A P ] 2 0 Se p 20 11 A global attractor for a fluid – plate interaction model

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier–Stokes equations in a bounded domain and a classical (nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We show that this problem generates a semiflow on appropriate phase space. Our main result states the existence of a compact finite-dimensional global attractor for this semiflow. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system. To achieve the result we first study the corresponding linearized model and show that this linear model generates strongly continuous exponentially stable semigroup.


Introduction
We consider a coupled (hybrid) system which describes interaction of a homogeneous viscous incompressible fluid which occupies a domain O bounded by the (solid) walls of the container S and a horizontal boundary Ω on which a thin (nonlinear) elastic plate is placed.The motion of the fluid is described by linearized 3D Navier-Stokes equations.To describe deformations of the plate we consider a generalized plate model which accounts only for transversal displacements and covers a general large deflection Karman type model (see, e.g., [24,25,26] and also [15] and the references therein).However, our results can be also applied in the cases of nonlinear Berger and Kirchhoff plates (see the discussion in Section 4.1).
This fluid-structure interaction model assumes that large deflections of the plate produce small effect on the fluid.This corresponds to the case when the fluid fills the container which is large in comparison with the size of the plate.
We note that the mathematical studies of the problem of fluid-structure interaction in the case of viscous fluids and elastic plates/bodies have a long history.We refer to [9,19,20,21,22] and the references therein for the case of plates/membranes, to [16] in the case of moving elastic bodies, and to [1,2,3,6,7,18] in the case of elastic bodies with the fixed interface; see also the literature cited in these references.
Our mathematical model is formulated as follows.
Let O ⊂ R 3 be a bounded domain with a sufficiently smooth boundary ∂O.We assume that ∂O = Ω ∪ S, where Ω ∩ S = ∅ and Ω ⊂ {x = (x 1 ; x 2 ; 0) : with the smooth contour Γ = ∂Ω and S is a surface which lies in the subspace R 3 − = {x 3 ≤ 0}.The exterior normal on ∂O is denoted by n.We have that n = (0; 0; 1) on Ω.We consider the following linear Navier-Stokes equations in O for the fluid velocity field v = v(x, t) = (v 1 (x, t); v 2 (x, t); v 3 (x, t)) and for the pressure p(x, t): where ν > 0 is the dynamical viscosity and G f (t) is a volume force (which may depend on t).We supplement (1) and (2) with the (non-slip) boundary conditions imposed on the velocity field v = v(x, t): v = 0 on S; v ≡ (v 1 ; v 2 ; v 3 ) = (0; 0; u t ) on Ω.
Here u = u(x, t) is the transversal displacement of the plate occupying Ω and satisfying the following equation (see, e.g., [8,24,25,26] and the references therein): where G pl (t) is a given body force on the plate, F(u) is a nonlinear feedback force which would be specified later and T f (v) is a surface force exerted by the fluid on the plate, T f (v) = (T n| Ω , n) R 3 , where n is a outer unit normal to ∂O at Ω and T = {T ij } 3 i,j=1 is the stress tensor of the fluid, Since n = (0; 0; 1) on Ω, we have that T f (v) = 2ν∂ x 3 v 3 − p.It also follows from (2) and (3) that ∂ x 3 v 3 = 0 on Ω and thus we arrive at the equation We impose clamped boundary conditions on the plate u| ∂Ω = ∂u ∂n ∂Ω = 0 (5) and supply (1)- (5) with initial data of the form We note that (2) and (3) imply the following compatibility condition Ω u t (x ′ , t)dx ′ = 0 for all t ≥ 0.
This condition fulfills when Ω u(x ′ , t)dx ′ = const for all t ≥ 0, which can be interpreted as preservation of the volume of the fluid.We also note that a similar class of models was considered before in [11,19,20,21].The main difference between (1)- (6) and models in these publications is that the papers mentioned deal only with longitudinal deformations of the plate neglecting transversal deformations (in contrast with the model (1)-( 6) which takes into account the transversal deformations only).This means that instead of (3) the following boundary conditions are imposed on the velocity fluid field: where u = (u 1 (x, t); u 2 (x, t)) is the in-plane displacement vector of the plate which solves the wave equation of the form This kind of models arises in the study of blood flows in large arteries (see the references in [19]).The model ( 1), ( 2), ( 8), ( 9) is simpler in several respects.One of them is related to the fact the force exerted on the plate by the fluid is more regular in the case (9) and does not contains the pressure in an explicit form.Moreover, the model ( 1), ( 2), ( 8), (9) does not require any compatibility conditions like (7), because the volume of the fluid obviously preserves in the case of longitudinal deformations.
In this paper our main point of interest is well-posedness and long-time dynamics of solutions to the coupled problem in (1)-( 6) for the velocity v and the displacement u.First we consider the linear version of this problem (i.e., the case when F(u) ≡ 0).For this linear version we prove well-posedness in the class of weak (energy) solutions and establish some additional properties of solutions which we need for treating the nonlinear problem.In particular, we show that in the homogeneous case (G f ≡ 0, G pl ≡ 0) the linear version generates strongly continuous exponentially stable semigroup.Then we consider a nonlinear version of this problem under rather general hypotheses concerning nonlinearity.These hypotheses cover the cases of von Karman, Berger and Kirchhoff plates.We show that problem (1)-( 6) generates a dynamical system in an energy type space.Our main result (see Theorem 4.8) states that under some natural conditions concerning feedback forces system ( 1)-( 6) possesses a compact global attractor of finite fractal dimension.
To establish this results we rely on recently developed approach (see [13], [14] and [15,Chapters 7,8] and also the references therein) which involves stabilizability estimates and notion of a quasi-stable system.
The paper is organized as follows.In Section 2 we introduce notations, recall some properties of Sobolev type spaces with non-integer indexes on bounded domains and collect some regularity properties of (stationary) Stokes problem which we use in the further considerations (see Proposition 2.2).Section 3 is devoted to a linear version of the problem.Our main result in this section is Theorem 3.3 on well-posedness of weak solutions.In Section 4 we deal with the nonlinear problem (1)- (6).First we prove well-posedness result in Theorem 4.3 and then show that in the case of autonomous forces the problem generates a gradient dynamical system.Our main result in this section states existence of a finite dimensional global attractor and describes some regularity properties of the trajectories from the attractor.The argument is based on the quasi-stability property established in Proposition 4.10.

Preliminaries
In this section we introduce Sobolev type spaces we need and provide with some results concerning to Stokes problem.

Spaces and notations
To introduce Sobolev spaces we follow approach presented in [33].
Let D be a sufficiently smooth domain and s ∈ R. We denote by H s (D) the Sobolev space of order s on a set D which we define as restriction (in the sense of distributions) of the space H s (R d ) (introduced via Fourier transform).We denote by • s,D the norm in H s (D) which we define by the relation We also use the notation (D) for every m = 0, 1, 2, . . ., and for s = m + σ with 0 < σ < 1 we have , where d(x, ∂D) is the distance between x and ∂D.The norm • * s,D is equivalent to • s,D in the case when s > −1/2 and s − 1/2 ∈ {0, 1, 2, . ..}, but not equivalent in general.
Understanding adjoint spaces with respect to duality between C ∞ 0 (D) and [C ∞ 0 (D)] ′ by Theorems 4.8.1 and 4.8.2 from [33] we also have that Below we also use the factor-spaces H s (D)/R with the naturally induced norm.
To describe fluid velocity fields we introduce the following spaces.
Let C (O) be the class of C ∞ vector-valued solenoidal (i.e., divergencefree) functions v = (v 1 ; v 2 ; v 3 ) on O which vanish in a neighborhood of S and such that v 1 = v 2 = 0 on Ω.We denote by X the closure of C (O) with respect to the L 2 -norm and by V the closure with respect to the H 1 (O)norm.One can see that We equip X with L 2 -type norm • O and denote by (•, •) O the corresponding inner product.The space V is endowed with the norm For some details concerning this type spaces we refer to [32], for instance.We also need the Sobolev spaces consisting of functions with zero average on the domain Ω, namely we consider the space

Stokes problem
In further considerations we need some regularity properties of the terms responsible for fluid-plate interaction.To this end we consider the following Stokes problem where g ∈ [L 2 (O)] 3 and ψ ∈ L 2 (Ω) are given.This type of boundary value problems for the Stokes equation was studied by many authors (see, e.g., [23] and [32] and the references therein).We collect some properties of solutions to (10) in the following assertion.
Proposition 2.2 With the reference to problem (10) the following statements hold. ( (2) In particular, we can define a linear operator by the formula for ψ ∈ L 2 (Ω) (N 0 ψ solves (10) with g ≡ 0).It follows from ( 11) and ( 12) that (3) Let g ∈ [H −1/2+σ (O)] 3 and ψ ∈ H σ * (Ω), with 0 < σ ≤ 1/2.Then we can define the trace of the pressure p on Ω, which possesses the property p| Ω ∈ H −1+σ (Ω)/R and Proof.Since the extension of elements from H σ * (Ω) by zero to the whole boundary ∂O do not change the smoothness Sobolev class, i.e., leads to elements from H s (∂O), we can use the regularity results available for the Stokes problem with the Dirichlet type boundary conditions imposed on the whole ∂O (see, e.g., [23,32] and also the paper [17] and the references therein).This observation leads to the following arguments.
1.The existence and uniqueness of solutions along with the bound in (11) follow from Proposition 2.3 and Remark 2.6 on Sobolev norm's interpolation in [32,Chapter 1].
3. We first represent v in the form v = v + v * , where v solves (10) with ψ ≡ 0 and v * satisfies (10) with g ≡ 0. Let p and p * be the corresponding representatives of the pressure (which are identified with an element in a factor-space).By the first statement we have that p ∈ H 1/2+σ (O) and thus by the standard trace theorem there exists p| ∂O ∈ H σ (∂O).This implies that p| In the case g ≡ 0 the pressure p * is a harmonic function in O which belongs H −1/2+σ (O).This allows us to assign a meaning to be the extension of φ by zero.Then by the trace theorem there exists a smooth function w φ on O such that The application of Green's formula yields (p * , ∆w φ Thus relation (14) follows from ( 15) and ( 16).

Linear problem
In this section we consider a linear version of (1)-( 6) which is obtained from (1)-( 6) by replacing equation ( 4) with its linear version.Thus we deal with the following problem which we supply with the initial data of the form To define weak (variational) solutions we need the following class L T of test functions φ on O: ) is said to be a weak solution to the problem in ( 17)-( 21) on a time interval [0, T ] if ) and u(0) = u 0 ; • for every φ ∈ L T the following equality holds: • the compatibility condition v(t)| Ω = (0; 0; u t (t)) holds for almost all t.
Remark 3.2 (1) It follows from the compatibility condition and the standard trace theorem that u t ∈ L 2 (0, T ; H (2) Taking in (22) where χ is a smooth scalar function and ψ belongs to the space one see that the weak solution (v(t); u(t)) satisfies the relation for almost all t ∈ [0, T ] and for all ψ = (ψ 1 ; ψ 2 ; ψ 3 ) ∈ W , where β = ψ 3 Ω .
Below as a phase space we use with the norm (u 0 ; We also denote by H a subspace in H of the form Our main result in this section is the following well-posedness theorem concerning the linear problem. Then for any interval [0, T ] there exists a unique weak solution (v(t); u(t)) to ( 17)-( 21) with the initial data U 0 .This solution possesses the property 27) and satisfies the energy balance equality for every t > 0, where the energy functional E 0 is defined by the relation Moreover, there exist positive constants M and γ such that for every initial data U 0 = (v 0 ; u 0 ; u 1 ) from H we have Remark 3.4 Let w 0 ∈ (I − P )H 2 0 (Ω), where the projector P is defined in Remark 2.1.Then one can see that the pair {v(t) ≡ 0, u(t) ≡ w 0 } solve problem ( 17)- (21) with the initial data (0; w 0 ; 0) and with G f ≡ 0, G pl ≡ 0. The pressure p is the constant determined from its boundary value on Ω: p| Ω = ∆ 2 w 0 (∆ 2 w 0 is a constant due to Remark 2.1).This observation gives us a relation between solutions with initial data from H and H, namely we have that for any U 0 = (v 0 ; u 0 ; u 1 ) ∈ H.This relation means that H is invariant with respect to dynamics governed by ( 17)-( 21) and explains why an exponential decay estimate of the form (30) cannot be true for every initial data U 0 = (v 0 ; u 0 ; u 1 ) from the space H.This remark allows us to derive from Theorem 3.3 the following assertion.21) with G f ≡ 0 and G pl ≡ 0 generates a strongly continuous contraction semigroup T t on H and on H by the formula T t U 0 = U (t), where U (t) is a weak solution to (17)- (21) with the initial data U 0 .This semigroup T t is exponentially stable on H, i.e., there exist positive constants M and γ such that Proof.Strong continuity of T t follows from (27).This semigroup is contractive and exponentially stable due to (28) and (30) with G f ≡ 0 and G pl ≡ 0.
We note that the generator of the semigroup T t defined via solutions to problem ( 17)- (21) in the space H has a rather complicated structure, see Appendix A in the end of the paper.This is why we avoid in the argument below calculations involving the explicit form of the generator.

Proof of Theorem 3.3
We use the compactness method and split the argument into several steps.
Step 1. Existence of an approximate solution.For the construction of Galerkin's approximations we use an idea of [9] in a slightly modified form.
As above one can also see that ∂ x 3 φ 3 i = 0 on Ω.We define an approximate solution as a pair of functions (33) for k = 1, . . ., n.This system of ordinary differential equations is endowed with the initial data where Π m is the orthoprojector on Lin{ψ j : j = 1, . . ., m, } in X and P n is orthoprojector on Lin{ξ i : i = 1, . . ., n} in L 2 (Ω).Since Π m and P n is are spectral projectors we have that We can rewrite system (32) and (33) as for some linear function g : R m+2n → R m+n and G ∈ L 2 (0, T ; R m+n ), where The first matrix in ( 35) is nonnegative and the second one is symmetric and strictly positive (since the functions {ψ i , φ j : i = 1, . . ., m, j = 1, . . ., n} are linearly independent).Therefore system (32) and ( 33) has a unique solution on any time interval [0, T ].It follows from (31) that where N 0 is given by ( 13).This implies the following boundary compatibility condition v n,m (t) = (0; 0; Step 2. Energy relation and a priori estimate for an approximate solution. It follows from ( 32) and ( 33) that the approximate solutions satisfy the relation for t ∈ [0, T ] and for every χ and h of the form where m ′ ≤ m and n ′ ≤ n.Therefore taking χ = v n,m we obtain the following energy balance relation for approximate solutions This implies the following a priori estimate (39) By the trace theorem from (36) we also have that Step 3. Limit transition.By (39) the sequence To obtain (42) we use the Aubin-Dubinsky theorem (see, e.g., [30,Corollary 4]).By (40) we can also suppose that One can see from (37) that (v n,m ; u n ; ∂ t u n )(t) satisfies (22) with the test function φ of the form where p ≤ m, q ≤ n and γ i , δ j are scalar absolutely continuous functions on [0, T ] such that γi , δj ∈ L 2 (0, T ) and γ i (T ) = δ j (T ) = 0. Thus using (41)-(43) we can pass to the limit and show that (v; u; ∂ t u)(t) satisfies ( 22) with φ = φ p,q , where p and q are arbitrary.By (34) and (42) we have u(0) = u 0 The compatibility condition (18) follows from (36) and ( 44), (45).
To conclude the proof of the existence of weak solutions we only need to show that any function φ in L T can be approximate by a sequence of functions of the form (46).This can be done in the following way.We first approximate the corresponding boundary value of b by a finite linear combination h of ξ j , then we approximate the difference φ − N 0 h (with N 0 define by ( 13)) by finite linear combination of ψ k .
Thus the existence of weak solutions is proved.One can also see from (38) and from (41)-( 43) that the constructed weak solution satisfies the corresponding energy balance inequality.
Step 4. Uniqueness.We use the same idea as in [28], but with a slightly modified test function, see (48).
Let U j (t) = (v j (t); u j (t); u j t (t)), j = 1, 2, be two different solutions to the problem in question with the same initial data.Then their difference for all φ ∈ L T , b = (φ| Ω ) 3 .Now for every 0 < s < T we take as a test function.We denote Substituting φ s into (47), we obtain (49) Integrating by parts the second term in (49) and using the relations ψ s (0) = 0 and φ s (s) = 0, we have s 0 Therefore (49) yields for almost all 0 ≤ s ≤ T .Therefore v(s) = 0 and u(s) = 0 for almost all 0 ≤ s ≤ T .Thus the uniqueness is proved.
Step 5. Continuity with respect to t and the energy equality.First we note that any weak solution (v(t); u(t); u t (t)) is weakly continuous in X × H 2 0 (Ω) × L 2 (Ω).Indeed, it follows from (24) that that any weak solution (v(t); u(t)) satisfies the relation for almost all t ∈ [0, T ] and for all ψ ∈ V = {v ∈ V : v| Ω = 0} ⊂ W .This implies that v(t) is weakly continuous in V ′ .Since X ⊂ V ′ , we can apply the Lions lemma (see [27,Lemma 8.1]) and conclude that v(t) is weakly continuous in X.The same lemma gives us weak continuity of u(t) in H 2 0 (Ω).Now using ( 24) again we conclude that (u t (t), β) Ω is continuous for β ∈ H 2 0 (Ω).The density argument yields weak continuity of u t (t) in L 2 (Ω).
To prove the energy equality, we follow the scheme of [28, Ch.1], see also [27,Ch.3].We first note that due to Remark 3.4 it is sufficient to consider the case when U 0 = (v 0 ; u 0 ; u 1 ) ∈ Ĥ.Then for every fixed 0 < s < t < T we introduce a piecewise-linear continuous function θ n (τ ) on R such that θ n (τ ) = 1 for s ≤ τ ≤ t and θ n (τ ) = 0 when τ < s − 1/n or ] and R ρ k (s)ds = 1.Now for k and n large enough we consider the function , where v is a weak solution to ( 17)-( 21), as a test function in variational equality (22).Substituting this φ into (22) and passing to the limit when k → ∞ we obtain that As in [28,Ch. 1] one can see that for every function h ∈ L 1 (0, T ) for almost all s and t.Therefore after the limit transition in (50) we obtain energy relation (28) valid for almost all s and t.Now using weak continuity of the solution (v(t); u(t)) and the energy inequality (which is valid for s = 0 and for every t) we can establish the energy equality.As in [27,Ch. 3] this also implies strong continuity of weak solutions with respect to t.
Step 6. Exponential stability.To prove the exponential stability estimate in (30), we construct a Lyapunov function using an idea from [11].Let where Ψ(v 0 , u 0 , u 1 ) = (u 0 , u 1 ) Ω + (v 0 , N 0 u 0 ) O with N 0 defined by (13), and ǫ > 0 is a small parameter which will be chosen later.We consider these functionals on approximate solutions (v n,m ; u n ) for which P u 0 = u 0 and thus P u n (t) = u n (t) for all t > 0. This allow us to substitute in (37) N 0 u n instead of χ and obtain that By Proposition 2.2, using the compatibility condition in (36) and the trace theorem we have that O , ∀ η > 0, and also Therefore it follows from (51) that Using the energy relation in (38) we also have that Therefore the function V n,m (t) ≡ V (v n,m (t), u n (t), ∂ t u n (t)) satisfies the relations for sufficiently small ε > 0 and with positive constants a i .This implies relation (30) for approximate solutions.The limit transition yields (30) for every weak solutions.This completes the proof of Theorem 3.3.

Nonlinear problem
In this section we deal with problem (1)-( 6) with a nonlinear feedback force.Fist we describe hypotheses concerning this force.Then we prove well-posedness (see Theorem 4.3) and construct the corresponding semiflow.
Our main result (see Theorem 4.8) states the existence of finite-dimensional attractor.

Structure of feedback force
We impose the following hypotheses concerning the nonlinear feedback force F(u) in the plate equation ( 4).

Assumption 4.1 (F1)
There exists ǫ > 0 such that F(u) is locally Lipschitz from for any , where Π ′ denotes the Fréchet derivative of Π.
(F3) The plate force potential Π is bounded on bounded sets from H 2 0 (Ω) and there exist η < 1/2 and C ≥ 0 such that The nonlinear feedback (elastic) force F(u) may have one of the following forms (which represent different plate models): where κ ≥ 0, q > r ≥ 0 and µ ∈ R are parameters, h ∈ L 2 (Ω), and where λ 1 is the first eigenvalue of the biharmonic operator with the Dirichlet boundary conditions.In this case the relation in (52) follows from the considerations given in [12,Sect.5].We also have that where F (s) = s 0 f (ξ)dξ is the antiderivative of f .Due to the second relation in (54) we obviously have (53).
Von Karman model: This model is well known in nonlinear elasticity and constitute a basic model describing nonlinear oscillations of a plate accounting for large deflections, see [28,15] and the references therein.The 1 We recall that according our definitions and the Airy stress function v(u) solves the following elliptic problem It is known (see, e.g., Corollary 1.4.5 in [15]) that for every η ∈ [0, 1], which implies (52).The potential Π has the form and possesses the properties listed in Assumption 4.1, see, e.g., [15,Chapter 4] for details.
Berger Model: In this case the feedback force has the form where κ > 0 and Γ ∈ R are parameters, h ∈ L 2 (Ω).One can see Assumption 4.1 is satisfied, for some details and references see, e.g., [10,Chapter 4] and [14,Chapter 7].
Then for any interval [0, T ] there exists a unique weak solution (v(t); u(t)) to (1)-( 6) with the initial data U 0 .This solution possesses the property where H is given by (25), and satisfies the energy balance equality for every t > 0, where the energy functional E is defined by the relation with the plate energy E(u, u t ) given by Moreover, there exists a constant a R,T > 0 such that for any couple of weak solutions U (t) = (v(t); u(t); u t (t)) and Û (t) = (v(t); û(t); ût (t)) with the initial data possessing the property U 0 H , Û0 H ≤ R we have The spatial average of u(t) is preserved.In particular, if U 0 ∈ H, then U (t) ∈ H for every t > 0. We recall that H is defined by (26).
Proof.The proof of the local existence of an approximate solution is almost the same, as in the linear case (see Theorem 3.3).We use approximate solutions of the same structure as in (31) which satisfy (32), (34) and also (33) with −F(u n (t)) + G pl (t) instead of G pl (t).Then using the standard argument we establish the energy relation in (56) for these approximate solutions.Now the positivity type estimate in (53) allow us to obtain the same a priori estimates as in (39) and (40).Therefore we can prove the global existence of approximate solutions and establish the existence of a weak solution U (t) = (v(t); u(t); u t (t)) by the same argument as in the linear case.
To make limit transition in the nonlinear term we use (52).Now we can consider the pair (v(t); u(t)) as a solution to linear problem with G pl (t) := −F(u(t)) + G pl (t).This allow us to obtain (55) and also derive energy balance relation (56) from (28) using the potential structure of the force F: F(u) = Π ′ (u).
Since the difference of two weak solution can be treated as a solution to the linear problem with G f ≡ 0 and G pl (t) := F(û(t)) − F(u(t)), we can obtain (57) from the energy equality (28).The uniqueness follows from (57).
Preservation of the spatial average of u(t) follows from the same property for approximate solutions.Remark 4.4 In the autonomous case we can suggest another form of energy relation (56).Let G pl (t) ≡ 0 and G f (t) ≡ G 0 ∈ V ′ be independent of t.Suppose that a pair (v * ; p * ) ∈ V × L 2 (O) solve problem (10) with g ≡ G 0 and ψ ≡ 0, i.e., Then the following form of the energy balance equation is valid: where with E * (u, u t ) given by Indeed, it follows from (58) that Substituting ψ = v * in (24) we also have that This and also the energy relation in (56) imply (59).
This remark allows us the derive from Theorem 4.3 the following assertion.
Then problem (1)-( 6) generates dynamical systems (S t , H) and (S t , H) with the evolution operator defined by the formula S t U 0 = (v(t); u(t); u t (t)), where (v; u) is a weak solution to (1)-( 6) with the initial data U 0 = (v 0 ; u 0 ; u 1 ).These systems are gradient with the full energy E * (v 0 , u 0 , u 1 ) as a Lyapunov function.This means that (a) U → E * (U ) is continuous on H, (b) E * (S t U 0 ) is not increasing in t, and (c) if E * (S t U 0 ) = E * (U 0 ) for some t > 0, then U 0 is a stationary point of S t (i.e., S t U 0 = U 0 for all t ≥ 0).Moreover, the set E R = {U 0 : E * (U 0 ) ≤ R} is a bounded closed forward invariant set for every R > 0.
Proof.We need only to check the properties of the functional E * .It is clear from Assumption 4.1(F2) that E * is continuous on H.By (59) We have that E * (S t U 0 ) ≤ E * (S τ U 0 ) for t ≥ τ ≥ 0. This gives the monotonicity of t → E * (S t U 0 ) and the invariance of E R .
Below we describe the set of stationary point of the evolution semigroup S t with more details.

Stationary solutions
As above we assume that G pl ≡ 0 and for any ψ ∈ W with ψ 3 Ω = β, where W is given by (23).Using (59) we have that v = v * , where v * solves (58).One can also see (∇v, ∇N 0 β) O = 0 for any v ∈ V 0 and β ∈ H 2 0 (Ω), where N 0 is defined in (13).Therefore from (60) with ψ = N 0 β we have the following variational problem for u ∈ H 2 0 (Ω): The following calculation performed first on smooth functions gives us Since the pressure p * in (58) is defined up to a constant, we can suppose that p . This provides us with the regularity of the pressure impact on the plate.
One can see that a function u ∈ H 2 0 (Ω) solves (61) if and only if u is a variational solution to problem for some constant C which may depend on u.Since every variational solution to (62) is an extreme point of the functional using relation (53) in Assumption 4.1 we can prove the existence of these solutions.Thus we obtain a family of solutions to (60) parameterized by the real parameter C. To fix somehow the constant C in (62) it is convenient to fix the average of u.In the case of the zero average we obtain the following assertion.
Proposition 4.6 In addition to Assumption 4.1 we assume that G 0 ∈ V ′ and there exist η < 1/2 and c ≥ 0 such that Then the set N 0 of solutions u to problem (61) with the property Ω udx = 0 is nonempty compact set in H 2 0 (Ω).
Proof.Restricting the functional Ψ on H 2 0 (Ω) we can prove the existence of its minimum point on H 2 0 (Ω).This means that N 0 is not empty.If u ∈ H 2 0 (Ω) is a solution, then taking β = u in (61) and using (63) we conclude that N 0 is bounded in H 2 0 (Ω).If {u n } is a sequence from N 0 , then from (61) we conclude that . This implies that the sequence {u n } is relatively compact.
Remark 4.7 A similar result can be obtain for the set N α of solutions u to problem (61) with the property u ≡ Ω udx ′ = α with a fixed α ∈ R, if instead of (63) we assume that there exist η < 1/2, c α ≥ 0 and a smooth function φ with the property φ = α such that Indeed, if we consider the functional Ψ on H 2 0,α = u ∈ H 2 0 (Ω) : u = α for some fixed constant C, then we can prove the existence of a solution u to (61) in H 2 0,α .Now substituting β = u − φ in (61) and using (64) we obtain the boundedness of the set N α in H 2 0,α .To prove the compactness of N α we use the same argument as in Proposition 4.6.
To obtain the result stated in Theorem 4.8 it is sufficient to show that the system is quasi-stable (in the sense of [15]).For this we use the stability properties of linear problem ( 17)-( 21) established in Theorem 3.3 to prove the following assertion.
Proof.We consider (v(t); u(t)) as a solution to to linear problem ( 17)- (20) with G f ≡ 0 and G pl (t) = −F(u 1 (t)) + F(u 2 (t)).Therefore it follows from (52) and (30) that Proof of Theorem 4.8 Lemma 4.10 means that the dynamical system (S t , H) is quasi-stable in the sense of Definition 7.9.2 [15].Therefore by Proposition 7.9.4 [15] (S t , H) is asymptotically smooth.Since the system is gradient, the boundedness of the set of the stationary points implies that there exists a compact global attractor.Moreover, the standard results on gradient systems with compact attractors (see, e.g., [5,10,31]) give us that A = M + (N ).
Since (S t , H) is quasi-stable the finiteness of fractal dimension dim f A follows from Theorem 7.9.6 [15].
To obtain the result on regularity stated in (66) and (67) we apply Theorem 7.9.8 [15].

A Appendix: Generator of linear semigroup
To find the structure of the semigroup T t generated by ( 17)- (21) in the space H we note that the evolution problem in (24) with G f ≡ 0 and G pl ≡ 0 can be written in the form for any ψ ∈ W .If we take now ψ ∈ V = {v ∈ V : v| ∂O = 0}, then we obtain that ṽ = v − N 0 w ∈ V solve the problem −ν∆ṽ + ∇p = f 0 , div ṽ = 0 in O; ṽ = 0 on ∂O.
Since f 0 ∈ X, this implies that ṽ ∈ V ∩ [H 2 (O)] We can also write the operator A in the form
• D = • 0,D for the corresponding L 2 -norm and, similarly, (•, •) D for the L 2 inner product.We denote by H s for s > 0 with the standard H s (Ω)-norm.