Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation

We consider a nonlinear, free boundary fluid-structure interaction model in a bounded domain. The viscous incompressible fluid interacts with a nonlinear elastic body on the common boundary via the velocity and stress matching conditions. The motion of the fluid is governed by incompressible Navier-Stokes equations while the displacement of elastic structure is determined by a nonlinear elastodynamic system with boundary dissipation. The boundary dissipation is inserted in the velocity matching condition. We prove the global existence of the smooth solutions for small initial data and obtain the exponential decay of the energy of this system as well.

1. Introduction. We consider a free boundary fluid-structure interaction model which consists of the viscous incompressible fluid and the nonlinear elastic structure. Both fluid and elastic body are contained in Ω which is a smooth bounded domain in R 3 . This domain is divided into two parts by the common interface of fluid and structure where the interaction takes place. The inner part is taken over by the elastic body, denoted by Ω e (t) ⊂ Ω while the fluid occupied the exterior part Ω f (t) = Ω \Ω e (t). We denote the fluid portion and the solid portion in the domain Ω at the start time by Ω e and Ω f , respectively. Both Ω e and Ω f are also smooth bounded domains in R 3 . Moreover, we use the symbol Γ c = ∂Ω e ∩ ∂Ω f to stand for the 1556 YIZHAO QIN, YUXIA GUO AND PENG-FEI YAO common boundary of fluid and solid at t = 0(For more details, see [2,9,10,11]). The motion of the fluid is described by the incompressible Navier-Stokes equations(see [14]): ∇ · u = 0 in Ω f (t), (2) in which the vector field u ∈ R 3 is the velocity of the fluid, while the displacement of elastic structure w ∈ R 3 is dominated by the following nonlinear elastodynamic system: which is derived by variational methods from the action functional where W (F ) : R 3×3 → R is the stored-energy function of the elastic material. The term |w| 2 in (3) is needed for the energy estimate consideration, see [8]. The interaction occurs on the common boundary Γ c (t) via the natural transmission boundary conditions matching the velocity and the stress. Fluid-structure interaction models have drawn considerable attention from both engineers and mathematical researchers. At beginning, these models were considered in a finite element framework (see ([4], [5], [6]) and reference therein). Recently, the topics about the mathematical theory of existence, uniqueness and stability of solutions for such models have been becoming quite attractive. For the linear elastic material, local in time well-posedness of the free boundary model was first established by Coutand and Shkoller in [2] and improved in [7,10] and [11] where there are no dissipative mechanisms on the interface. Then the global-in-time existence for the fluid-structure system with damping is established in [8] and [9] for small initial data and linear isotropic elastic material. In [8], the authors consider the case that there are boundary dissipative term on the common interface Γ c and internal dissipative term αw t , α 0 in the wave equations. And in [9], only internal dissipative term in wave equations are considered. Besides, [13], the global solutions and energy decay are also obtained for the linear wave equations with variable coefficients coupling with incompressible viscous fluid. For other topics on fluid-linear elastic structure system, see the short review in the introduction in [9].
As for the nonlinear elastic material, in [3], Coutand and Shkoller developed the short-time wellposedness theory for the system in which fluid couples with some specific quasilinear elastic material.
In this paper we assume that the elastic body is a general nonlinear material to consider the global smooth solutions and energy decay of the system where the fluid interacts the nonlinear elastic body determined by the action functional (3). The nonlinearity of the elasticity equations causes more difficulty and increase the complexity for us to establish our result and we need to do careful estimates to overcome this difficulty. For the fluid part of this system, we employ the method and the estimates obtained in [8]. For the elastic body, we use the multiplier methods and invoke the nonlinear estimates and compactness results derived in [16], [17], [1] and [18] to deal with our problem.
Let η(x, t) : Ω → Ω(t) be the position function with Ω(t) = Ω being the different states of the system with respect to different time. With the help of position function, the incompressible Navier-Stokes equations can be reformulated in the Lagrangian framework: in Ω f × (0, T ), i = 1, 2, 3, (4) where v(x, t) and q(x, t) denote the Lagrangian velocity and the pressure of the fluid over the initial domain Ω f , respectively. This means that v(x, t) = η t (x, t) = u(η(x, t), t) and q(x, t) = p(η(x, t), t) in Ω f . The matrix a(x, t) is defined as the inverse of the matrix ∇ x η(x, t), which means a(x, t) = (∇ x η(x, t)) −1 . Note that the Einstein summation convention is employed. The nonlinear elastodynamic equations for the displacement function w(x, t) = η(x, t) − x are formulated as the following: We seek a solution (v, w, q, a, η) to the system (4) and (5), where the matrix a = (a ij )(i, j = 1, 2, 3) and η | Ω f are determined in the following way: where the symbol " : " stands for the usual multiplication between matrices and I represents the identity matrix in R 3×3 . Making use of the notation a, we rewrite (4) as On the interface Γ c between Ω f and Ω e , we assume the transmission boundary condition where the constant γ > 0 and w ν N = DW (∇w + I)ν, and the matching of stress where ν = (ν 1 , ν 2 , ν 3 ) is the unit outward normal with respect to Ω e . On the outside fluid boundary Γ f = ∂Ω, we impose the non-slip condition We supplement the system (4) and (5) with the initial data v(x, 0) = v 0 (x) and (w(x, 0), w t (x, 0)) = (w 0 (x), w 1 (x)) in Ω f and Ω e , respectively. Let Based on the initial data v 0 , the initial pressure q 0 is determined by solving the problem Let the initial data v 0 ∈ V ∩ H 5 (Ω f ), w 0 ∈ H 4 (Ω e ), and w 1 ∈ H 3 (Ω e ) be provided. Moreover, v 0 , w 0 and w 1 are supposed to satisfy the compatibility conditions as follows: We now consider the hypothesis on the stored-energy function.
(H1) The elastic body of the system is in equilibrium with w = 0, i.e.
We need to impose the strong ellipticity condition at the zero equilibrium for the material function W .
(H2) There exists a positive constant µ > 0 such that The fourth order tensor D 2 W (F ) = ( ∂ 2 W ∂Fij ∂F kl (F )) is the Hessian of W with respect to the variable F = (F ij ) ∈ R 3×3 . Following the approach in [8] and [10], we construct the short-time solutions for the system (4) and (5) with boundary conditions (9)-(11) under the assumptions of Theorem 1.1 in Section 5, which is essential for the global existence of the solutions for the system we consider.
Our main results are given as follows.
Theorem 1.1. Let the assumptions (H1) and (H2) hold and Ω e be star-shaped with respect to a fixed point (Ω e ) are given small such that the corresponding compatibility conditions (13)-(21) hold and γ 2C > 0, where the constant C depends on the value of initial data. Then, there exists a unique global smooth solution

Remark 1.
Given initial data small the total energy of the system decays exponentially, where the total energy is defined in (124) later.
The proof of Theorem 1.1 will be given in Section 4. Next, we also list the result of short-time existence of this system and our result in Theorem 1.1 is based on this short-time existence results. We deal with the proof of the following assertion in Section 5. Theorem 1.2. Let the assumptions (H1) and (H2) hold and Ω e be star-shaped with respect to a fixed point x 0 ∈ Ω e . Suppose that initial (Ω e ) are given small such that the above listed compatibility conditions hold. Then, there exists a time T > 0 depending on initial data such that there is a unique solution (v, w, q, a, η) on the interval [0, T ] with the following regularity 2. Preliminaries. We list some lemmas and estimates in the literature which are needed in the proof of Theorem 1.1. Constant C may be different from line to line throughout this note.
Following Lemma 3.1 in [7], we derive a higher regularity version.
where C > 0 is large enough, the following statements hold: Similar to Lemma 3.2 in [7], we obtain a pointwise a-priori estimates with higher regularity for variable coefficient Stokes system.
From now on, for simplicity, we omit specifying the domains Ω f and Ω e in the norms involving the velocity v and the displacement w. But we still emphasize the boundary domains Γ c and Γ f . Now we turn to the nonlinear elastodynamic system. Let w be a solution of (5) on [0, T ] for some T > 0. Set We differentiate C(t) in time and havė Then, the derivatives of order j of C(t) for j = 2, 3, 4 with respect to t are listed as follows:C Define be a vector field on the interface Γ c , given by the formulas With the help of the above notations we introduce, we differentiate (5) or (23) in time for j 1 times and obtain where the remainder term caused by the nonlinearity of W is 1562 YIZHAO QIN, YUXIA GUO AND PENG-FEI YAO Therefore, with above preparations, we introduce the following various types of energy: the energy for (5) or (23) the first level energy of the whole system for j 1, the energy for (36) the j-th level energy of the fluid-structure interaction model , and the total energy of the whole system and the elastic body Moreover, we denote a remainder term which will appear by L(t) = 8 k=3 (E e (t)) k 2 . Now, with necessary notation introduced above, we give some lemmas which will be frequently utilized.
Suppose that the assumption (H2) holds. Hence, there exists a constant then for all b 1 , b 2 ∈ R 3 . Thus, it yields the following lemma, by Gärding's inequality.
is such that the condition (38) holds, then for some c γ0 > 0.
By slightly modifying the proofs of Lemma 2.1-2.3 and Theorem 2.1 in [16,17], we obtain the following lemmas. Before the statement of these lemmas, we collect a few basic properties of Sobolev space which we'll use often first.
(i) Let s 1 > s 2 0 and D be a bounded, open set with smooth boundary in R n (n 2). For any > 0 there is c > 0 such that (ii) If s > n 2 , then for each k = 0, ..., we have H s+k (D) ⊂ C k (D) with continuous inclusion.
Lemma 2.4. Let γ 0 > 0 be given and φ ∈ H 1 (D) be such that the condition (38) hold. Let f (·, ·) be a smooth function on D × R n and F (x) = f (x, ∇φ). Then there is c γ0 > 0, depending on γ 0 , such that Lemma 2.5. Let γ 0 > 0 be given and w satisfy the problem (5) on the interval [0, T ] for some T > 0 such that where the operator B w (t) is given by (34). Then, Thus, all the preparations for the proof of Theorem 1.1 have been made.

3.
A-priori estimates. We derive a priori estimates for the global existence of solutions to the dissipative fluid-nonlinear structure system when the initial data are sufficiently small.
We demand several auxiliary estimates involving different levels of energy.

3.1.
First level estimates. As defined above, V 0 (t) is the first level energy. We deal with this energy in this subsection. where is a dissipative term.
Proof. Take the L 2 -inner product of (4) with v and (5) with w t , respectively. Hence, by (27), we have Add (55) and (56) together and integrate in time from 0 to t. With the help of boundary condition (9) and (10), we arrive at the resulted inequality.
In order to proceed further, we have to derive the multiplier identities which will play a central role in our calculations. Denoteŵ = w (j) for 0 j 3. From (23) and (36),ŵ satisfies the following equations:ŵ where r(t) = r j−1 (t) as j = 2, 3; r(t) = 0 when j = 0, 1 and the operator A Letŵ be a solution of (57) and H be a vector field on Ω e with H(ŵ) = ∇ Hŵ . And let the scalar function ξ(x) ∈ C 1 (Ω e ). Then we have for > 0, t s Γc and Proof. For (58), we take 2H(ŵ) + ŵ as the multiplier. Multiply (57) by 2H(ŵ) + ŵ and integrate by parts over (s, t) × Ω e . Thus, we have t s Ωe with the help of Combining (60)-(62), we acquire (58). As for (59), we regard ξ(x)ŵ as the multiplier. Carry out similar calculations and we'll arrive at the multiplier identity (59).
Taking advantage of the above multiplier identities, we establish the following Lemma. Moreover, we intend to deal with the similar estimates in subsection 3.1-3.4 in a unified way.
The related boundary conditions of (57) are the following: wherev = v (j) for the related 0 j 3, and Now we estimate the terms in (66). By Cauchy-Schwartz inequality and (39) or Lemma 2.3, we have With the help of Lemma 2.5 and Young inequality, we derive that where 1 > 0 is sufficiently small and to be determined later on.
Applying the property of Stored-function W and the Sobolev inequality Using the Cauchy-Schwartz inequality, Young inequality and Lemma 2.3 again, we have in which 2 > 0 is also to be determined. By the star-shaped condition for the domain Ω e , it implies that there exists a constant ρ 0 > 0 such that x − x 0 , ν ρ 0 .
According to Lemma 3.3, for the first order energy, it implies that Next, we multiply (75) by sufficiently small 3 > 0 and add the resulted inequality to (53). Thus, we get the following lemma.
Remark 2. If the solution of this fluid-nonlinear structure interaction system can exist for all time and all the assumptions in Theorem 1.1 hold, then we may infer from the above lemma that the first level energy V 0 (t) decays exponentially.

3.2.
Second level estimates. As we have defined in Section 2, the second order energy and the corresponding dissipation term To start with deriving the similar estimates with (53) and (76) in above subsection, we differentiate the whole system in time. It follows that tr(a t : ∇v) + tr(a : Besides, the boundary conditions with respect to (77)-(79) are Lemma 3.5. Let the assumptions in Theorem 1.1 hold. The following energy inequality holds for t ∈ [0, T ] Proof. Take L 2 inner product with v t and w tt to (77) and (79), respectively. Utilizing the boundary conditions (80)-(82), we attain that and Add (85) and (86) together and integrate in time from 0 to t. Note that Due to the ellipticity of a(x, t), we get
According to Lemma 3.3, we find that After a similar procedure with subsection 3.1, we conclude that Lemma 3.6. Under the same hypotheses as in Lemma 57, we have for all t ∈ [0, T ] that

Third level estimates.
Here, we go further for the third level energy estimates. The third level energy is defined by and the dissipative term . Before the derivation of our estimates, we give the equations satisfied by v tt and w tt .
tr(a tt : ∇v) + 2tr(a t : ∇v t ) + tr(a : ∇v tt ) = 0 in Ω f × (0, T ), (92) Moreover, the boundary conditions with respect to (91)-(93) are the following: Lemma 3.7. Suppose that the assumptions in Theorem 1.1 hold, then the following energy inequality holds for t ∈ [0, T ] where 0 <¯ < 1 sufficiently small and Proof. Take Euclidean dot product to (91) with v tt and integrate over Ω f . After integrating by parts, we obtain 1 2 Next, we do the same operation to (93) with w ttt as above and also integrate by parts over Ω e . Thus it follows that 1 2 Adding (99) to (100) and integrating in time from 0 to t, it leads to By Lemma 2.5 and the Poincaré inequality, we have where 0 <¯ < 1 is small enough and to be determined.
Similarly, also by using Lemma 2.5, we obtain

From Lemma 3.3, we deduce that
Multiply (106) by > 0 with 2¯ < < min{ 1 2C , γ C } and add the resulted inequality to (97). Hence, it turns out that we arrive at the following lemma.

Fourth level estimates.
We move on to the Fourth level energy estimates and repeat what we do as above. The fourth level energy is defined by as in the previous section. Besides, the dissipative term for the fourth energy is . First of all, as before, we differentiate the whole system three times in time and obtain ∂ t v ttt − ∂ ttt div (a : a T : ∇v) + ∂ ttt div (aq) = 0 in Ω f × (0, T ), (108) And, the boundary conditions satisfied by the system (108)-(110) are as follows: Hence, we are ready to derive the energy estimates for the fourth order energy.
Lemma 3.9. Assume that the hypotheses of Theorem 1.1 hold, then the following energy estimate is true for t ∈ [0, T ] where 0 <˜ < 1 is sufficiently small and Proof. Take L 2 inner product with v ttt and w (4) to (108) and (110), respectively. From (111) and (112), we attain that Add (116) and (117) together and we have analogous estimates to (102)-(105) as well. Using the similar method with that in Lemma 3.7, we may arrive at (114) and conclude the proof.
As a consequence of Lemma 3.3, we have After the same procedure as that in Subsection 3.3, we acquire the following lemma.
3.5. Superlinear estimates. Our aim of this subsection is to deal with the perturbation terms in the second, third and fourth level energy estimates. They are The concrete presentation of the above three perturbation terms can be found in (84), (98) and (115), respectively. For the estimates of (84) and (98), we only list the results. For detail, refer to [8].
Now, we turn to (115), even though the computation is quite involved.
Proof. From (115), we have By Hölder inequality and Lemma 3.1, we get and where the Sobolev and interpolation inequalities are employed. Now we begin to treat R 37 . By the differentiated divergence-free condition (109), we deduce that Applying Lemma 2.1 and Corollary 1 along with Hölder's, Sobolev and interpolation inequalities, we have The sum of the first three terms on the right hand side of the above estimate is bounded by Therefore, combining (120)-(123), we conclude the proof of this lemma.

4.
Energy decay and global existence of the system. We aim at the global existence of solutions and the energy decay estimates in this section. Let the total energy of the whole system and its equivalent version where 1 > 0 is given sufficiently small and to be determined later. We make some preparations for the proof of Theorem 1.1.
We have Similarly, we obtain and ∇v tt (t) 2 In addition, it follows from Lemmas 3.1 and 3.4 that Combining Lemmas 3.5, 3.6 and 3.11, we have

YIZHAO QIN, YUXIA GUO AND PENG-FEI YAO
From Lemmas 3.7, 3.8 and 3.12, Moreover, according to Lemma 3.9, 3.10 and 3.13, we attain where the symbols P i , 1 i 7 denote the superlinear polynomials of their arguments, which are allowed to depend on 0 from Lemmas 3.12 and 3.13. Now multiply (125)-(127) by sufficiently small 1 , sum up (128)-(131) and then add them together to obtain Because of (49) in Theorem 2.7, we find that From (28), Lemma 2.4 and the properties of Sobolev space we list, we have Thanks to (29) and (134), similarly we obtain For (28) in case of s = 2, by (135) and Lemma 2.4, From (2) whereP is a polynomial with the degree of each term of it is at least 2. Submitting (134)-(137) and the boundary condition (63) into (133) and setting 0 small enough and γ 2C, where the constant C depends on X (0), by Lemma 2.5, it follows from (133) that where P is a superlinear polynomial as well. We rewrite (138) as where C 0 1, α 1 , ..., α m > 1 and β 1 , ..., β n > 1.

Construction of solutions.
In this section, we turn to the proof of Theorem 1.2. Following the method for linear wave equations coupled in fluid-structure interaction system in [8], we first construct solutions in Lemma 5.1 to the linear problem for the given matrix a = I, and for given nonzero forcing f in liquid equations andf in elasticity equations, nonzero divergence condition g and nonzero difference of stresses h and velocityh on the common boundary Γ c . Then, in the general case of given smooth elliptic matrix a(x, t), we apply a fixed point technique to the perturbed linear system, where Finally, we employ the retarded mollification technique for a to obtain the shorttime existence results for the system considered in Theorem 1.1 as that in [10]. Before we state the lemma for the linear system, we give the compatibility conditions satisfied by the initial data of the linear equations ∇v ttt (0) · ν, τ = ∂ ttt (l ∇w A(x, t))(0)ν, τ + h ttt (0), τ , on Γ c v ttt (0) = 0, on Γ f , and the regularity hypotheses of f, g,f , h,h f ∈ C([0, T ]; H 2 (Ω e )),f t ∈ C([0, T ]; H 1 (Ω e )),f tt ∈ C([0, T ]; L 2 (Ω e )), Lemma 5.1. Consider the linear coupled Stokes-elasticity system with the boundary conditions where w Aν = (l ∇w A(x, t))ν.
where¯ > 0 is a small parameter. Then, the map has fixed point in the Banach space X(T, R), where and T, R both are sufficiently small positive numbers, which depend on initial data and a(x, t).
Remark 3. By Proposition 3.2.1. and Theorem 1.7.4. in [1], we can infer that X(T, R) = ∅ and X(T, R) is a Banach space.
Proof. We begin the iteration by arbitrarily select an element in the space X(T, R), and denote it by (v (0) , w (0) , q (0) ). To obtain the result of this lemma, we need to prove that the mapping Γ maps from X(T, R) to X(T, R) and is a contraction in the norm of X(T ). By Lemma 5.1, we can infer that (v (n+1) , w (n+1) , q (n+1) ) = Γ(v (n) , w (n) , q (n) ) ∈ X(T ), if (v (n) , w (n) , q (n) ) ∈ X(T ). To finish the proof, we need the following estimates. First, we consider the four energy level estimates for the system (161)-(166). For the first level energy V (n+1) 0 which gives By Gronwall's inequality, we obtain (169) Then, we submit (169) into (168), which leads to Here, we have used the assumption of the given matrix a(x, t). Similarly, we attain the other level energy estimates as below.
where I 4 is defined similarly as above andP 1 ,P 2 are polynomials with the degree of each term of it is at least 2. The last term of (181) can be absorbed in the dissipation term This means that div DW (∇w δ + I) → div DW (∇w + I) in C([0, T ]; H 1 (Ω e )). Therefore, we recover the equations (4) and (5). For the boundary term and divergence condition, similar arguments also show their convergence. Thus, we complete the construction of short-time solutions of the system (4), (5) and (9)-(11). By Lemma 2.1 and Corollary 1, we arrive at the desired regularity of a and η. Hence, we finish the proof of Theorem 1.2.