LONG TIME BEHAVIOUR OF STRONG SOLUTIONS TO INTERACTIVE FLUID-PLATE SYSTEM WITHOUT ROTATIONAL INERTIA

. We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier–Stokes equations in a bounded do- main and a classical (nonlinear) full von Karman plate equations that accounts for both transversal and lateral displacements on a ﬂexible part of the bound- ary. Rotational inertia of the ﬁlaments of the plate is not taken into account. Our main result shows well-posedness of strong solutions to the problem, thus the problem generates a semiﬂow in an appropriate phase space. We also prove uniform stability of strong solutions to homogeneous problem.

1. Introduction. We deal with a coupled system which describes an interaction of a homogeneous viscous incompressible fluid which occupies a domain O bounded by the (solid) walls of the container S and a horizontal (flat) part of the boundary ∂O 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 use the full von Karman plate model. This is a rather general model based on Kirchhoff hypotheses that describes both longitudinal and transversal oscillations of the plate.
This fluid-structure interaction model assumes that large deflections of the elastic structure 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 [3,5,8,11,12,13] and the references therein in the case of plates, see also the literature cited in these papers.
We note that the global (asymptotic) dynamics in nonlinear plate-fluid models was studied before in [5,8,9]. The article [5] deals with a class of fluid-plate interaction problems, when the plate, occupying Ω, oscillates in longitudinal directions only. This kind of models arises in the study of blood flows in large arteries (see, e.g., [11] and the references therein). A fluid-plate interaction model, accounting for purely transversal displacement of the plate, was studied in [8]. A mathematical model, the most close to the current one, was considered in [9]. It takes into account both transversal and in-plane displacements. In contrast to the model in the present paper, it accounted for rotational inertia, and mechanical dissipation for the transversal component of the plate displacement was assumed.
Let O ⊂ R 3 be a bounded domain with a sufficiently smooth boundary ∂O. We assume that ∂O = Ω ∪ S, where with a smooth contour Γ = ∂Ω, and S is a surface which lies in the halfspace The exterior normal to ∂O is denoted by n. Evidently, n = (0; 0; 1) on Ω.
We consider the following linear Navier-Stokes equations in O for a fluid velocity field v = v(x, t) = (v 1 (x, t); v 2 (x, t); v 3 (x, t)) and a pressure p(x, t): where ν > 0 is the dynamical viscosity of the fluid and G f l is a volume force. We supplement (1) and (2) with the (non-slip) boundary conditions imposed on the velocity field v = v(x, t): where u = u(x, t) ≡ (u 1 ; u 2 ; w)(x, t) is the displacement of the plate occupying Ω.
Here w stands for transversal displacement,ū = (u 1 ; u 2 ) -for lateral (in-plane) displacements. These boundary conditions describe influence of the plate on the fluid. To describe influence of the fluid on the plate, we consider the surface force T f (v) exerted by the fluid, which is equal to T n| Ω , where n is the outer unit normal to ∂O at Ω and T = {T ij } 3 i,j=1 is the stress tensor of the fluid, T ij ≡ T ij (v) = ν v i xj + v j xi − pδ ij , i, j = 1, 2, 3.
Since n = (0; 0; 1) on Ω, we have that To describe the shell motion we use the full von Karman model which does not take into account rotational inertia of the filaments, but accounts for in-plane acceleration terms. Below for some simplification we assume that Young's modulus E and Poisson's ratio µ ∈ (0, 1/2) are such that Eh = 2(1 + µ), where h is the thickness of the plate. The corresponding PDE system is written in the form whereū = (u 1 ; u 2 ), G pl = (G 1 ; G 2 ; G 3 ) is an external load, and C(P (u)) is the stress tensor with This form of the full von Karman system was used earlier by many authors in the case when the plate does not interacts with other objects (v = 0) (see, e. g., [14] and the references therein).
We impose the clamped boundary conditions on the plate where n is an outer normal to ∂Ω in R 2 , and supply (1)-(7) with initial data for the velocity field v = (v 1 ; v 2 ; v 3 ) and the plate displacement vector u = (u 1 ; Here , j = 0, 1, are given vector functions subjected to some compatibility conditions which we specify later. We note that (2) and (3) imply the following compatibility condition This condition fulfills when and can be interpreted as preservation of the volume of the fluid. We emphasize, that even in the linear case we cannot split system (1)-(8) into two sets of equations describing longitudinal and transversal plate movements separately, i.e., we cannot reduce the model under consideration to the cases studied in [5,8]. The point is that the surface force T f (v) is not the sum of the corresponding loads in the models [5] and [8]. For the detailed discussion see [9], Remark 1.1 (C).
In this paper we deal with well-posedness of strong solutions to coupled system in (1)- (8) and their long-time dynamics. In the system under consideration rotational inertia is neglected, thus w t has lower regularity (L 2 (Ω)), then is the case of rotational inertia accounted for, which was considered in [9]. Such regularity still allows us to prove existence of weak solutions satisfying the energy inequality the same way as in [9], but uniqueness of these solutions is still an open question. Sedenko's method does not work here because the nonlinearity is strongly supercritical when rotational inertia is neglected.
Our main novelties are well-posedness of strong solutions and their uniform stability (in the case of zero external forces only). We face three main mathematical difficulties: (i) low regularity of time derivative of the transversal component of the plate displacement w t , (ii) a supercritical vector nonlinearity arising in the plate component, and (iii) a singular character of the surface force T f (v), which is defined via boundary traces of the pressure and spatial derivatives of the fluid velocity field. The last two ones do not allow us to apply the methods developed for shell/plates with subcritical and critical nonlinearities (see, e.g., [6,7] and the references therein), unlike [5,8]. These difficulties were overcame in [9] for the system like (1)-(8) but with rotational inertia accounted for. However, in most of the proofs there H 1 -regularity of w t is substantially used, that prevents application of such methods in this paper.
Our first main result (see Theorem 3.4) states well-posedness of problem (1)-(8) in the class of strong solutions. To achieve existence of strong solutions we prove additional smooth estimates for the first and second derivatives of the components of (1)- (8). Possibly they may be obtained by Sedenko's method (see [18]), however, we prefer to use smoothing effect due to the linearized Navier-Stocks equations. The other issues of well-posedness are rather standard by now in view of the smoothing effect. We also show additional regularity of the strong solutions.
Our well-posedness result shows that the strong solutions to (1)- (8) generates an evolution semigroup S t on an appropriate state space.
Our second main result (see Theorem 4.2) deals with long-time dynamics and states uniform stability of the evolution semigroup S t in the case of zero external forces. First we show exponential stability of S t with respect to the energy norm for zero external forces. Then, using possibility to pick initial data of arbitrary small energy norm, we first prove boundedness of strong solutions and then their uniform stability. Regularizing effect of Navier-Stocks equations is substantially used in the proof. It allows us to discard any mechanical damping in the plate component u and still have a stable system.

2.
Preliminaries. In this section we introduce Sobolev type spaces we need and provide some results concerning the Stokes problem.

Spaces and notations.
To introduce Sobolev spaces we follow approach presented in [21].
Let D be a sufficiently smooth domain and s ∈ R. We denote by H s (D) the Sobolev space of order s on the set D which we define as a restriction (in the sense of distributions) of the space H s (R d ) (introduced via Fourier transform) to the domain D. We define the norm in H s (D) by the relation We also use the notation · D = · 0,D and (·, ·) D for the corresponding L 2 norm and inner product. We denote by H s 0 (D) the closure of C ∞ 0 (D) in H s (D) (with respect to · s,D ) and introduce the spaces To describe fluid velocity field we introduce the following spaces. Let C(O) be the class of C ∞ vector-valued solenoidal (i.e., divergence-free) functions on O which vanish in a neighborhood of S and C 0 (O) be the class of C ∞ 0 (O) vector-valued solenoidal functions. We denote by X the closure of C(O) with respect to the L 2 -norm and by V the closure of C(O) with respect to the H 1 (O)-norm. Notations V 0 , X 0 are used for the closure of C 0 (O) with respect to the H 1 (O)-norm and L 2 (O)-norm, respectively. One can see that and We equip X and X 0 with L 2 -type norm · O and denote by (·, ·) O the corresponding inner product. We denote 0 are endowed with the norm · V α = ∇ · α,O , and V = V 0 . For the details concerning spaces of this type we refer to [20], for instance.
We also need the Sobolev spaces consisting of functions with zero average on the domain Ω, namely we consider the space and also H s (Ω) = H s (Ω) ∩ L 2 (Ω) for s > 0 with the standard H s (Ω)-norm. The notations H s * (Ω) and H s 0 (Ω) have a similar meaning. To describe plate displacement we use the spaces for weak solutions and for strong solutions. For weak solutions as a phase space we use with the standard product norm. We also denote by H a subspace in H of the form where w 0 is the third component of the displacement vector u 0 . Phase space for strong solutions will be defined later.

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)] 2 × L 2 (Ω) are given. This type of boundary value problems for the Stokes equation was studied by many authors (see, e.g., [20] and references therein). We collect some properties of solutions to (16) in the following assertion.
Proposition 1. The following statements hold.
In particular, we can define a linear operator N 0 : . It follows from two previous estimates that For the proof see [9].

2.3.
Tensors. For convenience we collect here tensor properties which will be used in the paper. They can be easily verified by direct calculations.
For the stress tensor C the following properties hold.
Lemma 2.2. Let A, B are two tensors of order 4. Then 3. Well-posedness theorem. To define weak (variational) solutions to (1)- (8) we need the following class L T of test functions φ on O: We also denote L 0 the following equality holds: where χ is a smooth scalar function and ψ belongs to the space for all t ∈ [0, T ] and ψ ∈ V with ψ Ω = β = (β 1 ; β 2 ; δ) andβ = (β 1 ; β 2 ). The following theorem on existence of weak solutions can be proved the same way, as in [9]. 3 . Then for any interval [0, T ] there exists a weak solution (v(t); u(t)) to (1)-(8) with the initial data U 0 . This solution possesses the following properties: The solution is bounded globally in t.
However, due to the strong supercriticality of the nonlinearity in full von Karman equation we didn't manage to prove uniqueness of weak solutions. Therefore we build dynamical system in a space of more smooth functions.
To describe behaviour of the fluid component, we will use the space For strong solutions we use the spaces Remark 1. The phase space for strong solutions is in agreement with the domain of the generator of the semigroup in the problem of fluid-structure interaction that encounts for in-plane displacements of the plate only, see formula (22) and Remark 2.3 from [5].
Then for any interval [0, T ] there exists a unique strong solution (v(t); u(t)) to (1)- (8) with the initial data U 0 . This solution possesses the following properties: • it is continuous with respect to t in the phase space, i.e.
• there exists C T > 0, depending on ||U 0 || Hs , such that for all ε > 0 • The solutions depends continuously (in strong topology) on initial data in the space H s . • The energy balance equality is valid for every t > 0, where the energy functional E is defined by for δ ∈ [0, 1/2), but not for δ = 1/2 [16]. Thus, extension by zero is not continuous from H 1 0 (Ω) H 3/2 (Ω) to H 3/2 (∂O) and the operator N 0 does not map (B) For an uncoupled full von Karman plate Sedenko [17] obtained existence and uniqueness of strong solutions of the smoothness where A is an operator generated by the form a(·, ·), defined by (27) on H 1 0 (Ω). But in our case, if we want to obtain v ∈ V 1 or more smooth, we need to haveū , and Galyorkin method cannot be used to prove, e.g., that Step 1. Existence. The estimates below are obtained for approximate solutions. Construction method for approximate solutions is the same as in [9] (see proof of Theorem 3.3), and we describe it here for the sake of convenience only. This method is inspired by [3]. Let {ψ i } i∈N be the orthonormal basis in X 0 consisting of the eigenvectors of the Stokes problem: Here 0 < µ 1 ≤ µ 2 ≤ · · · are the corresponding eigenvalues. Denote by {ξ i } i∈N the basis in H 2 0 (Ω) which consists of the eigenfunctions of the problem (∆ξ i , ∆w) Ω =κ i (ξ i , w) Ω , ∀ w ∈ H 2 0 (Ω), with the eigenvalues 0 <κ 1 ≤κ 2 ≤ . . . and such that (ξ i , ξ j ) Ω = δ ij . Further, let {η i } i∈N be an orthogonal basis in H 1 0 (Ω) × H 1 0 (Ω) such that (η j , η i ) Ω = δ ij , which consists of eigenfunctions of the problem , with the eigenvalues 0 <κ 1 ≤κ 2 ≤ . . . and ||η i || Ω = 1. The form a(η, w) is given by Letφ i = N 0 (0; 0; ξ i ) andφ i = N 0 (η i ; 0), where the operator N 0 is defined by (17). By Proposition 1 we have thatφ i ,φ i ∈ V 1−δ for every δ > 0.
Thus, we have for approximate (and weak) solutions (see [9]) In the arguments below we drop the subscripts for the sake of brevity.
(31) and after integration by t get For the proof of well-posedness we need the following lemmas.
• for every σ ∈ [0, 1] • ||w|| 2 3, • for every δ > 0 and γ > 0 The lemma formulated above gathers all the smoothing effects, that fluid produce on the plate. These estimates are essentially used for the proof of the existence of strong solutions.
We'll prove the lemmas later, now we proceed with (32). First we estimate the third term of the right hand side in it. This term is a combination of terms of the form with i, j, k, l = 1, 2. Using Schwartz inequality and interpolation we obtain Noting, that the norm of solution in H is bounded uniformly with respect to t and using Young inequality with p = 4/3, q = 4, we obtain that Now we turn to the last term of the right hand side in (32). It is a combination of terms of the form Using Hölder's inequality, Galiardo-Nirenberg inequality, and estimate (39), we obtain ||∆w|| Ω ||w|| Ω t 0 ||w|| 2 9/4(Ω) ≤ In a similar way, Choosing δ, γ small enough and substituting these estimates in (32), we get since every weak solution is globally bounded in H. Thus, using Gronwall' lemma, we obtain . Now we need to prove that E(0) is bounded from above by the norm of initial data in H s and this estimate does not depend on n, m. Since Π m , R n and P n are spectral projectors we have that w n (0) = ∂ t w n (0) → w 1 strongly in H 2 0 (Ω), n → ∞, u n (0) = ∂ tūn (0) →ū 1 strongly in H 1 0 (Ω), n → ∞. Substituting v n,m in the form (28) in equation (1) and setting t = 0, we get v n,m (0) = ∂ t v n,m (0) = −Av m (0) + Π m G f l + T n G pl , where A is a Stokes operator, T n is a projector onto Lin{φ j , j = 1...n}, and v m (0) = Π m v 0 . Thus, if v 0 ∈ H 2 (O) ∩ X 0 , theñ v n,m (0) → −Av 0 + G f l strongly in X 0 .
This estimate together with lemma 3.6 (after limit transition) give us existence of strong solutions.
Step 4. Uniqueness. The proof cannot be done the same way, as in [17], because of coupling with fluid. We use here modification of the idea.
Let (v;û) and (v;ǔ)(t) -two strong solutions with the initial data (v 0 ;û 0 ;û 1 ) and (v 0 ;ǔ 0 ;ǔ 1 ). Then their difference Multiplying the first equation by v, the second one by w t and the third one bȳ u t (since we consider strong solutions, we can perform this multiplication directly), we get d dt [||v(t)|| 2 O + ||w t (t)|| 2 Ω + ||ū t (t)|| 2 Ω + (C( 0 (ū)), 0 (ū)) Ω + ||∆w|| 2 Ω ] = − [νE(v, v) + (C(f (∇ŵ) − f (∇w)), 0 (ū t )) Ω + (C(P (û))∇ŵ − C(P (ǔ))∇w, ∇w t ) Ω ] Noting that after some calculations we obtain where In the following estimates of this step C T is a generic positive constant depending on time interval (0, T ) and H s -norms of strong solutions (v,û)(t), (v,ǔ)(t) on this interval. The second term in (50) consists of the terms of the form The third term consists of the terms of the form for which the following estimates are valid The similar way, for the last term holds Using interpolation, for the first term we obtain Thus, integrating (50) with respect to t and applying estimates (51)-(54), we obtain To estimate ||w(t)|| 2 Ω , we note that which together with the previous estimate gives Thus, uniqueness is proved.
Step 5. Continuity with respect to initial data. Estimate (55) gives us continuity with respect to initial data in H-norm. Thus, the evolution operator S t maps a convergent strongly in H s sequence to a convergent weakly in H s sequence. To obtain continuity in H s we can use method from [14], making use of "energy equality" (32). Let It follows from equations (1)- (8), that ∂ tt u n (t) ∂ tt u(t) weakly in L 2 (Ω), Thus, to prove continuity in the strong topology of H s , it is sufficient to prove E((ṽ n ,ũ n , ∂ tũn )(t)) → E((ṽ,ũ, ∂ tũ )(t)) a.e.
Using trace estimate and Theorem on intermediate derivative [16], for every δ > 0 we obtain which together with (35) implies (36). This estimate allows us to get better smoothness of v, and we can use (33) with σ = 3/4. Therefore the following estimate takes place: Noting, that and using subsequently (33) with σ = 3/4 and (36), we obtain (38). Now we estimate ||w|| 4,Ω . It follows from (5) that If we expand the last term, we'll see we need to estimate It is easy to see, that For K 1 we have for any ε > 0. This implies ||w(t)|| 4,Ω is bounded on [0, T ] provided E(t) is bounded on [0, T ]. However, the overall degree of smooth norms is too high to use this estimate in Gronwall inequality. Using interpolation, we get 1,Ω ||ū|| 4/5 9/4,Ω . Integrating (59) with respect to t from 0 to T , using Young inequality in the last estimate for K 1 and applying estimate (38), we obtain (37).
Remark 3. Due to preservation of mean of transversal displacement w (10) dynamical system (H s , S t ) cannot be dissipative. Therefore we consider DS on subspaces where w has zero mean, namely, ( H s , S t ). Spaces H and H s are defined by (15) and (23), respectively.
In this section we assume that In this case set of stationary points of the dynamical system (H s , S t ) is non-empty and bounded, see Proposition 4.2 [9]. We also modify energy for this section, setting 2 Ω + ∆w 2 Ω + (C(P (u)), P (u)) Ω + 2(G 3 , w) Ω For a proof of dissipativity we need the following estimates.
Let ||u|| W ≤ R. Then Proof. The lemma can easily be proved by direct calculations with the help of Korn inequality. Proof. Let's denote by H R the subset of H s such that E(v, u 0 , u 1 ) ≤ R 2 . We will prove, that dissipativity radius R 0 is the same for every DS ( H R , S t ), that give us the desired dissipativity. Like in [5], we use as a Lyapunov function where we chose appropriate η (depending on R). First we estimate full time derivative along the trajectory for the second term. Using Green formula and properties of N 0 , we obtain d dt ,Ω . To compensate positive ||ū|| 2 1,Ω we use (65) and boundedness of the trajectory. Thus, Evidently, we can choose δ 1 not depending on R and δ 2 depending on R to obtain proper signs of the terms. Now we need to choose η = η(R) to garantee Ψ is bounded from below by a function depending on the H-norm, which goes to infinity as the norm goes to infinity. Using (65) and properties of N 0 , we obtain Setting δ = 1/8 and choosing η < 1/2 such that η(1+c)c 3 < 1/2, η(1+c)(1+c 4 R) < 1/4, we obtain where constants C, D don't depend on R. Finally, using estimate (21), for every η : η(2γ 1 + C δ2 + C δ1 ) < ν/2, we get d dt Ψ((v; u; u t )(t)) + C 1 ηE((v; u; u t )(t)) ≤ ηC δ1 ||G 3 || 2 with C 1 , C δ1 not depending on R. This proves the lemma. The radius of dissipativity R d is of the form c 2 + C||G 3 || 2 Ω , where c 2 is from (64).
Corollary 1. Let G 3 = 0. Then the dynamical system ( H s , S t ) is uniformly exponentially stable with respect to H-norm.
In proof of existence of strong solutions we estimate integrals with respect to t of RHS. This prevents to build Lyapunov function in H s at once, like in [4]. Instead, first we prove global boundedness of a strong solution in H s and then prove stability of ( H s , S t ). Proof. Let B be a set of initial data bounded in H s , i.e., there exists R > 0 such that for all U 0 ∈ B ||U 0 || Hs ≤ R. Then due to Corollary 1 there exists t 0 = t 0 (B, R d ) such that for all t ≥ t 0 ||S t U 0 || H ≤ R d . Thus, we can chose initial data from the ball B R d in H of arbitrary small radius R d . Note, that H s -norm of S t U 0 may increase, but due to Theorem 3.4 ||S t0 U 0 || Hs ≤ C(t 0 , ||U 0 || Hs ).
In this proof we work with approximate solutions and make a limit transition in the very end of each step.

LONG TIME BEHAVIOUR OF STRONG SOLUTIONS 1263
Multiplying the first inequality by the second, we get Thus, we can chose R d small enough to guarantee existence of δ satisfying the second inequality.
Due to the energy equality (26) Finally, we arrive Since v t (t) = Av(t), we have similar estimate for v(t). Now we perform limit transition to justify estimates (70)-(74) for strong solutions with initial lata form B R d ⊂ H. Thus, we prove that strong solutions are bounded globally in t.