAr)£ + v • Vr?e = I in Q, ^+n0r,e = 0 on£, av 7jE(T,-) = 0 onO.

(15)

Using Ladyzenskaja, Solonnikov and Ural'ceva [11, Chapter 4, Theorem 5.3], problem (15) has a unique solution r)e € H2+a-1+a!2(Q) for a certain 0 < a < 1. In the next two lemmas, we give the L°°-estimate of {r)B(t)} in V and the boundedness of {bl/2AVe} in L2{Q). Lemma 4.1 There exists a positive constant Mi such that sup \r,e{s)\l+ 0<s
( Js

[ be\Ane\2dxdt < M1 f Ja

j v •V{rj2)dTdt

+M^^.y)

Js JT

for alls e [0,T] and e G (0,1]. Proof. Multiplying the first equation in (15) by Ane and integrating over Q with respect to x, with the help of Green's formula we have -|l»felv + 2 1

%\\AVe\2dx < (6|v| c l ( 5 ) 3 + l)|ife|», + \l\*v + 2n0 J

v • V(^)dT,

V i e [0,T].

(16) In fact, in order to get (16) we use the following equations and inequalities - / (v • Vr)s)Ar)£dx = V(v • Vr/£) • Vj)edx - f (v • Vr]e)^-dY J si J si \ r < 3|v| c l ( g ) , | n | V ^ d z + / n v • V ^ | V T ? E | 2 J dx+^J^v V(V2)dT and by virtue of (2) and (4) jf v • V (\\^vA

dx = Jdiv

Q | V % | 2 v ) dx = J i | V ^ | 2 ( v • v)dT = 0.

Finally we integrate (16) over [s, T] in time and apply Gronwall's inequality to get the conclusion. •

109 Lemma 4.2 There exists a positive constant M2 such that:

/r

v(<) ' V(77e2(i))dr < M2\nE(t)\2L2{T)

Vt e [0,71, Ve € (0,1].

Proof. Our geometric condition (Al) ensures that without loss of generality we may assume by taking an approximate local coordinate transformation and a partition of unity on r that: • Q c { x = ( i i , i 2 , x3); x3 < 0} and T contains a relatively open set T0 in K2 = {x | x3 = 0}. • supp (r/e) is compact infiU r 0 , Ve <E (0,1] and

In the above situation, we observe for any t £ [0, T] that /

v-V(7? 2 )dr

=

vV(V2)dXldx2

/ r

{vi,x! + v2,X2)ri2dx1dx2 + / 2v3ne^-dx1dx2 Jtf J R2 = - / (vi,xl+v2iX2+2n0v3)r]2dxidx2, J R2 where v := (^i, •02,^3)- Therefore, for any t € [0,T], =

-

/ vv -' Vv(?i ( r ;e2)ai )dr< ^ ( 2V^oi-^;|v|c.i n 0 + 2)|v| c l((Q)3|'/ Q ) 3| %E1|2L 2 (r) . Thus we have the conclusion of the lemma.

•

Now, on account of Lemmas 4.1 and 4.2 we have the uniform estimate 2 sup h(t)\ v+

f b£\Ane\2dxdt < M3, (17) Jo 0 00) and a function r\ 6 V such that 7]en —» 77, weakly in £ 2 (Q), Vrjen —> V77, weakly in L2(Q)3. Take r/En as a test-function in (14), and pass to the limit to obtain that [ (uj - u2)£dxdt = 0 WeV{Q). JQ

Thus Uj = u2 a.e. on
•

Remark 4.3 TTie idea of our uniqueness proof is due to Ladyzenskaja, Solonnikov and Ural'ceva [11, Chapter 3, Section 3], and this was also extensively used in Niezgodka and Pawlow [12], Rodrigues and Yi [15] and Rodrigues [14] for the uniqueness proof of generalized Stefan problems and continuous casting problems. A similar technique was recently applied for the uniqueness proof of Stefan problems arising in the Czochralski process of crystal growth (cf. Fukao, Kenmochi and Pawlow [6, 7]).

110 Remark 4.4 In particular, if Pi and P2 are strongly monotone in M, namely (A(fi) - Pi{ri)){rx - r2) > m0\ri - r 2 | 2

Vr-j, r 2 e R and i = 1,2,

(18)

Wiere m 0 is a positive constant, then the uniqueness of a solution of (5) is a direct application of [16, Theorem 3.1]. The main point of the proof is to use the estimate / (uj. - u 2 )v • \7{F 1(ul - u2))dx < mi|v| c i ( Q )3 |ui -u 2 |i!i(n)|ui

~u2\v.,

2

(19)

yUl,u2eL (Q), where mi is a positive constant so that I V F - ^ l ^ ^ a < mi\F-1z\v(=

mi\z\v.)

Vz e V*.

In fact, for two solutions «! and u2, we observe that {u[(t) - u'2(t),r,)v.y J Dn

+ (F(f3(Ul(t)) -

(ui{t) - u2{t))(v{t) • Vri)dx = 0,

p(u2(t))),V)v.y VT? e V, a.e. t € [0,T].

Now, take i ? _ 1 (ui(i) — u2(t)) as n and use (18) and (19). Then ~ K ( t ) - «2(t)| 2 ^ + (mo ~ S)\ui(t) - u2(t)\2LHn) < - ^ M C H Q J S M * ) " fora.e.te [0,T],

u2(t)\2v,,

where 8 is an arbitrary positive number. Choosing m0/2 as S and applying the Gronwall's inequality, we infer that Uj = u2. •

5

Application t o Czochralski crystal growth process

This work is motivated by modeling the Czochralski process of crystal growth. The Czochralski pulling method is widely used for the production of a column of single silicon crystal from the melt. The idea for pulling method due to Czochralski is quite simple. A crucible, equipped with a heater, contains the melt substance and a pul-rod with seed crystal, which moves vertically and rotates flexibly, is positioned above the crucible. The rod is dipped into the melt, and then lifted slowly with an appropriate speed so that a meniscus surface is formed below the crystal seed and the melt attached to the crystal solidifies continuously. By controlling some thermal situations in the process, one obtains the growth of a single crystal column with a desired radius as well as a desired growth pattern of the solid-liquid interface and temperature pattern in the crystal in order to improve the crystal quality (see [3, 5, 6, 7, 13]). In such a model of crystal growth, if we assume that the motion of the material (solid-liquid) domain is prescribed and a 3-dimensional convective vector field, which is caused by the deformation of the material domain, is prescribed too, at least satisfying condition (A2), then our approach seems applicable to the Czochralski process. In this

111 case, we consider two equations in the crucible. One of them is a degenerate parabolic equation which describe the solid-liquid phase transition in the material region. The other is an usual heat equation in the gas region. These two equations are combined by the transmission condition on the common boundary. In this case the transmission condition implies that the temperature is possibly discontinuous on the solid-gas and liquid-gas interfaces. This is one of the important points which should be furthermore discussed from the mechanical point of view.

References [1] H. Attouch and A. Damlamian, Application des methodes de convexite et monotonie a l'etude de certaines equations quasi lineaires, Proc. Royal Soc. Edinburgh, 97A (1977), 107-129. [2] H. Brezis, Operateurs Maximaux Monotones et Semi-groupes de contractions dans les espaces de Hilbert. Math. Studies 5, North-Holland, Amsterdam, 1973. [3] A. B. Crowley, Mathematical modelling of heat flow in Czochralski crystal pulling. IMA J. Appl. Math., 30 (1983), 173-189. [4] A. Damlamian, Some results on the multi-phase Stefan problem, Comm. Part. Diff. Eq., 2 (1977), 1017-1044. [5] E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems with change of phase. Arch. Rat. Mech. Anal., 123 (1993), 99-116. [6] T. Fukao, N. Kenmochi and I. Pawlow, Two phase Stefan problems in non-cylindrical domains. In preparation. [7] T. Fukao, N. Kenmochi and I. Pawlow, Transmission problems arising in Czochralski process of crystal growth. In Mathematical Aspects of Modeling Structure Formation Phenomena, GAKUTO Intern. Ser. Math. Sci. Appl., Vol.17, pp.228-243 Gakkotosho, Tokyo, 2001. [8] A. Ito, N. Yamazaki and N. Kenmochi, Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces. In Dynamical Systems and Differential Equations, Vol.1, ed. W. Chen and S. Hu, pp.327-349 Southwest Missouri State Univ., Springfield, 1998. [9] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Education Chiba Univ., 30 (1981), 1-87. [10] N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints. Nonlinear Analysis, 10 (1986), 1181-1202. [11] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type. Transl. Math. Monographs, Amer. Math. Soc, 23, 1968.

[12] M. Niezgodka and I. Pawlow, A generalized Stefan problem in several spaces variables. Appl. Math. Optim., 9 (1983), 193-224. [13] I. Pawlow, Three-phase boundary Czochralski model. In Mathematical Aspects of Modeling Structure Formation Phenomena, GAKUTO Intern. Ser. Math. Sci. Appl., 17, pp.203-227 Gakkotosho, Tokyo, 2001. [14] J. F. Rodrigues, Variational methods in the Stefan problem. In Phase Transitions and Hysteresis. Lecture Notes Math. Vol.1584, pp.147-212 Springer-Verlag, 1994. [15] J. F. Rodrigues and Yi-Fahuai, On a two-phase continuous casting Stefan problem with nonlinear flux. European J. Appl. Math., 1 (1990), 259-278. [16] K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials. In Recent Developments in Domain Decomposition Methods and Flow Problems, GAKUTO Inter. Ser. Math. Sci. Appl., 11, pp.287-310 Gakkotosho, Tokyo, 1998.

Existence of solutions of a segregation model arising in population dynamics Gonzalo Galiano Maria L. Garzon Departamento de Matematicas, Universidad de Oviedo, 33007 Oviedo, Spain, Email: [email protected] .uniovi.es, maria@or ion. ciencias. uniovi.es Ansgar Jungel Fachbereich Mathematik und Statistik, Universitat Konstanz, 78457 Konstanz, Germany, Email: [email protected]

Abstract A strongly coupled cross-diffusion model for two competing species is analyzed. The unknowns are the population densities of the species. An existence proof for the evolution problem in one space dimension is sketched. The proof is based on a symmetrization of the problem via an exponential transformation of variables and the use of a new entropy functional. The solutions can be shown to be nonnegative, by means of the exponential transformation of variables. Moreover, necessary conditions for segregation phenomena for the steady-state problem are given and numerically illustrated.

1

Introduction

For the time evolution of two competing species with homogeneous population density, usually the Lotka-Volterra differential equations are used as an appropriate mathematical model. In the case of non-homogeneous densities, diffusion effects have to be taken into account leading to reaction-diffusion equations. Shigesada et al. proposed in their pioneering work [27] to introduce further so-called cross-diffusion terms modeling interspecific influence of the species. Denoting by rii the population density of the i-th species (i = 1,2) and by J; the corresponding population flows, the time-dependent equations

113

114 can be written as d t ni + div Jx 9jn2 + div J 2 Ji h

= = = =

gi{nun2), 52 ("1,^2), - V ((ci + anni + ai2n2)ni) + diriiVU, - V ((c2 + oci\nx + a22n2)n2) + d2n2VU,

(1) (2) (3) (4)

in the bounded domain Q, C Rd (d > 1) with time t > 0. Here, £/ = U(x) is the (given) environmental potential, modeling areas where the environmental conditions are more or less favorable [24, 27]. The diffusion coefficients c* and a*, are non-negative, and d j £ l (i,j = 1, 2). The source terms are in Lotka-Volterra form: 9i{ni,n2)

= (fli-7n«i-7i2«2)ni,

02 ("l, n 2 ) =

{R2 - 721711 - 722"2)«2,

(5) (6)

where i?, > 0 is the intrinsic growth rate of the i-th species (i = 1,2), 711 > 0 and 722 > 0 are the coefficients of intra-specific competition, and 712 > 0 and 721 > 0 are those of interspecific competition. The above system of equations is completed with mixed Dirichlet-Neumann boundary conditions and initial conditions: n-i = n-D,i on To x (0,00), Jj • v = 0 ni(-,0) = n0,j

o n f ^ x (0,00), in fl, i = 1,2,

(7) (8)

where v denotes the exterior unit normal to d£l. This means that the population density is fixed at a part of the domain boundary (due to emigration and immigration processes), whereas no flux boundary conditions are prescribed at the remaining boundary parts. Eqs. (l)-(4) contain various types of reaction-diffusion models. Indeed, in the case OLij = 0 for i, j = 1,2, they reduce to the drift-diffusion equations, which has been studied in various fields of application, e.g. electro-chemistry, [1, 3], biophysics [7] or semiconductor theory [23]. When c\ = c2 = 0 and «i2 = a2i = 0, Eqs. (l)-(4) are of degenerate type. These types of problems arise, for instance, in porous media flow [17], oil-recovery [8], plasma physics [16] or semiconductor theory [15]. In chemotaxis, related models appear [9, 25]. For Q12 > 0 and a 2 i > 0, the problem becomes strongly coupled with full diffusion matrix A(ni,n2)

= , 1

Ci + 20:1171! + Q i 2 n 2

012^1

a2\n2 c2 + 2a22n2 + a 2 ini Nonlinear problems of this kind are quite difficult to deal with since the usual idea to apply maximum principle arguments to get a priori estimates cannot be used here. Furthermore, the diffusion matrix is not symmetric. Up to now, only partial results are available in the literature concerning the wellposedness of the problem. We summarize some of the available results for the timedependent equations (see [29] for a review) and refer to [20, 21] for the stationary problem. We also refer to [6, 13, 18] for related results on various cross-diffusion models. Global

115 existence of solutions and their qualitive behavior for a n = a 22 = a 2 i = 0 have been proved in, e.g., [2, 22, 26]. In this case, Eq. (2) is only weakly coupled. For sufficiently small cross-diffusion parameters a 12 > 0 and a21 > 0 (or equivalently, "small" initial data) and vanishing self-diffusion coefficients a n = a 22 = 0, Deuring proved the global existence of solutions [5]. For the case Ci = c2, a global existence result in one space dimension has been obtained by Kim [19], Furthermore, under the condition 8«n > a 12 ,

8a 22 > a 21 ,

(9)

Yagi [30] has shown the global existence of solutions in two space dimensions assuming a 12 = a2\. A global existence result for weak solutions in any space dimension under condition (9) can be found in [10]. Condition (9) can be easily understood by observing that in this case, the diffusion matrix is positive definite: £TA(nltn2)£

> min{ci,c 2 }|£| 2

for all £ e R2,

hence yielding an elliptic operator. If the condition (9) does not hold, there are choices of ci: atj, 7ij > 0 for which the matrix A(ni, n2) is not positive definite, and it is therefore unclear if the problem (l)-(8) can be solved for these data. More recently, Ichikawa and Yamada [14] have improved the results of Yagi, replacing condition (9) by 6 4 a n a 2 2 > aua21

or

6 4 a u a 2 2 = a12ot2i > 0.

(10)

They use the same techniques as Yagi combined with suitable energy estimates. From the view-point of mathematical biology, conditions like (9) and (10) mean that self-diffusion or diffusion is dominant over cross-diffusion. The aim of this paper is to show how the existence of solutions of problem (l)-(8) can be obtained, without assuming conditions like (9) or (10). In fact, these conditions are just technical restrictions, needed in [30, 14], since the existence of solutions of the steady-state problem can be proved without this condition (see [20]). More precisely, we are able to show that for any Cj, a; > 0 and in one space dimension d = 1, there exists a weak solution U\,u2 to (l)-(4), (7)-(8) such that U\ and u2 are nonnegative. We stress the fact that the non-negativity property is obtained without the use of the maximum principle. The idea of the proof is as follows: The system (l)-(4) is first symmetrized via an exponential transformation of variables. A priori estimates are then derived by using a new entropy functional yielding H1 bounds which are independent of the solutions. The non-negativity property is obtained from the embedding i3"1(f2) <-)• L°°(fi), which holds only in one space dimension, and the transformation of variables. We sketch the proof in Section 2. For the detailed proof, we refer to [11]. Before we state the results and sketch the method of proof, we perform (for a smoother presentation) the following change of unknowns: Ui = a2ini,

u2 = ot\2n2,

and

q = —VU.

We assume that ai2 > 0 and a 2 i > 0 which is no restriction since if a.\2 = 0 or a2\ = 0, at least one of the equations (1), (2) is weakly coupled, and the results of [26] apply. Eqs.

116 (l)-(8) can be reformulated as dtUi - div(cjVu, + 2a;i4;Vuj + V(uiu 2 ) + diU{q) = fi(ui, u2), onrDx(0,T), Ui = uDti (ciVui + 2aiuiVui + V{ulu2)+diuiq)-v = Q onTN x (0,T), u(-,0) = uOi infi, i = 1,2,

(11) (12) (13) (14)

where T > 0, uD)i = a2\nD^, uD}2 = ai2nDt2, «? = a2i«o,i, "2 = ai2«o,2 and ax = 0:11/021, o2 = a 2 2 /ai 2 . The source terms are given by fi{u\,u2) = {Ri — Ai"i — Pi2U2)ui, with /?K = 7ii/«2i, /?2i = 72i/ai2, i = l,2.

2

Existence of solutions of problem (11)-(14)

In this section we assume: (Al) n C R is a bounded interval, dQ = TD U TN, and TD ^ III. (A2) u° € L°°(n) satisfies M? > 7 > 0 in 0, uD,i = const. > 0 on TD,i = 1,2. (A3) ahCi > 0, di € R (i = 1,2) and 9 e L2(f2 x (0,T)). (A4) /; : [0, oo)2 —> R (i = 1,2) is continuous and it holds for all s, a > 0, p,q > 0: fi(s,a)
P

+ Ms,
and h(s,o~) (---)+ \p s)

h(s,o)

(- - ±) < C3, \q a)

for some d , C2{p,q), C3{p,q) > 0. We observe that the Lotka-Volterra source terms satisfy condition (A4) if /3a > 0, i = 1, 2 and £12 = hi > 0. The main result is the following theorem. Theorem 1 Let T > 0. Under assumptions (Al)-(A^) there exists a weak solution (ui,« 2 ) 0/ (ll)-(U) satisfying uuu2 € L 2 (0,T; ffx(n)) n W ^ O . T ; tf_1(^)) and ui{x,t),

u2{x,t) > 0

for {x,t) 6 O x (0,T).

The proof consists of several steps. Step 1. We work with unknowns which symmetrize the elliptic operator. Introduce w = (wi,w 2 ) by defining ux = ew\u2 = eW2 and set b{w) = (&i(u>),&2H) = {eWl,eW2). With the diffusion coefficients au(w) = c{eWi + 2aie2w- +

m+w e

*,

i = 1,2,

al2(w) = a2i(w) =

Wl+W2 e

,

Eqs. (11)-(14) are formally equivalent to dtbi(w)-d\v

lj2aij(w)VwJ

+ d

Mw)q)

=Fi(w),

infix(0,T),

(15)

I ^2 Oij(w)Vw.,- + dibi(w)q J • v = 0

on TN x (0,T),

(16)

w = u>D w(0) = w°

onrDx(0,T), in fi,

(17) (18)

where F^w) = / i (e w l ,e W 2 ), wD>i = log(« Dii ), and w° = log(uj), i = 1,2. Step 2. In order to solve the above problem, we introduce a semi-discrete problem. Let TV e N and let r = T/N be the time step. Given wk~x, approximating w in the interval ((k — 2)T, (k — l)r], we are seeking solutions wk of the elliptic problem

k(wk) -uwk-i) _ div / £ ai.{wk)Vwk+dA{w^A ^

= Fi(w*} in n>

a^iu^Vur) + dibi{wk)qk \-v

(19)

= Q onTN,

(20)

wk = w D on TD,

(21)

for A; = 1 , . . . , N. Here we defined qk = \ / (ife 1 1)T
ar)k,

where a — 2min{ci,c 2 }, f]k is the discrete "physical" entropy 2

r

k

v = X) / ( 6 iK)K f c - ti»D,0 - h{wk) + bi{wD))dx > o, and Tjk is another discrete entropy-type functional:

4 = E [(^-^

- (wk - wD,i))dx > 0.

In the original variables, r]k can be written as f(ui(logUi — logup^) — ut + up^dx, which justifies the notion of "entropy". We can prove the following entropy-type estimate, which holds in any space dimension.

118 Lemma 1 Let wk € i7 x (Q;R 2 ) be a weak solution of (19)-(21). constant C > 0 such that for any k = 1 , . . . , N and any T > 0, ^ + T J2 f (iH^I

+

a|VeTO-/2|2 + ai\Vew" A

Then there exists a

dx < n^1 + CT.

(22)

The key of the proof of this lemma is to use (wf — v>D,i) + a(6j(—IBJ) — 6j(—«);)) € HQ(£1 U FJV) = {t)£ i71(Q) : D = 0 on To} as a test function in the weak formulation of (19)-(21) (see [11] for the details). Step 4. We use the Lax-Milgram's lemma together with the Leray-Schauder's fixed-point theorem to prove the existence of weak solutions of the semi-discrete problem (see [11]). Lemma 2 Let K;* -1 G L°°(fl;R 2 ), k > 1. H^Q-R2) of (19)-(21).

Then there exists a weak solution wk €

We notice that, since the solution satisfies wk e i7 1 (fi;R 2 ) «-> L°°(Q;R2) in one space dimension, the unknowns uk = exp(iwf) are well defined and elements of H1(Q). Hence (uk, uk) = (eWl, e"*2), k = 1 , . . . , N, is a solution of the discrete problem corresponding to (11)-(14). Step 5. We define the piecewise constant functions u/T) by w{T){x, t) = wk{x)

if {x, t) G Q x ((fc - 1)T, fcr]

T

and g( ' in a similar way. Observe that assumption (A3) implies q(T) - > ?

in L2(QT) as r -> 0.

(23)

An immediate consequence of Lemma 1 is the following result: Corollary 1 Let r > 0. Then we have ||?/T'||L<>°(o,:r;i,i(n)) < C and 2

J2 (a||e w - M/2 ||^(o,r;^(n)) + ailleffi,M|U2(o,T;ifHn))) < C, iti/iere C > 0 is independent of T, a = 2min{ci,C2}, and 2

»>(T)(t) = E

.

/ (^(^M)K!r)-«'fl)-''i(^T)) + &i(«'i>)+aft(«'lT))-^T)+^,0)W^-

We also need an estimate for the discrete time derivative. For this we define ftW(.)t)

=

f l _ I (p(wk) - biw"-1)) + b(wk),

t > 0,

and introduce aT, the shift operator: aTw(T){-,t) = wk~l Then we have (see [11])

if t e {{k-i)T,kr],

k=

l,...,N.

119 L e m m a 3 It holds

||6(u/M) - KoM^h^Ty)

< CT,

Hft6W||1.(o,Tiv) + I I ^ I I L W ; ^ ) ) < C,

where C does not depend on T and V* = (HQ(Q U IV))* is the dual ofV = Hg(fl U TN). S t e p 6. We already have all the necessary estimates to pass to the limit r —> 0. Since the embedding H1^!) <-> L°°(Q) is compact in one space dimension, we can apply Aubin's lemma [28] to ¥T\ in view of the uniform bounds of Lemma 3, to obtain, up to a subsequence which is not relabeled, dt¥T) ->• dtz 6 ->• z Ur) ->• z

weakly in L 1 (0, T; V), weakly in L 2 (0,T; Hl{n)), strongly in L 2 (0,T; L°°(n)),

6(U/ T ) ) ->• u

weakly in L 2 (0, T; H^Q)).

(24) (25) (26) (27)

By Lemma 3 we have, as r —>• 0, ||6M - &(™ (T) )||LI(O,T ; V) < ||6(t«W) - 6(aTUJ(T))|UHo,r;v) -> 0,

and hence z = u. Finally, using the above estimates and convergences, we are able to prove that em-'T> = bi{w{T)) -» m strongly in L2{QT), i = 1,2. (28) Now we can let r -» 0 in the weak formulation of (19), i = 1,2, which reads for 0eL°°(O,T; (W^iil))*) : /" (dtb%(T),
-di f

(CiVera'T) + 2aiew'T)Ve,"-T).+ Ve",ST)+,"^)) • V^ctedt ew'r)q<-T'>-Vdxdt+ [

JQT

fi{ew^,ew^)cj)dxdt.

JQT

In view of (24)-(28), (23) and Assumption (A4) we obtain / {dtuh4>)dt+ / (ciVui + 2aiui'Vui + Jo JQT =

— di I JQT

u,q -V
V{u1U2))-V4>dxdt

Ji{u\,U2)dxdt,

JQT

i.e. u = («i,U2) is a weak solution of (11)-(12). Moreover, the initial condition (14) is satisfied in the sense of V. As a consequence of the above proof we obtain a positivity-preserving numerical scheme. This scheme will be employed for numerical simulations of the evolution problem in a future work. For numerical stationary solutions, we refer to, e.g., [10].

120

3

A simple example

We consider in this section, as an example, the one-dimensional version of the stationary problem corresponding to problem (11)-(14), with }i{u\,u2) = 0, i = 1,2, and non-flux boundary conditions, as stated in [27]. Notice that this choice of boundary conditions and source terms implies the conservation of the number of individuals of each of the species. This may be regarded as a simple equilibrium model of a dynamics in which population pressure effects are much faster than growth and competition effects. For the study of the long-time behaviour of solutions in the case in which interaction of pressure and growth-competition terms are present (no environmental potential), the reader is refered to [24]. We shall see that this rather simple example already exhibits the phenomena of segregation of populations. We obtain necessary conditions relating transport and crossdiffusion coefficients. Sufficient conditions either involves the size of the solution or are just a consequence of a very steep difference between the transport coefficients of both equations. Integrating the equations that one obtains when dropping the time derivative and the source terms of equations (11) in (0,:r), with i e ( l , and using the boundary conditions, we obtain the following problem. Find {ui,u2) : O x Q —>• R^_ such that («i(ci + a1u1 + u2))' + dxqui = 0

.

[U2(c2 +
and

Ja

/ u2 ~ u2.

(30)

in

Here, U\ and u2 stand for the mass of u\ and u2, respectively. We have the following existence result, which is independent of the size of the non-negative parameters a*. Therefore, condition (9) is not necessary in this example, too. Proposition 1 Let c\ and c2 be positive and assume q e L°°(fi). There exists a solution (ui,u2) of Problem (29)-(30) such thatui,u2 > 0 in CI andu\,u2 6 W1,a°(Q). In addition, ifqe H""'°°(fl) then the solution (uuu2) satisfies uuu2 6 W™+1'°°(n). The proof is straightforward. We just rewrite equations (29)-(30) as gfei(Mi,M2)

g{ui,u2)

u,

'

ni,

_

_qh2{ui,u2)

g(uuu2)

U,

(31)

with g(ui, u2) := (ci + 2a1u1 + u2)(c2 + 2a2u2 + ui) - uiu2 > 0, hi(ui,u2) := di(c2 + 2a2u2 + ui) - d2u2,

(32)

h2{ux,u2) := d2(ci + 2ai«i + u2) — diU^. Then, we use an iterative scheme on the representation formulae of the solutions given by i \ m\ [x -qhi(u u ) Ui{x) = «i(0) exp{f / ——; lt 2 xr - } , JO »l(«l>«2)

[ fx u2{x) = w2(0) exp{ / Jo

-qh2{ui,u2), ^ r^}, MMl.Ua)

(33)

121 with ui(0),u2(0) determined by (30) (see [10] for details). We now discuss the notion of segregation in a simplified framework. We shall assume, following [27], that the environmental potential U(x) is a smooth function such that the corresponding enviromental flux satisfies q(x0) = 0 for a single point x0 € fi. Definition 1 We say that the stationary problem (29)-(30) has the property of segregation if there exists a point XQ £ Q such that u\ and u2 have a local maximum and minimum at XQ, respectively (or vice versa). Observe that, since q(xo) = 0, from (31) we find u\(x0) = u'2(x0) = 0, so we deduce the existence of a critical point, x0, both for ui and u2. Taking derivatives in (31) and evaluating at Xo we obtain g(u1(x0),u2{x0))u"(x0) = g(u1(x0),U2(x0))u2(x0) =

-qJ{x0)h1(ui(xo),U2{x0)), -q'{xo)h2(u1(xo),U2(xo)).

Therefore, segregation will take place if hi(ui(x0),u2(x0))h2(ui(x0),u2{xQ)) (32) we see that this is equivalent to one of the following conditions: u2(x0)(di - 2o2di) > di(c2 + ui(z 0 )),

< 0. ^From

ui(x0)(di - 2aid2) > rf2(ci + u2(x0)).

(34)

In the trivial case in which d\d2 < 0, segregation does always occur. Notice that in this case x0 is an attractive point (from the environmental point of view) for one of the populations and a repulsive point for the other. We may then analyze the case di > 0, i = 1,2 (the case di < 0 is similar). From (34) we obtain a necessary condition involving the diffusion coefficients, namely dt > 2atdj, i,j = 1,2, i ^ j . Observe that if none of these conditions are satisfied (no segregation) then the coefficients oi, a2 satisfy a\a2 > 1/4, which is a condition in the same spirit as (9). On the other hand, since the L°° norms of «; are bounded when di —> 0, i = 1,2, see [10], we conclude that a sufficient condition for segregation is that di/d2 is large or small enough. We discretized problem (11)-(14), with no-flux boundary conditions and zero source terms, by a semi-implicit finite difference method and let tend t —> oo in order to obtain steady-state solutions. The numerical experiments conducted show that, for a wide range of model parameters and initial conditions, the numerical scheme is stable, which allows us to show some different behaviours of the two interacting species included in the model. In particular, it is interesting to observe when segregation of the two species, due to habitat hetereogenity, does appear. We also used an iterative scheme based on the formulae (33) and could verify that the solutions of (11)-(14), obtained in the large-time limit, coincide with the solutions of the iterative scheme. We show here numerical experiments corresponding to the following cases: (a) Large and smallcross-diffusion terms. Set c; = dt = 1 and constant initial conditions Mio = 10, «2o = 20. Then small a* correspond to large cross-diffusion and vice versa. Figure 1 shows the numerical solution of model corresponding to values of Oi = 0,0.1,10.

122 (b) Segregation effects due to different transport coefficients, i.e. di 3> d2. Set ct = d2 = 1 and tiio = 10, u2o = 10. Figure 1 shows the numerical solutions corresponding to a i = 0,2 = 1 and d\ = 4,8,20,40. As case (a) shows, large cross-diffusion enhance segregation effects, therefore we repeat the last simulation with a; = 0.1. Figure 2 shows the same results as before also for this case. (c) Set Ci = 1, ai = 1, a2 = 0.01, di = 40, d2 = 1, Mio = W20 = 10 (Figure 2). Here, a neat segregation of the two species can be observed.

Acknowledgements The authors acknowledge partial support from the TMR Project "Asymptotic methods in kinetic theory", grant number ERB-FMBX-CT97-0157 and from the DAAD Project "Acciones Integradas". The first and second authors were supported by the Spanish Ministerio de Ciencia y Tecnologia, project number MCT-OO-BFM-1324. The last author was supported by the Gerhard-Hess Program of the Deutsche Forschungsgemeinschaft, grant JU 359/3, and by the AFF Project of the University of Konstanz, grant 4/00.

References [1] H. Amann and M. Renardy. Reaction-diffusion problems in electrolysis. Nonlin. Diff. Eqs. Appl., 1:91-117, 1994. [2] N. Boudiba and M. Pierre. Global existence for coupled reaction-diffusion equations systems. J. Math. Anal. Appl, 250:1-12, 2000. [3] Y. Choi and R. Lui. Multi-dimensional electrochemistry model. Arch. Rat. Mech. Anal, 130:315-342, 1995. [4] P. Degond, S. Genieys, and A. Jiingel. Symmetrization and entropy inequality for general diffusion equations. C. R. Acad. Sci. Paris, 325:963-968, 1997. [5] P. Deuring. An initial-boundary value problem for a certain density-dependent diffusion system. Math. Z., 194:375-396, 1987. [6] Z. Duan and L. Zhou. Global and blow-up solutions for nonlinear degenerate parabolic systems with crosswise diffusion. J. Math. Anal. Appl, 224:263-278, 2000. [7] R. Eisenberg and D. Gillespie. A singular perturbation of the Poisson-Nernst-Planck system: application to ion channels. Preprint, 1999. [8] G. Gagneux and M. Madaune-Tort. Analyse mathematique de modeles non lineaires de I'ingenierie petroliere. Springer, Paris, 1996. [9] H. Gajewski and K. Zacharias. Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr., 195:77-114, 1998.

123

[10] G. Galiano, M. Garzon, and A. Jiingel. Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics. To appear in Revista de la Real Academia de Ciencias Exactas, Fis. y Nat, 2002. [11] G. Galiano, M. Garzon, and A. Jiingel. Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion system arising in population dynamics. Submitted for publication, 2000. [12] M. T. Gyi and A. Jiingel. A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Diff. Eqs., 5:773-800, 2000. [13] H. Hoshino. Nonnegative global solutions to a class of strongly coupled reactiondiffusion systems. Advances Diff. Equ., 5:801-832, 2000. [14] T. Ichikawa and Y. Yamada. Some remarks on global solutions to some quasilinear parabolic system with cross-diffusion. Funkcialaj Ekvacioj, 43:285-301, 2000. [15] A. Jiingel. Quasi-hydrodynamic Semiconductor Equations. Progress in Nonlinear Differential Equations and Its Applications. Birkhauser, Basel, 2001. [16] A. Jiingel and Y.-J. Peng. A hierarchy of hydrodynamic models for plasmas. Zeroelectron-mass limits in the drift-diffusion equations. Annales H. Poincare, 17:83-118, 2000. [17] A. S. Kalashnikov. Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russ. Math. Surveys, 42:169-222, 1987. [18] J. Kamel and M. Kirane. Pointwise a priori bounds for a strongly coupled system of reaction-diffusion equations with a balance law. Math. Meth. Appl. Sci., 21:12271232, 1998. [19] J.U. Kim. Smooth solutions to a quasi-linear system of diffusion equations for a certain population model. Nonlin. Anal, 8:1121-1144, 1984. [20] Y. Lou and W.-M. Ni. Diffusion, self-diffusion and cross-diffusion. 131:79-131, 1996.

J. Diff. Eqs.,

[21] Y. Lou and W.-M. Ni. Diffusion, vs. cross-diffusion: an elliptic approach. J. Diff. Eqs., 154:157-190, 1999. [22] Y. Lou, W.-M. Ni, and Y. Wu. The global existence of solutions for a cross-diffusion system. Adv. Math., Beijing, 25:283-284, 1996. [23] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser. Semiconductor Equations. Springer, 1990. [24] M. Mimura and K. Kawasaki. Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol., 9:49-64, 1980.

124 [25] K. Post. A non-linear parabolic system modeling chemotaxis with sensitivity functions. Preprint, Humboldt-Universitat Berlin, Germany, 1999. [26] M. Pozio and A. Tesei. Global existence of solutions for a strongly coupled quasilinear parabolic system. Nonlin. Anal., 14:657-689, 1990. [27] N. Shigesada, K. Kawasaki, and E. Teramoto. Spatial segregation of interacting species. J. Theor. Biol., 79:83-99, 1979. [28] J. Simon. Compact sets in the space V(0,T; B). Ann. Math. Pura AppL, 146:65-96, 1987. [29] Y. Wu. Qualitative studies of solutions for some cross-diffusion systems. In T.-T. Li, M. Mimura, Y. Nishiura, and Q.-X. Ye, editors, China-Japan Symposium on Reaction-Diffusion Eqautions and Their Applications and Computational Aspects, pages 177-187, Singapore, 1997. World Scientific. [30] A. Yagi. Global solution to some quasilinear parabolic system in population dynamics. Nonlin. Anal., 21:603-630, 1993.

125

Figure 1: Stationary density distributions u\{x) (solid line) and u2{x) (broken line) for x e (0,3). Left figure: case (a); the numbers correspond to the values of a,. Right figure: case (b) with ai = 02 = 1; the numbers correspond to values of d\.

Figure 2: Stationary density distributions u^x) (solid line) and u2(x) (broken line) for x e (0,3). Left figure: case (b) with di = a2 = 0.1; the numbers correspond to values of d\. Right figure: case (c).

The nonstationary Stokes and Navier-Stokes equations in aperture domains Toshiaki Hishida Fachbereich Mathematik, Technische Universitat Darmstadt D-64289 Darmstadt, Germany Email: [email protected] Abstract We consider the nonstationary Stokes and Navier-Stokes flows in aperture domains fi c R",TI > 3. We develop the Lq-U estimates of the Stokes semigroup and apply them to the Navier-Stokes initial value problem. As a result, we obtain the global existence of a unique strong solution, which satisfies the vanishing flux condition through the aperture and some sharp decay properties as t —> oo, when the initial velocity is sufficiently small in the Ln space. Such a global existence theorem is up to now well known in the cases of the whole and half spaces, bounded and exterior domains. In this article we study the global existence and asymptotic behavior of a strong solution to the Navier-Stokes initial value problem in an aperture domain SI C K" with smooth boundary dSl: dtu + u-Vu V-u u|an u\t=o

= Au-Vp =0 =0 =a

(xeQ,t>0), (xen,t>0), (t > 0), (x 6 SI),

y

,. '

where u(x, t) = (ui(x, t), • • • , un(x, t)) and p(x, t) denote the unknown velocity and pressure of a viscous incompressible fluid occupying SI, respectively, while a(x) = (ai(x), • • • , an(x)) is a prescribed initial velocity. The aperture domain Q. is a compact perturbation of two separated half spaces H+ U i?_, where H± = {x = (Xi, • • • , xn) € R"; ±xn > 1}; to be precise, we call a connected open set Q C Rn an aperture domain (with thickness of the wall) if there is a ball B C Rn such that Q. \ B = (H+ U #_) \ J3. Thus the upper and lower half spaces H± are connected by an aperture (hole) M C fl D B, which is a smooth (n — l)-dimensional manifold so that fi consists of upper and lower disjoint subdomains ft± and M: SI = Q+ U M U 0_. The aperture domain is a particularly interesting class of domains with noncompact boundaries because of the following remarkable feature, which was in 1976 pointed out by Heywood [20]: the solution is not uniquely determined by usual boundary conditions

126

127 even for the stationary Stokes system in this domain and therefore, in order to single out a unique solution, we have to prescribe either the flux through the aperture M N • uda, JM

or the pressure drop at infinity (in a sense) between the upper and lower subdomains Q± [p] =

lim p(x) lim p(x), |x|-»oo,xen+ |x|—Kx),xen_

as an additional boundary condition. Here, N denotes the unit normal vector on M directed to fi_ and the flux {u) is independent of the choice of M since V • u = 0 in Q. Later on, the observation of Heywood in the L2 framework was developed by Farwig and Sohr within the framework of LP theory for the stationary Stokes and Navier-Stokes systems [13] and also the (generalized) Stokes resolvent system [14], [12]. Especially, in the latter case, they clarified that the assertion on the uniqueness depends on the class of solutions under consideration. Indeed, the additional condition must be required for the uniqueness if q > n/(n — 1), but otherwise, the solution is unique without any additional condition; for more details, see Farwig [12], Theorem 1.2. The results of Farwig and Sohr [14] are also the first step to discuss the nonstationary problem (1) in the LP space. They showed the Helmholtz decomposition of the Lg space of vector fields (see also Miyakawa [29]) L"(Q) = L«(ft)eLJ(fi) for n > 2 and 1 < q < oo, where LI (Q) is the completion in LP (Q) of the class of all smooth, solenoidal and compactly supported vector fields, and £«(fi) = {Vp 6 Lq{Q);p € L9loc(Q)}. The space L«(0) is characterized as ([14], Lemma 3.1) Ll(Q) = {ue L"{Q); V • u = 0, v • u\Bn = 0, 4>{u) = 0},

(2)

where v is the unit outer normal vector on dfi,. Here, the condition (u) = 0 follows from the other ones and may be omitted if q < n/(n — 1), but otherwise, the element of LJ(n) must possess this additional property. Using the projection operator P — Pq from Lq{Q) onto LJ(Q) associated with the Helmholtz decomposition, we can define the Stokes operator A = Aq = - P , A on L«(Q) with domain D{Aq) = W2'"{Q) n W^q(n) n L%(Q). Then the operator —A generates a bounded analytic semigroup e~tA in each Lqa(Q), 1 < q < oo, for n > 2 ([14], Theorem 2.5). We are interested in strong solutions to the nonstationay problem (1). However, there are no results on the global existence of such solutions in the Lq framework unless q = 2, while a few local existence theorems are known. In the 3-dimensional case, Heywood [20], [21] first constructed a local solution to (1) for a prescribed either 4>{u(t)) or \p(i)], which should satisfy some regularity assumptions with respect to the time variable, when a G H2(Q) fulfills some compatibility conditions. Franzke [17] has recently developed the Lq theory of local solutions via the approach of Giga and Miyakawa [18] with use of fractional powers of the Stokes operator. When (u(t)) is prescribed, his assumption on initial data is for instance that a e Lq(Q),q > n, together with some compatibility conditions. The reason why the case q = n is excluded is the lack of informations about purely imaginary powers of the Stokes operator. In order to discuss also the case where

128 \p(t)] is prescribed, Franzke introduced another kind of Stokes operator associated with the pressure drop condition, which generates a bounded analytic semigroup on the space {u e Lq(fi); V • u = 0, v • u\gn = 0} for n > 3 and n/(n — 1) < q < n (based on a resolvent estimate due to Farwig [12]). Because of this restriction on q, the Lq theory with q > n is not available under the pressure drop condition and thus one cannot avoid a regularity assumption to some extent on initial data. It is possible to discuss the L2 theory of global strong solutions for an arbitrary unbounded domain (with smooth boundary) in a unified way since the Stokes operator is a nonnegative selfadjoint one in L\\ see Heywood [22] (n = 3), Kozono and Ogawa [26] (n = 3) and Kozono and Sohr [27] (n = 4, 5). Especially, from the viewpoint of the class of initial data, optimal results were given by [26] and [27]. In fact, they constructed a global solution with various decay properties for small a € D{A% ). Here, we should recall the continuous embedding relation D(A" ) C I J . For the aperture domain fl their solutions should satisfy the hidden flux condition (f){u(t)) = 0 on account of u(t) e Ll(Q) together with (2). In his Doktorschrift [16] Franzke studied, among others, the global existence of strong solutions in a 3-dimensional aperture domain when either <j){u{t)) or \p{t)\ is prescribed. Indeed, the local strong solution in the L2 space constructed by himself [15] was extended globally in time under the condition that both a £ #o(0) (with compatibility conditions) and the other data are small in a sense, however, his method gave no information about the large time behavior of the solution. This article provides the global existence theorem for a unique strong solution of (1), which satisfies the flux condition <j>(u(t)) = 0 and some decay properties with definite rates as t —>• oo when the initial velocity a is small enough in L"(Q),n > 3. For the proof, as is well known, it is crucial to establish the Lq-Lr estimates of the Stokes semigroup \\e~tAf\\L^)

< Cr°||/||L,(n),

(3)

l|Ve-"/||L,(n) < Cr-^H/H^n),

(4)

for all t > 0 and / e Lqa(Q), where a = (n/q - n/r)/2 > 0. Recently for n > 3 Abels [1] has proved some partial results: (3) for 1 < q < r < oo and (4) for 1 < q < r < n. However, because of the lack of (4) for the most important case q = r = n, his results are not satisfactory for the construction of the global strong solution possessing various time-asymptotic behaviors as long as one follows the straightforward method of Kato [24]. Our main result on the Lq-U estimates is: Theorem 1 Let n > 3. 1. Let 1 < q < r < oo (q ^ oo). There is a constant C = C(Q,,n,q,r) > 0 such that (3) holds for allt>0 and f e £«(fi). 2. Let 1 < q < r < n or\ < q < n < r < oo. There is a constant C = C(Q, n, q, r) > 0 such that (4) holds for allt > 0 and f G Lqa(Q). 3. Letl
fast->0 j

M M ( X )

ifq
129

]|Ve-M/||r = oft-"-1'2) as t —> 0 as t —» oo

if q < r < oo, ifq
oo,

where a = (n/q — n/r)/2. Furthermore, for each precompact set K in Ll(Q) every convergence above is uniform with respect to f € K. By use of the Stokes operator A, one can formulate the problem (1) subject to the vanishing flux condition 4>{u{t)) =

N • u{t)da = 0,

t>0,

(5)

JM

as the Cauchy problem dtu + Au + P{u • Vu) = 0,

t> 0; u(0) = a,

(6)

in LJ(Q). Given a e L"(Q,) and 0 < T < oo, a measurable function u defined on SI x (0, T) is called a strong solution of (1) with (5) on (0, T) if u is of class u e C([0, T); /£(«)) n C(0, T; D(A n )) n C\Q, T; Lna(tt)) together with lim^o \\u(t) — a\\n = 0 and satisfies (6) for 0 < t < T in L"(fi). We make use of (3) and (4) to obtain: Theorem 2 Let n > 3. There is a constant 5 = 6(0,, n) > 0 with the following property: if a e £"(£2) satisfies \\a\\n < 5, then the problem (1) with (5) admits a unique strong solution u(t) on (0,oo), which enjoys \\u(t) ||r = o (t-l'2+nl2r)

forn
\\Vu(t)\\n =

o{r^),

||ft«(t)|| B + ||Au(t)|| B = o ( r 1 ) , as t —> oo. The strong solution is constructed as the solution of the integral equation u(t) = e~tAa - [ e-{i-T)AP(u Jo

• Vw)(r)dT,

t >.0,

by means of a standard contraction mapping principle along the lines of Kato [24]. The obtained solution is unique within the class u € C([0, oo); Z£(Q)),

Vu G C(0, oo; L"(Q)),

without assuming any behavior near t = 0 as pointed out by Brezis [7]. When one prescribes a nontrivial flux (u(t)) = F(t) G Cue([0,T]) with some 6 > 0 and T > 0, there is T, 6 (0,T] such that the problem (1) with the flux condition admits

130 a unique strong solution on (0, Tt) provided that a € Ln{Q) satisfies the compatibility conditions V • a = 0, v • a\gn — 0 and (j){a) = F(0). This improves a related result of Pranzke [17] and can be proved in the same manner as the proof of Theorem 2 with the aid of the auxiliary function of Heywood ([20], Lemma 11), which is used for the reduction of the problem to an equivalent one with the vanishing flux condition (5). Up to now we have the same global existence result as Theorem 2 for the whole space (Kato [24]), the half space (Ukai [32]), bounded domains (Giga and Miyakawa [18]) and exterior domains (Iwashita [23]) since the Lq-U estimates (3) and (4) are well established for these four types of domains. Especially for exterior domains with n > 3, based on (3) for q = r due to Borchers and Sohr [5], some partial results were given by Iwashita [23], Giga and Sohr [19] and Borchers and Miyakawa [4]; in particular, Iwashita proved (3) for 1 < q < r < oo and (4) for 1 < q < r < n, which made it possible to construct a global solution. Later on, due to the following authors, (3) for n > 2,1 < q < r < co (q ^ oo) and (4) for 1 < q < r < n = 2 were also derived: Chen [8] (n = 3), Shibata [31] (n = 3), Borchers and Varnhorn [6] (n = 2, (3) for q = r), Dan and Shibata [10], [11] (n = 2), Dan, Kobayashi and Shibata [9] (n = 2,3) and Maremonti and Solonnikov [28] (n > 2). In the proof of the Lq-U estimates, it seems to be heuristically reasonable to combine some local decay properties near the aperture with the Lq-Lr estimates of the Stokes semigroup for the half space (Ukai [32]) by means of a localization procedure since the aperture domain n is obtained from H+ U H_ by a perturbation within a compact region. Indeed, Abels [1] used this idea that was well developed by Iwashita [23] and, later, Kobayashi and Shibata [25] in the case of exterior domains. We should however note that the boundary dQ is noncompact; thus, a difficulty is to deduce the sharp local energy decay estimate

\\e'tAf\\w^(nR) < Ct-n'"\\f\\ma),

t > 1,

(7)

for / e Ll(Q), 1 < q < co, where Q,R = {x G CI; \x\ < R}, but this is the essential part of our proof. The estimate (7) improves the local energy decay given by Abels [1], in which a little slower rate t~n/2li+e was shown. In [1], similarly to Iwashita [23], a resolvent expansion around the origin A = 0 was derived in some weighted function spaces. To this end, Abels made use of the Ukai formula of the Stokes semigroup for the half space ([32]) and, in order to estimate the Riesz operator appearing in this formula, he had to introduce Muckenhoupt weights, which caused some restrictions although his analysis itself is of interest. On the other hand, Kobayashi and Shibata [25] refined the proof of Iwashita in some sense and obtained the Lq-U estimates of the Oseen semigroup for the 3-dimensional exterior domain. As a particular case, the result of [25] includes the estimates of the Stokes semigroup as well. To prove Theorem 1 we employ in principle the strategy developed by [25] (without using any weighted function space) and extend the method to general n > 3. The proof is divided into three steps and the outline is as follows. The first step is the investigation of the Stokes resolvent for the half space H = H+ or H-. We derive some regularity estimates near the origin A = 0 of (A + A f f ) _ 1 P f f / when / e Lq(H) has a bounded support, where AH = — P#A is the Stokes operator for the half space H. Although the obtained estimates do not seem to be optimal compared with those shown by [25] for the whole space, the results are sufficient for our aim and the

131 proof is rather elementary: in fact, we represent the resolvent (A + A#) _ 1 in terms of the semigroup e~tA" and, with the aid of local energy decay properties of this semigroup, we have only to perform several integrations by parts and to estimate the resulting formulae. One needs neither Fourier analysis nor resolvent expansions. In the next step, based on the results for the half space, we proceed to the analysis of the Stokes resolvent for the aperture domain fi. To do so, in an analogous way to [23], [25] and [1], we first construct the resolvent (A + A)~1Pf near the origin A = 0 for / € Lq(Q) with bounded support by use of the operator (A + AH)~1PH, the Stokes flow in a bounded domain and a cut-off function together with the result of Bogovskii [2] on the boundary value problem for the equation of continuity. And then, for the same / as above, we deduce essentially the same regularity estimates near the origin A = 0 of (A + A)~lPf as shown in the previous step. In the final step we prove (7) and thereby (4) for q = r G (l,n] as well as (3) for r = oo, from which the other cases follow. Some of the estimates obtained in the previous step enable us to justify a representation formula of the semigroup e~tAPf in Wl'q(QR) in terms of the Fourier inverse transform of 9™(is + A)-1 Pf when / G Lq(Q) has a bounded support, where n = 2m + 1 or n = 2m + 2. We then appeal to the lemma due to Shibata ([30], [25]), which tells us a relation between the regularity of a function at the origin and the decay property of its Fourier inverse image, so that we obtain another local energy decay estimate \\e~tAPf\\wl„{nR) q

< Cr"/2+l/||L,(n),

t > 1,

(8)

for / G L (Q,), 1 < q < oo, with bounded support, where e > 0 is arbitrary. Estimate (8) was shown in [1] only for solenoidal data / G Z/'(0) with bounded support, from which (7) with the rate replaced by t~n/2q+s follows through an interpolation argument. But it is crucial for the proof of (7) to use (8) even for data which are not solenoidal (so that the support of Pf is unbounded). In order to deduce (7) from (8), we develop the method in [23] and [25] based on a localization argument using a cut-off function. In fact, we regard the Stokes flow for the aperture domain Q as the sum of the Stokes flows for the half spaces H± and a certain perturbed flow. Since the Stokes flow for the half space enjoys the L'-L°° decay estimate with the rate i~"/2« (Borchers and Miyakawa [3]), our main task is to show (7) for the perturbation part. In contrast to the case of exterior domains, the support of the derivative of the cut-off function touches the boundary dil; indeed, this difficulty occurs in all stages of localization procedures in the course of the proof and thus we have to carry out such procedures carefully. Furthermore, the remainder term arising from the above-mentioned localization argument involves the pressure of the nonstationary Stokes system in the half space and, therefore, does not belong to any solenoidal function space. Hence, in order to treat this term, (8) is necessary for non-solenoidal data, while that is not the case for the exterior problem. Acknowledgment. The present work was partially supported by the Alexander von Humboldt research fellowship.

132

References [1] Abels, H., Lq-Lr estimates for the non-stationary Stokes equations in an aperture domain, Z. Anal. Anwendungen (to appear). [2] Bogovskii, M. E., Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl. 20, 1094-1098 (1979). [3] Borchers, W., Miyakawa. T., L2 decay for the Navier-Stokes flow in halfspaces, Math. Ann. 282, 139-155 (1988). [4] Borchers, W., Miyakawa. T., Algebraic L2 decay for Navier-Stokes flows in exterior domains, Acta Math. 165, 189-227 (1990). [5] Borchers, W., Sohr, H., On the semigroup of the Stokes operator for exterior domains in L'-spaces, Math. Z. 196, 415-425 (1987). [6] Borchers, W., Varnhorn, W., On the boundedness of the Stokes semigroup in twodimensional exterior domains, Math. Z. 213, 275-299 (1993). [7] Brezis, H., Remarks on the preceding paper by M. Ben-Artzi "Global solutions of two-dimensional Navier-Stokes and Euler equations", Arch. Rational Mech. Anal. 128, 359-360 (1994). [8] Chen, Z. M., Solutions of the stationary and nonstationary Navier-Stokes equations in exterior domains, Pacific J. Math. 159, 227-240 (1993). [9] Dan, W., Kobayashi, T., Shibata, Y., On the local energy decay approach to some fluid flow in an exterior domain, Recent Topics on Mathematical Theory of Viscous Incompressible Fluid, 1-51, Lecture Notes Numer. Appl. Math. 16, Kinokuniya, Tokyo, 1998. [10] Dan, W., Shibata, Y., On the Lq-LT estimates of the Stokes semigroup in a 2dimensional exterior domain, J. Math. Soc. Japan 51, 181-207 (1999). [11] Dan, W., Shibata, Y., Remarks on the Lq-L^, estimate of the Stokes semigroup in a 2-dimensional exterior domain, Pacific J. Math. 189, 223-239 (1999). [12] Farwig, R., Note on the flux condition and pressure drop in the resolvent problem of the Stokes system, Manuscripta Math. 89, 139-158 (1996). [13] Farwig, R., Sohr, H., On the Stokes and Navier-Stokes system for domains with noncompact boundary in L'-spaces, Math. Nachr. 170, 53-77 (1994). [14] Farwig, R., Sohr, H., Helmholtz decomposition and Stokes resolvent system for aperture domains in L?-spaces, Analysis 16, 1-26 (1996). [15] Franzke, M., Strong solutions of the Navier-Stokes equations in aperture domains, Ann. Univ. Ferrara Sez. VII. Sc. Mat. 46, 161-173 (2000).

133 [16] Franzke, M., Die Navier-Stokes-Gleichungen in Offnungsgebieten, Technische Universitat Darmstadt, 2000.

Doktorschrift,

[17] Franzke, M., Strong L'-theory of the Navier-Stokes equations in aperture domains, Preprint Nr. 2139 TU Darmstadt (2001). [18] Giga, Y., Miyakawa, T., Solutions in Lr of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89, 267-281 (1985). [19] Giga, Y., Sohr, H., On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA 36, 103-130 (1989). [20] Heywood, J. G., On uniqueness questions in the theory of viscous flow, Acta Math. 136, 61-102 (1976). [21] Heywood, J. G., Auxiliary flux and pressure conditions for Navier-Stokes problems, Approximation Methods for Navier-Stokes Problems, 223-234, Lecture Notes in Math. 771, Springer, Berlin, 1980. [22] Heywood, J. G., The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29, 639-681 (1980). [23] Iwashita, H., Lq-Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces, Math. Ann. 285, 265-288 (1989). [24] Kato, T., Strong LP solutions of the Navier-Stokes equation in R m , with applications to weak solutions, Math. Z. 187, 471-480 (1984). [25] Kobayashi, T., Shibata, Y., On the Oseen equation in the three dimensional exterior domains, Math. Ann. 310, 1-45 (1998). [26] Kozono, H., Ogawa, T., Global strong solution and its decay properties for the NavierStokes equations in three dimensional domains with non-compact boundaries, Math. Z. 216, 1-30 (1994). [27] Kozono, H., Sohr, H., Global strong solution of the Navier-Stokes equations in 4 and 5 dimensional unbounded domains, Ann. Inst. H. Poincare Anal. Nonlineaire 16, 535-561 (1999). [28] Maremonti, P., Solonnikov, V. A., On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Sup. Pisa 24, 395-449 (1997). [29] Miyakawa, T., The Helmholtz decomposition of vector fields in some unbounded domains, Math. J. Toyama Univ. 17, 115-149 (1994). [30] Shibata, Y., On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain, Tsukuba J. Math. 7, 1-68 (1983).

134 [31] Shibata, Y., On an exterior initial boundary value problem for Navier-Stokes equations, Quart. Appl. Math. 57, 117-155 (1999). [32] Ukai, S., A solution formula for the Stokes equation in R™, Commun. Pure Appl. Math. 40, 611-621 (1987).

Global attractors for multivalued flows associated with sub differentials Noboyuki KENMOCHI Department of Mathematics Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan Email: [email protected] Noriaki YAMAZAKI Department of Mathematical Science Common Subject Division Muroran Institute of Technology 1-27 Mizumoto-cho, Muroran, 050-8585 Japan Email: [email protected]

Abstract In a real Hilbert space we consider a class of non-autonomous evolution equations including time-dependent subdifferentials and their perturbations. We are interested in the case when the Cauchy problems have in general multiple solutions, namely they may lose the uniqueness of solutions. In such a situation the associated flows (or dynamical systems) are multivalued, and there have not obtained so many theoretical results especially in the non-autonomous case. In this paper we investigate the large time behaviour of the multivalued flows from the view-point of attractors. In fact, we construct a global attractor for our multivalued flows, and discuss the relationship to the one for the flows associated with the limiting autonomous system.

1

Introduction

In this paper let us consider a nonlinear evolution equation in a real Hilbert space H of the form: u'(i) + dip*{u{t)) + g(t, u{t)) 3 f(t) in if, t > s ( > 0), (1.1)

135

136 where d(p* is the subdifferential of time-dependent proper l.s.c. and convex function tp* on H, g(t, •) is a single-valued operator in H which is rather small relative to ip* and / is a forcing term given in Lfoc([0, +00); H). We assume that ip*, g(i, •) and f(t) respectively converge to a convex function ip°° on H, an operator g°°(-) in H and an element /°° in H in appropriate senses as t —>• +00. Then, we have the limiting autonomous system (as s —>• +00): v'{t) + d
intf,

t > 0.

(1.2)

In this paper, assuming that the Cauchy problems for (1.1) and (1.2) have in general multiple solutions, we shall discuss the large-time behaviour of solutions of (1.1). In such a situation the solution operator E(t,s) (0 < s < t < +00) for (1.1) is multivalued. Namely, E(t,s) (0 < s < t < +00) is the multivalued operator from D(ips) into D{ipl) which assigns to each u0 G D(
There is a solution u of (1.1) on [s, +00) 1 ^ ^ ^(t) ^ j . u{s) = ^ ^

., „. (1.3)

Of course, the solution operator 5(i) (i > 0) for the limiting autonomous system (1.2) is similarly defined as a multivalued operator in Z?(
2

:= sup inf \x — y\H-

Global boundedness of solutions

In what follows, system (1.1) is referred as (Es) and its Cauchy problem associated with initial value u0 as (E S ;K 0 ), namely (E • u ) I U ' W + ^ ' M * ) ) + 9(t, u(*)) 3 f(t) in H, t > s (> 0), s ' 0> \ u(s) = MoDefinition 2.1. Let J be any interval in R+ with initial time s. Then (i) A function u : J -> H is called a solution of (E,) on J, if u E C(J; H) n W^(J; <^()(u(-)) e L}0C(J), u{t) e Didip*) for a.e. t e J and f(t) - u'(t) - g(t,u(t)) e V ( w ( t ) ) ,

a.e. t G J.

H),

137 (ii) A function u : J —> H is called a solution of the Cauchy problem (Es; w0) on J with given initial value u0 G H, if it is a solution of (Es) on J satisfying u(s) = u0. Now, let {o r } := {ar; r G R+} and {&,; r G R+} be two families of absolutely continuous (real) functions ar, br on R+, with parameter r G R+, such that a'r G Ll{R+) n L2{R+),

b'r G L^R+y,

(2.1)

by (2.1), the limits a r (+oo) := lim aJt) and 6 r (+oo) := lim br(i) exist, so ar and br t—>+oo

£—y+oo

are considered as continuous functions on R+ := [0,+oo]. With these families {ar} and {br}, we specify a class $({a r }, {6r}) of families {(yf) such that | 5 - « U < K ( t ) - 0 , ( 8 ) 1 ( 1 + 1^(2)1*) and V'(5)-^W<|6r(*)-6r(s)|(l

+ |^W|).

For {<£*} G $({a r }, {&,•}), d{-, •) and /(•), we further require the following conditions: ( A l ) There exists a positive constant C\ > 0 such that d\z\2H,

Vi G R~;, VZ G Z % ' ) .

(A2) For every finite r > 0 and t G i? + , the level set {z £ H ; tpt(z) < r} is compact in H. (A3) Diip1) C D(g(t, •)) C .ff for any t G R+, and (''(«(•)) G L}0C(J). (A4) There are positive constants C 2 , C3 such that lff(*,2)l!r < < V ( * ) + Cs,

Vi G S^, Vz G D(v»t)-

(A5) (demi-closedness) If z„ G D((ptn), {tn} C f? + , { £ (as n -4 +oo), then <;(£„, z„) -> #(t, 2) weakly in H. (A6) For each bounded subset B of if, there exist positive constants Ci{B) and C5(B) such that ¥>*(*) + (g(t,z),z-

b)E > C4(B)\z\j, - C5(B),

Vt eR~;,\/ze

Dtf),

V6 G B.

138 As to the existence and global boundedness of a solution for (E s ; u0) on [s, +00), we have the following theorem, which can be proved in the same way as in [8]. Theorems 2 . 1 . [8, Theorems 2.1, 2.2] Let f £ Lf0C(R+;H) and assume that { 0 and u0 e D( then, the solution u of (Es; uo) satisfies the following estimate: rt+l

sup \u(t)\2H + sup /

ipT(u(r))dT < JVi(l + S) + |u 0 |lf),

where Ni is a positive constant independent of f, s > 0 and «o 6 D(ips). Moreover, for each 6 > 0 and each bounded set B C H, there is a constant Ni{5, B) > 0, depending only on 5 > 0 and B, such that sup \u'\Wt+1.H) t>s+s

+ sup f\u{t)) t>s+e

< N2{5,B),

as long as s > 0 and «o € D(tps) D B.

3

Global attractor of the non-autonomous system

By Theorem 2.1 we see that the solution operator E(t,s) (0 < s < t < +00) for (Es) is well defined as a multivalued operator from D(tps) into D(ipl) which assigns to each Uo 6 D(tps) the set E(t, s)uo given by (1.3). Also, it is easy to check the following evolution properties (El) and (E2): ( E l ) E(s, s) =1 (the identity) on D( 0. (E2) E(t2, s)z = EfatjEtti, s)z for any 0 < s < tx < t2 < +00 and z G D(
as t -* +00.

(3.1)

l

Moreover, note that {

(3.2)

Then, by assumptions (A5), (3.1) and (3.2), as s —>• +00 the limiting system of (E„) is of the form: (Eoo)

i/(t) + 0 p o o ( i ; ( t ) ) + f l o > ( t ) ) 9 / " i n # ,

t > 0,

where g°° := g(oo,-). By (Eoo^o) we denote the Cauchy problem for (Eoo) associated with initial value v0.

139 The limiting autonomous system (E m ) can be considered as a special case of (Es) with
s(t)v0 := L e Dip*)

Theie

!\i f t r " o f ( E r ; S o n R+

w v ^ ' such that u(0) = v0 and ti(i) = z. Clearly, the following properties are satisfied: (51) S(0) = 7 o n % ° ° ) . (52) S{t + s)z = S{t)S{s)z, \/z € D{
Moreover, we can easily show the closedness of S(-) and E (•){•) in the following sense. Lemma 3.1. (i) Assume that tn,t G R+ with tn —> £, ^on, v0 6 D{ip°°) with v0n —> D0 in i? and an element zn 6 S(£„)i;on converges to some element z in H as n —> +oo. Then, z G S(t)vo. (M) Assume £fta£ s n , s, £„, t G -R+ tuitfj s„ —> s and i„ —>• i, won £ D((ps"), u0 G D(ips) with ^on -^ u0 in H and an element zn G S(i n + s n , sn)«on converges to some element z in H as n -¥ +oo. 27»en, z G £(£ + s, s)w0- J" particular, if s = +oo, tten z G S(£)itoNext, to construct a global attractor for the non-autonomous system (E s ), we need the concept of w-limit sets under E(t, s) or S(t). Definition 3.1. Let B(H) be a family of bounded subsets of H. Then, for each B € B(H) the sets w s ( B ) : = n U E(t + s,s)(D(^jnB) T>Ot>T,!>0

and

__=^___ us(B):=n\JS(t){D(

»)nB) T>Ot>T

are called respectively the w-limit sets of B under E(t, s) and S(t). Remark 3.1. (cf. [5, Lemma 1]) By the definition of the w-limit set U>E(B), it is easy to see that x € COE(B) if and only if there exist sequences {£„} C R+ with tn —> +oo, {s n } C R+, {zn} C B with zn G £)(• a; in if as n —> +oo. So far as the construction of a global attractor for {S(t)} is concerned, we can do it by modifying the standard technique established in [2] and [9]. The key is the following lemma: Lemma 3.2. (cf. [3, Lemma 4.1]) There exists a compact and convex subset BQ O/-D(
140

and for each bounded set B e B(H) there is a finite time Tg > 0 satisfying S(t) (D(ip°°) n f l j c B o

for all

t>TB.

By the above lemma, we can construct a global attractor for {S(t)}. In fact, the global attractor A o of {£>(*)} is given by Ax. = Us{B0). For a detail proof, see [9]. Here, by a global attractor A o of {£(*)} we mean that (i) A o is a non-empty and compact subset of H; (ii) for each bounded set B e B(H), distH(S(t)z, Aoo) —• 0

uniformly i n z e D{ip°°) n B

as t —• oo; (iii) Aoo = S(t)Ao for all t > 0. Now, let us mention our main result in this paper. Theorem 3.1. Let {'} e $({a r }, {br}) and assume that (A1)-(A6) and (3.1) hold. Then the set A:= U wE(B) (3.3) BeB(H)

satisfies that (i) At C Aoo and At is non-empty and compact in H; (ii) for each bounded set B G B(H), distn{E{t, s)z, At) —> 0

uniformly in s > 0 and z € D(ips) n B

as t ^r +co; (iii) A C S ( i ) A /or any t e R+. In our proof of Theorem 3.1 one of the important steps is to show the following lemma on the relationship between w-limit sets under E(t,s) and an absorbing set B0 for S(t). Lemma 3.3. Let Ba be the compact absorbing set for {S(t);t e R+} obtained in Lemma 3.2. Then, uE{B) C B0 for each B € B(H). In the proof of Lemma 3.3 the time dependence condition (*) on {
141

Next, let us show (iii) of Theorem 3.1. By Lemma 3.1, we easily see that for each XCB0, S(t)X C S(t)X,

Vt G R+.

(3.4)

Now, let x be any element of A*. By the definition of A,, there exist sequences {Bn} C B(H) and {xn} C H with xn G U>E{B„) such that x„ —>• a: in H

as n —> +oo.

(3.5)

It follows from Remark 3.1 that for each n, there exist sequences {tn,j} C R+ with tn,j -> +oo, { s n j } C R+, {znj} C B„ with znj G D(ips"J) and {«Jnj} C if with u„j G E(tnj + snj, snj)zn<j such that »„j —• a;n in H

as j —> +oo.

Let t be any time in R+. We note that

for j with tnJ- > t + 1. Hence there is a wnj G E(tnj + s„j — i, snj)znj

such that

Here by the global estimates obtained in Theorem 2.1, {w n j e ff ; j 6 iV with ^nj > £ + 1} is relatively compact in if, so that we may assume that the sequence wnj converges to some element wn G H as j —> +oo. Clearly, tun G LOE(Bn). Therefore from (ii) of Lemma 3.1 we observe that

xn e S{t)wn c S{t)uE(Bn), which implies that xn G (J S(t)u)B{Bn),

Vn > 1.

n>l

On account of (3.4) and (3.5), we see that x G [J S(t)cuE(Bn) = S(t) | J ujB(Bn) C S(t) (J uE{Bn) C S{t)A,. n>l

n>l

n>l

Hence A, is semi-invariant under S(t), namely A, C S(t)A, for all £ G R+.

Q

Theorem 3.1 says that the attracting set At is semi-invariant under S(t) associated with the limiting autonomous system (Eoo), in general. We now give a sufficient condition in order that A, is invariant under S(t). Definition 3.2. Let z G D((f°°). Then, we say that S(t)z is regularly approximated by E(t + s, s) as s —> +00, if for each finite T > 0 there are sequences {sn} C R+ with sn —> +00 and {zn} C H with zn G D((pSn) and zn —» z in i i satisfying the following property: for any function u G W l,2 (0, T; ii) satisfying u(t) G 5(i)z for all t G [0, T] there is a sequence {«„} C W 1,2 (0, T; ii) such that un(t) e,E(t + sn, sn)zn for all t G [0, T] and u n —> u in C([0, T]; i/) a s n - > +00.

142 Now, let us mention a result on the invariance of A, under S(t). Theorem 3.2 Assume that for any z of A*, S(t)z is regularly approximated by E(t + s, s) as s —> +00. Then, A, = S(t)A* C Aoo

for all T 6 R+.

(3.6)

Moreover, if S(t)z is regularly approximated by E(t+s, s) as s —• +00 for all z e A^, then A, = Aoo. (3.7) Proof. By Theorem 3.1, the attracting set At is semi-invariant under S(t). So we have only to show that S(t)A* C A* for any t 6 R+. Now let y e S(t)A». Then, by (iii) of Theorem 3.1 we see that y e S{t)A. C S{t + T)A»

for all r > 0,

which shows that there are sequences {T„} with T„ —> +00 and {xn} c A, such that y€S(t

+ Tn)xn.

Now by our assumption we observe that for each n there are sequences {s n j} C R+, {xn,j) C H and {ynj} C H such that snj->+00,

xnj- e D(), xnj->

xn in H

and )xn,j,

Vn,j —• 2/

in -ff

as j —> +00. Therefore, by the usual diagonal argument, we can find a subsequence {jn} of {j} such that xn := aJnj„> Vn '•— yn,j„ a n d s n := snjn satisfy \xn - XH\H < - , n

Vn € £ ( i + T„ + s„, s n )x„,

\yn - y\H < n

for every n = 1, 2,.... Since {£„} is bounded in # , say {£„} C i? for some B £ B(H), the above fact implies (cf. Remark 3.1) that y e uE{B) c -A». Thus 5(f).4„ C A*. Now here, we have At = S(t)A* for all t € i?+. Therefore it follows from the invariance of A* and ./"loo under S(t) that A, = S(t)A, C S(t)Aoo = Ax> /or aH t e JR+. Thus (3.6) holds. Assertion (3.7) is proved by a slight modification of that of (3.6).



We say in this paper that the set A*, having the properties in Theorem 3.1, is a global attractor for {E(t, s)}.

4

Examples

In order to illustrate our theorems we give two simple examples of non-autonomous flows in the one dimensional space H = R, which show that .4, ^ Aoo in general. For a given interval [a,b] we denote by I[a,b]{-) the indicator function on [a,b].

143 Example 1. We consider a family {ip1} of proper l.s.c. convex functions on R and a function g(t, u) on R defined as follows: f /[!,!](•)
for 0 < t < 2, for 2 < t < +oo for t = +oo

and g(t, u) := —\/u

for 0 < u < 1.

It is easily checked that they satisfy all the conditions in section 2. Now, consider the non-autonomous and autonomous evolution equations (Es) with / = 0 and (Eco) with f°° = 0, and the corresponding non-autonomous and autonomous systems E(t, s) and S(t), respectively. Also, let A* and A*, be the global attractors associated with them. Then, observing that E(t + s, s)u 0 = min{l, -At + c)2} with c = 2^j% e [-, 1] for s > 2, S(t)v0 = min{l, -(t + c)2} with c = 2 0 ^ G (0,1] and S(i)0 is the set [0,1] for all sufficient large t > 0, we have At = {1} and Aao = [0,1]. By the way, in this example, S(t)0 is not regularly approximated by E(t+s, s) as s —> +oo. Example 2. We next consider {
for

0 < t < 2,

! /[_!,!](•) for 2 < t < +oo, -y/u7[o,i](-) for 0 < fort t < +oo, = +co0 < u < 1, u 1 for 0 < t < 2, - - < u < 0, g(t,u) 2 - - ' 2 ~ u 1 for 2 < t < oo, < u < 0. Just as in Example 1, they satisfy all the conditions in section 2, and we can generate the non-autonomous and autonomous systems E(t, s) and S(t) associated with the evolution equations (Es) with / = 0 and (Soo) with /°° = 0, respectively. In this case it is easy to see that A, = A^, = [0,1] and for any z 6 [0,1], S(t)z is regularly approximated by E(t + s, s) as s —> +oo. It seems difficult to give any example showing that A* / S(t)A* in such simple situations as above.

and

References [1] H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.

144

[2] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, Amer. Math. Soc, Providence, R. I., 1988. [3] A. Ito, N. Yamazaki and N. Kenmochi, Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces, pp. 327-349, Dynamical Systems and Differential Equations, Missouri 1996 Volume 1, ed. W. Chen and S. Hu, Southwest Missouri State University, 1998. [4] A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstract and Applied Anal., 5(2000), 33-46. [5] V. S. Mel'nik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8(2000), 375-403. [6] U. Mosco, Convergence of convex sets and of solutions variational inequalities, Advances Math. 3(1969), 510-585. [7] M. Otani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46(1982), 268-299. [8] K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials, pp. 287-310, in Recent Developments in Domain Decomposition Methods and Flow Problems, GAKUTO Intern. Ser. Math. Appl., vol 11, Gakkotosho, Tokyo, 1998. [9] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Berlin, 1988. [10] N. Yamazaki, A. Ito and N. Kenmochi, Global attractor of time-dependent double obstacle problems, pp. 288-301, Functional Analysis and Global Analysis, ed. T. Sunada and P. W. Sy, Springer-Verlag, Singapore, 1997.

Quasiconvex extreme points of convex sets M a r t i n Kruzik C e n t e r of A d v a n c e d E u r o p e a n Studies a n d Research, Friedensplatz 16, 53 111 Bonn, G e r m a n y * E m a i l : [email protected] and I n s t i t u t e of I n f o r m a t i o n T h e o r y a n d A u t o m a t i o n , A c a d e m y of Sciences of t h e Czech Republic, P o d v o d a r e n s k o u vezi 4, 182 08 P r a h a 8, Czech Republic. E m a i l : [email protected]

Abstract Zhang [17] showed that if the quasiconvex hull of a compact set in K m x " is convex then also its rank-1 convex hull is convex. In this note we show that a reason for that is in a special structure of quasiconvex extreme points of compact convex sets. In particular, we show that compact convex sets are lamination convex hulls of their quasiconvex extreme points. We also give a clear geometric characterization of quasiconvex extreme points of compact convex sets. Finally, we show that quasiconvex exposed points of convex compact sets coincide with their lamination convex extreme.

1

Introduction

It was proved in [8, 16] that generalized Krein-Milman type theorems ([7]) hold for compact quasiconvex and rank-1 convex sets in TRmxn (the Euclidean space of real matrices m x n). In particular there exist minimal generators of those sets called quasiconvex and rank-1 convex extreme points, respectively. An example in [8] shows that quasiconvex and rank-1 convex extreme points differ in a general situation. In this short paper we give a simple geometric description of quasiconvex extreme points on compact convex sets. More precisely, we show that quasiconvex extreme points on compact convex sets are those points which cannot be written as a convex combination of two mutually different rank-1 connected matrices. In other words, for convex compact sets the notion of lamination convex extreme points coincides with that of rank-1 and quasiconvex extreme points. This gives a concrete description of this, generally a very abstract, notion. * address for correspondence

145

146 Further, we get that convex compact sets are generated by the sets of their quasiconvex extreme points via sequential lamination. An example in [8] shows that this does not hold for nonconvex sets, however. Moreover, our result implies that if the quasiconvex hull is convex then also the rank-1 convex hull must be convex, the fact which was proved in [17]. Finally, we show that recently defined quasiconvex exposed points ([18]) coincide with quasiconvex extreme points in case of compact convex sets. Proofs of our results heavily rely on separation theorems for convex sets (see e.g. [14]) and on Krein-Milman type theorems for quasiconvex and rank-1 convex sets. We say that a function / : IR mx " ->• IR is quasiconvex if for any ip G W1-°°(]Rn; IRm), (0, l)"-periodic, and any A G IR mx " f(A)<[

f(A + VV(x))dx

.

(1)

Quasiconvexity plays a crucial role in the calculus of variations. Namely, the sequential weak* lower semicontinuity of / : W1-00^; IR") -> IR, I(u) = / n /(Vu(i)) dx, Q C IR" a bounded domain is equivalent to the quasiconvexity of / : M m x n -» H, see [1, 10, 11]. Further we say that / : R m x " -> IR is rank-1 convex if for any A, B G H m x " , rank(A - B) < 1 and any 0 < A < 1 f(XA + (1 - X)B) < Xf(A) + (1 - X)f(B) .

(2)

Rank-1 convexity is a necessary condition for quasiconvexity but it is not a sufficient condition if m > 3 and n > 2 as shown in [15]. If min(m, n) = 1 then both the notions are equivalent to each other and the case m — 2, n > 2 still remains open. Let if be a compact set in ]R mxn . Besides the convex hull C(K) we define the quasiconvex hull and rank-1 convex hull Q{K) and R{K) of K, respectively, as follows (see e.g. [12]) Q{K) := {A G H m x n ; f{A) < s u p / V/ : R. mx " -> IR quasiconvex } ,

(3)

K

R(K) := {A G IR mxn ; f(A) < s u p / V/ : H m x " ->• E. rank-1 convex } .

(4)

K

Quasiconvex hulls play an important role in the theory of martensitic transformations. Namely, the quasiconvex hull of the zero set of a material stored energy density is the set of stress free configurations. The rank-1 convex hull is an inner approximation of this set. Finally, we define the so-called lamination convex hull L{K) of K as L(K) := [J L{(K), ielNu{0}

where L0(K) := K and Li{K) := {XA + (1 - X)B; 0 < A < 1 , rank(A -B)

= l,A,Be

L^K)}

, i e IN .

Clearly, we have L(K) C R(K) C Q(K) and contrary to quasiconvex and rank-1 convex hulls the lamination convex hull is relatively easy to construct. The notion of quasiconvexity is closely related to gradient Young measures. Let Q C IR" be a bounded domain and K C IR mx " compact. It is known ([3, 6]) that for any

147 sequence {uk}ke^ C W1'°°(ft;Irlm) such that, for almost all x e Q, Vuk(x) 6 K there exists its subsequence (here denoted by the same way) and a family of probability measures {vx}x&i, supported on K such that for any continuous function v : K —> ]R and any geLl(Q) lim / v(Vuk(x))g(x) k-HxJn

dx = / / v(A)i>JdA)g(x) dx .

(5)

JSIJK

The family of probability measures {vx}xm f° r which the above limit passage holds is called a gradient Young measure generated by {Vw^jtejN If {vx}xm is independent of x we call such a measure a homogeneous gradient Young measure. It follows from the analysis by Kinderlehrer and Pedregal [6] who found an explicit characterization of gradient Young measures that a probability measure v supported on a compact set K is a homogeneous gradient Young measure if and only if for any quasiconvex / : M m x " —> IR

/ (JK Au(dA)j < jK }{A)u{dA) .

(6)

A homogeneous gradient Young measures which satisfies (6) even for all rank-one convex functions will be called a homogeneous laminate ; cf. [13]. It is well known that the quasiconvex (rank-1 convex) hull of a compact set K c Mm> B = C = A and we denote their set by Kifi. The definition above is a particular case of D-convex extreme points defined in [9]. Finally, we denote by Ke the set of convex (usual) extreme points of K, by Aff K the affine hull of K, Aff K := A+ span(ii' - K) for any fixed A e K and the dimension of K, dim K := dim span(if — K). We say that A,B e JRmxn are rank-1 connected if rank(A — B) = 1. Aff K has rank-1 connections if it contains rank-1 connected matrices.

2

Properties of convex compact sets

From now on we suppose that K has more than one point. It holds for a compact set K c IR mx " such that Q(K) = K that K = Q{Kqfi), cf. [16], and, in general, L{Kqfi) C K, strictly. Surprisingly, if K is convex we have the following result.

148 Proposition 2.1. Let K c R m x " be a compact convex set. Then K = L ^m

j^(Kqye).

Proof. We proceed by induction. If dim K = 1 and Aff K has rank-1 connections then the assertion clearly holds due to the Krein-Milman theorem because Ke C Kqfi; cf. [16]. If Aff K has no rank-1 connections then K = ifg,e by [4, Th. 4.1] and K = Li(K). Suppose that the assertion holds for sets of the dimension less than d and that dim K = d. Take p £ dK and let H be a supporting hyperplane to K through p. KnH is compact, convex and its dimension is less than d. By the induction hypothesis KnH = L ^ ^ j ^ n jj((Kf] if),, e ). But if A G {K n H)q>e c K n H then A G if,,e because K is in one of the half spaces determined by H. Thus p E K n H C i ( j j m ft n ^-(if,, e ) C L d _i(if,, e )- If Aff A" has rank-1 connections then Lrf(if) = Li(dK). Therefore K C Li(I/
•

The proposition above shows how convex sets are generated by means of their quasiconvex extreme points and it answers a question raised in [16] in this special case. Corollary 2.2. Let K C R m x " be compact, dim K = d and such that Q(K) is convex. Then Ld{Kq,e) = Q{K). Proof. We have Q(if) = Ld{Q(K)qfi) Q(if),, e c if,,e; see [16]. '

because <2(if) is convex On the other hand, •

The following result has been first obtained by Zhang [17] and a new proof has recently been found in [5]. We get it here as a simple consequence of Corollary 2.2. Corollary 2.3. Let K c IR mx " be compact, dim K = d and such that Q{K) is convex. Then Ld(K) = Q{K). Proof. We have L d (if,, e ) C L d (if) c Q(K). But due to Corollary 2.2 Z,d(if„,c) = Q(if) and therefore Lrf(if) = Q(if). ' • We can give a simple geometric description of quasiconvex extreme points of convex sets. Proposition 2.4. Let K c M m x n be a compact convex set. Then Kqfi = Kre = Kifi. Therefore, A £ if is a quasiconvex extreme point of K if and only if there is no interval in K with distinct rank-1 connected endpoints and with A as an inner point. Proof. As in general Kqe C ifr,e C if|,e we must only show that K^e C if,jC. We proceed by induction. If dim if = 1 and Aff if has rank-1 connections then the assertion clearly holds. If Aff if has no rank-1 connections then if = if,je by [4, Th. 4.1] and the assertion holds again. Suppose that the assertion holds for sets of the dimension less than d and that dim K = d. If Aff if has no rank-1 connections then the argument is the same as for the dimension one, so suppose that Aff if has rank-1 connections. Take A G ifj,e. Then A must be in dK and let H be a supporting hyperplane to if through A. KnH is

149 convex, compact and its dimension is less than d. Moreover, by the induction hypothesis and lamination extremality A € (K n H)i,e C (K n H)qfi. If z/ is a homogeneous gradient Young measure supported on K with the first moment A it must be supported on KCiH as K entirely stays in one of the half spaces defined by H and we have that v = 5A, i.e., A e Kqfi. This yields K\fi c if,ie.

• There is a geometric description of quasiconvex extreme points by means of quasiconvex exposed points given in [18]. Recall that argmax M / := {A e M; f(A) = maxBgM f(B)} for a compact set M and a continuous function / defined on M. Definition 2.5. (see [18]) Let K c Wnxn be quasiconvex. A e K is called a level k quasiconvex exposed point of K if there is a sequence of compact subsets {lfj}*_0 such that K0 := K and K^ := {A} and quasiconvex functions {/i}*=1 such that Ki+l := {Ae K; Ae argmaxK.fi+1}

,i = 0,...,k-l

.

The set of all quasiconvex exposed points of finite levels is denoted by Kq>ex. Zhang showed that for a compact set K the set of level 2 quasiconvex exposed points is dense in Kq<e. It is not clear whether in general Kqje = Kq<ex. If we restrict ourselves to functions which are quasiconvex only on K, i.e., which satisfy the Jensen inequality (6) for all gradient Young measures supported on K then it is really true that Kqfi = Kqfix. Namely, as Kqfi is the Silov boundary of K, cf. [2, 8], we easily see that each point from Kq^e is even a level 1 quasiconvex exposed point. In general, for convex sets we can prove the following. Proposition 2.6. Let K c M m x n be compact and convex. Then Kq<e = Kqex = Kifi. Proof. It suffices to prove the first equality. We again proceed by induction. If dim K = 1 then if K has rank-1 connections it has only two extreme points and the assertion follows from the theorem [18, Th. 1.1] by Zhang. If K has no rank-1 connections then we can translate it such that Aff K is a subspace without rank-1 matrices and the result again follows by Zhang [18, Th. 1.3]. Suppose that dim K = d and that the assertion holds for sets up to the dimension d — 1. If K has no rank-1 connections we are done again; cf. [18, Th. 1.3], as all point in K are quasiconvex exposed points. Otherwise if A 6 Kq<e it must be at the boundary of K. Take a supporting hyperplane H through A to K. Then A e (H fl K)q<e and by the induction hypothesis A € (H C\ K)q>ex. By the definition above it means that there is a sequence of sets M 0 := (H n K),... M^ := {A} and a sequence of quasiconvex functions / i , . . . . / * such that Mi+1 = {P G Kti /i+i(P) = maxfi+l{B),B e #;}. But H n K = argmax^/, where / is an affine function such that H = {x; f(x) = c} and K is in the region where / < c. Therefore, if we define the sequence of sets N0 := K, Ni = M;_i, i = 1 , . . . , k + 1 and a sequence of functions v{ := f, vt = / i + i , we get that A is quasiconvex exposed point because the definition above is satisfied for {iV,} and {u;}. Q

150 Corollary 2.7. Let K C H m x n be compact. Then {Kq>e n C(K)„,e) C K,,CT. In particular, Ke c i f , ^ as Ke C {Kqfi n C^if),,,,). Proof. By the previous proposition C(K)qfi = C(K)qfix and, clearly, .K^nC^-fiT),^ c Remark 2.8. The assertion that K,>e = KqfiX for convex sets is implicitly contained in [16, Th. 1.5]. The fact that Ke C Kqfix has been pointed out in [18]. Note that here we extended definition 2.5 to nonquasiconvex sets. Acknowledgment: Partial support by the grant A 107 5005 (Grant Agency of the Academy of Sciences of the Czech Republic) is gratefully acknowledged.

References [1] E. ACERBI, N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1986), 125-145. [2] E.M. ALFSEN, Compact convex sets and boundary integrals. Springer, Berlin, 1971. [3] J.M. BALL, A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transition. (M.Rascle, D.Serre, M.Slemrod, eds.) Lecture Notes in Physics 344, Springer, Berlin, (1989), pp. 207-215. [4] K. BHATTACHARYA, N . B . FIROOZYE, R.D. JAMES, R.V. KOHN, Restriction on

microstructure. Proc. Roy. Soc. Edinburgh 124A (1994), 843-878. [5] G. DOLZMANN, B . KIRCHHEIM, J. KRISTENSEN, Conditions for equality of hulls

in the calculus of variations. Arch. Rat. Mech. Anal. 154 (2000), 93-100. [6] D. KINDERLEHRER, P . PEDREGAL, Characterizations of Young measures generated by gradients. Arch. Rat. Mech. Anal. 115 (1991), 329-365. [7] M. KREIN, D. MILMAN, On extreme points of regularly convex sets. Studia Math. 9 (1940), 133-138. [8] M. KRUZI'K, Bauer's maximum principle and hulls of sets. Calc. Var. and PDEs 11 (2000), 321-332. [9] J. MATOUSEK, P . PLECHAC, On functional separately convex hulls. Discrete Cornput. Geom. 19 (1998), 105-130. [10] C.B. MORREY, J R . , Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25-53. [11] C.B. MORREY, J R . , Multiple Integrals in the Calculus of Variations. Springer, Berlin, 1966. [12] S. MULLER, Variational models for microstructure and phase transitions. Lecture Notes of the Max-Planck-Institute No. 2, Leipzig, 1998. [13] P . PEDREGAL, Laminates and microstructure. Europ. J. Appl. Math. 4 (1993), 121149.

151 [14] R.T. ROCKAFELLAR, Convex Analysis. 2nd ed., Princeton, New Jersey, 1972. [15] V. SvERAK, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh 120 (1992), 185-189. [16] K. ZHANG, On the structure of quasiconvex hulls. Ann Inst. H. Poincare, Ann. nonlin. 15 (1998), 663-686. [17] K. ZHANG, On various semiconvex hulls in the calculus of variations. Calc. Var. and PDEs 6 (1998), 143-160. [18] K. ZHANG, On the quasiconvex exposed points. ESAIM: Control Optim., Calc. Var. 6 (2001), 1-19.

The Richards equation for the modeling of a nuclear waste repository Olivier Lafitte and Christophe Le Potier CEA/DM2S, Centre d'Etudes de Saclay 91191 Gif sur Yvette Cedex, France E-mail : [email protected] Abstract We study a nonlinear degenerate parabolic equation (called the generalized Richards equation) modeling the water pressure in a nuclear waste repository. We show that this degenerate parabolic-elliptic equation has a unique solution, when considering inhomogeneous Dirichlet boundary conditions induced by the surrounding geological material. We describe this solution (as in [2]) as the limit of a piecewise linear function. When studying a more general model taking into account the mechanical behavior of the material, we show that the new equation on the water pressure h is a new Richards equation, but no longer degenerate. We deduce that any model that will take into account the mechanical behavior will be more regular than the Richards equation.

1

Introduction

Modeling t h e saturation processes in a high level waste deep geological repository is a crucial problem in t h e nuclear industry. This situation corresponds to t h e evolution of t h e water flow in a saturated-instaturated medium. T h e equation used in this set-up is t h e generalized Richards equation : dt{9(h))

= C{h)dth

= div(K{h)Wh),

(1)

which comes from t h e generalized Darcy's model U dt9{h) + div{U)

= =

-K(h)Vh, 0,

( ) (

'

where h stands for t h e water pressure in t h e medium, 9(h) is t h e saturation and K(h) is t h e permeability coefficient, see for example [8], [6], [4] for t h e justification of this model. This model does not take into account t h e mechanical behavior of t h e material, which is t h e aim of t h e third section of this paper.

152

153 The region h > 0 corresponds to the saturated region, where the function 9(h) is constant, equal to 6S. Hence, since C(h) = &, C(h) is equal to 0 in the saturated region and the elliptic-parabolic equation is degenerated. The first section contains classical results. We show, applying an easy extension of the results of Benilan and Wittbold [2] and that of Alt and Luckhaus [1], that the equation (1) with the initial Cauchy condition h(x,0) = hi(x), and the boundary conditions h(0) = h0, h(l) = hi, has a unique solution h(x,t) for (x,t) e [0,1] x K + , and that, for this unique solution, the quantity Q(x,t) = 8(h(x,t)) can be computed as the limit of a sequence (0„ (x, t))n, where Q„ is the solution of an elhptic problem. We give the method which builds the sequence hn(x,t) from the elliptic equation and we deduce 0„(x,£) = 8(hn(x, t)). It is interesting to notice that the scheme used by the CEA to solve numerically this generalized Richards equation is the same as the one used for the construction of the sequence hn. 1.00 .80 -SO

-to

.00 -.20 -.ao

-.80 -1.00 .00

.50

1.00

Function h ( x , t > , t - j * 1 0 0 0 a , j - 0 . , 6 , b l | « ) - 0 . B m

1.50

2.00

2.50

3.00

| 0 . , 1 . 5 a [ , b l U l — 0.8 on ]1.5m,3n)

Figure 1: Evolution of h„ (x,nAt).

2

Uniqueness of t h e solution of t h e generalized Richards equation

We assume that h0{0) = M0) and h0(l) = hj(l). We have : T h e o r e m 2.1 Assume that 9(hi(.)) belongs to ^([0,1]) andht belongs to W2'1^, 1). The equation (1) with the above Dirichlet and Cauchy conditions has a unique solution h (t, x) in W1'™ {[0,oo};I?{0,l)).

154 The sketch of the proof is presented in [7]. We introduce : h0(x) = h0 + (hi - h0)x. Under the assumption K(h) > C > 0, let us introduce Kirchoff's transform F of K : rv+tm(x)

F(x,v)=

/

K(s)ds

The function v \-> F(x,v) is strictly increasing. We denote by R(x,)) =
- h0).

Hence the equation (1) is equivalent to the following equation on v(x,t) = h(x,t) — h0(x) : dt(0(v(x,t)

+ h0(x))) = d2x2(F(x,v(x,t)))

+ (hi -

h0)-^(K{h0(x)))

and v satisfies the homogeneous Cauchy condition and the initial condition : v(x,0) = hi(x) - h0(x). We first show that the equation : -^2(F(x,v(x)))

~ (h - h0)^(K(h0(x)))

= /(*),

(3)

has a unique solution for / G i^QO, 1]) and that / H-> V is increasing and continuous from L 1 to W2'1. We thus introduce, for u e L^O, 1), the set A u =

[ ~^2{F{x,v)) X \

- (In - h0)±[K(h0(x))], v e W^ n Wo1'1, 1 u = 9(h0(x) + v(x)). J

For / in Au, we check that there exists v such that 9(ho(x) + v(x)) = u(x) and

f(x) = - ^ ( ^ ( * . « ) ) + dx(K(h0(x))). This partial differential equation on v has a unique solution satisfying the homogeneous Dirichlet boundary condition. Hence v constructed by the procedure is unique. This proves the invertibility of A. The accretivity of A comes from the monotonicity of the operator / — i > v. We thus prove the invertibility and accretivity of the multi-valued operator A (in the sense described by Crandall and Ligget [3]). For all / , there exists (through [3]) a unique (through Lemma 2.2) u in L x (0,1) such that / belongs to u + eAu, e>0. The main tool is the following Lemma. L e m m a 2.2 For v0 G W0' , v'0 G BV, the following assertions are equivalent :

155 • existence of v in L°°([0, oo[; W2-1 n W01,2) such that 9{h0(x) + v(x,t)) belongs to iy1'°°([0, oo[; L1) and h0(x) + v(x,t) is a weak solution of (1), with the initial condition : v{x, 0) = vQ (x) (that is 9(h0(x) + v(x, 0)) = 0(fc/(x)) = 0i{x)), • existence of a solution u in W1,oo([0,oo[;Z/1) such that 0 € u{t) + Au(t), a.e. and u{0)=8(hi{x)). The proof of this Lemma comes from the construction of a sequence approximating the solution. We thus construct the sequence unE{x) = (I + eA)-n{Bi{x))

; u°(x) = Oi(x).

We notice that this construction can be done as follows : • find v\(x,t) solution of 8{vl{x) + h0(x)) - ei{x) •

d2x2 (F (x,vl(x)))

+ (h, - h0)£

(K (ho(x)))

• compute u\ (x) = 9{v\{x) + h0{x)) ; • consider u\ instead of 8i(x) in the first step. For e = 1/iV, we build uN(x,i) uN(x,t)

as the continuous function defined for t e [^, ^ - ] by = u$/N{x) + {Nt -

k)v*%{x).

As A is accretive, (u^(x,t))N converges, when N goes to infinity, to the same limit as (7 + jjA)~ (9i). This limit is denoted by u(x,t). Denote by v(x,t), for almost every t, the unique (through Lemma 2.2) solution of : -d2x,(F(x,v(x,t)))

- (hi - ho)^{K{ho(x)))

= ft«(a:.*)-

As0(u(x,t)+ho(a;)) = u(x,t) (using the definition of the limit), we check that h(x,t) = v(x,t) + ho(x) is a solution of Richards' equation (1). As v(0,t) = v(l,t) = 0, by construction, and v(x,0) = hi(x) — h0(x), h is a strong solution of the equation (1) with h(0,i) = ho, h(l,t) = h\ and the initial condition 9(h(x,Q)) = 9(hi(x)). We thus constructed h as this limit, and it is enough to construct v as the unique solution of (3) to obtain the result of Lemma 2.2. • Note that another initial condition gj(x) such that 9{hi{x)) = 9(gi(x)) leads to the same solution h(x, t) of the problem on ]0, +oo[, through the uniqueness result. We may also notice that the solution of this problem can be constructed in the following way : introducing 0n+1(a;) = F(x,vn+i(x)), we know that n+1(0) = <j>„+i(l) = 0 and that :

156 -e^+1(x)

+ 8(h0(x) + R(x, n+1(x))) = 9n(x) + efa - h0) —

(K(h0(x))).

We have thus a fixed-end nonlinear elliptic equation. For each e, we build the sequence (n{x))). As before, let e = 1/N. The function 8N(x,t), which is continuous, piecewise linear and equal to 6n(x) at t = n/N, converges to 8(x,t) and the unique solution of (1) is the solution of Lemma 2.2 for f(x) = dt9(x,t). This indeed proves the existence of the solution of the problem, and the uniqueness comes from the results of Benilan and Wittbold for regular initial data. • Note again that, in the above Figure 1, we gave a representation of hn(x, nAt) for n = 0,..., 8 for a Van Genutchen model for 8(h) (see [6]).

3

Coupling with t h e mechanical behavior

The Richards equation was well studied by physicists, but it appears that it is not the best equation to model a porous medium which is deformed under the pressure of a liquid. Coupling Daxcy's law and the permeability law with the mechanical behavior of the medium yields interesting results; in particular, the generalized Richards equation obtained with this process is no longer a partial differential equation but can be seen as a pseudo-differential equation, and it is no longer degenerate. Recall that the displacement tensor e is given in terms of the stress tensor a through a linear relation with constant coefficients, which is (i and j denote one of the coordinates) e

u = ^ ( <>ii ~ v zl aii J 2(1 + 1/)

It is then easy to deduce the stress tensor in terms of the displacement (matrix D)

(l + */)(l-2z/) 0ii =

' (l-^eu + ^^T/r, ) ,

(1+ !/)(!-2«/) (2 " " J 6 *

When we have air and water pressure in the medium (pa and pw) as well as other forces 6, the linear relation between the stress tensor, the water pressure pw and the deformation tensor e is

157 da = D(de - hsd(pa - pa)), 1 (l\ where hs = — I * • We use the fundamental law of dynamics : H

° \ 1 /

div (dtcr) = -dtb, dk , dk : dlj to obtain the system on e and h = pw —pa. Using now e« = -^— and e;,- = —— + -—?-, we OTj era., oxi obtain t h e following equation for t h e displacement (lx, ly, lz):

+ Lwdtpw = -dtb -

hsdtVpa,

where KT is a matrix differential elliptic operator of order two and Lw is a vector field acting on dtpw- We use the relation between the porosity and the mechanical displacement (lx, ly, lz), given by u) = dxlx + dyly + djj. The total system is thus ' KTdtl + Lwdth dt(8(h))

= =

dt~b, ft(wS(/i))

=

S(h)dtui +

=

div{K(h)Vh),

u—oth ^

8tu = ftTt(e) = MB{x,V)dtl = dt(djx + dyly + dzlz). Note in this system that the function S(h) is the function 9(h) of the first part, the notation considered depends on the physical problem studied. We have two ways of rewriting the problem: • The first approach uses the pseudo-differential calculus because the matricial linear operator leads to a matricial symbol, and the calculus of K^1, where KT is elliptic, is easier in this set-up. This system leads to the equation:

"IK ~ S^MB(<X'

v

= div(K(h)Vh)

-

) ^ F l L " ) dth

,5)

S(h)MB(x,V)K^dt~b.

In general, the operator R = UJ^ — S(h)MB(x, V)K^ 1LU1 is a pseudo-differential operator of order 0. When the Lame coefficients depend on the position (which may be the case of a complicated geological structure), the pseudo-differential calculus easily gives the leading order term of this zeroth-order operator. 'Note that it is not exactly the porosity as physically introduced, because in tid we have u = gTre and we omitted, for simplicity, the coefficient \ in the porosity given here.

158

The second more classical approach relies on the particular form of the operators and of the constitutive relation of the material. In fact, considering the three equations of the fundamental law of dynamics, and taking the divergence, we easily obtain

A(

J l ^ f c , -

{l + u)(l-2u)

E dth {l-2v)Hs

which implies that the function (1+J)7112l/)^ai hence the equality

_

+ dAPa

+ 9tdiv(6) = 0,

(i-L)H,^tn ^ o e s

no

^ depend on h,

dt{e{h)) = s{h)dtw + Lo~dth an becomes, when d — 3

ww-l-f + irr^)**-^ Note that this equality becomes dt{6{h)) = L^f + because we have

D

1 0 0 0

Ht(1_„)ff(fe)]

dth+F when d = 2,

/ 1 \ 1 {l +

E v){\-2v)

0 0 0

d=2

v°/ D

1 1 0 0

w

E \-2v

/ 1 \ 1 1 0 0

d = 3.

Even in the region where S'(h) = 0 (which was the region where the Richards equation was degenerate when the mechanical behavior was omitted) the coefficient of dh in the equation (5) is greater than a constant equal to -=— s~£7~> where Ssat is the constant E (1 — v) Hs value of the function S(h) in the saturated region, equal to the minimum of S(h). We note that the Richards equation without the mechanical behavior is u)—dth = div(i;sr(/i)V/i) + F, where a; is a constant. Hence the coefficient of dth in this equation vanishes in the saturated region, because S(h) is constant. Hence, when the mechanical behavior is considered, u> is non longer a constant as we saw and the elliptic-parabolic equation is ' dS 1 1+v . dth = div (K (h) V/i) + G,

159 where G is a source term. Introduce the function 0(h) such that :

"W-i+ ££>>• As ©' > Co, the function © is strictly monotone. Hence we can consider U = 0(h) and h = 0 _ 1 ((7), with (0 _ 1 )'(/t) < 1/C0 and the equation (5) writes dtU = d&(B(U)) + G, which is a non-linear non degenerate elliptic parabolic equation. A similar idea was used by Le Potier [6] in the numerical computation of the solution of the coupled problem, splitting the term S(h)dtu> into two terms, one being positive and

dth (

leading to a non-degenerate Richards equation. Precisely, he writes dte — —

l

\

1 I =

D ldt<J. Hence using to = Tr(e) and dtcr = -MB (z,V) K^dtib), he obtains a new system, where the modified Richards equation has a "mechanical" source term and a non vanishing coefficient for the term dth, which is w— + —S(h). It is still a coupled system ah H3 (because a depends on h and is still in the equation for h), but the coefficient of dth is positive, and a usual numerical method for the heat equation is convenient in most cases.

References [1] H.W.Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311-341. [2] P. Benilan, P. Wittbold, Sur un modele parabolique-elliptique. Math. Mod. Num. Anal. M2AN 33(1) (1999), 121-127. [3] M.G. Crandall, T. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971), 265-298. [4] P.S. Huyakorn, G. F. Pinder, Computational methods in Subsurface Flow. Academic Press, New York, 1983. [5] J. Kacur, Solution of some free boundary problems by relaxation schemes. SIAM J. Numer. Anal. 36(1) (1999), 290-316. [6] C. Le Potier, F. Cany, Couplage hydromecanique et transferts de gaz dans les milieux poreux. Rapport DMT/SEMT/MTMS/RT/00-115+ du 5/12/2000. [7] O. Lafitte, C. Le Potier, Quelques remarques sur le couplage du transfert hydrique et du comportement mecanique en milieu poreux. Rapport DM2S/SFME/MTMS/01005Mars 2001. [8] G. de Marshy, Quantitative Hydrogeology. Academic Press, New York, 1986.

Trace identities and universal estimates for eigenvalues of linear pencils* Michael Levitin Department of Mathematics, Heriot-Watt University Riccarton Edinburgh EH14 4AS, U. K. Leonid Parnovski Department of Mathematics, University College London Gower Street, London WC1E 6BT, U. K. Email : [email protected] ; [email protected]

Abstract We describe the method of constructing the spectral trace identities and the estimates of eigenvalue gaps for the linear self-adjoint operator pencils A — XB.

1

Introduction

Since 1950's, mathematicians have devoted a lot of efforts to the construction of the so-called universal eigenvalue estimates for elliptic boundary value problems. As an illustration, we recall the original result of Payne, Polya and Weinberger [5], who have shown in 1956 that if {A.,} is the set of (positive) eigenvalues of the Dirichlet boundary value problem for the Laplacian in a domain fi C M*1, then A

m

Am+i — Am < — y \j, ran '—'

(i)

3=1

for each m = 1,2,.... The term universal estimate is used in this context since (1) does not involve any information on the domain Q apart from the dimension n. The Payne-Polya-Weinberger inequality has been significantly improved over the course of several decades; similar universal estimates have been also obtained in the spectral problems for operators other then the Euclidean Dirichlet Laplacian (or Schrodinger "The research of M.L. was supported by the EPSRC grant GR/M20990. Both authors were also partially supported by the EPSRC Spectral Theory Network

160

161 operator), e.g. higher order differential operators in R", operators on manifolds, systems like Lame system of elasticity, etc. We refer the reader to the important paper [3] and the recent survey [1] for more details. In the recent paper [4], we have shown that the majority of the universal eigenvalue estimates known so far can be easily obtained from a certain operator trace identity which is valid not only for boundary value problems for PDEs but, under minimal conditions, for a general self-adjoint operator acting in a Hilbert space. Namely, we have proved the following Theorem 1.1 Let H and G be self-adjoint operators such that G(DH) C DH. Let \j and 4>j be eigenvalues and eigenvectors of H. Then for each j :

E KIM&M£ k

Xk X

,

.l[WM,^{M 2

~ i

= 5>*-Ai)|(Gfc,&>|2.

(2)

k

This result implies Theorem 1.2 Under conditions of Theorem 1.1, m

m

-(A m+1 - Am) £ ( [ [ # , G ] , G ] ^ ) < 2 ^ | | [ t f , G ] ^ | | 2 . 3=1

(3)

3=1

The inequalities (3) may be viewed as a family of estimates of the spectral gap A m+ i — Am for the operator H, with different estimates being obtained for each choice of the operator G. The particular choice of G depends, of course, on the problem, and cannot be prescribed. In [4], we give numerous examples of concrete estimates for boundary value problems for elliptic PDEs (and systems of PDEs) obtained using Theorem 1.2. In particular, the classical estimate (1) follows from Theorem 1.2 by taking G to be an operator of multiplication by the coordinate X[, summing the resulting equalities over I, and using some elementary bounds. The aim of this paper is to extend Theorems 1.1 and 1.2 to the spectra of linear self-adjoint operator pencils of the type A — \B. The statement of the problem and the main results are collected in Section 2; the proofs are in Section 3.

2

Statement of t h e problem and main results

We consider a linear operator pencil V{\) = A-XB,

(4)

acting in a Hilbert space H, equipped with the scalar product (.,.) and the corresponding norm ||.||. Here, A and B are self-adjoint operators such that DA C DB- We identify the domain of the pencil D-p with DA- By the spectrum of V we understand the set of

162 complex values A for which V(X) is not invertible. Throughout this paper we assume, for simplicity, that the spectrum consists of isolated real eigenvalues Ai < A2 < . . . (counted with multiplicity) which may accumulate only to +00, and that the system of corresponding eigenfunctions Uj such that V(\J)UJ = 0, is complete in H (the case of a pencil with a continuous spectrum can be treated as well, cf. Remark 2.5 in [4]). We refer to [2] for a general discussion of the spectral theory of operator pencils. An important property, which we shall use later on, is the fact that, under a technical condition (BUJ,UJ) ^ 0 (which is satisfied automatically if, e.g., B > 0), the eigenfunctions can be assumed to be normalised by the relations (Buj,uk) = Sjk.

(5)

Indeed, multiplying the equality V{\j)Uj = 0 by uk in H, and using the fact that A and B are self-adjoint, we get {Auj,uk) = (uj,Auk) = \j (Buj,uk)

= Afc

(uhBuk),

which implies (5). By completeness, any element / e K can be expanded in a convergent series / =

^{Bf,uk)uk, k

and for any

f,g£H,

{Bf,g) =

Y,{Bf:Uk)-{uk,Bg). k

Thus, by the density argument,

(f,g) = J2(f>u*)-(u*>B9)-

(6)

k

In what follows, [H, G] denotes the standard commutator HG — GH. Our main result is the following spectral trace identities for 'P(A). Theorem 2.1 Let V(X) = A — \B be a self-adjoint linear operator pencil with discrete spectrum \j and eigenfunctions Uj. Let G be an auxiliary self-adjoint operator with domain DQ such that G(D-p) C D-p C DQ. Then for each j

pmXi^U^=-lm^GhG]u3,u]}.

(7)

and £ ( A * - \,)\ (BGUj,uk)

| 2 = - i <[[7>(A,-),G] , •

(8)

163 Remark 2.2 Instead of the condition G(D(A)) C D(A) we can impose weaker conditions GUJ e D(A), G2Uj e D(A), j = 1,.... Moreover, the latter condition can be dropped if the double commutator is understood in the weak sense, i.e., if the right-hand side of (7) and (8) is replaced by ([V(X),G]UJ,GUJ) (see (13) below). The trace identities (8) imply the following universal estimate, which generalises the Payne-Polya-Weinberger inequality for linear operator pencils. Theorem 2.3 Under conditions of Theorem 2.1 m

771

-(Am+i-Am)^([p(A3),G],GK,U,)<2^||B-1[P(A3),GK.||2. J=l

3

(9)

3=1

Proof of t h e main results

We start with establishing (8). Obviously, we have [A - XjB, G) UJ = (A - \jB)Guj.

(10)

Therefore, ([A - XjB, G] Uj,GUj)

= {(A - XjB) GUJ,GUJ) .

(11)

Since A, B and G are self-adjoint, we have, using (6) {{A - XjB) GUJ,GUJ)

=

] T ({A - XjB) GUj,uk)

(uk,BGUJ)

k

= Y,^Gui^A~XiB">Uk^u>"BGu^ =

(12)

2

^(Afc-A^IBGuj.u*! . k

Using the fact that [A - XjB,G] is skew-adjoint, the left-hand side of (11) can be rewritten as (G [A - XjB, G] Uj,Uj)

= - {{[A - XjB, G], G] Uj,Uj) + {[A - XjB, G] GUj,Uj) = - ({{A - XjB,G],G]UJ,UJ) + (UJ, [A - XjB,G]Uj),

so (G[A - XJB,G]UJ,UJ)

= - - {[[A~XjB,G],G]uj,Uj)

(13)

(notice that (G [A - XjB, G] Uj,Uj) is real, see (11) and (12)). This proves (8). Since (10) implies {[A - XjB,G]Uj,uk) this also proves (7).

= (GUJ, [A - XjB,G]uk)

= {Xk - Xj)

{BGuj,uk),

164 We now proceed to the proof of (9). Let us sum the equations (7) over j = 1, ...,m. Then we have

^ g w^rf...£r(1)Wi%i„,

(14)

Parceval's equality implies that the left-hand side of (14) is not greater than

"m+1

—

Am

=1

This proves (9).

References [1] M.S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Edinburgh, 1998), E.B. Davies and Yu Safarov, eds., London Math. Soc. Lecture Notes, vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 95-139. [2] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nons elf adjoint Operators. Transl. Math. Monographs, vol. 18, AMS, Providence, 1969. [3] E.M. Harrell II, J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Amer. Math. Soc. 349 (1997), 2037-2055. [4] M. Levitin, L. Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues. J. Funct. Anal, (to appear), preprint at http://www.ma.hw.ac.uk/~levitin/research.html. [5] L.E. Payne, G. Polya, H.F. Weinberger, On the ratio of consecutive eigenvalues. J. Math. Phys. 35(1956), 289-298.

Universal estimates for the blow-up rate in a semilinear heat equation Julia M A T O S « , Philippe SOUPLET (2-3) W Departement de Mathematiques, Universite d'Evry Val d'Essonne Boulevard des Coquibus, 91025 Evry Cedex, France (2)

(3)

1

Departement de Mathematiques, INSSET, Universite de Picardie, 02109 St-Quentin, France Laboratoire de Mathematiques Appliquees, UMR CNRS 7641, Universite de Versailles, 45 avenue des Etats-Unis, 78302 Versailles, France Email : [email protected] ; [email protected]

Introduction

We consider the semilinear heat equation ut = Au + \u\P~\ u (x, 0) = u0 (x)

0<(
ieR"

, . >

(

where p > 1 and u0 e L°°(RN). Let u be the unique classical solution of (1) which is denned on a maximal time interval [0, T). This solution u is bounded on [0, T"] x R ^ , for all 0 < T" < T. When T < oo, we have ||u(t)||oo —> oo as i —> T, where || • H^ denotes the norm in L°°(HN), and we say that u blows up in finite time T. Denote ps = ^ | if N > 3 and ps = oo if N = 1,2. It is well-known that if p < ps and u is a nonnegative solution of (1) which blows up in finite time T then there exists a constant C > 0, depending on u, such that

||«(t)lloo
(2)

Giga and Kohn [8] obtained this upper bound on the blow-up rate of u(t) and proved that (2) still holds for sign-changing solutions provided (3N-A)p < 3N+8. Furthermore, with no restriction on p > 1, the same estimate holds (cf. Bricher [3] and Matos [14]) for any nonnegative solution of (1) which is nondecreasing in time and such that its blow-up set is compact. (Recall that a G RN is in the blow-up set if u(tn,xn) —• oo for some tn —» T and xn —> a.) However, it is known that one may have a faster blow-up rate in dimensions N > 11 for sufficiently large values of p > ps (see Herrero and Velazquez [12])-

165

166 On the other hand, for global positive solutions of (1) on a smoothly bounded domain fi C R w , with homogeneous Dirichlet boundary conditions, a universal a priori bound of the form suplluWHoc^C^.p.r), f o r a l l r > 0 , (3) t>T

was recently established by Fila, Souplet and Weissler [4] under the assumption (N— l)p < N + 1, and by Quittner [19] provided iV < 3 and p < p$. The first goal of this note, motivated by (2) and (3), is to establish a global a priori bound for the blow-up rate of nonnegative solutions of (1) which is universal, that is independent of the solution u.

2

Main results

Theorem 2.1 Let p > 1 and e € (0,1). Let w0 G L°°(RN), u0 > 0, and assume that one of the following conditions holds: (i)p
+ %,

(ii) UQ is radially symmetric and nonincreasing as a function of r = \x\, and (N-2)p
+2

ifN<3, ifN>3.

{

'

There exists a constant C = C(p, N, e) > 0, independent ofu, such that if the classical solution u of (1) exists on [0,T) x HN, then lh(t)||oo
eT
(5)

It is unknown whether the upper blow-up rate (2) can fail for all N > 3 and all P > Ps- However it is interesting to note that when N > 3 and p > ps, the universal global blow-up rate (5) can be true for no value of e £ (0,1), even if one restricts to radially symmetric nonincreasing initial data. (Indeed this would imply that Theorem 2.2 below holds for such p - see after that Theorem.) Let us note that Theorem 2.1 holds whenever u exists on [0,T) x R " and that we do not need to assume that u blows up at t = T. Actually, Theorem 2.1 has several interesting consequences, not only related to blowing-up solutions. The first one concerns the decay rate in large time of global nonnegative solutions of (1). Theorem 2.2 Letp > 1 satisfy (4)- Assume thatuo € L°°, u0 > 0, is radially symmetric and nonincreasing as a function ofr = \x\, and that the classical solution u of (1) is global (i.e., T = oo,). Then there exists a constant C = C(p,N) > 0, independent ofu, such that \\u(t) ||oo < C f ^ , t>0. (6)

167 When p < ps and u 0 , Vu 0 satisfy a fast decay condition (namely, / \f(x)\2e^2/4 dx < oo), the estimate (6) was obtained by Kavian [13], with a constant C depending on u. The result of [13] relied on the method of forward self-similar variables. Interestingly, although a result on global solutions, Theorem 2.2 relies on the use of backward self-similar variables (and on some new ideas - see section 3). Later in [21], the second author proved that when p < ps, any global nonnegative solution of (1) (with w0 6 L 2 nL°°) must satisfy lim t _ 00 ||w(t)|| 00 = 0. On the other hand, we see that (6) cannot be true unless p < p$. Indeed, when N > 3 and p > Ps, there exist positive, classical stationary solutions of (1), which are radially symmetric nonincreasing (see e.g. Haraux and Weissler [10]). Also, note that the self-similar solutions to (1) (constructed by [10] for p > 1 4- 2/JV) decay precisely like ||«(*)lloo = C r ^ . For other sufficient conditions ensuring global existence and decay of positive solutions of (1), we refer to the recent paper by Gui, Ni and Wang [9] and to the references therein. In particular it is observed in [9] (see p. 590) that all previous results concerning the decay in time provide rates no slower than t~p^ when p < ps- (Some solutions with slower decay rates are constructed in [9] for N > 11 and sufficiently large values of p > p§.) From the works [13, 10, 9] and the references in [9], it thus seems natural to make the following conjecture: Conjecture 2.3 When p < p$, all global nonnegative classical solutions of (1) decay at least like i" 1 /^" 1 ' as t -> oo. Theorem 2.2 proves this conjecture in dimensions N < 3 in the radial symmetric case. Moreover, the estimate is global on (0, oo), with a universal constant. As a consequence of Theorem 2.2, we obtain a new kind of parabolic Liouville type Theorem, concerning solutions of (1) that are defined globally on (—00,00). Corollary 2.4 Let p > 1 and let u be a global nonnegative classical solution of ut = Au + up,

— 00 < t < 00,

x € RN,

Assume that (4) holds and that for all t, u(t,.) is radially symmetric and nonincreasing as a function of r — \x\. Then u = 0. Furthermore, we partially improve a parabolic Liouville Theorem of Merle and Zaag, concerning solutions of (1) that are defined globally in the past (see Corollary 1.6 in [16]). Corollary 2.5 Assume p > 1, T < 00 and let u be a global nonnegative solution of ut = Au + up,

- 0 0
i6RN,

with u(t,.) £ L°° for all t < T. Moreover, assume that one of the following conditions holds: (i) p < 1 + | ,

168 (ii) for all t
K

Then u = 0 or there exists T0 > T such that u(t,x) = (p-l)-i/(p-D.

= K(T0 — t ) _ 1 ^ _ 1 \ where

In Corollary 1.6 of [16], the same conclusion was obtained under the additional assumption that there exists C > 0 such that u(t,x)
- o o
leR",

But instead of (i) or (ii), it was only assumed that (N — 2)p < N + 2. We prove Corollary 2.5 as a consequence of Theorem 2.1 and of [16, Corollary 1.6]. Finally, we obtain an estimate for the initial blow-up rates of local solutions to ut = Au + up. Corollary 2.6 Assume p > 1, 0 < T < oo and let u be a nonnegative classical solution

of

ut = Au + uF,

0
ie

RN,

with u(t,.) e L°° for all t < T. Moreover, assume that (i) or (ii) in Corollary 2.5 is satisfied. Then \Ht)\\oo < Ct~&, 0 0. Results on initial blow-up rates for equation (1) were first derived by Bidaut-Veron [2]. Namely, it was proved there that (7) holds under the condition p < N(N + 2)/(N — l) 2 , without radial symmetry restriction. Note that this condition is better than ours when N > 4, worse when N = 2,3 and identical when N = 1. The method of [2] is completely different from ours and quite involved (relying on Bernstein type gradient estimates, Aronson-Serrin Harnack inequalities and multiplier arguments). Also for the CauchyDirichlet problem on a bounded domain, initial blow-up rates up to the boundary are studied by Quittner and the second author in [20]. Remark 2.7

a. Theorem 2.1 fails for e = 0.

b. It is well-known that all blowing-up solutions of (1) satisfy the following universal global lower bound:

H^lloo^er-i)"^ 1 , o
(8)

where K = (p - 1 ) - V ( P - I ) .

c. Assuming p > 1 and (N — 2)p < N + 2, and w0 £ i? 1 (R i v ), Merle and Zaag [17] derived the following uniform estimates of order one for the blow-up rate of a

169 nonnegative solution u of (1): for all e > 0, there exists t0 = t0(e) € (0,T) such that K

^(T-t)7blWu(t)\^^K+{^

+ £)l]og{T_t)l

forallte[t0,T).

N_K

The constant ^2 appearing in the term of order one is optimal and t 0 (e) depends on the H2 norm of the initial data UQ- However this estimate only holds on a neighbourhood of the blow-up time T ofu(t), whereas estimate (5) is global in the existence time interval of the solution. f

d. It seems that when p < 1 + 2/iV, the estimate (5) can be obtained by the methods of [1], where, among other things, the authors prove upper blow-up rates for a reactiondiffusion system which generalizes equation (1). The arguments of [1] are rather involved and rely on a Moser's iteration scheme. Our approach here is completely different and simpler, especially in the case p < 1 + 2/N. Throughout this note, C denotes various positive constants which depend only on the indicated arguments. In the next sections, we present the main tools and ideas used in the proofs of Theorems 2.1 and 2.2. For the detailed proofs the reader is referred to [15].

3

Main tools: self-similar variables and convolution Lebesgue spaces

The study of the blow-up rate of u(t) is done by using the method of backward self-similar variables, introduced by Giga and Kohn [7] and Galaktionov and Posashkov [5]. For each a G R/^, we rescale u by similarity variables around {T,a) by setting S=

"l0g(T-t)'

wa{s,y) =

y= x

W=t

(9)

(T-t)^u(t,x).

This function w = wa is defined in (s 0 , +oo) x R w , with s 0 = — logT, and satisfies

w, = Aw-\yVw

w (so,y)

= {v),

+ \w\!'~1w-f^,

s>s ,

0 w 2/eR ,

jeR",

,1Q.

where (y) = T^u0(a + y/Ty). The study of the boundedness of (T - t)r-^ IKOIU when t approaches T is equivalent to the study of the boundedness of the global solution wa. Theorem 2.1 has then the following equivalent formulation: Let 8 > 0. Assume either (i) or (ii). Then there exists a constant C = C(p,N,8) > 0 such that for any global nonnegative classical solution w of (10), it holds II^MIloo < C for all s > so + 6.

170 Let us briefly sketch our basic ideas to establish (5). We start from the simple fact that, for each a £ R N , the rescaled nonnegative global solution wa of (10) has to satisfy /

!"«(«, y)p(y) dy < (p- \)~7^ = K for all s > s0,

with p(y) = (47r)-T e - V4 .

Since wa(s, y) = WQ{S, y + ae as follows

S//2

(11)

), this can be restated in terms of convolution product

H/o* w0(s)||oo = sup / w0(s,r])p(r] — aes'2)dn < K for all s > s0. o e R « JR."

(12)

This estimate leads us to consider the problem (10) in convolution Lebesgue spaces. Definition 3.1 For all 1 < q < oo and any function f e L]oc(RN), we set \\f\\lP = IIP* l/l'll^* = f sup j where || • ||, denotes the norm in Lq = Lq(RN). LqpA(RN) is then defined by

\}{y)\"P{a - y) dy)

*,

(13)

The convolution Lebesgue space L9pA =

Ll* = {feLlc(RN)--\\P*\f\qC<™}These are Banach spaces with the norm \\-\\*p- Also, for q = oo, we define L^

(14) =

L°°(RW).

The central part of our arguments is then the analysis of the smoothing properties of the linear problem zs — Az - \y • Vz in (0,+oo) x R w , . . z(0,y) =
171

4

Linear and nonlinear smoothing effect in convolution Lebesgue spaces

Let A be the operator defined in the Hilbert space L2 by V {A) = {/ e L2p : V / G [iff A(f)=Af-\yVf,

,

Af-^y-Vf£L2

in the distributional sense! ,

feV(A),

where L2p is the weighted Lebesgue space defined on the usual way corresponding to the Borel measure p(y)dy. The operator —A is self-adjoint with dense domain in L2 and it is the generator of a semigroup of contractions on that space which we denote by (5(s))s>oIt is easily proved that (5(S)^)(2/) = ( T ( l - e - ^ ) ( e - ^ ) i

0 e L

J,

yeR",

where T(t) = (Gt)* denotes the linear heat semigroup in RN and G t (x) = ( 4 7 r i ) - ? e - ] ^ ,

t > 0.

(16)

Note that d = p. We prove that the linear semigroup (S(s)) s > 0 denned in L2 admits a uniquely, densely defined extension to L* still denoted by (<S(s))s>o. In particular, if <> / £ L}pik and <j> > 0, then S(s)cj) > 0. Moreover, for each 1 < q < oo, (S(s))s>o restricts to a semigroup of contractions on LqPii,. Our main result in the linear theory is a smoothing effect of the semigroup («S(s))s>0 on convolution Lebesgue spaces L* . Theorem 4.1 . Let <j> e L*^, tuifft 1 < g < oo. Then S(s)<j> 6 L ^ /or all q < r < oo, and ||S( S M|;„ < (1 - e - s ) - ^ ( i - » ||0||; ip , /or a// s > 0. We should point out that this estimate does not hold if we replace the Lqpik norms by L? norms (see e.g. [11]). Next, we are concerned with the nonlinear problem (10) (where we assume that so = 0) with initial data <j> e L9 , 1 < q < oo. Note that, if w is a classical solution of (10) (that is, w E C1-2{{0,oo) x R>) and w e L°°{{0,T),L°°(RN)) for each T > 0), then it solves the integral equation w(s)=S(s)
J'S{s

- T) (\W{T)\P-^T)

- ^Yj

dr,

(17)

for all s > 0. An essential ingredient to the proof of our main result Theorem 2.1 and its consequences is the following smoothing property for the equation (10) in L9 spaces.

172 Theorem 4.2 . Let qc = N(p — l)/2 and let q > 1 satisfy qc < q < oo. Let M > 0. For any T > 0, any <j> € L°° with \\<j)\\qtP < M, and any solution w e L°°((Q,T),L':o) of (17), it holds

IMs)IU < C8-%M\\*qiP,

0 < 8 <min(T, Sl ),

where C > 0 depends only on N, p, q and Si > 0 depends only on N, p, q, M. The proof of Theorem 4.2 relies on the linear estimates of Theorem 4.1. Actually, from these estimates, we can also derive a local existence-uniqueness result for problem (10) in LP spaces (in the spirit of Theorem 4 in [4] and based on arguments of [22]).

5

Universal bounds a n d uniform decay rates

The result of Theorem 2.1, case (i) (p < 1 + 2/N) is a direct consequence of the estimate (12), which can be restated in terms of convolution Lebesgue norms as lko(s)Hi,p < « for all s > s 0l and of Theorem 4.2 applied with q = 1 > N(p — l)/2. In view of case (ii) of Theorem 2.1, we also need to derive a uniform a priori bound for the blow-up rate of nonnegative solutions of (1). More precisely, for any (N — 2)p < N + 2, we prove that the constant C in (2) depends only on the L°° norm of the initial data and on an upper bound of the blow-up time T. This result was already obtained in [18] for the vector-valued problem associated to (1) under the condition (3N — 4)p < 3iV + 8 (in the scalar problem no sign restriction on the solution was made). In terms of the rescaled problem (10), we prove the following theorem. Theorem 5.1 . Assume p > 1 and (N - 2)p < N + 2, and let C0 > 0. Let 6 L°° be nonnegative, satisfying |||oo < Co, and such that the solution w of (10) (with SQ = 0) is global. Then, IHs)||oo0. To conclude the proof of Theorem 2.1, case (ii) when (N — 2)p < N, we then proceed similarly as in [4]. Multiplying the equation (10) by p(y) and integrating in space on R w and in time over (s0,s0 + 5), one shows the existence of sx e (0,6/2) such that JRJv wp(si,y)p(y)dy < C(p,N,S). Using the fact that the Lq and Lq norms are equivalent in the class of nonincreasing nonnegative radially symmetric functions, we obtain \Hsi)\\;,P
(18)

The conclusion then follows by combining (18) with Theorem 4.2, applied for q = p > N(p - l)/2, and Theorem 5.1. The proof in the case (N — 2)p < N + 2, N < 3 is inspired from the method of [19] and is somehow more involved. Beside convolution Lebesgue spaces, it relies on rescaling and contradiction arguments and on energy estimates similar to those in [6] and [8].

173 Finally, Theorem 2.2 follows easily from Theorem 2.1. Since the solution u of (1) exists globally on [0, oo) x RN, by applying Theorem 2.1 with e = 1/2 for each T > 0, we deduce that K * ) l | » < C ( J V , p ) ( T - t ) - 1 / < ' ' - 1 \ T/2
For each fixed t0 > 0, applying (19) with T = 2t0 and t — t0 yields

and the result follows.

References D. Andreucci, M. A. Herrero, J. J. L. Velazquez, Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. Inst. Henri Poincare 14, No. 1 (1997), 1-53. M.-F. Bidaut-Veron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, in: Equations aux derivees partielles et applications, articles dedies a Jacques-Louis Lions. Gauthier-Villars, Paris. (1998) pp. 189-198. S. Bricher, Blow-up behaviour for nonlinearly perturbed semilinear parabolic problems, Proc. Royal Soc. Edinburgh 124A (1994), 947-969. M. Fila, P. Souplet, F. Weissler, Linear and nonlinear heat equations in Lp6 spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87-113. V. A. Galaktionov, S. A. Posashkov, The equation ut = uxx + u13. Localization, asymptotic behavior of unbounded solutions. Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 1985, n°97, 30 pp (in Russian). Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), 415-421. Y. Giga, R. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319. Y. Giga, R. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40. C. Gui, W.-M. Ni, X. Wang, Further study on a nonlinear heat equation, J. Differ. Eq. 169 (2001), 568-613. A. Haraux, F. B. Weissler, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167-189. M. A. Herrero, J. J. L. Velazquez, Blow up behaviour of one dimensional semilinear parabolic equations, Ann. Inst. H. Poincare, Anal. Nonlin. 10 (1993), No. 2, 131-189.

174 [12] M. A. Herrero, J. J. L. Velazquez, Explosion de solutions des equations paraboliques semilineaires supercritiques, C. R. Acad. Sci. Paris, t. 319 (1994), 141-145. [13] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincare, Anal. Nonlin. 4 (1987), 423-452. [14] J. Matos, Convergence of blow up solutions of nonlinear heat equations in the supercritical case, Proc. Royal Soc. Edinburgh 129A (1999), 1197-1227. [15] J. Matos, Ph. Souplet, Universal blowup estimates and decay rates for a semilinear heat equation, Preprint. [16] F. Merle, H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139-196. [17] F. Merle, H. Zaag, Refined uniform estimates at blow-up and applications for nonlinear heat equations, GAFA, Geom. Funct. Anal. 8 (1998), 1043-1085. [18] F. Merle, H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations, Math. Ann. 316 (2000), 103-137. [19] P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), 299-305. [20] P. Quittner, Ph. Souplet, Initial blow-up rates for a nonlinear heat equation, to appear. [21] Ph. Souplet, Sur I'asymptotique des solutions globales pour une equation de la chaleur semi-lineaire dans des domaines non bornes, C. R. Acad. Sci. Paris, Serie I 323 (1996), 877-882. [22] F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in V, Indiana Univ. Math. J. 29 (1980), 79-102.

Asymptotic behavior of curves evolving by forced curvature flows Hirokazu Ninomiya Department of Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194 JAPAN. The motion of an interface is one of most important and attractive topics in applied mathematics. Especially, the mean curvature flows have been studied by many researchers. In the two dimensional case, it is called a curve shortening flow. Asymptotic behaviors of curve shortening flows are well-known. If a curve is a simply closed one, it becomes convex eventually and is shrinking to a point (see [7, 8]). If it possesses two asymptotes, Huisken has proved that it converges to a self-similar solution (see [5, 10]). In this paper, we consider how the behavior changes, if we add a constant driving force. First we will define an interface. A pair (T(t),v) is called an interface provided that there exists a family of connected open sets D(t) in RN such that F(i) is a smooth connected boundary of D(t) and v is a outer normal vector on T(t) pointing from D(t) to D(t)c. By the definition, an interface is not allowed to have any self-intersection points. We consider here an interface (T(t),u) which satisfies V =H +k

(1)

where V is a normal velocity in the direction of u, H is the curvature, and k is a given constant. This equation appears in the several fields of applied mathematics. For examples, this system represents the motion of transition layers of the Allen-Cahn equation [6], and the filamentray vortex of the Ginzburg-Landau equation confined in a plane [4], the BZ reaction [12]. In the curve shortening flow (i.e., k = 0 and N = 2), the property of non-existence of self-intersection points preserves in time (see [7, 8]). In the case k ^ 0, unfortunately, some interfaces may possess some self-intersection points eventually, even if the initial interface has none. Actually, in the case k > 0, take the initial interface T(t) and D(t) as in Figure 1, i.e., £>x(0) C D(0) C D2{Q), where

£>!(*) :=

s((°),-Ri(*))U5((_°0),fli(*)).

D2(t) := B((J),fl 2 (i))UB((~ 0 6 ),W) ,

175

176

§1'<%gm VAP*

§L

W

(®?\

y//y/z

IS

Figure 1: Example of an initial interface which will possess self-intersection points.

== -J^t)~k>

i^W

RM = R°,

where B(x, r) is a disk of radius r centered at x. It follows from the comparison principle (see [14, Lemma 4.1]) that Dx{t) C D(t) C D2{t) as long as D(t) possesses no selfintersection points. We can choose positive numbers a, Ri,R% and b as follows: a > R° > -, k kR\ - log \kR\ + 1| > k{a - E?) + log b2>(Rl

+

ka — 1

kRl-

a)2-a\

(e.g. a = Z/k, b = 7/k, Rf( = 2/fc, R\ = 4/fc). It follows from these assumptions that two circles in D?(t) are not extinct and the circles in Di(t) never touch those of D2(t) before two disks in D\(t) collide. Therefore the interface T(t) becomes self-intersection points in finite time. It is natural to think that D(t) possesses two holes if the solution can be extended after the singularity (see [2]). However, one can also consider the self-crossing interfaces (see [1]). It is also remarked that the self-intersection may occurs in even the mean curvature flow (k = 0) in R 3 (see [9]). If the interface is represented by the graph y = u(x,t), the equation (1) is reduced to the following Cauchy problem ut

l + ul

+ fc^/l + u\

i£R,i>0,

(2)

with initial condition u(x, 0) = u0(x)

iGR.

(3)

177 The existence and uniqueness of solutions of this equation is studied in [3, 4] (also see [11, 13]). Note that the maximum principle holds for this equation (cf. [15]). We will give the definition of the traveling front of (1). As an example, we take the line y = xta,n9 with v = t(—sm6,cos6) where 0 < 9 < ir/2. It is easily seen that this interface moves not only with speed k in the direction i/, but also with speed k/cos9 in the direction '(0,1). So, it is natural that the velocity of the traveling front should be specified. The interface (T(t),v) is called a traveling front with velocity v, if r(i) = 17(0) + vt. The above example is regarded as the traveling front with velocity kv or velocity '(0, fc/cos#„). Deckelnick et al in [4] proved the existence of the traveling curved front and studied the stability of the front under some restricted assumptions for «o- The author and Taniguchi relaxed the assumption for the initial data and classified all the traveling fronts in [13, 14]. They proved the following (see [13, Proposition 1.1 and Theorem 1.2]). Theorem 1 Any traveling front of (1) with velocity '(0, c) is one of the three, after appropriate translations, (i) lines y = xta,u9t, and y = —xtan6» (ii) a traveling curved front Tc(t) which possesses two asymptotes y = irrtanft,, (iii) stationary circles with radius l/\k\ only in the case c = 0, where 9„ = arctan(\/c 2 — k2/k). Moreover the explicit form of the traveling curved front Tc(t) with speed c(> k) is given in 1 +

x{9; c)

y(9;c)

cVc 1

k2

log

1/

c+ k 9 r tan -

\j c-

k

2

/c + fc 9 ! " 1 \j/ c-k r tan 2fccos9 — k

•cl0Eb^r

for9e (-0.A)The traveling curved front Tc(t) is "V-shaped", which connects two asymptotes. The existence of this traveling front is also reported in [1,4]. The proof of this theorem consists of two parts. First it was proved that the traveling front with non-zero speed should be an entire graph (see [13, Lemma 2.4]). Next all the traveling front of (2) are classified (see [13, Lemma 2.3] for the details). In the case (ii) where the initial interface possesses two asymptotes, we denote the exterior angle between two asymptotes by 9", which is equal to 7r — 29,, if their inclinations are ±tan#„. It is checked that the speed c of the traveling curved front Tc(t) is uniquely determined by the angle 9* (or 9„), i.e., c = k/ cos 9*. This traveling front can be observed in a liquid BZ reaction (see [12]). The asymptotic stability of the curved traveling front in (2) is discussed in [4, 14].

178 T h e o r e m 2 The traveling curved front is asymptotically stable, if the initial perturbation is restricted to BCl := {v e CX(R)

sup

(|n(o;)| + |vT(x)|) < oo, lim v(x) = 0}.

-0O<X<00

|l|->0O

Furthermore, it is shown in [14] that the interface which have two asymptotes with angle 9* converges to the corresponding traveling curved front as t tends to infinity, provided that 9* < IT. This means that Tc(t) is globally asymptotically stable. If we do not take BCg as the perturbation space, the situation changes. In the class BC1 :={ve

C^fR) |

sup

(\v{x)\ + \vx{x)\) < oo},

—oo<:r
the traveling curved front is not asymptotically stable. Namely, for any e > 0, there exists an interface T(i) of (1) such that

v(t)n (rc(t)- ( ° ) ) ^ 0

dist(r(o),rc(o))<4e, T{t)n (rc(t) + ( ° \\?9,

for all t > 0 (see [14, Theorem 4.1]). This interface oscillates at infinity around Tc(i). The very interesting result [16] by Yagisita should be remarked. He considered nearlyspherically expanding fronts of the Allen-Cahn equations. Though the front is known to converge to the sphere after the rescaling of the radius to unity, he proved that the difference between the front and the expanding sphere does not decay as t tends to infinity. Next we will explain the case where the angle between the two asymptotes is greater than n. Using the method of functional analysis, Deckelnick et al also proved in [4] that, for u0(x) = — |x|tan0*, u(x, t) - tQ ( -

+ 0,

(4)

as t tends to infinity, where \s\ < fcsinft,,

x/F Q(s) :=

\s\ tan#* -

COS0*

\s\ > ksin9t,

(5)

where 9, = (9* — 7r)/2. This result indicates that the top order of the solution is tQ{x/t). We can expect the difference between u and tQ(x/t) does not decay as in [16].

References [1] P . K. BRAZHNIK, Exact solutions for the kinematic model of autowaves in twodimensional excitable media, Physica D, 94, (1996) 205-220. [2] Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom. 33, (1991), 749-786.

179 [3] K.-S. Chou and Y.-C. Kwong, On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth, Calc. Var. Partial Differential Equations 12-3, (2001), 281-315. [4] K. Deckelnick, C. M. Elliott, and G. Richardson, Long time asymptotics for forced curvature flow with applications to the motion of a superconducting vortex, Nonlinearity 10 (1997), 655-678. [5] K. Ecker, and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 130-3 (1989), 453-471. [6] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces (CBMS-NSF Reg. Conf. Ser. Appl. Math. 53), SIAM (1988). [7] M. Gage, and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom. 23 (1986), 69-96. [8] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Diff. Geom. 26 (1987), 285-314. [9] M. A. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J. 58 (1989), 555-558. [10] N. Ishimura, Curvature evolution of plane curves with prescribed opening angle, Bull. Austral. Math. Soc. 52-2 (1995), 287-296. [11] O.A., Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs 24, Providence RI, (1968), American Mathematical Society. [12] V. Perez-Munuzuri, M. Gomez-Gesteira, A. P. Munuzuri, V. A. Davydov, and V. Perez-Villar, V-shaped stable nonspiral patterns, Physical Review E 51-2 (1995), 845-847. [13] H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, in " Free boundary problems: theory and applications, i" GAKUTO Internat. Ser. Math. Sci. Appl. 13 (2000), 206-221. [14] H. Ninomiya and M. Taniguchi, Stability of traveling curved fronts in a curvature flow with driving force, to appear in Methods and Application of Analysis. [15] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, (1984) Springer-Verlag. [16] H. Yagisita, Nearly spherically symmetric expanding fronts in a bistable reactiondiffusion equation, J. Dynam. Differential Equations, 13-2 (2001), 323-353.

Existence of non-steady flows of an incompressible, viscous drop of fluid in a frame rotating with finite angular velocity M. Padula and V.A. Solonnikov Department of Mathematics, University of Ferrara, Via Machiavelli 35, 44100, Ferrara, Italy Emails : [email protected] ; [email protected] Abstract We give an existence theorem of global non-steady solutions of the Navier-Stokes equations governing flows of a incompressible viscous fluid totally bounded by free surface. Specifically, we prove that, in the presence of surface tension, if a rigid rotation admits a stable configuration in the sense of capillarity theory, then to any suitably small initial perturbation to the shape and to the velocity field there exists a unique global non-steady solution which eventually goes to the stable rigid rotation as time goes to infinity, with exponential decay rate. Our proof is based on a existence theorem of local regular solutions proved by [13], and on some a priori estimates on the solution, uniform in time for weak norms of the solution in the wake of [7]. The result does not assume smallness on angular velocity of the drop, only smallness on initial data!

1

Introduction

Plateau's work inaugurated in 1863 t h e study of shape and stability of a drop driven by rotation of surrounding liquid. Plateau's drop was pierced by a shaft and immersed in a tank of liquid having almost the same density of the drop. T h e shaft was mounted vertically and turning it, Plateau could bring the drop into rotation. T h e drop is driven to rotate at an imposed angular velocity UJ. W h e n increasing the rate of this rotation, the drop progresses through a sequence of shapes, axisymmetric at first, then ellipsoidal and two lobed, and then a most remarkable toroidal ring t h a t remains intact for a short time. Shapes and stability of isolated, self-gravitating masses rotating freely in space have been much analyzed, e.g. by Maclaurin 1742 (perfect spheroids), Jacobi (1834) (ellipsoids), and t h e equilibrium shapes are analogous to those found by Plateau.

1.1

Results on Equilibrium Shapes

In theory, t h e shape r t of the drop in rigid rotation would have been strictly set by the balance of the capillary force t h a t comes from surface tension of t h e curved drop surface,

180

181 and the gyrostatic pressure force which is exerted on the surface by the rotating liquid owing to ordinary centrifugal force, under the constraint to have constant given volume \ilt\. Mathematically, this balance is expressed, in spherical coordinates, by the functional

G(R,t) = a\rt\-^j

R5{y/ t)

^' ^n2edS1-Po\Qt\,

(1)

where a represents the surface tension, p0 is a Lagrange multiplier and Si is the unit surface. For steady flows : G(R, t) = G(R). The basic flow is a steady flow characterized by a shape f2(,, a gyrostatic pressure pt,, and a uniform rigid rotation with angular velocity cj = u;e3, which is considered as a parameter. The fluid is supposed to be in equilibrium in the frame rotating with the drop, therefore the shapes are also called equilibrium shapes. Several equilibrium shapes of rotating liquid drop are theoretically possible. The equilibrium shape physically realizable is that which renders minimum the potential G. In capillarity theory, the range of "stability" of the different shapes is investigated when LU is varying, and it is shown that rigid rotational stable shapes exist, in correspondence of a finite value of angular speed w, may be they are not unique ! Some interesting analysis can be found, e.g., in [3], [1], [2]. The aim of this paper is to prove that the above said configurations, which are stable in the class of small fluctuations in shape, are also asymptotically stable in the class of non-steady flows with respect to small initial fluctuations in shape and in velocity inside the drop. We quote the papers [4], [6], [7], and [8]- [13] concerning the existence problem for the full free boundary problem. In particular in [7], it is proved, for the first time, existence of an unsteady regular motion of rotating liquid drop, exponentially decaying to a rigid steady configuration corresponding to a given angular momentum. In that paper, it is assumed smallness on the size of angular momentum. Here we intend to remove such a restriction, at least for star-shaped domains. Precisely we prove the following. Theorem 1.1 Let T0 = 3fio be given by equation R = Ro (y) with R(y,0) = Ro(y), y e Slt Ro e C ^ S i ) , a e (0,1), and letv(x,0) = v0{x), with v 0 £ C 2 + a ( n 0 ) . Moreover, let Rf, (w) satisfy the stability hypothesis of capillarity theory, that is Ri, is an isolated local minimum for G (R) at fixed u>, and set v<, = UJ x x. Assume additionally that VQ satisfies the compatibility condition

/

x x v0dx = na

x x vbdx, J nb

and let |vo - v t l c a + a ^ ) + |.Ro - Rb\c*+°(Si) - e>

(2)

with e sufficiently small. Then the fluid drop has a unique solution (R,v,p) defined in an infinite time interval t > 0 and possessing the following properties : R (.,t) € C3+a (Si), v(.,t) € C 2 + a (ft t ), p(.,t) 6 C 1 + Q (fi ( ), Vt > 0, where Tt = dQt is given by equation

182 .R = R(y,t),

y € Si. The solution satisfies the inequality

sup | w t ( . , t ) | c , „ ( n ) + sup | w ( . , t ) | c 2 + „K( n ) + sup te{0,T) te(0,T) ' te(o,T) +

™P ^ (••*) "

ftlc+«(n,)

^

c

\p{.,t)-pb\cl+a(n) l "

(l w o| C 2 + « ( n o ) + |#o - ifc| c »+.(s 1 )) c ~"»

V T G

(0. ° ° l . (3)

where w = v — V&, iwrfi some constants 6 > 0 and c independant

ofT.

We give the proof for star-like configurations. As it will be clear from the proof, our result heavily relies on the assumption that the basic configuration Rb renders minimum the energy functional of capillarity theory. Therefore, in principle, our proof can be enlarged to more general equilibrium configurations, like the toroidal ones.

2

Position of t h e problem

Let a drop of incompressible viscous fluid occupy a star-shaped domain Q. = {y G R 3 : \y\ < R(y/\y\)}, r = 90. We wish to determine the configuration fit C R 3 , of prescribed volume 47r/3, admissible with a steady rigid rotation of angular velocity u;e3.

2.1

Equilibrium results

We remind that the energy functional, given by (1), must assume a minimum in Rb. Denote by Vgj the derivatives along the unit sphere S\, in spherical coordinates, and set g = R2 + \VSlR\ , OR 1 OR Vsj = e 9 — ++ ee,f ,

~de

' '^e'^^

n = (-VSlR,R)

l

y/R* + \VSlR\ 2

Also, it holds : frdS = Js Ry/gdS\. The condition that Rb is a minimum for G implies, in particular, that it must be a stationary point for G, that is, for R = Rb + tp, the first derivative of G(R) with respect to t at t = 0 must vanish, i.e. : 0 = 5G(Rb) = ^-\ =
pR*sin26dSi\

+ -p0

k

=°

Sl

VR-VpildStl h ) =° [ J Si

pR2dS1\

l4

(4)

. =°

Integrating by parts, and owing the arbitrariness of p, we deduce the classical eigenvalue problem in unknown Rb and po : H{Rb)+pb(Rb)+p0

= 0,

(5)

where the double mean curvature of the surface H(R) has the following expression :

H

1„

(VSlR\

2

™=a** (-%-)-v?

(6)

183

Also, weset: K{R) = H(R)+pb{R).

Thus the first variation of: SG(R+tp)\t=0

=

^

is given by : 6G(R) = JSipR2(K.(R) + p0)dSi, and the hypothesis that G assumes a minimum at Rb implies that SG(Rb) must vanish, i.e. :

SG{Rb) = J PRl{K{Rb) + Po)dSl = 0, Vp. Si

Hence we find, at the equilibrium position RflC(Rb) = R2b[H{Rb) +Pb(Rb)] =

-p0Rl

(7)

Since, in the sequel, we need a precise computation of the second variation of G, here we make some exact computations. Set p = R — Rb, we have

f

G(R) - G(Rb)

dG\ dt \Rb+tp

dt.

(8)

J o

Also, we set

d2G

SG(Rb) = ^§\ , 62G(Rb) = ds, 22 |\R . dt \R b

(9)

b

(8) implies 1 G(R)-G(Rb) = Jf fdG\ ( dt \R +tp _dG\ dt \R b

I I I. r (£2. J o

=

dt + 5G{Rb) b

ds dt + 6G{Rb) - p J 2 (d G\ d G\ ]dg 2

Rb+sp

ds2 \Rb

dt + -82G{Rb) + SG(Rb)

2 b) + nl, SG{Rb) + -82G{R

J o V ds

(10) where nl denote nonlinear terms. We now compute the second variation of G at Rb. Set F = R^/R? + \VSlR\2, V e = de, V

+2

/ F)2 F

/ ORdVeR

Sl

{avj?^2

pVeP + 2

,),

Vp,

f)2F

+

d dVeRpVvP

B2 R.

WJT2^2 +

^dp2

+

2

e^RV°pv*p

~ 2Lj2R3b Sln2 9p2 ~ 2poRbp)

dSl

that is a norm for p in W^iSi) if Rb is a minimum for G. In particular, from (10), at the minimum configuration Rb, we also have [6G(R) - SG(Rb)\ = 62G(Rb) + / f ^ ? | - &2G(Rb[ ds. J o _| as \ Rb+sp

184

This yields SG(R)-SG(Rb)

=

/ Q

=

2.2

2

P[R

K.(R) - R2bIC(Rb)]dS1 = S2G{Rb) fill v /

+ nonlinear terms IIPIIM'1(S ) + nonlinear terms.

Equation of motion

Let a drop of incompressible viscous fluid occupy a star-shaped domain Qt = {y € R 3 : \y\ < R(y/\y\)}- We wish to determine the domain Qt C R 3 , the velocity vector field (y,t) and the scalar pressure field p(y,t), satisfying the problem ' v t + (v • V)v - i/V2v + Vp = 0 V•v = 0 Tn(v,p)n =

0, y € fit, t > 0,

w2 aH{R)n + — R2 sin2 6 - pe,

nrRt(y/\y\,t) = vn, R(y/\y\,0) = R0(y/\y\), v|t=o = v 0 (y)

^

y G Q0,

where T(v,p) = —pI+2vS(v), v is the constant kinematic viscosity (the density has been choosen equals one), S is the symmetric part of the gradient of velocity. Furthermore, Yt is the boundary of Qt, n is the exterior normal to Tt, u: is the angular velocity of the tank, and pe is the constant external pressure. Moreover, since the pressure is defined up to a constant we can fix uniquely the pressure by setting pe = —po in (12)3.

2.3

Conservation laws

Let us observe that problem (12) admits the conservation laws of mass, momentum, angular momentum, and energy expressed by

ij^vMv

= 0,

(13)

AC

jtJ

[ y x v ( s , f s = 0,

d

-{T[y](t) + G(R)} + V(t) = 0, where

G(R) = a|rt| - £ j

s

«

TM(t) = f ^ d l V(t) = 2v I 8(v(y,t))dy. J fit

M

sin' 6dSl -

PM

185 Through a suitable choice of measure unity, we fix the origin 0 of the frame at the center of mass of tob, and assume that the center of mass of to0 is still in 0. Furthermore, we fix the initial data such that

lai = l^o| = y , l /

ydy = / vdy = 0, «« JJk v{y,t)dy = / v0(y)dy = 0,

[y x v{y,t)]dy

=

J fit

/

(14)

[yxv0(y)]dy=

J f!0

[yxvb{y)]dy. J f!b

All conservation laws are expressed in terms of the flow v, p, not in terms of the perturbation w = v — V(„ r = p — pb ! All conservation laws are independent of the choice of reference frame where problem is written ! In the frame rotating with angular velocity w, call it Rb, the basic flow reduces to the rest for the velocity field with a mere perturbation of the spherical surface for the shape of the domain !

2.4

Basic flow

The system (12) admits as steady solution the rigid rotationVb(y) = w x y, Pb(y) = \ W.'|2 where \y'\ denotes the distance of y with respect to the 3/3-axis, in the domain Qb = {y : \y\ < Rb}, with Rb solution of the Young-Laplace equation of interface configuration (5).

3

T h e equations of p e r t u r b a t i o n

We now look for the best reference frame where to write the perturbed problem.

3.1

Choice of t h e reference frame

Consider the noninertial frame 1Z rotating with angular velocity o;e3, and set v(j/,t) = w(x, t) + vb(x). In 7?., the fluid has velocity w, and problem (12) is written as follows:

w t + (w • V)w - i/V2w + Vp = V•w Tn(w,p)n nrRt{x/\x\,t) R{x/\x\,0) w| t = 0

= = = = =

V ( — ~ - j - 2toe3 x w, 0 {aH{R)+pe)n, w n Ro{x/\x\), w0(a;) = v 0 (x) - V|,(x)

x € tot, t > 0, (15) leT,, x € too,

where \x'\ denotes the distance of x from the a;3-axis, and the terms — 2ue3 x w, V ( " '% ) represent the Coriolis and the centrifugal force, respectively.

186

3.2

Initial D a t a

Notice t h a t in 71 t h e basic velocity is zero, therefore w is the perturbation t h a t will be assumed initially small. Concerning the shape Qt it will be the perturbation (a slight deformation) of the basic stable shape Q,b (ui), and it will be supposed initially close to fit, say R0(x/\x\) — Rb is small. Concerning t h e initial data, due to t h e conservation laws (13) they must statisfy some conditions. Precisely, we remind t h a t in "R, the center of mass, t h e volume, angular momentum are constants of the motion, see (14). Therefore, we choose initial d a t a as small perturbations of t h e basic flow, paying attention to choose f2p such t h a t it has the same center of mass O (origin of the reference frame) as flb, and same volume f2o = 47r/3. Due to t h e invariance of relations (14) 1 , we have the following relations 47T

L

xdx

/

x d x = 0, O.0

vt(x,t)dx

/

= =

fit

/ [xx w(x,t)]dx J «t

/

w0(x)dx

= 0,

fi§\

J «o — / [xxvb(x)}dx J sib

— / J nt

=

vb(x]r3drdSi.

/ [e r x J SiJ R

[xxvb(x)]dx

In particular from (16)4 if follows t h a t J"n [x x w(a;,t)]d3; < c \\R — RbW^ts y

4

T h e equation of energy

Notice t h a t equations (15) are not yet the perturbation equations, because b o t h p and R denote the pressure and t h e radius of the flow in the rotating frame. However, they are enough to write energy identity. Actually, multiplying (15)i by and integrating over fi( by transport theorem we deduce /

\w\2dx + v [

|S(w)|2dr (17)

= [

w

n{oH(R)

+ Pe)dS

+ f

w •V ( ^ L \

dx.

It is well known t h a t

/ wn(aH{R)+pe)dS=-4-

JTt 1

f dS.

dt JTt

Denoting by at, eit i = 1,2,3 two basis respectively of R and 7£, for any vector x we have Y^i=\ £ , a « : Yli=i ZtkVkZl'ei = S i = i Vk^k- Hence, by the orthogonality of Z we have Jn x x wdx = Ja y x vfdy.

187 Hence from the identity .,2

(17) yields

d_ [r ^j-dx iwi2.

dt Jat

d r ._ w 2 /• R5

+ 1 f dS-1^f ^-sin2edSl at JTt I JSl 5

*

+ v

f | S (w)| 2 dx = 0. Jnt

(18)

We obtain easily a norm for the perturbation p = R-Rb, to the surface Rb = Rb(x/\x\) if we assume that G(R) assumes a local isolated minimum in Rb. Indeed,, for regular function R(x/\x\,t), in a small neighborhood of Rb, it holds G(R)-G(Rb)

= \\R-Rb\\2wUSi).

(19)

Finally, since G(Rb) is constant in time we deduce the following energy equation for the perturbation |

{llwllL(ft) + G(R) - G(Rb)} + H|S(w)|| 2 2 ( n t ) = 0.

(20)

From Korn's inequality, we know that the gradient of w is increased by S(w) plus the square of momentum and angular momentum of w. The momentum is zero, cf. (16)3, while the angular momentum is increased by R — Rb, cf. (16)4. This allows us to state c

ll Vw lli2(nt) ^ l|S(w)|| 2 2(nt) + H-R- RbW^Si)-

(21) Till now we have obtained a classical energy estimate for the perturbation, with a dissipative term only for the deformation of the velocity field (not even for the velocity !). In the next section we shall provide a dissipative term for p = R — Rb and for w.

5

Auxiliary equation

In this section we wish to find a "dissipative" term for R — Rb, to this end we use the method introduced in [5] and already applied to free boundary problems in [6], [7]. We notice that problem (15) can be rewritten as follows w t + (w • V)w - i/V2w + Vr = -2we3 x w, V•w = 0 Tn(w,r)n = [a{H(R) - H{Rb)) +Pb{R) - Pb(Rb)]n, nrRt(x/\x\,t) = w-n

R(y,o) = Mv) w| t = 0

= w 0 (i) —^—

(22) x(zTt,

yeslt

x S fi0,

where r = p-[

x e fit, t > 0,

= V ~ Pb-

188 We solve an auxiliary problem. Find a solenoidal vector function w in W^fij) satisfying the problem V-v(x,t)

llvllwjtn,) + l|0 t v|| i2( no

= 0

x e fit,

< c(||p|| W; i (rj) + ||w • n|| L 2 ( r t ) ).

The compatibility condition is satisfied because the volume is constant. The construction of w can be done by reducing fit to a fixed domain, thus following the method outlined in [7]. Furthermore, we select v to satisfy the additional constraint

/ v-#ote=0,

V * = e* x x, i = 1,2,3. Jn This is possible. Indeed, we choose A = ]T}i=i c'(t)a^(x), with a^ G Cg°(Q.t), and since fit C -02, where Bi is a ball of radius 2, we can also assume ||ai||w2(nt) < 1, J n s^dx = 1. Then, if V satisfies (23), the vector field V = V + rotA, A ' S C£°(fit),'satisfies : J n V • tydx = 0, provided that — / V • * d x = / rotA • *cte = / rofr& • Adx. Jilt

JQt

Jilt

In particular, we have / rot(ei x x) • Adx = —2 / At = —2at = / e; • (x x V')da;, Jnt Jat Jnt which tells us that the coefficients a^t) are bounded by the L2-norm of V , and finally, V is bounded by \& and w in W^fi). Let us multiply (22)i by V, orthogonal to rigid motions, and integrate over fit. Using (23)2, (22)2 we get (w«, V) + ((w • V)w, V) + i/(S(w), VV)

L

V • n(a{H(R) - H(Rb))+Pb(R)

-Pb{Rb))dS

= -2(w x w, V),

(24)

rt where (.,.) denotes the scalar product in L2(fij). We notice that, by transport theorem, it yields (w t , V) + ((w • V ) w , V ) 4 ( w , V) - (V t ,w) - ((w • V ) V , w ) . at Thus, we rewrite (24) in the form ±(w,V)-J

(25)

V-n(a(H(R)-H(Rb))+Pb(R)-Pb(Rb))dS

= -i/(S(w), VV) + ((w • V)v, w) + (v4, w) - 2(w x w, V). Our aim is to increase the right hand side of (26) with terms of the kind either : ll s ( w )lli 2 («t) + H s ( w )lli2(r!oll-R _ ^HwjHsi)) o r nonlinear terms. Now we notice that, by

189

(23)3, embeddings theorems and Poincare's inequality, the first two terms at the right hand side are increased by ||S(w)||| 2 ( n 0 + ||S(w)|| i a ( n t )P - Rb\\w}(Sl), plus nonlinear terms as we wish. The term F(w) = (vj,w) - 2(w x w , V ) , presents some problems since it is increased only by the quadratic term ||Vw||i2(nt)||.R - Rb\\wi(Sly To overcome this difficulty, we remind that w is orthogonal to rigid motions and that the vector field w can be always decomposed into a sum of two vectors w' + h with w' such that ll Vw 'lli 2 (nt) — llS(w)ll!2(fit)> a n d h a r i 8 i d motion. The linearity of F(yv) implies that f ( w ' ) < ||S(w)||| 2 ( n t ) + ||S(w)|| I , 2(nt) ||fl-fl t || w , 2 i (Sl) as we wish. Therefore it remains the term F(h), with h = (f>xx. This will be handled employing the orthogonahty of w to rigid motions. Let R1 be a reference frame rotating with angular velocity —2u> in 1Z. Let us call e; a basis in TZ, and a^ a basis in R'. Poisson's formula implies : ^a^ = —2w x aj. Therefore, since the same vector w has different components V1, W', in the two frames, related by the identity V\t)ei = Wj{t)a.j(t) = V(t), we have

£v-™.-|c*<.M0>-^W-*x<*<.*,M.-| where §^V, |^V = |^V mean the time derivatives of the same vector w in the different frames 1Z. and R', and where ~V = |£V + 2w x V is the derivative in the frame rotating with angular speed 2u>. In this way, we write F(h) = ^ V + 2 ( w x V , h ) = ( ^ V , * x x Since we are in Eulerian coordinates, it holds dr ~

/ (it

f

-

dr

dT f

—V • x xdx at J nt dt dt J n t - [ (V • 4> x x ) w ndS = / V • [("V • <j> x x) w]dx.

ir,

Jn,

^

'

Therefore, F(h) is a nonlinear term. The latter term at left hand side in (24) requires some calculation because we don't want require smallness on the derivatives of Rb. Let us observe that

V • n{K{R) - K(Rb))dS

=

h - / (IC(R) V ' 03 y i s

= - / J Si

^~^(IC(R)

KiR^R^gdS, (27)

- ICiR^dS^

*

Our aim is to prove that (27) is equivalent to the norm of R — Rb in W^iSi). By hypothesis, Rb is a minimum configuration for G(R). Noting that R3 — Rb = (R — Rb){R2 + R2b + RRb), since we can choose p = R - Rb, by (11), (7), we rewrite (27) as follows

190 R3

/

_ P3 r =—*(/C(/Z) - IC{Rb))dSl = /

S l

2

22 T

1 ^ p (fl

fRtRRb

R* + Rl + RRb ftp + Rl3iP +

I. a

2

R2

_L

' , \ , ,-,21--/ r>\

-It-Hf,

+

p{R2K.{R) o2l-/

RltCiR^dS, nn j n

~ l) (#£(*) - RtlC(Rb))dS_ _

(28)

RRb

{Rl-tf)K{Rb)dS,

i ^ „ , f„™„,„.

= l|p||^ 2 i (5l ) + non linear terms +p0 //

2 n.\ p„2fn (.R +, .R 6)

R2

+

R2

+

RR

b ^—

t>1

R\dSi

= P\\pW^i,s •. + non linear terms, where the nonlinear term in p is at least of grade three and can be rendered small for small initial data as we shall see from estimates in the existence theorem. Gathering all informations we deduce - | ( w , V) +/?|M|^ 1 ( S l ) < c (||HI^(ft)IIS(w)|U 2(ni) + ||S(w)||i, ( n t ) ) , (29) where c is also function of p and S(w), because of nonlinear terms. Therefore, adding (29) multiplied by a suitable small constant e to (18) we obtain the inequality

|

{ll w IIL(n t) + G(R) - G(Rb) - e(w, V ) } + H|S(w)||| 2 ( n i ) + eP\\p\\w,(Sl)

(30)

< ce||p||W2i(Sl)l|S(w)|U2(f2t) +ce||S(w)||| 2 ( n t ) , and, for e small enough we get

Jt

£ + H|S(w)||! 2 ( n t ) + eP\\p\\wHSl) < 0,

(31)

with £ = 2 (l|w||i 2(ft[) + G(R) - G(Rb) - e(w, V)) = ||w||£ 2(nt) + | 11^(51)Applying Korn's and Poincare's inequahties to the term H|S(w)||w n > + ^IMIw'fSi)' we finally deduce ^-S + bS < 0, at and by Gronwall's lemma we obtain the desired decay estimate S(t) < e-bt£(0).

(32)

(33)

We remark that (33) has been obtained under regularity assumption on the basic flow, but not under smallness hypotheses on ui !

191

6

Reduction t o initial-boundary value problem in a fixed domain

We intend to prove that if u satisfies the hypothesis of stability required by capillarity theory, then for small perturbations of the rest state there exists a regular global solution to (12) that decays exponentially to zero as time goes at infinity. To this end, in our proof will play an essential role only smallness of perturbation, in this statement the gradient of the basic shape is not infinitesimal ! For this reason the perturbation equation will be linearized around basic flow carefully, say we write the equations in reference frame rotating with angular velocity u> = oie3. Of course, in Tt the velocity w is directly the perturbation to the rest, only we must consider a suitable linearization of Tt around i \ . We remind that the surface r ( is given by the equation |x| = R(x/\x\,t) and the origin O coincides with the center of mass of the liquid, i.e., Jn x^ds = 0, k = 1,2,3. Assume that Ot contains the ball BR^O) = {|x| < R{\ and is contained in R^R^O) = {\x\ < CQR\}, define a smooth function x(\x\) x such that x{\ \) = 0, if |x| < -Ri/2, xd^l) = 1) if W > -Ri. a n < i consider the coordinate transformation

—C 1 • * « © $ - . ) ) - • < • • « .

(34)

or, in spherical coordinates

This transformation is invertible, if R — Rb is not too large, and it maps Qb onto fi( and Tt onto Vb leaving BRl/2 invariant, and in some neighbourhood of I\ it holds : x = j^y. Let J = (I 32 -)

be the Jacobian matrix of the transformation y = y(x, t) inverse to

(34) and J = det J. It is clear that J is an inverse matrix to J - 1 = I -£?•) V

'

whose

V°W/i,j=l,2,3

elements are easily calculated by differentiation of (34). The vector field u = J w / J , with w(j/,t) = -w(x(y,t),t), is solenoidal2 and it holds W(x(y,t),t)

and

=

J(y,t)J-1(y,t)u(y,t)=J(y,t)u(y,t),

192 Hence :

wt = (ju)t - x ( M ) ^ J f f ( y ' JTvH)(J(y,t)u) and the equations (22) take the form J u t + J ( u - X{\y\) o/./L',V(y • J r V,)(J(2/,i)u)

Wlwl)

+ J ( u • V,,) J u - v{3TVy • J T Vj,)Ju + J T V,,r = - w e 3 x j u , V - u ( y , t ) = 0, where r = p — pb, p = R — Rb. The initial conditions are u| (= o = u0{y) = J 0 w 0 ,

p| t = 0 = po(y).

Further, we separate the tangential and the normal components in (22)3 which gives IlbIlS{u, p)n = 0,

-r + vn-S{u,p)n

= K,{R)-]C(Rb),

yeTb,

(35)

where nf = f — n(n • f) is a projection of the vector field f(x), x € F t , onto the tangent plane to r t , (analogously for Hb) and <S(u, p) = J T V ® J u + (J T V g> J u ) T , is a transformed matrix S(w) = V w + (V ® w ) r . We also transform the expression : K,{R)-K,{Rb) = a{H{R)-H(Rb))+pb{R)-pb{Rb), using the formula : U{R)-H{Rb) = n • A(i)x — ri;, • A 6 y, where A(t) and Ab are Laplace-Beltrami operators on r ( and Tb, respectively. We have K.(R) - IC(Rb) = anb • Ab (p(y/\y\,t)er) + T\ (p) + Ti (p), where : Ji(p) F2{p)

= =

a((n-nb)-A{t)x + nb-(A(t)-Ab)y)+pb{R)-pb(Rb), anb-{A(t)-Ab)(erp(y/\y\,t)).

We note in particular that the expression rif,- (A:(£) — A 2 (t)) y

depends only on the

first derivatives of Rj and R2, because nb-dy/d^j = 0 (£1, £2 are arbitrary local coordinates on Td)- Therefore, Ti (p), i = 1,2 denote the sums of lower order and nonlinear terms with respect to p, respectively. Let us consider kinematic boundary conditions. The normal n (x(y, t)) to Tt and the normal n^ to ilb are related to each other by

»(*(*')) = | 5

(36)

and in the neighbourhood of Tb the elements of J are given by Rbf-i , td (Rb Jij = -^6} + y'-. 6 +y -R > dy-\~R. so, J = (Rb/R)

. Therefore condition (22)4 : (n • eT)Rt(y,t) n

„

Rl(y/\y\)

„

n

= w • n, is transformed into ,,,,*

193 Our free boundary problem reduces to the problem of finding u(y,t), r(y,t), y e flb, and p{y,t), y e Si, satisfying the relations

J U t + J4U -

. 3TVy)(3(y,t)u)

x{\y\)PMMA{y T

-v(J Vy

T

• 3 Vy)Ju

+ J(U • V , ) J u

T

+ J Vj,r = -2we 3 x Ju, V-u(y,t) = 0 yeflb,t>0, n 6 nS(u, p)n = 0, -r + vnS{u, p)n = K.{R) - K.{Rb), R2 (nb-eT)pt{y,i) = j±u-nb(y,t) y e Si,

u(y, 0) = 3(y, 0)v0(x(y, 0)) = u0(y) y € Qb, p{y, 0) = Ro(y) - Rb{y) = p0{y) y £ Fb, (38) or, which is the same u ( — i/V2u + Vr V u IL,S(u)n[, -r + vnb • S(u)n(, -cmb- Ab{p{y/\y\,t)er) nb • erpt u(2/,0) p(y,o)

= f (u, r, p) — 2o;e3 x u, = 0, = II 6 b(u,p), = = = =

d{u,p) + F{p), u • nb + h(u, p) u 0 (y) po(y)

(39)

2/ e r 6 , t > o, y e Qb, y e Si,

where f(u,g,p)

(I - J)ut + xM^/V'fhy

Mv/\y\)

• 3T^y)(3(y,t)u) - J t u - J(u • V„)Ju

= -2we 3 x (J - l)u + v ( ( J r V „ • J r V „ ) J u - V 2 )uJ + (I - J T )V. b(u, p) = rif,S(u)ni, - ILS(u, p)n, d(u,p) = v(nb-S(u)nb-n-S(u,p)n), T(p) = T1{p)+^2(p), (R? h(u,p) = u - n J - ^ - l (40)

7

Existence theorem

In this section we prove Theorem 1.1. For this we need to consider the linear problem

194 v t - ! / V 2 v + Vp V-v n»S(v)ni -p + vnb • S(v)n 6 -crnb- Ab{p{y/\y\,t)eT) nb • erpt v(y,0) p(y,o)

= f(y,t), = 0, = b(y,t), (41)

= d(y,t), = v • nb + h(y, t) y£Tb,t£ = v0(y) yeUb, = Po(y) yeSu

with some given i{y,t), h(y,t), d(y,t),

(0, T ) ,

h(y,t).

Theorem 7.1 Let f(y,t), d(y,t), h(y,t) be continuous in y, t, f(.,i) 6 Ca(flb), d e C1+a{rd), h e C2+a{Td), Vi e (0,T), T < oo, b e c,1+a-<1+0!>/2(rf> x (0,T)), v 0 e C 2 + a (0 6 ), po e C 3 + a (5i), and let the conditions V-v0 = 0,

n 5 S(vo)n 6 | S l =b(2/,0),

b • n b = 0,

(42)

be satisfied. Then problem (41) has a unique solution p(.,t) e C 3 + a (5i), v(.,£) € C2+a{ttb), p(-,i) £ C1+a(ttb) withvt(;t) e Ca{Qb) definedfort e (0,T), and this solution satisfies the inequality sup (\vt(.,t)\c°(sib) + |v(.,t)|cs+-(n,,) + b(-,*)|cJ+-(nt) + |p(-,<)|c^( S l ) < c(T) (|v0|c2+»(nt) + |po|c3+«(Si) + |b(-,*)lc1+».(1+»>/2(rbx(o,r))) +c(T)sup (|f(.,t)|c. ( n,) + \d(,t)y+°(rb) + |A(-,*)lc+-(rJ where c(T) is a non-decreasing function

(43)

ofT.

Proof. Due to (41)5, the boundary condition (41)4 can be written in the form -p + vnb • S(v)nb - anb • / A 6 e r Jo

v • nb + h(y,

T)

dr = anb • Aberp0(y) + d(y,t),

nb-eT

-p + vnb • S(w)nb - anb • / Abw(y,r)dr

+ / Zx(w)dr

J 0

J 0 A Ab-

= a m • A 6 e r p 0 (y) + d(y, t) + anb • 0

r n

'

dr.

6 ' er

where Ji(v) = nf, • Aber

n 6 • Abv, a.b - e r is the first order differential operator applied to v. The solvability of the initial-boundary value problem

195 v t - i / V 2 v + Vp = f(y,i) V - v = 0, n 6 S(v)n t = b{y,t), - p + vnb • S(v)n 6 -o-ni • / Abv{y,T)dr+ Jo

/ h(v (y,r))dT Jo

=

yeSlb,t€(0,T),

b(y,t)

+ [ B{y,T)dT Jo v(y,0) = v„(y)

yerb,te{0,T), ye Slb, (44)

and the estimate sup {\vrt(;t)\c°(n,,) + \w(;t)\c*+°(nb) + |p(-,t)|c+»(n6)) t
< c(T) (|w0|c2+»(n,,) + |polc3+«(S!) + |ty^*)lci+°.(i+°)/2(rtx(o,r))) (45) +c(T) sup (|f(-,0lo»(n,) + \b(;t)\c^(rb) + |S(-,t)|c-(r t ) ) , t
< c(T) (|w 0 | C 2+«(n 6 ) + |po|c3+=(Si) + |b(-i*)lc1+«.(1+«)/a(rbx(o,T))) +c(T) sup (|f(-,i)| c «(^) + \d{;t)\c^{Tb) + \h(;t)\C2+a(Tb)). Since p satisfies the elliptic equation (41)4 on Si, Schauder's estimate yields the same inequality for |p(-,t)|C3+«(Sl). The theorem is proved. • Now, we turn our attention to the nonlinear problem (39). We start with the estimation of the nonlinear functions (40) (major part of these estimates is obtained in ([7]) Let j(*' = (jjj\y,t)) be the Jacobian matrices of the transformations (34) with R = Rk(y,t),

k = 1,2. The differences J>}' — j\f satisfy the inequalities

W - 7? (1) C H . ( „ b ) I 7(2)

dt(

'3

*J ' c"=+»(nb)

< c\Pl - P2|o*«+-(tu)

k

<

A; = 0 , 1 .

c(\p2t - Pw|c*+>+«(ni) + IPI - P2|c*+i+«(n6))

= 0,1,2,

In particular : \Jij> — ^ij|c*+°(fib)

at^lc^tn,,)

c

^

-

c

lp|c*+»+»(ni,) c c+1+

(^'l "

'*(«i.)

A; = 0 , 1 , 2 , +

p c !+1+<

ll"

"(^))

k = 0 1

>-

The same inequahties hold for the elements of the matrices J(*', ( J ^ ) _ 1 , as well as for the functions j(*> = det J(*> and vector fields n
196 (but of course these vectors are evaluated on Tt). T h e coefficients of t h e difference of the Laplace-Beltrami operators A ^ ' ( t ) — A ( 2 ) (t) on the surfaces |x| = Rk(x/\x\,t) can be estimated in a similar way (see [12]). Omitting lengthy but elementary technical details we present the estimates we are going to use below. L e m m a 7.2 / / Y((u,r,p) ssup|us(-,s)|c«(n(,)+sup|u(-,s)|c.2+»(nl) s
k = 1,2,

then s u p | f ( u 2 , g 2 , P 2 ) - f ( u i , ? i > P i ) | c - ( n b ) + | b ( u 2 , p 2 ) -b(u 1 ,p 1 )| c .i + <,,(i+.)/2 ( r b X ( 0 i t ) ) + s u p | d ( u 2 , p 2 ) - d ( u i , p i ) | c i + « ( r b ) + s u p | / i ( u 2 , p 2 ) - ^(ui,Pi)lc 2 +»(r b ) +

SU

d

P ^ ( - ^ ( P z ) - -^i(Pi))lci+»(r t ) + sup \T2(p2) - ^2(pi)|c2+«(r b ) s
(46)

< criYt(u2 - U ! , r 2 - r1}p2 - pi) + c|p 2( - Pit|ci+°(r b )In particular, ifYt(\i,r,p)
c i+°(r b )

P K u > P)|d+»(r b )

+ ssup|^ 2 (p)|c2+«(r b ) <*

(47)

< cr?y t (u,r,p) + c|p f ci+°(r b )-

-sup | ( v^ ( P )

at

!<*<">

SU

c«(r„)

These estimates and Theorem 7.1 enable us to prove t h a t t h e problem (39) is uniquely solvable in a finite time interval. We restrict ourselves to the case of small initial data. T h e o r e m 7.3 Assume that the datauo g C2+a(Q.b), p € C 3 + a ( 5 i ) satisfy the compatibility conditions V - u 0 = 0, II(,S(uo)n(,|r 6 = n 6 b(u 0 ,po) and that |u 0 |c 2 +»(n b ) + |Po|c3+'"(s1) < E < 1. Then, there exists T = T(e) such that the problem (39) has a unique solution {u{y,t),r{y.t),p{y,t)) such thatue C2+a{nb), ut e C{Slb), r e C1+a{Qb), p 6 C 3 + a ( S i ) for arbitrary t 6 ( 0 , T ) , and YT(u,r,p)

=

sup(|u ( (-,t)|c«(n b ) + |u(-,i)| C 2+c ( « b )

<

C T

+ |P(->*)|C3+"(.91))

( ) (|uo|c 2 +»(n b ) + |po|c3+«(Si)) •

Proof. We write the problem in the form

"

K-,*)l ci+»(n„)

197 u f — vV2u

+ Vr V u rii,S(u)n(, -r + vnb • S ( u ) n 6 -anbAb{p{y/\y\,t)eT)

n;, • e r p f u(?/,0) p(y,o)

= = =

f (u, r, p) — 2cje 3 x u, 0, Ili,b(u,p),

=

d{u,p) + JF^po)

r d_ Txd,T J odt

= = =

(48)

+

Ti{p),

u • ni, + h(u, p) u 0 (y) po{y)

y£Tb,t>0,

and apply the method of successive approximations. We p u t u ' ° ' ( y , t ) = u 0 , r ' 0 ' = 0, p (0) (z/,<) = Po(y), and define ( u ( m + % , i ) y m + 1 > ( i / , i ) , p ( m + 1 % , t ) ) , m = 0 , 1 , . . . as a solution to the linear problem (m+l)

u,

• vV2u(m+1)

+ Vr( m + 1 > v.u(m+i) n 6 s(u( m+1 ')n b _ r (m+l) + ^ . S(u< m + 1 ')n 6 -anh-kb{p^+lKy/\y\,t)eT)

=

f(u( m >, r<m>, p(m>) - 2we 3 x u,

=

0)

=

d(u<™\pM) + ^i(po)

= n 6 b(u( m ),p( m )), -/'J^i(p ( m ) )dT + ^2(p("

v.

Q ^(m+1)

n;, • e r p £

„(m+l)

„

— u v T ' • rij, „ ( - + ! % , 0) P(m+1)(y,0)

= =

h(u( m ',p( m > , > u0(y) po(y)

yeTb,t>0, y e fifc,

2/GSi, (49)

By virtue of (43), for arbitrary t G (0,T) : y ^ u W . r 1 ' , ^ 1 ' ) < ci (|u 0 | c »+«(n t ) + |Po|c3+«(Sl)) Let us estimate y ( ( u ( m + 1 ' , r m + 1 ) , P ( m + 1 ) ) assuming that yr(u(*V(A0,P(i;))<<S,

fc

(50)

= l,..,m.

For k = 1 this inequality holds, if Cj£ < 5. T h e differences ,(*+!)

=

„(fc+l)

satisfy the relations

,(*0

p(fc+i)

= r (fc+i)

_rW

£(*+!)

=

p(fc+l)

_ p< ,(*)

198 vf+1)

- i/VM fc+1 ) + Vp(*+1) = f (uO, r « , p « ) - 2ve3 x v ^ -f(u(fc-i)i7.(fc-i)iP(fc-i))i V-v(fc+1> = 0,

iuscv^ 1 ))^ = n6b(u(fc),PW)-ni,b(u(fc-1),p(*-1)),

_p(*+i) + j . n , . S(v(fc+1))n6 -anb

• Ab (V<=+D (lL,t\

er\

= d(u«,p«) - ^u^pC*"1))

n„ • e r d fc+1) - v 0, v(fc+1)(2/,0) = 0 yeQb, Z{h+1)(y,0) = 0 ye St. (51) We estimate (v* +1 , pk+1, r^"1-1') by inequality (43) taking account of (46) and of the elementary relation |v«(-,*)lc«(n t )< f

\y{k)(;r)\c^b)dr.

J 0

As a result, we arrive at y t (w<* +1 \p t+1 \f<* +1 >)
/"V T (wW,p fc) ^ (fc) )d 1

Hence m+l k=i

+c2 f*xiyt(ww,p*))ew)+ rx)VT(ww,p*),ew)dT

It follows that, if c25 < 1/2, then, by virtue of (49) and of Gronwall's inequality

y((u(™+Vm+1),P(m+1)) < Et^Yt(^k),^\ek)) <

(52)

C{T) (|u0|C2+«(S2t) + |Po|c'+"(Si)) •

Hence, condition (50) is satisfied also for k = m + 1, if c(T)e < 8\. We see that the parameters 5 and e should satisfy the condition : max(ci,c(T))e<(5< —-. 2c2

(53)

In this case we can repeat the above argument and show that (50) holds for all k = 1,2,3, the sequences (u' m ' l r'" l ',p' m ') are convergent in the norm Yr(-> •> •) to the solution of the problem (39) and the solution satisfies (52). The uniqueness of the solution also

199 follows from the inequality (43) applied to the difference of two solutions of the problem (39) (we omit the details). Next, we obtain uniform estimates for the solution of the problem (39) defined in an arbitrary time interval (0,T). To this end, we evaluate the norm Y

toAu,r>P)

=

SU

P

|u s (-,s)|c«(n 6 )+

to — T<S
+

sup

|u(-,s)|C2+„(fi6)

tQ—T<S
sup

|r(-,s)| c i+« ( n i ; ) +

(0— T<S<*0

sup

|p(-,s)| C 3+« (Sl) ,

to — T<S
with a fixed r € (0,1] and with i 0 > 2T. Theorem 7.4 If sup

|u s (-,s)| c =(n 1 ,)+

*0 — T<S
+

sup

|u(-,s)|c2+«(nb)

to~T<S
sup

|r(-,s)| c i+«(n t )

to—r<s
(^A.}

sup

|p(-,s)|C3+«(Sl) < <5,

to—T<s
with some sufficiently small 5 > 0, then the solution of the problem (39) satisfies the inequality y"to,r(u,r,p) < c ( sup ||u(-,s)|| L2( n b )+ sup \\p{-,s)\\wi(Sl) ) , \t1<s
t1<s
(55)

J

where t\ = t0 — 2T, T is a small positive number and c = C(T) is a constant independent

of t0. Proof. We set w =

q = r(x,

UCA,

where (,\(t) is an infinitely differentiable function such that CA(*) = 1 for£>£1 + 2A,

CA(*) = 0 for t < h + A,

\('x{t)\ < cA"\

and A € (0, r/2). The functions w and q satisfy the relations V w w ( - !/V2w + Vq noS(w)n 6 u| t = t 0

= = = =

0, f + u& y e Qb, * > h bCA, 0.

Moreover, integrating by parts in [n6 • T(u, r)nb - anb • Aberp\(x -

[nb • T(u,r)n b - an • Aberp]Cx{t')dt' J ti

= {d + F)Cx{t)- f (d + f)(x(t')dt', J t-l

(56)

200 we obtain nb-T(w,q)nb-anb-

Abwdt'+ J ti

= d(x+

j

h{Mv)dt' J ti

[nb • T ( u , r)nb - d]('x(t')dt'

+ f ^-F(p)(,(t')dt' + *nb- f

(57)

Ab^^df.

We consider w and r as a solution of the problem (55)-(56). Inequality (45) implies

sup

|ws(-,s)|c«(nt)+

tl<S
sup

|w(-,s)|C2+Q(nii)+

tl<S
< c

• • (

sup

|f|c-(n k ) +

sup

\ti+X<s
sup

\r(-,t)\ci+a(Qb)

ti<S
\h\C2+a(rb)

ti+A<s«o

sup ~^T + lblc'+-.<'+-)/*(r,,x(*1,t„)) + A-( 1+a >/ 2 sup |b(y,s)| t1+x<s
+

h+X<s
+A M

sup

ti+\<s
|u(-,s)| C i+a ( r ,,)+

\*i+A<s
sup

\r(;s)\c-(Th)

ti+A<s<4 0

j

(58) To estimate t h e norms of r and u in the right hand side, we use the boundary conditions and t h e interpolation inequalities. We have sup \b{y,s)\ rtx((i+A,fo) \r(-,s)\c°(rb)

< <

sup |Vu(2/,s)|, Six(ti+\*o) c(\u(;s)\ci+«(rb) + \d\c°(rb) +

< <

e|u(-,s)| C 2+»(n l ) + c e - 5 / 2 - a | | u ( - , s ) | | L 2 ( n 6 ) , £\p{-,s)\c3+°(s1) + ce~~5/2~a\\p(-,s)\\w}(s1)

and |u(-,s)| C i+a ( n i ) \p(-,s)y+-(Sl)

c

Finally, when we consider (39)4 as Schauder's estimates, we obtain

a n

(59) \p{;s)\e'+<'(s1))

elliptic equation for p on Si and make use of

\p{-, s)| c «+«(s 1 ) < c [\d + r - unb • S(w)n 6 )(-, s)| c i+a ( S l ) + \\p(-, s)\\wi(Sl)j Hence, t h e right hand side of (59)2 does not exceed ce(\d(-,s)\cl+a{rb)

+ \r{-,s)\ci+«{Tb)

+ |w(-,s)| C 2+«(n,,)) + \d\c°(rb) +c(e)(l|w|| i 2 (n 1 > ) + ||p(-,s

)ikc*>)Now, when we choose e = As' and taking account of (47), we obtain

.

201 sup

|u s (-,s)| C a (n ,,)+

ti+2A<s
+

sup

|u(-,s)| C 2+a (nt)

ti+2X<s
sup

|r(-,s)| C i+«(n k )+

sup

ti+2A<s<( 0

|p(-,s)| C 3 + « (Sl)

ti+2A<s
/ +Cl(e'+ 6) [

sup °

h+x<s
|u(-,s)| c 0 + « ( n i , ) +

I \

+

sup \r{-,s)y+^nb) ti+A<s<*o

SUP |p(-,S)|C3+a(Sl) tl+A<s
+c( £ ')A~ 7/2 - Q ( sup ||v(-, s )|U 2 ( s i t ) + sup Vi<S<«0

\

,finx 160)

I /

IIT/MH^SA

«l<S
/

We define the function of A /(A) = A 7 / 2 + a (

sup

|u 5 (-,s)| c «(n 6 )+

Vi+2A<s
+

sup

sup

|u(-,s)|C2+«(ni>)

ti+2X<s
|r(-,s)| c i+a (f!b )+

ti+2A<s
sup

|p(-,s)|C3+o.(Sl;

ti+2X<s
and we multiply (60) by \7l2+a. Since A < T/2, (60) implies f(X)
+ S)f(\/2)

+ K,

where K = c{e')[

sup ||u|| i2 (o 6 )+ sup \\p{-,s)\\wi{Sl) \tl<S
tl<S
i, /

is independent of A. We assume that c'27/2+a6 < 1/4, and, choosing e' such that c'e'2 7 / 2+Q < 1/4, we obtain

which furnishes (51), once we take A = T / 2 . The proposition is proved.

•

The following Corollary is a consequence of (33) and (55). Corollary 7.5 For any arbitrary solution (u,r,p) of the problem (39) with the data satisfying |u0|c2+<»(ftt) + |Po|c3+«(Si) < £ and the condition / x x wocte = / / x x vbdSi, Jfio JSiJ.R which is defined in time interval (0,T), T < oo and satisfies (53) there holds the uniform estimate swpYt{u,r,p) < ce~bt (|u0|c2+«(ni,) + |po|c3+=(Si)) . with the constants c and b independent of T.

202 Repeated application of the local existence theorem (Theorem 7.2) leads to our main result Theorem 1.1. • Acknowledgment The authors are grateful to professor Taddia for his valuable comments. The authors thank the GNFM-INDAM group for financial support. Padula thanks also the 60% contract MURST.

References [1] R.A. Brown and L.E. Scriven, Proc. R. Soc. London, A 371 (1980). [2] F. Brulois, Var. Meth. Free Surf. Int. P. Concus and R. Finn Eds. Springer-Verlag, Berlin. [3] S. Chandrasekhar, Proc. R. Soc. London, A 286 (1965). [4] I.Sh. Mogilevskii & V.A. Solonnikov, On the solvability of an evolution free boundary problem for the Navier-Stokes equations in Holder spaces of functions, Mathematical problems related to the Navier-Stokes equations, 11 (1992), 105-181. [5] M. Padula, On the exponential decay to the rest state for a viscous isothermal fluid, J. Fluid Mech. and Anal., 1 (1998). [6] M. Padula & V.A. Solonnikov, On the Rayleigh-Taylor Stability, Annali dell'Universita' di Ferrara, (sez.VIII, Sci. Mat.) 46 (2000), 307-336. [7] M. Padula & V.A. Solonnikov, On the global existence of nonsteady motions of a fluid drop and their exponential decay to a uniform rigid rotation Quaderni di Matematica, to appear. [8] V.A. Solonnikov, Solvability of the problem of evolution of an isolated amount of viscous incompressible capillary liquid, Zapiski Nauchn. Semin. LOMI, 140 (1984), 179-186. [9] V.A. Solonnikov, Unsteady motion of a finite mass of fluid bounded by a free surface, Zapiski Nauchnykh Sem. LOMI, 152 (1986), 137-157. [10] V.A. Solonnikov, On an evolution of a isolated volume of viscous incompressible capillary liquid for large values of time, Vestnik Leningrad Univ., ser.l, 3 (1987), 49-55. [11] V.A. Solonnikov, On non-steady motion of a finite isolated mass of self-gravitating fluid, Algebra and Analysis, 1 (1989), 207-249. [12] V.A. Solonnikov, On the justification of the quasistationary approximation in the problem of motion of a viscous capillary drop, Interfaces and free boundaries, 1 (1999), 125-173.

[13] A. V. Solormikov, Evolution free boundary value problems, Summer course in Punchal, 2000. To appear.

On a convection-diffusion equation with partial diffusivity * Andrea Pascucci Dipartimento di Matematica, Universita di Bologna ^

Abstract We consider the Cauchy problem for the nonlinear degenerate equation in RJV+1 div(AVu) + u(b • Vti) - dtu = f(-,u), where A > 0 is a constant symmetric matrix and ker(A) is generated by b. We prove the existence of a local viscosity solution u and we study the interior regularity of u in the framework of Hormander type operators.

1

Introduction

We consider the Cauchy problem for the nonlinear convection-diffusion equation div(AVu) + u(b • Vu)-dtu

= f{-,u),

in ST = R w x]0,T[,

(1.1)

with initial datum u{-,0)=g,

inR*.

(1.2)

We assume that / , g are globally Lipschitz continuous functions. Moreover we assume that the matrix A is constant, symmetric and positive semidefmite. The convection direction b is constant and generates ker(A). Equations of form (1.1) were studied by Escobedo, Vazquez and Zuazua [11] in order to describe the asymptotic behaviour as t —)• oo of solutions to a related parabolic equation with complete diffusion. We remark that, without loss of generality, by performing a suitable change of variables we may assume that A is diagonal so that b points along a coordinate axis, for instance b = eN. Then it is convenient to denote a point in M.N by (x, y) with x = (xi,..., x^-i) and y € R. Hence (1.1) becomes Lu = Axu + udyu-dtu

= f(-,u),

in ST,

where A x denotes the Laplace operator acting in the x variables. 'Keywords: nonlinear degenerate parabolic equation, interior regularity, Hormander operators. ^Piazza di Porta S. Donato 5, 40127 Bologna (Italy). E-mail: [email protected] 'Investigation supported by the University of Bologna. Funds for selected research topics.

204

(1.3)

205 Equation (1.3) also arises in mathematical finance, when studying agents' decisions under risk. The classical approach for this financial problem is based on the representation of agents' preferences in the framework of the utility theory and various models have been proposed, aiming to taking into account many aspects of the dynamics of the economy. Epstein and Zin in [10] propose a utility functional which is the solution of a backward stochastic differential equation. Recently Antonelli, Barucci and Mancino [1] propose a more sophisticated utility functional that considers some other aspects of decision making, such as the agents' habit formation, which is described as a smoothed average of past consumption and expected utility. In that model the couple of processes utility and habit is described by a system of backward-forward stochastic differential equations. In [1] is proved that there exists a unique solution u of such system, that satisfies some suitable initial and final conditions, and which is a viscosity solution, in the sense of the User's guide [9] of Cauchy problem (1.3)-(1.2). Moreover, in [1] it is proved that the solution u is defined in a suitably small interval of time [0, T[ and satisfies \u(x,y,t)-u{x',y',t)\

< c0(\x - x'\ + \y - y'\),

\u(x,y,t)-u{x,y,t')\
+

\(x,y)\)\t-t'\^

for every (x, y), (x1, y') e R", t, t' € [0, T], where Co is a positive constant that depends on the Lipschitz constants of / and g. Related problems also arise in stochastic control theory. For instance, the value function v of a suitable control problem is a semiconcave solution of the following Cauchy problem dxxv + ^(dyu)2-dtv

= ip,

v{-,0) = il>,

inR 2 x]0,T[, inR 2 ,

for some continuous functions