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Stability of the equilibria for spatially periodic ows in porous media

Joachim Escher Georg Prokert

Universitat Gesamthochschule Kassel Fachbereich 17 Mathematik/Informatik Heinrich-Plett-Str. 40, 34109 Kassel

escher@mathematik.uni-kassel.de prokert@mathematik.uni-kassel.de

Abstract

We prove exponential stability of the equilibria for two moving boundary problems describing ows in porous media which are of Hele-Shaw and Stefan type, respectively. The main tool is the principle of linearized stability for fully nonlinear parabolic evolution equations. Crucial points are the assumption of spatial periodicity for the ows and the identi cation of conserved quantities. Key words: fully nonlinear parabolic equation, exponential stability, moving interface

1 Introduction

The investigation of various mathematical models for ows in porous media has recently attracted a lot of interest from di erent points of view. This article is concerned with two such models that share the following basic properties: the ow is governed by Darcy's law, the porous medium is either completely saturated by the liquid or dry, and both phases are separated by a sharp interface, the main interest is in the determination of the motion of this interface, i.e. we are concerned with a moving boundary problem. The use of Darcy's law together with linear constitutive equations leads to linear governing equations for the velocity potential u. From a mathematical point of view, the moving boundary problems that are formed by these equations together with appropriate initial and boundary conditions are equivalent to a (multidimensional) Hele-Shaw ow problem and a Stefan-type problem, respectively. We will consider a geometric setting and driving forces describing an in nite layer of liquid which moves under the in uence of gravity; no other driving forces are considered. Moreover, the liquid domain is bounded from below by a xed impermeable horizontal bottom. The corresponding free boundary problems have been discussed in 7, 8, 10, 11] for the case of an incompressible porous 1

medium (the Hele-Shaw problem) and in 6] in the case of a compressible porous medium (the Stefan-type problem, see also 4, 5]). Our analysis is based on the methods used in 6, 7, 8] which can be brie y and informally sketched as follows: 1. The problem is reformulated as a nonlinear initial boundary value problem on a xed reference domain, using a transformation via an a priori unknown, timedependent di eomorphism. 2. By discussion of the linearization, this problem is identi ed as an abstract parabolic Cauchy problem. 3. Well-posedness of the problem and smoothness of its solution can be shown using the theory of maximal regularity for fully nonlinear parabolic equations, see e.g. 2, 3, 12]. In general, these methods can also be used to investigate the stability of stationary solutions on the basis of the \principle of linearized stability", i.e. by studying the spectrum of the linearized evolution equation near a stationary solution. However, even in the simplest two-dimensional case of the Hele-Shaw type problem it turns out that the L2 -realization of the corresponding linearized evolution equation has a pure continuous spectrum, given by 0; 1). Moreover, there is a continuous family of equilibria given by all liquid layers of constant density. It is the aim of this article to prove exponential stability for a modi ed version of the two moving boundary value problems above, overcoming these di culties in the following way: By demanding periodicity of the ows in space, the unbounded domain Rn is replaced by the torus Tn , and by the identi cation of a conserved quantity, the evolution is a priori restricted to a submanifold of the phase space which intersects the set of equilibria in a single point. It is shown that the equilibrium of these restrictions is exponentially stable.

2 Flows in incompressible porous media

In the description of the problem as well as in the notation, we closely follow 8]. Let 2 (0; 1) and de ne Given f 2 U , let

U := f 2 BUC 2+ (Rn ) j xinfn f (x) > 0 : 2R

~ f := f(x; y) 2 Rn R j 0 < y < f (x); x 2 Rn g and denote the components of its boundary by ~ ~ ?f := f(x; f (x)) j x 2 Rn g; ?0 := Rn f0g: We consider the elliptic boundary value problem (BVP) 9 n+1 u = 0 in ~ f ; = ~ @n+1 u = 0 on ?0 ; ; ~ u = y on ?f ; 2

(2.1)

where n+1 is the (n + 1)-dimensional Laplacian, @n+1 denotes the (n + 1)st partial derivative and y is the (n + 1)st coordinate function on Rn+1 = Rn R. This BVP is a simple model for the motion of an incompressible liquid in a rigid porous medium under the in uence of gravity. The liquid occupies the domain ~ f over the impermeable bottom ~ ?0 , and u is the velocity potential. (The well-posedness of this problem will be shown under additional assumptions later.) One is interested in the evolution of the interface separating the saturated from the dry part of the porous medium which is represented by ~ the boundary component ?f (t) , where now f : 0; 1) ?! U . This evolution is described by the nonlinear Cauchy problem

@t f (t; ) + 1 + jrn f (t; )j2 @ u( ; f (t; )) = 0; f (0; ) = f0

p

(2.2)

on Rn , where f0 corresponds to the initial interface, rn denotes the gradient on Rn , u is the solution of (2.1), and @ u denotes the derivative of u with respect to the outer unit ~ normal on ?f . We remark that problem (2.1), (2.2) is mathematically equivalent to a multidimensional one-phase Hele-Shaw problem with mixed boundary conditions. In our stability analysis we shall make the following crucial periodicity requirement on f and on u:

f (t; x + 2k ej ) = f (t; x) 8x 2 Rn ; j = 1; : : : ; n; k 2 Z; t 0 (2.3) u(x + 2k ej ; y) = u(x; y) 8(x; y) 2 ~ f (t) ; j = 1; : : : ; n; k 2 Z; t 0: (2.4)

In other words, instead of (2.1), (2.2) we discuss = = u = p @t f (t; ) + 1 + jrn f (t; )j2 @ u( ; f (t; )) = f (0; ) = where

f (t) ?f (t) n+1 u @n+1 u

0 0 0;

y

f0

in f (t) ; on ?0 ; on ?f (t) ; on Tn ; on Tn

t 0; t 0; t 0; t > 0;

9 > > > > = > > > > ;

(2.5)

:= f(x; y) 2 Tn R j 0 < y < f (t; x); x 2 Tng; := f(x; f (t; x)) j x 2 Tn g; ?0 := Tn f0g;

for t 0 and Tn denotes the n-dimensional torus. For the sake of simplicity, we identify periodic functions on Rn and functions on Tn . Remark: From a modeling point of view, it is unsatisfactory to demand periodicity both for f and for u. Actually, it seems reasonable to conjecture that (2.3) and u 2 BUC 2+ ( ~ f ) imply (2.4) . To show this, however, one needs to know whether the solution of (2.1) is unique in BUC 2+ ( ~ f ). Unfortunately, we are only able to prove this under the additional assumption that f is constant or n = 1. As in 7, 8], problem (2.5) is transformed to a problem on a xed reference manifold. Let := Tn (0; 1), Z c := (2 1)n f0 (x) dx; (2.6)

Tn

3

set g := f ? c, and de ne 2 Di

2+

( ;

f)

by (x; y) 2 :

g,

(x; y) := g (x; y) = (x; (1 ? y)(c + g(x))); De ning the push-forward and pull-back operators induced by

g g

: :

C ( f ) ?! C ( ); u 7! u g ; C ( ) ?! C ( f ); v 7! v ?1 ; g

g; n+1 g g ( i rn+1 ( v ) ni ); g

we introduce the transformed operators A(g) and Bi (g), i = 0; 1, acting on C 2 ( ) by

Bi (g)v :=

where

0

A(g) := ?

i = 0; 1;

n0 := (?rn g; 1) and n1 := (0; : : : ; 0; ?1). Observe that n0 is normal to ?c+g . For an

explicit coordinate representation we refer to 8], Lemma 2.1. Although an unbounded geometry is considered there, the expressions are formally the same an in 8]. Transformation of (2.5) to and ?0 , respectively, yields

and

1

stand for the trace operators on ?c+g and ?0 , respectively. Moreover,

A(g)v = 0

g(0) where v := g u and g0 := f0 ? c.

B1 (g)v @t g + B0 (g)v

v

in 0; 1); = g + c on ?0 0; 1); = 0 on ?1 0; 1); = 0 on ?0 (0; 1); = g0 ;

9 > > > > = > > > > ;

(2.7)

In the following, we will employ the so-called little Holder spaces which are de ned as follows: For k 2 N , let hk+ (Tn ) denote the closure of C 1 (Tn ) in BUC k+ (Tn ) and let hk+ ( ) denote the closure of BUC 1 ( ) in BUC k+ ( ). The spaces hk+ (Tn ) and hk+ ( ) are Banach algebras. Hence, by arguments analogous to the proof of Lemma 2.3 in 8], it follows that for some open neighborhood O of 0 in h2+ (Tn ), we have

A 2 C ! O; L h2+ ( ); h ( ) ; ? ? Bi 2 C ! O; L h2+ ( ); h1+ (Tn) ; i = 0; 1; where we use the obvious identi cation Tn fig = Tn , and where C ! denotes analytic

dependence. Moreover, given g 2 O, we have

?

?

?

(A(g); 0 ; B1 (g)) 2 Isom h2+ ( ); h ( ) h2+ (Tn ) h1+ (Tn ) :

(2.8)

This can be proved analogously to Theorem 3.5 in 7], the only necessary modi cation being the use of the compactness of to ensure that any bounded and uniformly continuous function v on attains its maximum on . In view of (2.8) we can de ne for any g 2 O the solution operator

T (g) := (A(g); 0 ; B1 (g))?1 jf0g h2+ (Tn ) f0g;

4

i.e., given h 2 h2+ (Tn ), we write T (g)h = v for the unique solution of the BVP 9 A(g)v = 0 in ; = v = h on ?0 ; B1 (g)v = 0 on ?1 : ; The same arguments as in 8], Lemma 2.5, show that ? ? T 2 C ! O; L h2+ (Tn); h2+ ( ) : Using this operator, we can reduce (2.7) to a single nonlocal Cauchy problem on Tn: @t g + (g) = 0; (2.9) g(0) = g0 ; where (g) := B0 (g)T (g)(g + c): It follows immediately from the de nitions that T (g) commutes with the operator h 7! h + c and that B0 (g) vanishes on constants. Hence (g) = B0 (g)(T (g)g + c) = B0 (g)T (g)g: (2.10) From the analyticity of T and B0 we get ? (2.11) 2 C ! O; h1+ (Tn ) : The following result on the solvability of (2.9) holds: Theorem 2.1 (Existence, uniqueness, and smoothness of solutions 7, 8]) Given g0 2 O, the Cauchy problem (2.9) has a unique maximal solution g 2 C (J; O) \ C ! (J_ Tn ); where J = 0; t+ (g0 )) is the maximal interval of existence. Remark: As we are concerned here with stability of the equilibrium only, we refrain from choosing O maximally large. We refer to 7, 8] for this. The proof of Theorem 2.1 can be given in analogy to 7, 8], based on the theory of maximal regularity for fully nonlinear parabolic equations, see also 2, 3]. The only necessary changes are due to the fact that instead of an unbounded domain in Rn+1 we consider here a bounded (n + 1)-dimensional manifold. However, both the parameter-dependent estimates for the occurring elliptic boundary value problems and the interpolation results for little Holder spaces can be carried over to this case without essential di culties. We start our discussion of the equilibrium g = 0 of (2.9) with an observation expressing essentially the conservation of volume for the domain saturated by the liquid. On L2 (Tn ) we introduce a continuous projection P by 1 Z u dx: Pu = u ? (2 )n n T k+ (Tn ) is continuous on these spaces, too. On all Obviously, the restriction of P to h these spaces, ker P is given by the constants and Im P is given by the functions with ~ ~ mean 0. For the latter we will write L2(Tn ), hk+ (Tn ) etc. 5

Lemma 2.2 (Conservation of volume) Given g 2 O, we have (g) 2 ~ 1+ (Tn ). h Proof: Let f := g + c and denote by u the solution of (2.5)1{(2.5)3. By de nition, we have T (g)g = g u and Gauss' theorem and (2.10) give

Z

Tn

(g) dx = = =

Z

Z Z

Tn f0g Tn

f

B0 (g)T (g)g dx =

g

p Z

Z

f0g

@u 1 + jrn gj2 @n dx = @u d = ? Z @n ?

0

Tn f0g

g ( 0 (rn+1 u)

Z

n0 ) dx

u dx ?

?0

@u @n d ?f @n+1 u d = 0:

Observe that due to (2.6) we have

~ and due to Lemma 2.2 the evolution proceeds completely in h2+ (Tn ). Next, we calculate the Frechet derivative ? A := 0 (0) 2 L h2+ (Tn ); (h1+ (Tn ) of at 0. One nds 0 (0) h] = B0 (0) h]T (0)0 + B0 (0)T 0 (0) h]0 + B0 (0)T (0)h 0 = B0 (0)T (0)h; 0 where we have used the linearity of the mappings T (0), B0 (0) h], and T 0 (0) h]. More explicitly, ! @u 0 (0) h] = ; 0 @n where u is the solution of

?c

~ g0 2 h2+ (Tn );

(2.12)

i.e. A is essentially the Dirichlet-Neumann operator on Tn fcg for the Laplacian with homogeneous Neumann boundary condition on Tn f0g. Due to the product structure of Tn (0; c), (2.13) is solvable by separation of variables. Thereby one obtains complete information on the closure AL of A to L2 (Tn). Lemma 2.3 (The linearized operator on L2) The operator AL has a complete orthogonal system of eigenfunctions ek , k 2 Zn, given by ek ( ) = eik ; 2 Tn : These eigenfunctions correspond to the eigenvalues k := jkj tanh cjkj, where jkj = jk1 j + : : : + jkn j.

2 2

u = 0 in Tn (0; c); u = h on Tn fcg; @u = 0 on Tn f0g; @n

9 > = > ;

(2.13)

6

The proof rests on the well-known completeness of the system fek j k 2 Zng in L2 (Tn ) and on the explicit solution of (2.13) for h = ek , cf. (2.18) below. Observe that, in particular, the operators AL and P commute, D(AL ) = H 1 (Tn ), 0 2 (AL ), and all eigenvalues of AL are nonnegative. By standard arguments from functional analysis together with earlier results on A, the information on the spectrum of AL can be carried over to A. Lemma 2.4 (The linearized operator in h1+ ) Consider A as an unbounded, densely de ned operator in h1+ (Tn ) with D(A) = h2+ (Tn ). The spectra of A and AL coincide. Proof: It can be shown as in the proof of Corollary 6.3 in 7] that ? A 2 H h2+ (Tn ); h1+ (Tn ) : (2.14) In particular, the resolvent set of A is not empty. Moreover, the embedding h2+ (Tn ) ,!,! h1+ (Tn ) is compact. Thus, A is an operator with compact resolvent and therefore ( 9] Theorem III.8.29) its spectrum consists only of eigenvalues having nite multiplicity. Because of ek 2 h2+ (Tn ) for all k 2 Zn, these functions are eigenfunctions of A corresponding to the eigenvalues k . On the other hand, due to the embedding h2+ (Tn) ,! H 1 (Tn ), all eigenfunctions of A are also eigenfunctions of AL . This implies the assertion. The only obstacle for a proof of exponential stability of g = 0 in (2.9) is the fact that 0 2 (A). However, the eigenspace E0 corresponding to this eigenvalue consists precisely ~ of the constants, and by Lemma 2.2 it is su cient to consider the restriction A of A ~ 1+ (Tn ) which is a complement of E0 in h1+ . The following lemma provides the to h ~ ~ ~ necessary results on A. To begin, observe that h2+ (Tn) is dense in h1+ (Tn ). ~ Lemma 2.5 (The linearized operator on h1+ (Tn)) ~ as an unbounded, densely de ned operator on ~ 1+ (Tn ) with D(A) = ~ Consider A h ~ 2+ (Tn ). Then h ~ ~ ~ A 2 H h2+ (Tn ); h1+ (Tn) (2.15) and ~ (A) = f k j k 2 Znnf0gg: Proof: Using the direct decomposition ~ hk+ (Tn ) = hk+ (Tn) E0 ; k = 1; 2; and the fact that AP PA we can interpret A as a matrix operator ~ A= A 0 0 0 ~ and conclude (2.15) from 1], Corollary 1.6.3. For k 2 Znnf0g, one has ek 2 h2+ (Tn ) ~ ~). On the other hand, given 2 (A) and f 2 h1+ (Tn), the and hence k 2 (A equation (A ? I )u = f (2.16) 2+ (Tn ). Applying P to both sides yields is uniquely solvable in h P (A ? I )u = (A ? I )Pu = Pf = f;

2 2 2 2 2 2 2

7

~ ~ i.e. Pu solves (2.16), hence u = Pu 2 h2+ (Tn ). Thus 2 (A), and consequently ~) (A). (A ~ ~ It remains to show that 0 2 (A). Recall that the equation Au = f is equivalent to 2+ (Tn (0; c)) such that the existence of a v 2 h

v @v @n @v @n v

= 0 = f

in Tn (0; c); on Tn fcg;

9 > > > > > = > > > > > ;

= 0 on Tn f0g; = u on Tn fcg:

(2.17)

If f = 0 then it follows from the maximum principle that any v 2 h2+ (Tn (0; c)) satisfying (2.17)1{ (2.17)3 is a constant, hence u is a constant and thus u = 0 since u 2 ~ ~ ~ h2+ (Tn ). This proves the injectivity of A. To show surjectivity, we x f 2 h1+ (Tn ) and P approximate f in the BUC 1+ -norm by smooth f j , j 2 N . We have f j = k2Znnf0g aj ek k where the aj are rapidly decreasing in k for any xed j . Setting k

vj ( ; y) :=

X

k2Znnf0g

jkj sinh cjkj ek ( ) cosh jkjy

in Tn (0; c); on Tn fcg; on Tn f0g; on Tn fcg;

9 > > > > > > = > > > > > > ;

aj k

(2.18)

~ and uj := vj jTn fcg we nd by standard arguments that uj 2 C 1 (Tn ) and

vj @vj @n @vj @nj v

= 0 = fj = 0 = uj

(2.19)

~ hence Auj = f j . For j; l 2 N , the well-known Schauder estimate for solutions of (2.19)1 { (2.19)3 yields

follows from 14], Theorem 12.12, that both formulations are equivalent.) Clearly, one has Moreover, the obvious estimate

2

C kf j ? f l kBUC (Tn) + kvj ? vl kL (Tn (0;c)) : (Usually, in the last term on the right the BUC 0 (Tn (0; c))-norm is used. However, it

2+

kvj ? vl kBUC

(Tn (0;c))

?

1+

2

kuj ? ul kBUC

2+

(Tn)

C kvj ? vl kBUC

2

2+

(Tn (0;c)) :

~ limit in (2.19) yields Au = f . We are ready now for the proof of the main result of this section: 8

kvj ? vl kL (Tn (0;c)) C kf j ? f l kL (Tn) C kf j ? f l kBUC (Tn) ~ shows that fvj g and fuj g are Cauchy sequences in h2+ (Tn (0; c)) and h2+ (Tn ), 2+ (Tn (0; c)) and u 2 h2+ (Tn ), respectively. Passage to the ~ converging to some v 2 h

1+

Theorem 2.6 (Exponential stability) For any ! < 1 = tanh c, there are positive constants r and C such that for any g0 2 O with kg0kBUC (Tn) < r we have t+ (g0 ) = 1 and for the solution g of (2.9) the

estimate

2+

kg(t)kBUC

holds.

2+

(Tn) +

kg0(t)kBUC

1+

(Tn)

Ce?!t kg0 kBUC

2+

(Tn)

8t 0

~ ~ ~ Proof: Setting X := h1+ (Tn), D := h2+ (Tn), G := A ? , !0 := ? supfRe j 2 ~ (?A)g = 1 , we nd that all assumptions of Theorem 9.1.2. in 12] are satis ed because of (2.11), (2.12), and Lemmas 2.2 and 2.5. Our result follows from this theorem.

3 Flows in compressible porous media

In the description of the problem as well as in the notation, we will follow 6] in this section. We consider a free boundary ow problem in a deformable porous medium. The geometric setting is the same as before, i.e. we assume that for xed time the liquid occupies the domain ~ f := f(x; y) 2 Rn R j 0 < y < f (x)g ; where f 2 C 1 (Rn ) is a uniformly positive function whose graph ~ ?f := f(x; y) 2 Rn R j y = f (x)g represents the interface between the saturated and the dry part of the porous medium. Accordingly, we de ne ~ A := f 2 C 1 (Rn ) j xinfn f (x) > 0 2R ~ and for f 2 C 1 ( 0; T ]; A) ~ f;T := (t; z ) 2 0; T ] Rn+1 j z 2 ~ f (t) : Note that ~ f;T is an unbounded striplike domain in 0; T ] Rn+1 . We are looking for ~ a pair (u; f ) of (su ciently smooth) functions u : ~ f;T ?! R, f : 0; T ] ?! A satisfying the system of equations @t u ? u = 0 in ~ f (t) ; t > 0; 9 > > ~ > @n+1 u = 0 on ?0 ; t 0; > > = ~ f (t) ; t 0; > u = f on ? p (3.1) ~ > @t f + 1 + jrn f j2 @ u = 0 on ?f (t) ; t > 0; > > > > u(0; ) = u0 in ~ f ; > ; n f (0) = f0 on R

0

n

o

9

with given initial data (u0 ; f0 ). As in the previous section, @ u denotes the derivative ~ of u(t; ) with respect to the outer unit normal on ?f (t) and the Laplacian involves the spatial coordinates only. The crucial di erence of (3.1) to the problem (2.1),(2.2) is the fact that the governing equation (3.1)1 is parabolic, hence it is not possible here to determine u(t; ) for given f (t) at the xed time t 2 0; T ]. Instead, one has to consider a pair of coupled Cauchy problems for u and f . The system (3.1) occurs as a model for the gravity-driven motion of a compressible Newtonian liquid in a linearly elastic porous medium, where u represents the velocity potential of the ow. For the derivation of this model we refer to 6] and the references therein. Note, moreover, that (3.1) can also be interpreted as a one-phase Stefan problem with mixed boundary conditions. As in the incompressible case, we demand spatial periodicity of f and u, i.e. (2.3) ~ ~ and (2.4), and consider (3.1) with A, ~ f (t) , ?f (t) , ~ f;T replaced by

A :=

?f (t)

f;T f (t)

f 2 C 1 (Tn ) j x2Tn f (x) > 0 ; min := f(x; y) 2 Tn R j 0 < y < f (t; x)g ; := f(x; y) 2 Tn R j y = f (t; x)g ; := (t; z ) 2 0; T ] Tn R j z 2 f (t) :

Let us transform (3.1) to a problem on the xed reference domain := Tn (0; 1) using a di eomorphism depending oni the unknown function f . Following 6], we x h 1 1 p > n + 2, s 2 0; p , 2 p ; 1 ? n+1 . Furthermore, we write ?i := Tn fig, i = 0; 1, p and denote by i the trace operator from to the boundary component ?i . Throughout this section we write Wpt ( ) and Wpt (?0 ) for the Sobolev-Slobodeckii spaces over and ?0 , respectively. It is shown in 6], Section 2, that there is an extension operator

0

2 L Wpk+s? p (?0 ); Wpk+s ( ) ; k = 1; 2;

1 1

such that 0 0 is the identity on Wpk+s? p (?0 ) and 0 g vanishes near ?1 for all g 2 Wp1+s? p (?0 ). (Here and in the following, references to results in 6] are to be understood with the necessary changes due to our periodicity assumptions, i.e. with Rn replaced by Tn . The results remain valid, and there are no essential new di culties in the proofs.) We set Z Z 1Z (3.2) Q0 := u0 dx ? 2 n f02 dx0 + n f0 dx0 : T T f It will be shown later that for our purposes it is natural to demand Q0 > 0. Making this assumption, we de ne d > 0 as the (unique) solution of (2 )n d(d + 2) = Q (3.3) 0 2 (see the remark after Lemma 3.3 below). Moreover, with

1 0

D :=

g 2 Wp2+s? p (?0 ) j (x;y)2 (d + 0 g) > 0; inf

1

10

(x;y)2

sup

p

jrx ( 0 g)(x; y)j2 + j(1 ? y)@n+1 ( 0 g)(x; y) ? ( 0 g)(x; y)j2 < (1 ^ )

and 2 (0; d) xed, we de ne for any g 2 D the mapping g : ?! g+d by g (x; y ) := (x; (1 ? y )(d + ( 0 g )(x; y )): The corresponding pull-back and push-forward operators will be denoted by g and g , respectively. It is shown in 6], Lemma 2.2.a,b and Corollary 2.3, that D is an open neighborhood of 0 in Wp2+s? p (?0 ), and g 2 Di 1+ ( ; g+d ) with g 2 L ?W k+s ( ); W k+s ( k = 0; 1; 2; g+d ) ; p p for any g 2 D. This justi es the following approach: Using the obvious identi cation of ?0 and Tn we demand f (t; ) ? d 2 D, u(t; ) 2 Wp2+s ( f (t) ) for t 2 0; T ] and de ne functions g on 0; T ] ?0 and v on 0; T ] by g := f ? d; v(t; ) := g(t) (u(t; ) ? d): The problem (3.1) for (u; f ) on the unknown domain f;T can be transformed now to a problem for (v; g) on the xed cylindrical domain 0; T ] , namely, 9 @t v + A(g)v + R(v; g) = 0 in (0; T ] , > > > v = g on 0; T ] ?0 , > > > B1 (g)v = 0 on 0; T ] ?1 , = (3.4) @t g + B0 (g)v = 0 on (0; T ] ?0 , > > > > v(0; ) = v0 in , > > ; g(0; ) = g0 on ?0 ,

1

(u0 ? d) 2 Wp2+s ( ) are the given initial data, ? g ( g v); g g ( g+d r( v ) n0 ); g g ( 0 r( v ) n1 ); 1)@ R(v; g) := (y ?G(g)n+1 v 0 (B0 (g)v); G(g) := d + 0 g + (y ? 1)@n+1 ( 0 g); n0 := (?rx g; 1) and n1 := (0; ?1) denote outer normals on ?f and ?0 , respectively, and g+d is the trace operator from g+d to ?g+d . Note that g 2 D implies that G(g ) has a positive lower bound on and G(g) = j det D g j: (3.5) For an explicit coordinate representation of the transformed operators A(g) and Bi (g) we refer to 6], Lemma 2.5. By transformation to g+d and Gauss' theorem one has where g0 := f0 ? d 2 D and v0 := g A(g)v := B0 (g) := B1 (g) :=

0

? A(g)vj det D g j dx =

Z

Z

g+d

(

g v ) dx =

Z

@ g+d

@ ( g v) d

11

= =

Z Z

?g+d ?0

1 + jrx gj ? Z B0(g)v dx0 + B1(g)v dx0

?1

g+d r( p

g v)

n0 d + Z 2

0

0

r( g v) n1 dx0

(3.6)

for v 2 Wp2+s ( ), g 2 D. In order to discuss (3.4) in a suitable abstract framework we set

E1 := (v; g) 2 Wp2+s ( ) Wp2+s? p (?0 ) j 0 v = g; 1 @n+1 v = 0 ; D1 := f(v; g) 2 E1 j g 2 Dg and de ne : D1 ?! E0 by (v; g) := (A(g)v + R(v; g); B0 (g)v): Following 6], one nds that D1 is open in E1 and 2 C ! (D1 ; E0 ). With these de ni1

E0 := Wps ( ) Wp1+s? p (?0 );

1

tions, we reformulate (3.4) as an abstract Cauchy problem @t z + (z ) = 0; (3.7) z (0) = z0 := (v0 ; g0) in E0 and it is clear that any solution z 2 C ( 0; T ]; D1) \ C 1 ( 0; T ]; E0) yields a solution of (3.4) via z = (v; g). Moreover, for > 0, > 0 we de ne

W ; := (v; g) 2 D1 kvkWp

2+

s( )

< ; k@n+1 vkC

( )

< ; k 0 @n+1 vk

It is easily seen that W ; is an open neighborhood of 0 in D1 . The crucial result on the parabolic character of (3.7) is the following ( 6] Corollary 3.6): Lemma 3.1 (Parabolicity of (3.7)) For any > 0 there is a > 0 such that @ (z ) 2 H(E1 ; E0 ) 8z 2 W ; : This is the crucial fact in the proof of the classical solvability of (3.7). More precisely, the following results are proved in 6], cf. Theorem 3.8.: Theorem 3.2 (Solvability of (3.7) and time regularity of the solution) Let > 0 be given. There exists > 0 such that for each z0 2 W ; there is a t+ := t+ (z0 ) > 0 and a unique maximal solution z 2 C ( 0; t+ ); W ; ) \ C 1 ( 0; t+ ); E0 ) of (3.7). Moreover,

s s z 2 C 1 (0; t+ ); Wp2+~( ) Wp2+~? p (?0 )

1

Wp

1+

1 s? p

(?0 )

<

:

for any s 2 (0; s). ~

12

It is easily checked that z = (0; 0) is a stationary solution of (3.7). As in the previous section, the rst step in the investigation of its solvability is the identi cation of a conserved quantity. Lemma 3.3 (A conserved quantity) Let z = (v; g) be a solution of (3.7) in the sense of Theorem 3.2. Then the functional Z 1 Z (g + d)2 dx0 + Z (g + d) dx0 I (z ) := (v + d)j det D g j dx ? 2 ? ?

0 0

is constant on 0; t+ ). Proof: Due to the continuous embeddings E1 ,! BUC 1+ ( ) s s Wp2+~( ) Wp2+~? p (?0 ) ,! BUC 1+ ( ) BUC 1+ (?0 ) we have

1

BUC 1+ (?0 ) and

z 2 C 0; t+ ); BUC 1+ ( ) BUC 1+ (?0 ) \ C 1 (0; t+); BUC 1+ ( ) BUC 1+ (?0 ) ; and the mapping t 7! I (z (t)) is continuous on 0; t+ ) and di erentiable on (0; t+ ) with

derivative

?

?

I_(z ) =

Z

?

=

Z

Z

@t vj det D g j dx +

?0

Z Z

Z 1

@t g(g + d) dx0 +

Z 1

?0 0 ?0

(v + d)@y (y ? 1) 0 (@t g)] dydx0

@t g dx0 @y (v + d)(y ? 1) 0 (@t g)] dydx0

0 0

@t vj det D g j dx +

Z

0

Z

Z 1

@y v(y ? 1) 0 (@t g) j det D j dydx0 ? Z @ g(g + d) dx0 + Z @ g dx0 ? t t g G(g) ? 0 ? ? Z Z (@t v + R(v; g))j det D g j dx ? (g + d)B0 (g)v dx0 =

?0 0

?

= ? =

Z

Z

Z

?0

@t g(g + d) dx0 +

Z

Z

?0

@t g dx0

Z

?0

A(g)vj det D g j dx + B0 (g)v dx0 +

?1

?0

B1 (g)v dx0 +

?0

@t g dx0

Z

?0

@t g dx0 = 0;

where (3.5) and (3.6) have been used. Remark: It is natural to reinterpret Lemma 3.3 in terms of the original problem (3.1). For solutions of this problem, the quantity Z Z 1Z Q(u; f ) = u dx ? 2 n f 2 dx + n f dx T T ft

( )

is conserved during the evolution. For positive Q(u; f ), this fact determines to which of the possible equilibria (u; f ) = (c; c), c > 0, a solution with given initial data (u0 ; f0 ) 13

may tend. In view of (3.2), (3.3), and Q(c; c) = (2 2) c(c +2) we nd c = d. On the other hand, for initial data with Q(u0; f0 ) < 0 (3.8) one obtains that a corresponding solution cannot tend to any equilibrium. An elementary calculation shows that (3.8) can hold only if @n+1 u0 is relatively large. This gives rise to the conjecture that solutions of (3.1) with (3.8) will cease to exist at a nite time at which the strict positivity of f is lost, i.e. at which the upper boundary component of the liquid domain will touch the bottom. (Of course, it is by no means clear whether there are further blowup mechanisms.) Note, moreover, that if Q0 > 0 then for any > 0 there is a > 0 such that I (v; g) > 0 for all (v; g) 2 W ; . In a next step, we derive the necessary information on the spectrum of the operator A := @ (0). Due to the dense embedding E1 ,! E0 , A can and will be considered as an unbounded, densely de ned operator on E0 with D(A) = E1 . Lemma 3.3 in 6] applied to z = (v; g) = 0 shows that

n

A(w; h) = (A(0)w; B0 (0)w):

The spectrum of A consists only of nonnegative real eigenvalues. Proof: From Lemma 3.1 we know that A 2 H(E1 ; E0). As in the proof of Lemma 2.4, it follows from this together with the compactness of the embedding E1 ,! E0 that (A) consists only of eigenvalues. On E1 we de ne the scalar product ( j ) by

?

Lemma 3.4 (The linearized operator)

(v; g) (w; h) := d

Z

vw dx +

Z

?0

gh dx0 =

Z

d

0 v 0 w dx +

Z

?d

0 g 0 h dx0 :

The operator A is symmetric and positive semide nite with respect to this scalar product. Indeed, for arbitrary (v; g); (w; h) 2 E1 ,

?

A(v; g) (w; h) = ?

= ? + = ?

Z

Z d Z d Z

(

0v

? 0

0 v ) 0 w dx +

Z

(

0 w) dx + 0 w)

Z

?d

@ ( 0 v) 0 h dx0

? 0

?0

d

v@ (

?@

?d ( 0 v) 0 w

Z

v@ ( 0 w) ? @ ( 0 v) 0 w dx0 dx0 +

Z

0v

( 0 w) dx +

0 v ) 0 v dx + 0 v ) 0 v dx0

?d

Z Z

0 g@

( 0 w) dx0 = (v; g) A(w; h)

?d

@ ( 0 v) 0 h dx0

?

and

?

= =

Z

A(v; g) (v; g) = ?

Z d

(

Z d d

jr(

0 v ) 2 dx

j

?

Z

jr( 0 v)j2 dx 0;

?d

@(

?

?d

@ ( 0 v) 0 g dx0 @(

0 v ) 0 v dx0 +

Z

?0

?d

@ ( 0 v) 0 g dx0

14

where we have used that 0 v = g, 0 w = h, 1 @n+1 v = 1 @n+1 w = 0. On the complexi cation of E1 we introduce the scalar product j ] by ? z1 jz2 ] = (x1 jx2 ) + (y1 jy2 ) + i (y1 jx2 ) ? (y2 jx1 ) ; where zj = xj + iyj , xj ; yj 2 E1 , j = 1; 2. Suppose now is an eigenvalue of A with eigenfunction z 6= 0. This implies z jz ] = Az jz ] = z jAz ] = z jz ]; hence = 2 R, and z can and will be chosen to be real. Then (Az jz ) = (z jz ) 0, hence 0. As in the incompressible case discussed above, it is not hard to see that 0 is an eigenvalue of A. Therefore, by exploiting the conservation of I , one again has to consider a reduced system in which this eigenvalue does not occur. In the present situation, however, the evolution does not proceed in an invariant subspace but, according to Lemma 3.3, in the invariant subset M0 := fz 2 E1 j I (z ) = Q0 g: Using (3.5), it is straightforward to calculate the Frechet derivative of I 2 C ! (D1 ; R) at (v; g) 2 D1 as

@I (v; g) w; h] =

Z

+

Z

wj det D g j dx +

?0

Z

Z 1

h(1 ? (g + d)) dx0 ;

?0 0

(v + d)@y (y ? 1) 0 h] dydx0

(w; h) 2 E1 . This linear functional is nondegenerate, and it follows that M := M0 \ D1 is a smooth hypersurface in E1 . Note that 0 is the only stationary point of (3:7) in M. Our strategy is to describe the restriction of the dynamical system given by (3.7) to M using a ~ local di eomorphism mapping M near 0 onto its tangent space E1 := T0 M = ker @I (0; 0) where

@I (0; 0) = d

= d

Z

w dx + d w dx +

Z

Z

Z 1

Z

?0 0

@y (y ? 1) 0 h] dydx0 + (1 ? d)

Z

Let us de ne the projection P 2 L(E0 ) \ L(E1 ) by ; 1) ? P (v; g) := (v; g) ? (2 )(1(d + 1) (v; g) (1; 1) : n ~ ~ Clearly, P (E1 ) = E1 and ker P = f(c; c) j c 2 Rg. De ning E0 := P (E0 ), we obtain the direct decomposition ~ Ei = Ei ker P; i = 0; 1: The operator A vanishes on ker P , hence A = AP . Furthermore, ; 1) ? ; 1) ? (I ? P )A(w; h) = (2 )(1(d + 1) A(w; h) (1; 1) = (2 )(1(d + 1) (w; h) A(1; 1) = 0 n n 15

?0

h dx0 :

?0

h dx0

~ ~ for (w; h) 2 E1 , hence A = AP = PA on E1 . We de ne the restriction A := AjE1 and ~ that will be important for our purpose. collect the properties of A Lemma 3.5 (The linearized operator on the restricted space) ~ ~ ~ ~ A 2 H(E1 ; E0 ) and the spectrum of A consists only of positive eigenvalues. ~ ~ ~ Proof: By the same arguments as in the proof of Lemma 2.5, we obtain A 2 H(E1; E0 ) ~) ~ 2 Isom(E1 ; E0 ). This will be done by ~ ~ and (A (A). It remains to show that A discussing the Neumann BVP

R

@u = @u = 0 R u dx + ?d u dx0 = 0: d

Z

? u =

in d, on ?d , on ?0 ,

1

9 > > = > > ;

(3.9)

We will show that that for any ( ; ) 2 Wps ( d ) Wps+1? p (?d ) satisfying

d

dx +

Z

?d

dx0 = 0

(3.10)

there exists precisely one solution of (3.9) in WRs+2 ( d ). To show uniqueness, we set p ( ; ) = 0 and obtain from (3.9)1 { (3.9)3 that d jruj2 dx = 0, hence u is constant, and it follows from (3.9)4 that u = 0. To show existence, we rst remark that ( ; ) 2 Lp ( d ) Lp (?d ) ,! L2 ( d ) L2 (?d ), and a discussion of the weak formulation shows that (3.9) has a unique solution u 2 H 1 ( d ) satisfying an estimate

kukL (

2

d)

C k kL (

2

?

d)

+ k kL (?d )

2

C k kWps (

d) +

k kW s

p

+1

1 ?p

(?d )

: (3.11)

Consider the case where and are smooth. In this case, it follows from standard elliptic regularity results that u is smooth, and the a priori estimate given in 13] Theorem 5.3.4 together with Theorem 12.12 in 14] and (3.11) yields

kukWps

+2

( d)

C k kWps ( C 0 k kWps(

d)

+k k

Wp

1 s+1? p

(?d )

+ kukL (

2

d)

d) +

k kW s

p

+1

1 ?p

(?d )

:

The solvability of (3.9) in Wps+2 ( d ) in the general case follows now by a standard approximation argument. ~ To complete the proof, we set ( ; ) := ( 0 w; 0 h) for (w; h) 2 E0 . Then (3.10) is ~1 if and only if v = 0 u, g = 0 v, i.e. this satis ed, and A(v; g) = (w; h) with (v; g) 2 E operator equation is always uniquely solvable.

Lemma 3.6 (A local chart for M) ~ There are neighborhoods U1 and V1 of 0 in E1 and M, respectively, and a di eomor1 (U1 ; V1 ) such that phism 2 Di P = idV and @ (0) = idE . ~

1 1

16

~ ~ Proof: De ne the function F 2 C 1(E1 E1; E1 R) by F (x; z ) := (Pz ? x; I (z ) ? Q0 ): Clearly F (0; 0) = 0, the partial Frechet derivatives in (0; 0) are given by @1 F (0; 0) h] = (?h; 0); @2 F (0; 0) k] = (Pk; @I (0; 0) k]); and @2 F (0; 0) is invertible with ~ ?(@2 F (0; 0))?1 @1 F (0; 0) h] = h; h 2 E1 : By this, all assertions follow from the Implicit Function Theorem. Let > 0 be given and choose > 0 according to Theorem 3.2. Furthermore, we ~ choose open neighborhoods V V1 and U U1 of 0 in M and E1 , respectively, such that V W ; \ V1 , U = P V ] and is Lipschitz continuous on U . Assume z0 2 V and de ne ~ t+ := supft 2 0; t+) jz (t) 2 Vg; ~ z (t) := Pz (t); t 2 0; t+ ); ~ ~ := P ~ ~ ~ with z and t+ from Theorem 3.2. Note that ~ 2 C 1 (U1 ; E0 ), z 2 C ( 0; t+ ); U ) \ 1 ( 0; t+ ); E ) and due to ~ ~0 C P (z (t)) = P ( (Pz (t))) = ~ (~(t)) z it satis es @t z + ~ (~) = 0; ~ z (3.12) z(0) = Pz ~ on

Theorem 3.7 (Exponential stability for the projected problem)

0; ~+). t

0

There are positive constants r, M , and ! such that for any z0 with kPz0kE < r the following holds: ~ ~ ~ i) z is the only solution of (3.12) in C ( 0; t+ ); U ) \ C 1 ( 0; t+ ); E0 ), ~ ~ ii) z can be uniquely extended to a solution of (3.12) in C ( 0; 1); U ) \ C 1 ( 0; 1); E0 ), ~ still denoted by z, ~ iii) z satis es the estimate ~ ~ kz(t)kE + k@tz (t)kE Me?!tkPz0kE : ~

1 1 0 1

~ Proof: Using Lemma 3.6 we nd @ ~ (0) = P@ (0) = A: As (A) is closed, we obtain from Lemma 3.5 that ~ !0 := inffRe j 2 (A)g > 0: ~ ~ ~ The results follow immediately from this and A 2 H(E1 ; E0 ) by 12], Theorem 9.1.2. 17

Corollary 3.8 (Exponential stability for the original problem) There are positive constants r, M , and ! such that for any z0 with kz0 kE < r one has t+ = 1 in Theorem 3.2 and the estimate kz (t)kE Me?!tkz0 kE ; t 0;

1 1 1

holds.

r ~ Proof: Let r := kP kL E where r denotes the constant r from Theorem 3.7 and assume ~ ~ kz0 kE < r. This implies the statements of Theorem 3.7, in particular, t+ = t+ and thus fz (t) j t 2 0; t+ )g = fz(t) j t 2 0; t+ )g]; ~

( 1) 1

i.e. the trajectory of the maximal solution of (3.7) is the image of a relatively compact set under a continuous map, hence it is relatively compact. By 12], Proposition 8.2.1 and Lemma 8.2.2., this implies limt!t z (t) 2 @W ; or t+ = 1. The rst possibility is excluded because V W ; and W ; is open, hence t+ = 1. The estimate of the corollary is an immediate consequence of Theorem 3.7 and the Lipschitz continuity of and P . Due to the smoothness of z in t, the stability results can also be translated back to the original problem (3.1).

+

References

1] Amann, H.: Linear and Quasilinear Parabolic Problems I: Abstract Linear Theory, Birkhauser 1995. 2] Angenent, S.: Nonlinear analytic semi ows, Proc. Roy. Soc. Edinburgh 115A (1990) 91{107. 3] Angenent, S.: Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature, Annals of Math. 132 (1990) 451{483. 4] Bizhanova, G.I., Solonnikov, V.A.: On the solvability of the initial boundary value problem for a parabolic equation of second order with dynamic boundary conditions in weighted Holder spaces (in Russian), Algebra and Analysis 5 (1993) 109{142. 5] Bizhanova, G.I.: Multidimensional Stefan and Florin problems in weighted Holder spaces of functions (in Russian), preprint, 1994. 6] Escher, J.: On moving boundaries in deformable media, Advances in Mathematical Sciences and Applications 7 (1997) 1 275{316. 7] Escher, J., Simonett, G.: Maximal regularity for a free boundary problem, Nonlin. Di . Eq. Appl. 2 (1995) 467{510. 8] Escher, J., Simonett, G.: Analyticity of the interface in a free boundary problem, Math. Annalen 305 (1996) 439{459. 9] Kato, T.: Perturbation Theory for Linear Operators, Springer Berlin etc. 1976.

18

10] Kawarada, H., Koshigoe, H.: Unsteady ow in porous media with a free surface, Japan J. Indust. Appl. Math. 8 (1991) 41{82. 11] Kawarada, H., Koshigoe, H., Sasamoto, A.: The behaviour of free surface appearing in the ow through porous media, Avd. Math. Sci. Appl. 1 (1992) 157{174. 12] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser 1995. 13] Triebel, H.: Interpolation Theory, Function Spaces, Di erential Operators, NorthHolland 1978. 14] Wloka, J.: Partial Di erential Equations, Cambridge University Press 1987.

19

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