Bemelmans J. et. al. (editor) Elliptic and Parabolic Problems: Rolduc and Gaeta 2001

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if a = h

[x] = x x

V<p,

we get

a + VAf(V- a,) = /i

[x] = * x Vy, (44)

or at the boundary (|a;| = 1),

[ip]

i—s-

[p].

(45)

if a = /ii[x]||s|=i = Vx - x£x, we get

a + VAA(V-a„) = /»i[x]||»|=i + hi [~X + \£{£ + s)

l|.|=i

= /.i[|£(£ + i)-

x]lw=i. (46)

or at the boundary (|ai| = 1),

hxM _>/»![!£(£

+1)"

*].

(47)

Hence, given the boundary condition a, we can solve the boundary operator equation (42)

for a

. For example, given a = fix, with Afi = 0, i?(0) = 0, we have

{a-n)

= £~

fi, a-V(a-n)

= fix-V£-

fi = h

[-£-

fi]\

lxl=1

. (48)

Therefore, according to (46), a

= hi [-2(£ + \)£~

fi]-

Below, we present the solution of the boundary operator equation (42) for various

boundary condition a.

boundary condition (a)

x x Vip

Vx - x£x

fix

Oi-H

x x Vtp

hi [2(£ + \)£-

]

[-2(£ +

\)£~*fi]

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 (Q

) on which

the Euler Operator £ acts as a simple multiplication (£Q

= mQ

). Using solutions

(24o,6),

v = a

+ VAf{V-a

) + Vj

, p = -V-a

, (49)

we obtain a table of solutions as following

boundary condition (a)

x x VQ

- mQ

WmX

velocity (v)

x x VQ

- mQ

x + \{m + 3)(|ai|

- l)VQ

QmX - (*j£) {\xf - l)VQ

pressure (p)

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

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

'Any' solution

(v,p) of the

homogeneous SBVP

in a

ball

is a

linear superposition

the elementary solutions. Remarks similar

those after

the

table

section

3.1.1.

apply

here

well.

For neat (functional) analytic considerations (also

for the

general case)

refer

to our

forthcoming paper

(see [5]).

References

[1] X.X.

SHENG

and

W.M.

ZHONG,

General Complete Solutions

of the

Equations

Spatial

and AxiSymmetric Stokes Flow. Quaterly Journal

Mechanics

and

Applied Mathematics

44 (1991), p.537-548.

[2] B.S.

PADMAVATHI,G.P. RAJA

SEKHAR,

and T.

AMARANATH,

A Note on Complete General

Solutions

the Stokes Equations. Quaterly Journal

Mechanics

and

Applied Mathematics

51 (1998), p.383

- 388.

[3]

TEMAM,

Navier-Stokes Equations. Amsterdam: Elsevier Science Publishers B.V. (1984)

526pp.

[4]

STAKGOLD,

Green's Functions

and

Boundary Value Problem. Toronto: John Wiley

Sons,

Inc. (1998) 689pp.

[5]

DE GRAAF,

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 cylin-

drical 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

- Au

= f(t, x

) in (0, T) x Q

{t, x) = 0 on (0, T) x dtt

, (1.2)

(0, x) = u

) in tie,

(we denote by x = (£1,2:2) the points in R

, dSli denotes the boundary of

Clt).

Our data

/ =

f(t,x

= uo(s

) 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

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:

- aP.tioo = f(t, x

) in (0,T) x (-1,1),

(f,a;) = 0 on (0,T)

x{-l,l},

(1.3)

Uoo(0,

x) = ^0(2:2) in (-1,1).

More precisely, we have proved (see [4] Theorem 2.1 or [2] Theorem 9.2) that, for any

given £

> 0,

Woo

in L°°(0,T;L

{Qe

)) n L

(0,T;H\Ct

)),

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

u) - AuJ = ft(t, x) in

(0,

T) x Q

u\{t,x) = 0 on (0,T) x dn

, (1.4)

u\(0,x) = uo

(x) in Q(,

and uf be the solution to ,

f d

u\ - !\u\ = f

{t, x) in

(0,

T) x Q

{t,x) =

on (0,T)x

{-£,£)

x {-1,1},

{t,x) =0 on (0,T) x {-£,£}. x (-1,1),

u\(0,x) = Mo,(

) in n^.

Then we will prove, under certain assumptions, that

u\ - u? -> 0 in L°°(0,r;L

)) nL

(0,r;if

£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,x

), u

(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 u

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 The linear case

Let us consider a bounded open set of R" given by

=(-£,£)P xu) (2.1)

(1.5)

where w is a bounded open subset of

R™

, 1 < p < n. For x = (x\, ...,x

i, ...,a;

) €

R" we will set

Xi = (xi,

•••,x

= (x

...,x

We denote by

du>

a measurable subset of du and we will assume altough it is not essential

that

|dw

| > 0. (2.2)

Then we set

(-e,ey

xduo,

= r

ud(-£,£)

xu>

(2.3)

and

H^(Qi;Ti) = {ve H

^) ; v = 0 on I\} i = 1,2.

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

2ini

= {j[

\Vv(x)\dx}

1/2

(2.4)

Let Vf, Vf be two closed subspaces of

HQ(QI;

) equipped with the norm (2.4) such that

H^Qf, r\) C V

\ V? C Hl(Sl,; T

). (2.5)

Note that V} and Vf contain functions not vanishing on (—£,t)

xflw\ <9w

fteL^TiH^Slt-.To)'), (2.6)

where ^(fi^; T

)' denotes the dual of Hg(Q

; F

), 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

(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) ^

(

+ !)

W>0

- (

)

where c and a are positive constants. Remark that (2.7)-(2.8) imply that it holds that

I/*ILW;V?')<

)

W>0,Vt

= l,2. (2.9)

Moreover, as already explain, we allow also a dependence of the initial condition u

with

respect to £ and X\ but we must add the assumption

Kkn

(

<c(£

+ l) W>0. (2.10)

We consider then coefficients

aij = aij(t,x) eL°°((0,T) x R

x ui) Vi,j = 1,

...,n.

(2.11)

To simplify the notation we set

A = A(t,x) = (

aiJ

)

iJ=l n

(2.12)

and we suppose that for some A, A > 0 it holds that

|A(t,a;)|<A a.e. t 6

(0,T),

a.e. x e R

x w, (2.13)

^(*,z)£-£> A|£|

a.e. te (0,T), a.e. i6f x w, V£ e R". (2.14)

(|£| denotes the euclidean norm of f, | | the norm of matrices subordinated to the euclidean

norm of K.".) Let us set

ffHCri^.V?') = {ve L

(0,T;Vi)

«* £ L

(0,T;V7')}

•

(2.15)

For /f G L

(0, T;

.Hp

(fi<; r

)'), u

6 L

(f2^) there exists a unique u\ solution to

( uieH\0,T;Vi,Vi>),

(u\,v)

+ J A(x,t)Vu\Vvdx =

(fe,v) mV'(0,T),VveVi,

I u\{0,-)=u

{-) mL

(We refer the reader to [7], [1] for a

proof.

X>'(0,T) denotes the usual space of distribu-

tions on (0,T), (•, -)

the usual L

(f^)-scalar product.) Remark that if

du>

= dus then

H^QfJi) = H£(£li). Hence, if we choose p = 1, V? =

H^{Q

= ^(f^To) 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

> 0, any r > 0, there exists a constant C independent of I such that

\u\

U?|L°°(0,T;L2(J1

))

+ \u\ ~ U?Ua(o,T;ffi(n<

)) ^ J

(

)

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).

Then there exists

a constant C

—

C(w) independent of

and v such that

\v\

<C\\V

v\\

2flt

- (2-18)

(V^

= {d

x +1

, •••jC'x„).) See [3] for a

proof.

We can now estimate u\. We have indeed

Lemma 2.2 Leti G

{1,

2} andu\

the solution to (2.16). Then there exists C = C(\,p)

a constant independent of

such that

L»

(

„,

T;1

(nt))

[P4\\ln

(

!)•

(

)

•il^

+ / AVu\Vu\dx = </«(0,«j> < l/J(*)|.||Vujl|

J SI/

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

1 d.

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

5|I«JIU

A||V«JI|^<^I/,I;

^|IV«JI|^

Integrating on (0, i) we obtain for a.e. t

\v-m\ln

+ *J

||V*4l|2,„,d*<

luoM^

+ lflMl^dt- (2.21)

With (2.9), (2.10) we obtain

min(l,

||ui(t)|^ + ^IIV^HU*} ^

^ +

(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\ ~

e\l°o(o,T;L2(n

l/2k+1

))

+ J

||V(MJ

- u\)\

on dt

<T2[\W«-<)\\l

ni/

Jt- (2-23)

Proof of lemma 2.3. In R

we consider a smooth function p = p{X{), 0 < p < 1 such that

P=l on

(~\'l)

(

24)

p = 0 outside (-1,1)", (2.25)

XlP

< C, (2.26)

where C is some positive constant. Then, for a.e. t, for any t\ € [0,£\, it holds that

x = (X

) H. (u\ -u

)(t)p

{f) e H^QcT,). (2.27)

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 ~

^' ^

)

{(

< "

e)p

}

°-

(

)

According to [4], {u\ - uj)p € #H0,T;#

(^,r

),.ff

(ft6r

)') and it holds that (in

(0,T))

^(uj-u?)p,«) = (-(ui-u?),t;p) Vue^cn/.ri). (2.29)

Thus (2.28) becomes

l_d_

2~dt

\{u\

- u\){t)p\l

+ J AV{u\ - u

)V{{u\ - u

}dx = 0 (2.30)

Jn,

in,

in L

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

(u\

- uj)(t)p\l

+ f AV(u\ - u?)V(u] - u

dx =

Jn,

- / AV{u\ - u\)Vp{u\ - u

)pdx. (2.31)

1 Jn,

2d* • .in,

In,

Using (2.13), (2.26) and the the Young inequality ab < ^o

+ eb

, this last quantity

bounded by

2AC

— / \V{u\ - uj)\\u] - uj\pdx < —-5- / \V(uj - uj)\

dx + e \u\ - uj\

1 Jn, "1 Jn,, Jn,,

in,

«i Jn

and, with the Poincare inequality of Lemma 2.1, also by

^f / \V{u\-u

dx + C'ef \V

{u\-u

"1 Jn

where C" is a constant independent of

£—

recall that p is independent of X^. Going back

to (2.31) and using the ellipticity condition (2.14), we obtain choosing e = A/2C"

±-\{u\ - u

)(t)p\l

+ X f |V(uJ - uj)\

dx < J / |V(uJ - u

dx.

Jn,

<-i

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

fi^/2,

«?)(*)ll,n,

l/a

f [

«/)l

2(fa

^ S f I

^ " ^l^'

Jn,

l/2

i Jo Jn

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

. This completes the proof of

Lemma 2.3.

•

Proof of Theorem 2.1. From (2.23) we derive

\u\

- u

Al°°(o,T;mn,

/2k+1

))+ J ||V(uJ - i

= J

{l««

Al~

t/2k))

||V(ui

t*?)!!^*]

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

\\V(

Mln

lf2k+1

Now, by Lemma 2.2,

iiv(«j-«?)ii*

ini

dt<2jf

iiv«iii^dt+2jr

iivu?ii^dt

<C(^

+ l).

Then we derive from (2.32) that it holds that

2 l

l«J " u

e\l~(o,T;mn

l/2k+l)

) + ^ ||VuJ - u

dt < 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*

> £

. We obtain

+ \u

- u

0tT

im )) < -^r (2.33)

^lL~(0,T;L2(!!,

)) "I" !"£ ~ '"/ lL

(0,r

;

if»(«/„)) - ^

where C = C(A, A,p,n). Note that, since uj

—

u| is vanishing on a lateral part of f^

, the

-norm of the gradient is equivalent to the i7

(Q£

)-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,

(-«,<fxucr. (3.1)

We set

= {-£,£)" xdu, (3.2)

Hl(n

;

) = {ue H

) ; v = 0 on T

} (3.3)

and we consider

Vl, V

two

closed subspaces

Hg(Q,f,

)

equipped with

the

norm

(2.4)

such that

H^Qe)

c V

\ Vl C

Hl{Q,

- r„).

(3.4)

Let

denote

by atj =

aij(t,x,u),

i,j =

l,...,n

Caratheodory's functions

i.e.

functions

such that

(t,

x) i->

j(t,

measurable

R, (3.5)

H->

j(t,x,u)

continuous

a.e. (t,x)

(0,T) xW x

ui.

(3-6)

We write

the

matrix

the coefficients under

the

form

)=A = A(t,x,u

)

=^2u))

™

where

Ai is a p x n

matrix

and A

is a (n

—

p) x n

matrix.

assume that

satisfies

the following usual hypothesis,

^••?>A|f|

V£GR",VUGR,

a.e. {t,x)

(0,T) xW x u, (3.8)

\A(t,x,u)\

< A

VaeR,

a.e.

(i,a;) G

(0,T)

xl'xu,

(3.9)

for

> 0.

Furthermore

suppose that

\/u,v

R, a.e. (t,x) G (0,T)

xfxu

the

following holds

|A(t,s,u)-i4(t,a;,i;)| <Cw(|u-?;|), (3.10)

where w

is a

nondecreasing positive function

on R

such that

(s)

(3.11)

Under

the

assumptions (3.5), (3.6), (3.8),

(3.9) one can

show that,

for i = 1,2,

there

exists

solution

f uleH'(0,T;Vl,Vl'),

(u'

,v)2+

A(t,x,u'

)Vu'lVvdx

Jn,

i(-i,-,*

• ,^-

-.-.-«--f-~

(312)

,v) mV'(0,T),Vv€Vi,

{ u\(0,-)=u

(-) mL

for

)

and fe G

(0,T; ^(fi^To)')-

For an

existence proof

refer

for

instance

[1].

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

(t,x,u)\<A'

a.e. (t,x)

(0,T) x W x u, \/u e R.

(3.13)

Under these assumptions

let us

show