Approximate Kelvin-Voigt Fluid Driven by an External Force Depending on Velocity with Distributed Delay

Abstract

We consider the approximate 3D Kelvin-Voigt fluid driven by an external force depending on velocity with distributed delay. We investigate the long time behavior of solutions to Navier-Stokes-Voigt equation with a distributed delay external force depending on the velocity of fluid on a bounded domain. By a prior estimate and a contractive function, we give a sufficient condition for the existence of pullback attractor of NSV equation.

1. Introduction

In this paper, we consider 3D Navier-Stokes-Voigt (NSV) equation with a distributed delay external force depending on the velocity of the fluid: where is the velocity field of the fluid, is the pressure, is the kinematic viscosity, is the length scale parameter of the elasticity of the fluid, the external force and initial velocity field are defined in the interval of time , where is a fixed positive number and is a bounded smooth domain of .

The NSV equation was introduced by Oskolkov [1] to give an approximate description of the Kelvin-Voigt fluid and was proposed as a regularization of 3D Navier-Stokes equation for the purpose of direct numerical simulations in [2]. Since the term changes the parabolic character of the equation, the NSV equation being well posed in 3D, many authors have studied the long time dynamics of this model. Kalantarov and Titi [3] investigated the existence of the global attractor, the estimation of the upper bounds for the number of determining modes, and the dimension of global attractor of the semigroup generated by the equations. By a useful decomposition method, Yue and Zhong [4] proved the asymptotic regularity of solution of NSV equation and obtained the existence of the uniform attractor; they also described the structure of the uniform attractor and its regularity. García-Luengo et al. [5] investigated the existence and relationship between minimal pullback attractor for the universe of fixed bounded sets and universe given by a tempered condition.

Partial differential equations with delays arise from various fields, like physics, control theory, and so on (see, e.g., [6–10]); the unknown functions depend on not only present stage but also some past stage. The existence and stability of solution and global attractor for Navier-Stokes equation with discrete delay were established in [11–13]. The existence of pullback attractors in and was proved for the processes associated with nonclassical diffusion equations with variable bounded delay in [14, 15]. Delay effect has been considered on an unbounded domain in [16]. The existence of pullback attractor for a Navier-Stokes equation with infinite discrete delay effect was studied in [17].

The aim of this paper is to investigate the NSV equation with a distributed delay, instead of the discussions with finite delays in the references. Our purpose is twofold. We first show the existence and uniqueness of solution to NSV equation (1) with a distributed delay; then we prove the existence of pullback attractor for the process generated by the NSV equation (1).

This paper is organized as follows. In Section 2, we give some preliminary results and prove existence of solution to NSV equation with a distributed delay. In Section 3, we derive the existence of pullback attractor by prior estimates and contractive functions.

2. Existence of Solutions

In order to prove the existence of solutions to problem (1), we define the function spaces is the closure of in with the inner product and associate norm , is the closure of in with scalar product and associate norm , where it follows that , where the injections are dense and compact. We will use for the norm in and for the duality pairing between and .

Define the linear continuous operator as We denote ; one has that and , for all is the Stokes operator, where is the orthoprojector from onto ; also denote and .

Define the trilinear form on by and the operator as and denote .

The trilinear form satisfies that We also recall that there exists a constant depending only on such that

For the term containing the time delay, satisfies that  is a measurable function,  for all ,  there exists a positive constant , such that ; if and , then

Remark 1. Hypotheses - imply that , so we have for .

Problem (1) can be rewritten as then we get the existence of solution to problem (10).

Theorem 2. Let , let satisfy the hypotheses , and let . Then, ; there exists a unique weak solution to (10) such that Moreover, if , then problem (10) admits a strong solution.

Proof. Consider the Galerkin approximations for problem (10): where , , and and are the corresponding orthonormal eigenfunctions and eigenvalues of operator , respectively; then, We now derive a prior estimate for the Galerkin approximate solution. Multiplying (12) by , summing from to and using the fact we obtain that, for a.e. , Integrating (15) from to , we obtain that Remark 1 implies that Then, and So, we have The Gronwall inequality implies that Putting (20) into the right-hand side of (18), we have This implies that Now, multiplying (12) by and integrating over , we have since then integrating the above inequality from to , by (17), (20), and (22) we have Since is bounded in , we obtain that By the Faedo-Galerkin scheme, for example, see [14, 18], according to the estimates (22) and (27), we can get existence of the weak solution; here we omit the details.
We next consider the uniqueness of solution. Let be two solutions to problem (10) corresponding the initial data and , respectively.
Denote ; then, we have Multiplying (28) by and integrating over , we obtain Notice that Substituting (30) into (29) and integrating from to , we get implies that As the property of operator and Poincaré, we have Substituting (32) and (33) into (31), we get The last inequality comes from Poincaré inequality and the boundedness of . Therefore, the Gronwall inequality implies the uniqueness of the solution. The proof is complete.

3. Existence of Pullback Attractor

In this section, we will prove the existence of pullback attractor to problem (10). First we give existence of pullback absorbing set for the process generated by the global solution to problem (10).

Lemma 3. Assume hold and ; then, the process is pullback dissipative, where .

Proof. Multiplying (10) by and integrating over , we obtain where is a constant determined later.
By Poincaré inequality, we have Since then Integrating (38) from to , we get Assumptions imply that Substituting (40) into (39), we have Let , choosing . implies that then, which implies Now, if we take , then, for , we have We denote by the set of all functions such that Then, the closed ball in defined by is pullback absorbing set for . The proof is complete.

We next prove the asymptotic compactness of solution to problem (10) by contractive functions; see [19, 20].

Let be a Banach space and let be a bounded subset of . We call a function which, defined on , is a contractive function on if for any sequence there is a subsequence such that Denote all such contractive functions on by .

Theorem 4 (see [19]). Let be a semigroup on a Banach space and have a bounded absorbing set , ; there exist and , such that where depends on . Then, is asymptotically compact in .

Lemma 5. Assume that hold; the process generated by the global solution to problem (10) is asymptotically compact.

Proof. Let be the solution to problem (10) with initial data (), respectively. Denote ; then, satisfies the equivalent abstract equation with the initial condition , .
Set an energy function Multiplying (50) by and integrating over with , , we have then we have Using Poincaré inequality and (51) and (53), we have Integrating (52) from to with respect to , we obtain Substituting (54) into (55), we get Set then, by (56), we have One has ; we choose large enough such that . By Theorem 4, it suffices to prove that is a contractive function, since is a bounded positive invariant set.
If (), we have the limits By , we have Combining (60)-(61) with (58), we have that is a contractive function.
The proof is complete.

Theorem 6. Assume that hold and ; then, the process generated by the global solution for problem (10) has a pullback attractor.

Proof. Lemma 3 implies has a pullback absorbing set and Lemma 5 implies is asymptotically compact; we obtain the conclusion immediately.

According to Theorem 6, and . Under the assumptions , the NSV equation (1) with a distributed delay has a pullback attractor.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Yantao Guo carried out the long time behavior of solutions. Shuilin Cheng carried out the pullback attractor. Yanbin Tang carried out the distributed delay. All authors read and approved the final paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant nos. 11471129 and 11272277).