On a nonlinear fluid-structure interaction problem defined on a domain depending on time

Nonlinear Analysis 38 (1999) 549 – 569

On a nonlinear uid-structure interaction problem de ned on a domain depending on time Fabien Flori ∗ , Pierre Orenga Centre de Mathematique et de Calcul Scienti que, URA 2053, Universite de Corse, Quartier Grossetti, 20250 Corte, France Received 30 June 1997; accepted 10 October 1997

Keywords: Fluid-structure interaction; Compressible Navier–Stokes equations; Nonlinear partial dierential equations; Functional analysis; Partial dierential equations in a non-cylindrical domain

1. Introduction In the present work we study a uid-structure coupling problem for which the structure occupies a portion of the uid domain boundary. Coupling is expressed by the hypothesis on normal velocity continuity at the boundary and by introducing a parietal pressure term in the equation of structure behaviour. Under conditions of weak disturbances it is classically assumed that the uid occupies a xed domain termed [5, 6]. For general cases in which we consider nonlinear conservation equations, however, this assumption leads to diculties which do not allow us to prove that the problem is well posed. If we consider the structure’s displacement within the uid domain, however, we can then show that the problem is well posed for a bidimensional situation which is the object of this study. The main diculty comes from the nonlinearity of the mass conservation equation. Indeed, if we consider the structure to be xed then the conservation of the uid mass is not valid and we do not obtain an energy-type estimate such as found in [2]. The results given below are for a bidimensional uid domain coupled to a plate. To simplify the problem, the velocity eld is sought out under the form of a potential and we present the results as a linear momentum equation. ∗

Corresponding author. E-mail: { ori, orenga}@lotus.univ-corse.fr.

0362-546X/99/$ - see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 1 2 4 - 2

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

Fig. 1.

Fig. 2.

We write Qp = p ×]0; T [ where p is an open of R which physically represents the plate at rest,S 1 = 1 ×]0; T [ where 1 is a part of the uid boundary assumed to be xed, 2 = t∈]0;T [ 2 (t) where 2 (t) is the deformation of the plate at time t. p and 1 are assumed to be suciently smooth. In the Euclidean space R3 , we consider the band B = {x; t=x = (x1 ; x2 ) ∈ R2 ; 0 ≤ t ≤ T ¡∞}; and Q =

S

t =

t∈[0; T ]

[

t is a bounded open of this band with

]u(x; t); 1[;

x1 ∈ p

where u is the displacement within the structure. t is the intersection of Q with the hyperplane t = constant. We set  as the lateral boundary of Q. We de ne 0 (resp.

T ) as being the interior in R2 of the intersection of Q with t = 0 (resp. t = T ). We assume that section s = Q ∩ {t = s} continuously depends on s and is never empty.

s represents the domain occupied by the uid at time s. We write v; p and , the velocity, pressure and density in the uid. We consider an equation of the type p = k [1]. If we set all of the constants, with the exception of

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

uid viscosity, equal to 1 then problem (P) under study is as follows Fluid equations:  @v     − 
v + ∇ = 0 in Q;      @t       rot v = 0 in Q;        v · n = 0 on 1 ;    (F)    v(t = 0) = v0 in 0 ;          @  + div(v) = 0 in Q;    @t   (t = 0) = 0 in 0 : Coupling condition:  @u (C) − =v · n @t

on 2 :

Plate equations:  2 @2 u @ u   −
+
2 u +  = f   @t 2 @t 2     u = ∇u · n = 0 on @Qp ; (S)  u(t = 0) = u0 on p ;        @u (t = 0) = u on : 1 p @t

551

(1.1)

(1.2)

in Qp ; (1.3)

Remark 1. When solving (F), if we assume that the structure occupies a xed domain

p ; then the conservation of mass does not hold. Indeed; integrating Eq. (F)5 we obtain Z Z Z @u d  6= 0; = v · n = − dt t

p

p @t if we take into account the displacement of the structure; however; the uid occupies the domain [ ]u(x; t); 1[

t = x∈ p

and by using the derivation laws under an integral; we obtain Z Z Z Z @ @u d + v · n; = = dt t

t @t

p @t

p according to Eq. (C) we then arrive at Z d  = 0: dt t

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

In addition; if the domain is xed we must set  where v · n¡0 on 2 [2, 3] in order to obtain estimates but to set the density on the plate has no physical meaning. Conversely; we reveal that it is possible to obtain a priori estimates on uid velocity and on  log  if the motions of the structure and the uid geometry are taken into consideration. Remark 2. The de nition of the domain leads us to work with functional spaces de ned over a family of opens dependent on time. We must demonstrate that these spaces conserve a certain number of the classical functional space properties when the domain is not time dependent. We will de ne these spaces as Lp (Lq ( t )) or W m; p (W r; q ( t )) by analogy with the spaces Lp (0; T ; Lq ( )) and W m; p (0; T ; W r; q ( )). These spaces Lp (Lq ( t )) are endowed with the norm (Z

T

0

Z

q

p=q )1=p

|v|

t

from which we deduce the norms of the spaces W m; p (W r; q ( t )). As a general rule; when t is suciently smooth we use the classical projection operators of W ; (Q) in W ; (R3 ) to deduce properties of the spaces W m; p (W r; q ( t )). For example; for the space H 1 (H 2 ( t )) we use the projection operator which at v ∈ H 3 (Q) associates v ∈ H 3 (R3 ) where v has a compact support in R3 . In an analogous manner; to de ne the trace spaces, we return to R2+ by using local sharts and can subsequently; via projection operators; nd the trace results. Remark 3. It should be noted that to de ne the trace of a function v ∈ H s ( t ); the domain t must be of the class C s−1; 1 regardless of the value of t. The boundary must therefore be suciently smooth and u must belong to the class C s−1; 1 . In addition; 2 (t) can be de ned by applying u : p → 2 (t). If u is of the class C k ; then the boundary 1 ∪ 2 (t) of t is C k in terms of time. Also; as v · n is de ned in 2 (t) and ut in p ; we can thus assume that the function v · n is de ned in p where n; the normal of 2 (t); is the vector  n=

 @u ; −1 : @x1

In particular; we can verify that Z

t

@ @v = @t @t

t

Z v+

Z

Z

t

Z

div v =

p

v · n:

p

v

@u ; @t

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

553

Remark 4. As the problem is de ned for a time-dependent domain; it is solved using the Galerkin method but with test functions de ned on Q. In this way; by using the derivation formulas; when we write the energy equality; we obtain the following time derivatives: Z Z TZ 1 1 1 vt v = kv(T )k2L2 ( T ) − kv(0)k2L2 ( 0 ) − |v|2 g; 2 2 2 p

t 0 in order to obtain solutions for any given time; we elliptically smooth the momentum equation. We now give an existence result to problem (P). 2. An existence theorem We set W = {’ = ∇h ∈ H 1 ( t )2 ; ’ · n = 0 sur }. Thus, if the data are small enough 0 ∈ H 1 ( 0 );

0 ≥ 0;

v0 ∈ H 2 ( 0 ); u0 ∈ H02 ( p );

(2.1) (2.2)

u1 ∈ H01 ( p );

(2.3)

f ∈ L4 (Qp );

(2.4)

v0 · n0 = −u1 ;

(2.5)

then we obtain the following result. Theorem 2.1. If 0 ; v0 ; u0 and u1 satisfy the assumptions (2.1) – (2:5); then the problem (P) admits a unique solution (v; u) which satis es v · n=−

@u @t

in W 1; 4 (0; T ; H02 ( p )):

The proof of this theorem is performed in four steps. In Section 3 we solve problem (F) by setting v · n = g ∈ W 1; 4 (0; T ; H02 ( p )): The continuity equation is solved using the characteristics method, this last method being well adapted to this equation for a variable domain. Conversely, the momentum equation is solved by using an elliptic smoothing technique [8] coupled to the Galerkin method. By applying Kakutani’s xed point theorem between the continuity and momentum equations, we show that problem (F) admits a solution (v; ) in L∞ (L2 ( t )2 ) ∩ L2 (H 1 ( t )2 ) × L∞ (L1 ( t )):  is not smooth enough to give meaning to the trace of  nor does it demonstrate the uniqueness. Smoothness results are introduced in Section 4 (see also [3, 7]) which

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

establish that () has a meaning in W 1; 4 (W 3=4; 4 ( 2 (t))). These results allow us to solve problem (S) in Section 6 and nally, by reusing the Kakutani theorem between the problems (F) and (S), we prove in Section 7 the existence of a unique xed point (v; u) such that v · n=−

@u @t

in W 1; 4 (0; T ; H02 ( p )):

3. Study of uid equations 3.1. Homogeneous problem In order to obtain initial and boundary conditions which are homogeneous we introduce the problem (G)  @wg   − ∇ div wg = 0 in Q;   @t    wg · n = g ∈ W 1; 4 (H02 ( 2 (t))) on 2 ; (3.1) (G)   · n = 0 on  ; w  g 1     wg (t = 0) ∈ H 2 ( 0 )2 : To solve (G) we use the problem (R)  −∇ div Rg = 0 in Q;       Rg · n = g ∈ W 1; 4 (H 2 ( 2 (t))) 0 (R)    Rg · n = 0 on 1 ;    Rg (t = 0) = v0 :

on 2 ;

It should be noted that this last problem, which is easier to solve than Eq. (G), could have been used to homogenize (F) but this would have led to certain diculties and, in particular, in the obtainment of a priori estimates. In addition, as the injection of W 1; 4 (H02 ( t )) is continuous in C 0 (H02 ( t )), we show that Eq. (R) admits a solution in W 1; 4 (H 5=2 ( t )). Indeed, we can solve problem (R) for all values of t and we obtain a solution in H 5=2 ( t ). Also, the application at which g(t = s) ∈ H02 ( s ) associates Rg (t = s) ∈ H 5=2 ( s ) is continuous allows us to conclude as to the smoothness of solution Rg . Finally, in order to demonstrate that Eq. (G) admits one solution, we set w˜g = wg −Rg and we solve  @w˜g   − ∇ div w˜g = F in L4 (H 5=2 ( t ));    @t   rot w˜g = 0 in Q; ˜ (G)   w˜g · n = 0 on ;      w˜g (t = 0) = 0 in 0 ;

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

˜ 1 , that is to say by elliptically smoothing Eq. (G)  @2 w˜g @w˜g    − − ∇ div w˜g = F  → F +  2  @t @t        rot w˜g = 0 in Q;     w˜g · n = 0 on ;   (G˜ ) @w˜g   = 0 on ;   @t     w˜g (t = 0) = 0 in 0 ;       @w˜    g (T ) = 0 in T : @t

555

in L4 (H 5=2 ( t ));

We solve problem (G˜  ) by the Galerkin method and we show that the solution w˜g ˜ converges in L∞ (L2 ( t )) ∩ L2 (H 1 ( t )) towards the solution to problem (G). To examine the smoothness of w˜g we introduce a smooth base of X = {’ ∈ H 1 (Q); ’ = ∇h; ’(0) = 0; ’ · n = 0} ˜ in nite dimension and we consider problem (G)  @w˜gn   − ∇ div w˜gn = Fn → F dans L4 (H 5=2 ( t ));    @t   rot w˜gn = 0 dans Q; (G˜ n )   w˜gn · n = 0 sur ;      w˜gn (t = 0) = 0 dans 0 : We multiply Eq. (G˜ n )1 by
w˜gn |
w˜gn |2 and by w˜ gn; t |w˜gn; t |2 , respectively, and we then integrate with respect to Q. Young inequalities successively leads to k
w˜gn k4L4 (Q) ≤ C kFn k4L4 (Q) + k
w˜gn k4L4 (Q) + kw˜gn; t k4L4 (Q) ≤ C kFn k4L4 (Q) + kw˜gn; t k4L4 (Q) +

3 1 kw˜gn; t k4L4 (Q) + k
w˜gn k4L4 (Q) ; 4 4

1 3 kw˜gn; t k4L4 (Q) + k
w˜gn k4L4 (Q) : 4 4

By adding these two estimates and by setting  = 0:5, for example, we obtain (0:59 − )k
w˜gn k4L4 (Q) + (0:59 − )kw˜gn; t k4L4 (Q) ≤ C; thus if  is small enough we arrive at w˜g ∈ L4 (W 2; 4 ( t )) ∩ W 1; 4 (L4 ( t )): In conclusion, we nally obtain wg = Rg + w˜g ∈ W 1; 4 (H 5=2 ( t )) + L4 (W 2; 4 ( t )) ∩ W 1; 4 (L4 ( t )):

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

It should be noted that wg ∈ L4 (W 2; 4 ( t )) since H 5=2 ( t ) is included algebraically and topologically in W 2; 4 ( t ). We now set v = w + wg in Eq. (F) which gives us the following homogeneous problem (F0 )   @w   − 
w + ∇ = 0 dans Q;      @t      rot w = 0 dans Q;        w · n = 0 dans  = 1 ∪ 2 ;  0   (3.2) (F )     w(t = 0) = v0 − wg (0) = 0 dans 0 ;     (  @    + div(w) + div(wg ) = 0 dans Q;    @t (t = 0) = 0 ≥ 0 dans 0 : 3.2. An existence result for weak solutions of Eq. (F0 ) The existence of a weak solution to problem (F0 ) is obtained if 0 ∈ L1 ( 0 ); 0 log 0 ∈ L1 ( 0 ) ∈ L1 ( 0 );

(3.3)

0 ¿0 and if the value of g is small enough, i.e kgkL∞ (0; T ; L2 ( p )) ¡C;

(3.4)

where C is de ned in the proof. We then give the following result. Theorem 3.1. If 0 and g satisfy assumptions (3:3) – (3:4); then problem (F0 ) admits a unique solution (w; ) satisfying (w; ) ∈ L∞ (L2 ( t )2 ) ∩ L2 (W ) × L∞ (L1 ( t )) and the following estimates: 1 kwk2L∞ (L2 ( t )2 ) + A1 kwk2L2 (w) + 2

Z

t

 log  ≤ C

with A1 =  − C1 kg(t)kL∞ (0; T ; L2 ( p )) ¿0: The proof is made up of three points: • construction of approximate solutions and smoothing of the problem. • A priori estimates. • passage to the limit.

(3.5)

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

557

3.3. Approximate solutions Let us consider a base of L2 (W ) and of X = {w ∈ H 1 (Q) | w · n = 0; w(0) = 0} which shall be given the notation {w1 ; : : : ; wi ; : : :} and for which each function belongs to H 4 (Q). We de ne Xn as being the set of linear combinations of the n rst functions belonging to this base. We introduce the following approximate problem: determine wn ∈ Xn and n ∈ C 1 (Q) which satisfy  R  an (wn ; v) = T (n ; div v) ∀v ∈ Xn ; 0 (Vn )  w (0) = 0 in

n 0 with Z an (wn ; v) =

wn =

n X



T

0

 Z T @wn ;v +  (div wn ; div v); @t 0

ai wi (x; t)

i=1

and

(Rn )

 @n   + div((wn + wg; n )n ) = 0;   @t     n (t = 0) = 0n ∈ Cc1 ( 0 );       ¿0; 0n

  0n → 0 in L0 ( 0 );       0n log 0n → 0 log 0      wgn → wg in C 2 (Q):

in L1 ( 0 );

To obtain approximate solutions, we disturb problem (Vn ) by introducing an elliptic smoothing operator. We thus obtain problem (Vn ): determine wn ∈ Xn and n ∈ C 1 (Q) which satis es n RT an (wn ; v) = 0 (n ; div wn ) ∀v ∈ Xn ; (Vn ) where Z an (wn ; v) = 

0

T



@wn @v ; @t @t



Z +

0

T



 Z T @wn ;v +  (div wn ; div v) @t 0

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

and

(Rn )

 @n   + div((wn + wg; n )n ) = 0;   @t     (t = 0) =  ∈ C 1 ( );  ¿0; n

0n

c

  0n → 0 in L1 ( 0 );      0n log 0n → 0 log 0

0

0n

in L1 ( 0 ):

This problem is solved by a xed point method, (Rn ) being solved by the characteristics method such that (wn ; n ) satis es Eq. (3.5). Indeed, we shall show that in order to obtain an estimate on n log n , we use the continuity equation which must be exactly veri ed. As a result, the characteristics method is used. This would not be the case if the Galerkin method was used to solve the continuity equation. Lemma 3.2. The smoothed problem admits a unique solution (wn ; n ) ∈ Xn × C 1 (Q) and n ≥ 0. Proof. We eliminate the index n for this demonstration. We begin by setting w = w∗ in (Rn ) where w∗ is a given function in Xn . We then introduce T1 , the map which at w∗ ∈ Xn associates  ∈ L∞ (L 2 ( t )), the solution to problem (Rn ), and the map T2 which at  ∈ L∞ (L 2 ( t )) associates w ∈ Xn , a solution of problem (Vn ). We then de ne the map to be T = T2 ◦ T1 and we will show that T satis es the hypotheses necessary to the application of Kakutani’s xed point theorem. We solve problem (Rn ) using the characteristics method in the following manner:

The curve t 7→ (X (x; t); t) are termed the characteristics of origin x. Two types of characteristics exist, those originating from a point x in 0 and, since we take the plate motions into account, those originating from a point of the boundary and which remain on the boundary. We can then write ∀x ∈ 0  Z t  ∗  (X (x; t); t) = 0 exp − (div w + div wg; n )(X (x; t); t) dt ; 0

which demonstrates that if 0 ≥ 0 then  ≥ 0.

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

559

In addition, by writing the energy inequality we obtain Z Z @ div( (w∗ + wg; n )) = 0  +

t @t

t that is to say 1 1 d k k2L 2 ( t ) − 2 dt 2 hence, d k k2L 2 ( t ) + dt

Z

t

Z

p

| | 2 g −

1 2

Z

t

(w∗ + wg; n )∇| | 2 +

Z

p

gn | | 2 = 0;

div(w∗ + wg; n )| | 2 = 0:

Owing to the smoothness of the base functions we obtain div(w∗ + wg; n ) ∈ C 1 (Q) ⊂ L (Q) which, according to [4], implies  ∈ C 1 (Q). We can then apply the Gronwall lemma which leads to Z T kdiv(w∗ + wg; n )kL∞ ( t ) : (3.6) k k2L∞ (L 2 ( t )) ≤ k0 kL2 2 ( 0 ) exp ∞

0

at

In order to solve problem (Vn ), we replace the test functions v by w and we arrive

@w 2 1 2 2



@t 2 + 2 kw kL∞ (L 2 ( t )) + kdiv w kL 2 (Q) L (Q) Z Z Z TZ 3 T =  div w + |w | 2 gn : 2

t

p 0 0

The right-hand side terms are estimated in the following manner: Z TZ T2 k k2L∞ (L 2 ( t )) ;  div w ≤ kw kX2n + 

t 0 Z TZ |w | 2 gn ≤ C1 kgn kL∞ (0;T ; L 2 ( p )) kw kX2n : 0

p

We thus arrive at the estimate (min(; ) −  − C1 kgn kL∞ (0;T ; L 2 ( p )) )kw kX2n ≤

T2 k kL2∞ (L 2 ( t )) 

and by taking into consideration estimate (3.6) on  we obtain Z T kdiv(w∗ + wg; n )kL∞ ( t ) ; kw kX2n ≤ K1 T 2 exp 0

hence, kw kX2n ≤ K10 T 2 exp

Z 0

T

kw∗ kXn + K20 ;

(3.7) (3.8)

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

K10 and K20 being positive constants which depend on the problem’s data. We can then apply Kakutani’s theorem for which we refer the reader to [2]. Thus, the problem w = T (w ) possesses a solution in Xn . As the functions of the base of X belong to H 4 (Q), we have div w ∈ C 1 (Q) and  ∈ C 1 (Q). We then demonstrate that when  approaches 0, 

@ 2 wn →0 @t 2

in L 2 (Q);

wn → wn

in L 2 (W ) weak;

n → n

in L∞ (L 2 ( t )) weak star:

In addition, as we are dealing with a nite dimension, the passage to the limit of div(n wn ) towards div(n wn ) does not present any diculties. Remark 5. The conservation of the uid mass is valid. Indeed Z Z Z Z d @n = n + gn n = wg; n ·nn ; dt t

t @t

p

p hence, d dt

Z

t

n = 0:

3.4. A priori estimates We have the following result. Lemma 3.3. The solution to the approximate problem satis es the estimate Z n log n ¡C: kwn kL2∞ (L 2 ( t )) + A1 kwn kL2 2 (W ) + sup t

t

(3.9)

Proof. We can multiply Eq. (F0n )1 by wn to the scalar product of L 2 (Q) sense. We then obtain Z Z Z tZ 3 t 1 n div wn + |wn | 2 gn : (3.10) kwn kL2∞ (L 2 ( t )) + kdiv wn kL2 2 (Q) = 2 2

t

p 0 0 The main diculty lies in the estimation of the rst right-hand side term. To overcome this diculty, we use the continuity equation. We have Z tZ Z 1 Z tZ Z 1 ∇n n div wn = − (wn n ) n

p u

p u 0 0 =

Z tZ 0

p

Z u

1

log n div(wn n );

(3.11)

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

561

by recalling that div(wn n ) = −@n =@t − div(wg; n n ); however, we arrive at Z tZ 0

Z

p

1

u

n div wn = − =−

+

Z tZ

Z

p

0

Z

p

Z

p

0

1

Z

p

0

1

u

div(wg; n n ) log n

@ (n log n − n ) @t

u

Z tZ

Z tZ

@n log n − @t

u

Z tZ 0

1

1

wg; n n ∇(log n ) −

u

Z tZ

p

0

gn n log n : (3.12)

By using derivation formulas and Remark 5, Eq. (3.12) becomes Z tZ 0

Z

p

1

u

Z n div wn = − −

t

0

Z

0

=−

p

Z

p

1

u

Z tZ 0

Z

p

Z tZ Z

−

@ @t

Z n log n +

n gn +

Z tZ 0

u

1

Z u

Z

Z

p

0

1

p

n log n + Z

p

u

1

t

@ @t

Z

Z

p

u

1

n

wg; n ∇n

1

u0

0n log 0n

n div wg; n ;

(3.13)

where we note that no trace terms appear. Eq. (3.10) is therefore written as Z 1 2 kwn kL∞ (L 2 ( t )) + kdiv wn kL 2 (Q) + sup n log n 2 t

t =

3 2

Z t

0

Z

p

|wn | 2 g +

Z

p

Z

1

u0

0n log 0n −

Z tZ 0

p

Z u

1

n div wg; n :

(3.14)

The rst term of the right-hand side of the equation is estimated as in Eq. (3.8) while the last term is estimated as follows: Z Z |n div wgn | ≤ Ckdiv wg kL1 (L∞ ( t )) 0 : (3.15) Q

t

Finally, Eq. (3.14) becomes 1 kwn k2L∞ (L 2 ( t )) + A1 kdiv wn kL2 2 (Q) + sup 2 t where C is an independent constant of n.

Z

t

n log n ≤ C;

(3.16)

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

It should be noted that we can obtain this estimate only if A1 =  − C1 kgkL∞ (L 2 (

2 (t)))

¿0;

which imposes a constraint on the data. 3.5. Passage to the limit We present a lemma which allows us to pass to the limit in the continuity equation. This lemma has been demonstrated by [9] for a time-independent domain but it, nevertheless, satis es the conditions of the present study. If n ; wn and wg; n , three sequences which satisfy the following conditions: wn ∈ H 4 (Q) and converges weakly towards w in L 2 (H 1 ( t ) 2 ); wg; n ∈ H 4 (Q) and converges strongly towards wg in L 2 (H 1 ( t ) 2 ); n and n log n are bounded in L∞ (L1 ( t )) and @n + div(n wn ) + div(n wg; n ) = 0: @t We thus obtain the following lemma: Lemma 3.4. We can extract sub-sequences from and n and wn which satisfy Z Z n  dx dt →  dx dt ∀ ∈ L1 (L∞ ( t )); Qf

Qf

wn n is bounded in L2 (L1 ( t )2 ); wn n → w in L1 (Q): Remark 6. Let us note that the estimate  bounded in L∞ (L1 ( t )) does not allow us; once the approximate solution have been constructed; to pass to the limit. The estimate  log  bounded in L∞ (L1 ( t )) is very useful as it prevents the occurrence of diculties linked to the weak topology of L1 ( t ). The passage to the limit in the momentum equation does not present any diculties. In the next paragraph, we present a smoothness result which gives a meaning to the trace  and which demonstrates the uniqueness of the solution. 4. A smoothness result We have the following result: Lemma 4.1. If (@0 =@xi=1; 2 ) ∈ L4 ( 0 ); under the assumptions of Theorem 2.1  ∈ L∞ (W 1; 4 ( t )) ⊂ L∞ (Q); w ∈ L4 (W 2; 4 ( t ));

() ∈ L4 (W 3=4; 4 ( (t))):

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

563

Proof. Let us take the derivative of the continuity equation with respect to xi with i = 1; 2 and then let us multiply by 3 @ @ : @xi @xi By integrating with respect to Q, we obtain

4 4 Z 4 Z Z 4 @

@ @ @

@ g−4 − g=3 v∇ − 4

@t @xi L4 ( t ) @xi

p @xi

t

p @xi 2   Z @v @ @ div  ; × @xi @xi @xi

t hence,

4   Z 4 Z

@ @v @ @

@ =3 div  div v − 4 @t @xi L4 ( t ) @xi @xi

t @xi

t

2 @ : @xi

(4.1)

If we write the last term of Eq. (4.1) for i = 1 and i = 2, and then if we add the expressions thus obtained we arrive at 2 2 !     Z @v @ @ @v @ @  div + div A = A 1 + A2 = @x1 @x1 @x1 @x2 @x2 @xi

t 2 2 ! Z @v @ @ @v @ @ : ∇ + + @x1 @x1 @x1 @x2 @x2 @xi

t We estimate A1 in the following manner: !







@ 3

@ 3

@v @v





+ div A1 ≤ kkL∞ ( t ) div ; @x1 L4 ( t ) @x1 L4 ( t ) @x2 L4 ( t ) @x2 L4 ( t ) but kkL∞ ( t ) ≤ CkkW 1; 4 ( t ) ≤ C thus, A1 ≤ C

!



@

@



+ ;

@x1 4 @x2 L4 ( t ) L ( t )

!





@v @v

div

+

@x1 4

div @x2 4 L ( t ) L ( t )

!



@ 4

@ 4

+ :

@x2 4

@x1 4 L ( t ) L ( t )

In much the same manner, term A2 leads to ! !





@ 4

@v

@v

@ 4



: + + A2 ≤ C

@x1 4

@x1 4

@x2 4

@x2 4 L ( t ) L ( t ) L ( t ) L ( t )

564

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

Thus, by using these estimates in Eq. (4.1) and by setting:



@ 4

@ 4

(t) + (t) ; y(t) =

@x2 4

@x1 4 L ( t ) L ( t ) we obtain

@v

+ @x2 L4 ( t ) L4 ( t ) !





@v @v

+ + y:

div @x1 4

div @x2 4 L ( t ) L ( t )

@y ≤C @t



@v

kdiv vkL4 ( t ) + @x1

(4.2)

By proceeding in the same manner as was seen for w˜ g in Section 3, however, we demonstrate that k
wk4L4 ( t ) ≤ Ck∇k4L4 ( t ) ≤ C 0 y(t):

(4.3)

Finally, by bringing estimate (4.2) to inequality (4.3), and by recalling that
wg ∈ L4 (W 2; 4 ( t )), we obtain Z t 0 y()(1 + log(1 + y()); y(t) ≤ C + C 0

which using the Gronwall lemma, leads to  bounded in L∞ (W 1; 4 ( t )). Moreover, this last result reveals that the trace of  is de ned in L∞ (W 3=4; 4 ( (t))) and that, using (4.3), w ∈ L4 (W 2; 4 ( t )). 5. A uniqueness result for the solution to problem (F) Let us recall that, for reasons of simplicity, we seek the velocity under the form of a potential, that is to say that w = ∇h. The momentum equation is thus written as follows: ∇(ht − 
h + ) = 0

in Q;

hence ht − 
h +  = (t)

in Q

(5.1)

with (t) being a function which depends only on time. Theorem 5.1. Based on the hypotheses of Theorem 2.1 and on Lemma 4.1, problem (F) admits a unique solution (v; ) which satis es: (v; ) ∈ L4 (W 2; 4 ( t )) × L∞ (W 1; 4 ( t )):

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

565

Proof. We consider two solutions to problem (F0 ), (∇h1 ; 1 ) and (∇h2 ; 2 ). These solutions satisfy hi; t − 
hi + i = ; i; t + div(i wi ) + div(i wg ) = 0: The functions w = ∇h1 − ∇h2 = ∇h and  = 1 − 2 then satisfy the equations ht − 
h +  = 0;

(5.2)

t + div(1 w + w2 ) + div(wg ) = 0:

(5.3)

We de ne the auxiliary function problem:
=

which is the solution to the following Neumann

in t ;

∇ · n=0

t:

in

By multiplying Eq. (5.2) by h and Eq. (5.3) by respect to t , we arrive at Z Z Z h ht −  h
h + h = 0;

t

Z

Z

t

t +

t

t

t

and then by integrating with (5.4)

Z

div(w1 + w2 ) +

div(wg ) = 0:

t

(5.5)

The rst term of Eq. (5.5) can be written as follows: Z Z Z Z Z Z @ @ @ @ 1
= ht =
− g
+ (
)− |
|2 @t @t @t 2 @t

t

t

t

t

p

t =−

1 @ 2 @t

Z

t

|∇ |2 −

Z

p

g
−

1 2

Z

g

g|∇ |2 :

(5.6)

In much the same way, the last term of Eq. (5.5) is written as Z Z Z div(wg ) =
wg ∇ − g
−

t

t

p

Z =−

t

(∇ · ∇)wg ∇ −

1 2

(∇ · ∇)wg ∇ +

1 2

Z =−

t

−

1 2

Z

p

g|∇ |2 −

Z

p

Z

t

Z

t

wg ∇|∇ |2 −

Z

p

g


div wg |∇ |2

g
;

(5.7)

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F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

nally, using expressions (5.6) and (5.7), equality (5.5) becomes 1 @ 2 @t

Z

t

|∇ |2 =

Z

Z div(w1 + w2 ) +

t

1 − 2

Z

t

t

(∇ · ∇)wg ∇

div wg |∇ |2 ;

(5.8)

where no trace terms appear. By adding Eq. (5.4) and (5.8) we thus obtain 1 1 d (khk2L2 + k∇ k2L2 ) + k∇hk2L2 = 2 dt 2

Z

g|h|2 −

p

Z

t

Z − Z +

h
−

1 2

Z

t

div wg |∇ |2

Z

t

t

∇ w1 +

t

div (w2 )

(∇ · ∇)wg ∇ :

(5.9)

The rst term of Eq. (5.9) is estimated in the following manner: 1 2

Z

p

g|h|2 ≤

Cs kgkL∞ k∇hk2L2 : 2

(5.10)

If we set y = khk2L2 + k∇ k2L2 , then the three following terms lead to: Z

h
≤ k∇hk2L2 + C k∇ k2L2 ≤ k∇hk2L2 + C y;

t

1 2

Z

t

Z

t

div wg |∇ |2 ≤ Ckdiv wg kL∞ k∇ k2L2 ≤ Cy;

∇ w1 ≤ k∇hk2L2 + C k1 k2L∞ k∇ k2L2 ≤ k∇hk2L2 + C y:

The second to last term of Eq. (5.9) can be written as Z

Z

t

div(w2 ) = − Z

t


w2 ∇

1 (∇ · ∇)w2 ∇ − = 2

t

Z

t

div w2 |∇ |2 :

(5.11)

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

567

However, as  is a bounded function, then ∇ satis es |∇ | ≤ K. Consequently, the term ((∇ · ∇)w2 ; ∇ )L2 can be estimated by kdiv w2 kL1= ≤ K 2 kdiv w2 kL1= y1− ; K 2 k∇ kL2(1−) 2 where  is a positive number. In addition, by multiplying Eq. (5.1) by (
h)(1−)= and if  is small enough, we obtain the following estimate: kdiv w2 kL1= ≤ C; which allows us to estimate the second term of Eq. (5.11) under the form Cy1− . Finally, the last element of Eq. (5.9) is estimated in the same manner as two previous elements: Z (∇ · ∇)wg ∇ ≤ Cy1− :

t

Finally, relation (5.9) becomes   CS dy  2 2 ≤ C1 y + C2 y1− + 2  − 2 − kgkL∞ k∇hk L dt 2 and if  and the data value g are small enough, we obtain dy  ≤ C1 y  + C2 : dt Thus, by using the Gronwall–Bellmann lemma we obtain y(t) ≤ (C2 )1= exp(C1 t) and, as the right-hand term converges towards zero when  converges towards zero, this proves that y(t) is equal to zero which completes the proof of this theorem. 6. Study of the plate equations Classically, the weak formulation associated with the plate equations is: nd u ∈ W 1; ∞ (0; T ; H01 ( p )) ∩ L∞ (0; T ; H02 ( p )) which satis es (utt ; ) + (∇utt ; ∇ ) + (
u;
) = −(; ) + (f; )p

∀ ∈ H02 ( p ):

Since the trace of  has a meaning in L∞ (0; T ; W 3=4; 4 ( p )), it is easy to demonstrate that if u0 ∈ H02 ( p ), u1 ∈ H01 ( p ) and f ∈ L2 (0; T ; H −1 ( p )), then this problem admits a unique solution in W 1; ∞ (0; T ; H01 ( p )) ∩ L∞ (0; T ; H02 ( p )). To demonstrate the existence of a xed point such as wg · n = −u, we need smoothness results on u. To this end, we search u under the approximate form u=

n X i=1

ui (t) i ;

568

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

where i is a solution to the spectral problem
2 i = i i i =

in p ;

@ i =0 @n

on p :

2 2 Let us note that that { i }∞ i=1 is a base of H0 ( p ) orthonormal in L ( p ).

Lemma 6.1. If f ∈ L4 (Qp ); then u satis es the estimate kuk4w1; 4 (0; T ; H 2 ( p )) ≤ M0 + T 4 C 0 (1 + kgk4w1; 4 (0; T ; H 2 ( p )) ); 0

(6.1)

0

where M0 is a constant which is solely dependent on the data. Proof. Let us successively multiply the plate equation by utt |utt |2 ,
utt |
utt |2 and
2 u|
2 u|2 and then let us integrate with respect to Qp . We thus obtain 3 kutt k4L4 (Qp ) ≤ C (kk4L4 (Qp ) + kfk4L4 (Qp ) ) + (k
utt k4L4 (Qp ) + k
2 uk4L4 (Qp ) ) 4   1 kutt k4L4 (Qp ) ; + 2 + 2 3 k
utt k4L4 (Qp ) ≤ C (kk4L4 (Qp ) + kfk4L4 (Qp ) ) + (kutt k4L4 (Qp ) + k
2 uk4L4 (Qp ) ) 4   1 k
utt k4L4 (Qp ) ; + 2 + 2 3 k
2 uk4L4 (Qp ) ≤ C (kk4L4 (Qp ) + kfk4L4 (Qp ) ) + (kutt k4L4 (Qp ) + k
utt k4L4 (Qp ) ) 4   1 k
2 uk4L4 (Qp ) : + 2 + 2 By adding these inequalities and by setting  = 0:75, for example, we obtain (0:1224 − 2)(kutt k4L4 (Qp ) + k
utt k4L4 (Qp ) ) ≤ C (kk4L4 (Qp ) + kfk4L4 (Qp ) ) and we can always nd a value of  small enough to render positive the left-hand member. Thus, kut k4W 1; 4 (0; T ; H 2 ( p )) ≤ M0 + T 4 kk4L4 (W 1; 4 ( t )) ; 0

Eq. (5.4), however, leads to kk4L4 (W 1; 4 ( t )) ≤ C(1 + k
wg k4L4 (Q) ) ≤ C 0 (1 + kgk4W 1; 4 (0; T ; H 2 ( p )) ); 0

therefore, kut k4W 1; 4 (0; T ; H 2 ( p )) ≤ M0 + T 4 C 0 (1 + kgk4W 1; 4 (0; T ; H 2 ( p )) ): 0

0

F. Flori, P. Orenga / Nonlinear Analysis 38 (1999) 549 – 569

569

7. Theorem proof We apply the Kakutani theorem. To this end we note that if the data and T are weak enough and if kgkW 1; 4 (0; T ; H02 ( p )) ≤ K with K being well chosen, the map  : g ∈ E = W 1; 4 (0; T ; H02 ( p )) → ut ∈ E applies the ball B(0; K) of E with centre 0 and radius K within itself. That is to say, (B(0; K)) ⊂ B(0; K). E is a re exive and separable banach space. Consequently, B(0; K) is metrizable for the weak topology and, nally, compact for this same topology. We are therefore able to work with ordinary sequences. We will demonstrate that if we consider gn as a weak convergent sequence towards g in B(0; K) and such that weakly

un; t = (gn ) −→ t ; then t = (g) = ut . If (w; n ) is a solution associated to Eq. (F) with w(0) = wn (0) and (0) = n (0), then (wn ; n ) and (n ) weakly converge towards a limit ! in L∞ (0; T ; W 3=4; 4 ( p )). As problem (F) admits at most one solution, then ! = (). As a result for problem (S) we have (un; tt ; )p + (∇un; tt ; ∇ )p + (
un ;
)p = −(n ; )p + (fn ; )p ; which converges weakly towards (tt ; )p + (∇tt ; ∇ )p + (
;
)p = −(; )p + (f; )p = (H; )p ; weakly

when gn −→ g in B(0; K). Finally, as problem (S) has only one solution for a given second member H ∈ L2 (Qp ), then  is the unique solution associated with Eq. (S) and, hence, t = ut . This last result demonstrates that we indeed have a unique xed point and that w · n = g = −ut . References [1] M. Bruneau, Introduction aux theories de l’acoustique, Publications de l’Universite du Maine, 1983. [2] F.J. Chatelon et P. Orenga, On a non-homogeneous shallow water problem, M2 AN, 1997. [3] F.J. Chatelon et P. Orenga, Some smoothness and uniqueness results for a shallow water problem, Adv. Diernetial Equations 1997, a paraˆtre. [4] R.J. Di Perna et P.L. Lions, Ordinary dierential equations, transport theory and sobolev spaces, Invent. Math. 98 (1989) 511–547. [5] F. Flori et P. Orenga, On a uid-structure interaction problem in velocity–displacement formulation, M3 AS 8(4) (1998) 543–572. [6] F. Flori et P. Orenga, Analysis of a nonlinear uid-structure interaction problem in velocity–displacement formulation, Nonlinear Anal. TMA (1999) to appear. [7] A. Kazhikov, Some new statements for initial boundary value problems for Navier–Stokes e quations of viscous gas, Lecture notes in Math. 1995 In Progress in Theoretical and Computational Fluid Mechanics, ed. G.P. Galdi, J. Macek and J. Necas. Longman, Haulow, 1994. [8] J.L. Lions, Sur certaines e quations paraboliques non lineaires, Bull. Soc. Math. France 93 (1965) 155–175. [9] P. Orenga, Un theoreme d’existence de solutions d’un probleme de Shallow water, Arch. Ration. Mech. Anal. 130 (1995) 183–204.