Research Article | Open Access

Yantao Guo, Shuilin Cheng, Yanbin Tang, "Approximate Kelvin-Voigt Fluid Driven by an External Force Depending on Velocity with Distributed Delay", *Discrete Dynamics in Nature and Society*, vol. 2015, Article ID 721673, 9 pages, 2015. https://doi.org/10.1155/2015/721673

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

**Academic Editor:**Luca Gori

#### 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 asWe 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 thatWe 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 asthen 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 thatThen, andSo, we have The Gronwall inequality implies thatPutting (20) into the right-hand side of (18), we have This implies thatNow, multiplying (12) by and integrating over , we have sincethenintegrating the above inequality from to , by (17), (20), and (22) we haveSince is bounded in , we obtain thatBy 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 haveMultiplying (28) by and integrating over , we obtainNotice thatSubstituting (30) into (29) and integrating from to , we get implies thatAs the property of operator and Poincaré, we haveSubstituting (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 SincethenIntegrating (38) from to , we getAssumptions imply thatSubstituting (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 byis 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 equationwith the initial condition , .

Set an energy functionMultiplying (50) by and integrating over with , , we havethen we haveUsing Poincaré inequality and (51) and (53), we haveIntegrating (52) from to with respect to , we obtainSubstituting (54) into (55), we getSetthen, by (56), we haveOne 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 limitsBy , we haveCombining (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).

#### References

- A. P. Oskolkov, “The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers,”
*Zapiski Nauchnykh Seminarov Leningrad Otdel Mathematics Institute Stekov (LOMI)*, vol. 38, pp. 98–136, 1973. View at: Google Scholar - Y. Cao, E. M. Lunasin, and E. S. Titi, “Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models,”
*Communications in Mathematical Sciences*, vol. 4, no. 4, pp. 823–848, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. K. Kalantarov and E. S. Titi, “Global attractors and determining modes for the 3D Navier-Stokes-Voight equations,”
*Chinese Annals of Mathematics, Series B*, vol. 30, no. 6, pp. 697–714, 2009. View at: Publisher Site | Google Scholar | MathSciNet - G.-C. Yue and C.-K. Zhong, “Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations,”
*Discrete and Continuous Dynamical Systems Series B*, vol. 16, no. 3, pp. 985–1002, 2011. View at: Publisher Site | Google Scholar | MathSciNet - J. García-Luengo, P. Marín-Rubio, and J. Real, “Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations,”
*Nonlinearity*, vol. 25, no. 4, pp. 905–930, 2012. View at: Publisher Site | Google Scholar | MathSciNet - J. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 1977. View at: MathSciNet - Y. Tang and M. Wang, “A remark on exponential stability of time-delayed Burgers equation,”
*Discrete and Continuous Dynamical Systems. Series B*, vol. 12, no. 1, pp. 219–225, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Y. Tang and L. Zhou, “Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay,”
*Journal of Mathematical Analysis and Applications*, vol. 334, no. 2, pp. 1290–1307, 2007. View at: Publisher Site | Google Scholar | MathSciNet - J. H. Wu,
*Theory and Applications of Partial Differential Equations*, Springer, New York, NY, USA, 1996. - L. Zhou, Y. B. Tang, and S. Hussein, “Stability and Hopf bifurcation for a delay competition diffusion system,”
*Chaos, Solitons & Fractals*, vol. 14, no. 8, pp. 1201–1225, 2002. View at: Publisher Site | Google Scholar | MathSciNet - T. Caraballo and J. Real, “Navier-Stokes equations with delays,”
*The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences*, vol. 457, no. 2014, pp. 2441–2453, 2001. View at: Publisher Site | Google Scholar | MathSciNet - T. Caraballo and J. Real, “Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays,”
*The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences*, vol. 459, no. 2040, pp. 3181–3194, 2003. View at: Publisher Site | Google Scholar | MathSciNet - T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes models with delays,”
*Journal of Differential Equations*, vol. 205, no. 2, pp. 271–297, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Hu and Y. Wang, “Pullback attractors for a nonautonomous nonclassical diffusion equation with variable delay,”
*Journal of Mathematical Physics*, vol. 53, no. 7, Article ID 072702, 17 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Y. Li and Y. M. Qin, “Pullback attractors for three-dimensional navier-stokes-voigt equations with delays,”
*Boundary Value Problems*, vol. 2013, article 191, 2013. View at: Publisher Site | Google Scholar - P. Marín-Rubio and J. Real, “Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 67, no. 10, pp. 2784–2799, 2007. View at: Publisher Site | Google Scholar | MathSciNet - P. Marín-Rubio, J. Real, and J. Valero, “Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 5, pp. 2012–2030, 2011. View at: Publisher Site | Google Scholar | MathSciNet - P. Marín-Rubio and J. Real, “Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,”
*Discrete and Continuous Dynamical Systems. Series A*, vol. 26, no. 3, pp. 989–1006, 2010. View at: Publisher Site | Google Scholar | MathSciNet - A. K. Khanmamedov, “Global attractors for wave equations with nonlinear interior damping and critical exponents,”
*Journal of Differential Equations*, vol. 230, no. 2, pp. 702–719, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Yang and C.-K. Zhong, “Global attractor for plate equation with nonlinear damping,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 11, pp. 3802–3810, 2008. View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2015 Yantao Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.