Random Attractors for Stochastic Retarded 2D-Navier-Stokes Equations with Additive Noise

Jia, Xiaoyao; Ding, Xiaoquan

doi:https://doi.org/10.1155/2018/3105239

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Research Article | Open Access

Volume 2018 | Article ID 3105239 | https://doi.org/10.1155/2018/3105239

Random Attractors for Stochastic Retarded 2D-Navier-Stokes Equations with Additive Noise

Xiaoyao Jia¹and Xiaoquan Ding¹

Academic Editor: Maria Alessandra Ragusa

Received04 Jan 2018

Accepted29 Aug 2018

Published13 Sept 2018

Abstract

In this paper, the existence and the upper semicontinuity of a pullback attractor for stochastic retarded 2D-Navier-Stokes equation on a bounded domain are obtained. We first transform the stochastic equation into a random equation and then obtain the existence of a random attractor for random equation. Then conjugation relation between two random dynamical systems implies the existence of a random attractor for the stochastic equation. At last, we get the upper semicontinuity of random attractor.

1. Introduction

In this paper, we study the asymptotic behavior of the solutions for stochastic retarded Navier-Stokes equations with additive noise on a bounded domain in :where is a bounded set in , is a positive constant, is the delay time of the system, are given functions, is the strength of noise, and are two-sided Wiener processes on a probability space which will be specified later.

Navier-Stokes equations are basic equations of describing uncompressible viscous fluids, which are very important problems in fluid mechanics. Hence, Navier-Stokes equations have attracted many authors’ attention. As all of us know, the asymptotic behavior of dynamical systems, which can be studied by attractor, is an important problem. Regarding the asymptotic profile of the solution, using similarity variables, some interesting results can be found in [1, 2]. The random attractor for stochastic system was introduced in [3, 4] as an extension of the attractor for deterministic system in [5]. The attractors for dynamical systems have been studied by many authors (see [3, 4, 6–19]). The attractors are also used to study deterministic Navier-Stokes equations (see [9, 20–22]) and stochastic Navier-Stokes equations (see [4, 23, 24]). For the stochastic Navier-Stokes equations with additive noise and without time delay, the pullback attractors have been investigated in [23, 24]. However, as far as we know, there is no result on the random attractors for the stochastic retarded Navier-Stokes equation on bounded domains. So the main task of this paper is to study the asymptotic behavior of random attractor for stochastic Navier-Stokes equation (1).

To study the asymptotic behavior of random attractors, we shall establish a random dynamical system for (1) first. Then, we prove the existence and uniqueness of pullback random attractor. To achieve this, we derive the uniform estimates of solutions for Navier-Stokes equation with delays and additive noise. Finally, we study the convergence of random attractors as the noise intensity goes to zero.

The paper is organized as follows. In Section 2, we recall some notations and results on pullback attractors for random dynamical systems. And we show that the stochastic retarded Navier-Stokes equations generate a random dynamical system. In Section 3, we obtain some uniform estimates of solutions for system (14). These estimates are used to prove the existence of bounded absorbing sets and the asymptotic compactness of the solutions. In the last section, we obtain the existence of a pullback random attractor.

2. RDS Generated by Stochastic 2D-Navier-Stokes Equation

In this section, we introduce some notations and recall some results about random attractors for random dynamical systems (RDS). The readers can refer to [3, 4, 6, 8, 10] for more details.

Suppose that is a continuous RDS of over . And suppose is a collection of subsets of . We refer to [4, 7] for the existence of random attractor for continuous RDS.

Proposition 1. Let be a random absorbing set for the continuous RDS in and be -pullback asymptotically compact in . Then has a unique -random attractor which is given by

Next, we refer to [19] for the upper semicontinuity of random attractors.

Proposition 2. Suppose the following conditions are satisfied.
(1) For given , suppose is a random dynamical system over a metric system . For -a.e. , , , and with , there holds(2) Assume that has a random attractor and a random absorbing set such that for some deterministic positive constant and for -a.e. ,with .
(3) There exists such that for -a.e. , is precompact in .
Then, for -a.e. ,

In the rest of this paper, we use the following notation. Let be an open bounded set with regular boundary . Setand let be the closure of in with norm and inner product . Let be the closure of in with norm , and inner product , where for ,Let and be dual space of and . Then one has , and the injections are dense and compact. Let be the duality pairing between and . For , define , byLet . Then , , and is a orthogonal projection from to . Next, we define a trilinear form on byThen, is a continuous bilinear form and satisfies the following condition (see [5, 22]):By the Poincaré inequality and the homogeneous Dirichlet boundary, there is a constant such thatThe norm of a general Banach space is denoted by . For , we denote by the function defined on by the relation , . Let denote the Banach space of all continuous functions endowed with the supremum norm .

In this section, we consider the following stochastic Navier-Stokes equation on with additive white noise:Here is a positive constant, is the delay time of the system, are independent two-sided Wiener processes defined on a probability space on a probability space , withthe Borel -algebra on is generated by the compact open topology [6] and is the corresponding Wiener measure on ; and is a given function in and are given functions in with , and is a function that satisfies the following conditions.(A1) is continuous with , and, for all , and ,(A2).

Let

Then is an ergodic metric dynamical system. Notice that the above probability space is canonical; one hasTo study the random attractor for system (14), we first transform that system into a deterministic system with random parameter. For , we introduce the Ornstein-Uhlenbeck process:Then, one has that solves the following equation (see [11] for details):Moreover, the random variable is tempered, and is -a.e. continuous. It follows from Proposition 4.3.3 [6] that there exists a tempered function such thatwhere satisfies, for -a.e. ,Here is a positive constant which will be fixed later in (43) and (44). Then, it follows that, for and for -a.e. ,By [3, 25] and Itô formula, one has thatIt follows from (26)-(28) that there exists a positive constant large enough, and a constant , such that, for all ,Let ; then (21) implies thatLet . Then satisfieswith the boundary conditionand with the initial conditionHere .

For the deterministic problem (31)-(33) with random parameter, under the assumptions (A1) and (A2), by the Galerkin method (using a similar method to Theorem 2.1 [26]), one has that for all , , and , it has a unique solution . Define a mapping bywith for . Then, it is easy to verify that is a continuous random dynamical system associated with the stochastic Navier-Stokes equation. Notice that . DefineThen is a continuous random dynamical system. Moreover, and are conjugated random dynamical system. Therefore, in the following sections, we mainly consider the existences of random attractors for .

3. Uniform Estimates of Solutions

In this section, we prove the existence of the random attractor for the random dynamical system associated with the stochastic Navier-Stokes equation (14). We first establish the existence of the random attractor for its conjugated random dynamical system ; then the existence of a -random attractor for follows from the conjugation relation between and .

From now on, we assume that is the collection of all tempered subsets of with respect to . In the rest of the paper, we use notation to denote positive constant which is independent of .

Lemma 3. Let and suppose that conditions (A1) and (A2) hold. Then there exists a random ball centered at 0 with random radius such that is a random absorbing set for in ; that is, for every and -a.e. , there is , independent of , such that for all .

Proof. Taking the inner product of (31) with , we obtain thatNow, we estimate each term on the right-hand side of (36). Notice that are given functions in . By the definition of , the first term can be estimated as follows:Here is a positive constant depending only on . By condition (A2), we can choose small enough such that . Then, Young’s inequality implies thatChoose small enough such that . By condition (A1) and Young’s inequality, we have thatThen it follows from (36)–(39) thatNotice that and . Hence there exists a constant , such thatwith independent of . By (24) and (25), we have that, for -a.e. , for any , there exists a constant , such thatChoose small enough such thatThen by (40), (41), and Poincaré inequality (13), we obtain thatwith , . By Gronwall’s inequality, one has that, for all ,Now, we estimate the fourth term on the right-hand side of (46). Note thatThen, by (44) and (47), we haveTherefore, (46) and (48) implywith . Therefore, for fixed , we have that, for ,and for ,Thus, we deduce from (50) and (51) that, for all Replacing by in the last inequality, we get that, for all ,By (29) and (42), one has that, for ,andSetThen, there exist a , independent of , such that, for all ,This ends the proof.

Lemma 4. Let and suppose that conditions (A1) and (A2) hold. Then there exists a tempered random variable such that, for any and , the solution of (31) satisfies, for -a.e. , for all ,Here is the random variable defined in Lemma 3.

Proof. By (49) we have thatReplacing by , and by (54) and (55), one has that, for all ,On the other hand, we have thatEquations (60) and (61) imply that, for all ,This ends the proof.

Lemma 5. Let and suppose that conditions (A1) and (A2) hold. Then, there exist random variables , , for any , and ; the solution of (31) satisfies the following estimates: for -a.e. , for all , and Here, is the random variable defined in Lemma 3.

Proof. Taking the inner product of (31) with , we obtain thatNow, we estimate each term on the right-hand side of (65). By (12) and Young’s inequalityandConsequently, it follows from (65)–(67) and Poincaré inequality (13) thatwithReplacing by and replacing by in (68), we get that, for all ,Now, we use uniform Gronwall inequality to estimate . By (57) and (22), we can get that, for all ,andConsequently, it follows from (58), (71), and (72) that, for ,By condition (A1), we get that, for ,It follows from (22), (57), and (74) that, for ,Similarly as (73), we have that, for ,By (58), (70), (73), (76), and uniform Gronwall inequality, we have that for -a.e. , for all ,Consequently, we obtain that, for all ,Integrate (68) over ; then one has thatReplacing by in (79), we find thatWe need to estimate each term on the right-hand side of (80). By (78), one has that, for ,Similarly, we can deduce that, for , Then, it follows from (81)–(83) thatandThen, it follows from (80), (84), and (85) thatThis ends the proof.

We use Ascoli theorem to prove the existence of a pullback random attractor. Before that, we need to prove the following result.

Lemma 6. For all , and for , there exists a random variable such that

Proof. By (31),Lemma 5 is used to estimate the first term on the right-hand side of (88). One has that, for ,For the second term on the right-hand side of (88), we have thatNext, we use Lemmas 3 and 5 to estimate the third and fourth terms on the right-hand side of (88). For , we can obtain thatandIn (92), we use condition (A1). For the last two terms on the right-hand side of (88), by (22), we haveThis completes the proof.

Theorem 7. Let and suppose that conditions (A1) and (A2) hold. Then, the random dynamical system has a unique -random attractor in .

Proof. By Lemma 3, we know that, for fixed , has a closed random absorbing set in . Next, we use Ascoli theorem to prove that is -pullback asymptotically compact in . That is, we need only to show that, for -a.e. , the sequence has a convergent subsequence in , as . Because , there is , such that , as . By Lemmas 3 and 5, we obtain that, for all and ,It follows from that for each , is relatively compact in . By Lemma 6, one has that is equicontinuous. Ascoli theorem [27] implies that is relatively compact in . Thus, Proposition 1 implies the existence of a unique -random attractor in .

Notice that and are conjugated random dynamical system. Hence, it follows from Theorem 7 that has unique random attractors in .

4. Upper Semicontinuity of Random Attractor

In this section, we consider the upper semicontinuity of random attractors of Navier-Stokes equation (31), when the intensity of noise tends to zero. We write the solution and the random dynamical system of (31) as and , respectively. LetwithSet In Section 3, we prove that has a unique -pullback random attractor . It follows from the invariance of the random attractor that, for , . SetwithThen, for all ,This implies thatWe will show that, as , the solution of (31) converges to the limiting deterministic system:

Lemma 8. For , let and be the solution of (31) and (101), respectively. Then, for , , there exists a positive constant which is independent of , such that for all

Proof. Let . By (31) and (101), we have thatTaking the inner product of (103) with , we obtain thatNow, we estimate each term on the right-hand side of (104). For the first and second termsBy (12), (24), and Young inequality, one hasFor the third term on the right-hand side of (104), by condition (A1) and (25), one has thatandFor the fourth and fifth terms on the right-hand side of (104),It follows from the above inequalities that we have Using Gronwall inequality, we get thatNote thatThen, it follows thatBy (49), one has that, for all , (46) and (48) implyNotice that, for all ,By (29) and (42), one has that, for all ,Here and are constants depending on . Then by (116)–(119),It follows from (115) and (120) that one has that, for all ,which implies (102).

Theorem 9. Suppose (A1) and (A2) hold. Let be random attractor of , respectively. Then, for -a.e. ,

Proof. Let and in . Then, by Lemma 8, we can obtain that, for -a.e. , and ,Assume that is defined by (95). From the proof of Lemma 3, we find that is an absorbing set of in . It is easy to see thatBy Theorem 7 and the invariance of the random attractor , we have thatis precompact in . Then, (122) follows from Proposition 2.

By the conjugated relation between and , it follows from Theorem 9 that the random attractor of converges to the random attractor of , as .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China under Grants 11301153 and 11271110, the Key Programs for Science and Technology of the Education Department of Henan Province under Grant 12A110007, and the Scientific Research Funds of Henan University of Science and Technology.

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Copyright © 2018 Xiaoyao Jia and Xiaoquan Ding. 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.

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