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

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.


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  = ( 1 ,  2 ,  3 ) is the velocity field of the fluid,  is the pressure, ] > 0 is the kinematic viscosity,  > 0 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 [−ℎ, 0], where ℎ is a fixed positive number and Ω is a bounded smooth domain of  3 .
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 − 2 Δ  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][7][8][9][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][12][13].The existence of pullback attractors in   1 0 and   1 0 ∩ 2 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.
Proof.Consider the Galerkin approximations for problem (10): where   = ∑  =1      ,   = ∑  =1        , 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  = 1 to  and using the fact we obtain that, for a.e. > , Integrating (15) Remark 1 implies that The Gronwall inequality implies that Putting ( 20) into the right-hand side of ( 18), we have This implies 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 , V be two solutions to problem (10) corresponding the initial data  and , respectively.
(30) Substituting (30) into (29) and integrating from  to , we get As the property of operator  and Poincaré, we have 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.

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).
Proof.Multiplying (10) by  and integrating over Ω, we obtain where  is a constant determined later.By Poincaré inequality, we have Since then Substituting (40) into (39), we have which implies Now, if we take  ≥  + ℎ, then, for  ∈ [−ℎ, 0], we have We denote by R the set of all functions  : (−∞, +∞) → (0, +∞) such that lim 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 Φ(⋅, ⋅) which, defined on  × , is a contractive function on  ×  if for any sequence Denote all such contractive functions on  ×  by ().
The proof is complete.
1, 2) is a contractive function, since  is a bounded positive invariant set.If   →  ( → ∞), we have the limits lim