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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 267328, 23 pages
Ergodicity of Stochastic Burgers’ System with Dissipative Term
1College of Mathematics and Statistics, Chong Qing University, Chong Qing 401331, China
2Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
Received 9 October 2013; Revised 6 November 2013; Accepted 6 November 2013
Academic Editor: Hamid Reza Karimi
Copyright © 2013 Guoli Zhou 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.
A 2-dimensional stochastic Burgers equation with dissipative term perturbed by Wiener noise is considered. The aim is to prove the well-posedness, existence, and uniqueness of invariant measure as well as strong law of large numbers and convergence to equilibrium.
The paper is concerned with the 2-dimensional Burgers equation in a bounded domain with Wiener noise as the body forces like this where is the velocity field, is viscid coefficient, denotes the Laplace operator, represents the gradient operator, stands for the -Wiener process, and is a regular bounded open domain of . Burgers equation has received an extensive amount of attention since the studies by Burgers in the 1940s (and it has been considered even earlier by Beteman  and Forsyth ). But it is well known that the Burgers’ equation is not a good model for turbulence since it does not perform any chaos. Even if a force is added to equation, all solutions will converge to a unique stationary solution as time goes to infinity. However, if the force is a random one, the result is completely different. So, several authors have indeed suggested to use the stochastic Burgers’ equation to model turbulence, see [3–6]. The stochastic equation has also been proposed in  to study the dynamics of interfaces.
So far, most of the monographs concerning the equation focus on one-dimensional case, for example, Bertini et al.  solved the equation with additive space-time white noise by an adaptation of the Hopf-cole transformation. Da Prato et al.  studied the equation via a different approach based on semigroup property for the heat equation on a bounded interval. The more general equation with multiplicative noise was considered by Da Prato and Debussche . With a similar method, Gyöngy and Nualart  extended the Burgers equation from bounded interval to real line. A large deviation principle for the solution was obtained by Gourcy . Concerning the ergodicity, an important paper by Weinan et al.  proved that there exists a unique stationary distribution for the solutions of the random inviscid Burgers equation, and typical solutions are piecewise smooth with a finite number of jump discontinuities corresponding to shocks. For model with jumps, Dong and Xu  proved that the global existence and uniqueness of the strong, weak, and mild solutions for a one-dimensional Burgers equation perturbed by Lévy noise. When the noise is fractal, Wang et al.  get the well-posedness.
Burgers system is a well-known model for mechanics problems. But as far as we know, there are no results about the long-term behavior of stochastic Burgers’ system. We think that the difficulty lies in the fact that the dissipative term cannot dominate the nonlinear term . However, in many practical cases, we cannot ignore the energy dissipation and external forces, especially considering the long-term behavior. Therefore, we introduce dissipative term and study the ergodicity of the following equation: where , denote the absolute value or norm for the real number or two-dimensional vector, respectively.
We believe that our work is new and is worth researching. The methods and results in this paper can be applied to stochastic reaction diffusion equations and stochastic real valued Ginzburg Landau equation in high dimensions. But we cannot extend our result to dynamical systems with state-delays. Since in order to show the existence of an invariant measure, we should consider the segments of a solution. In contrast to the scalar solution process, the process of segments is a Markov process. We show that the process of segments is also Feller and that there exists a solution of which the segments are tight. Then, we apply the Krylov-Bogoliubov method. Since the segment process has values in the infinite-dimensional space , boundedness in probability does not automatically imply tightness. For solution processes of infinite-dimensional equations, one often uses compactness of the orbits of the underlying deterministic equation to obtain tightness. For an infinite-dimensional formulation of the functional differential equation, however, such a compactness property does not hold. For ergodicity of stochastic delay equations, we can see . We believe that stochastic Burgers’ system with state-delays is a very interesting problem.
In order to study ergodicity of problem (2), we use a remarkable dissipativity property of the stochastic dynamic to obtain the existence of the invariant measure. For uniqueness, we try to use the method from  to prove that the distributions induced by the solution are equivalent. It is well known that the equivalence of the distributions implies uniqueness, a strong law of large numbers, and the convergence to equilibrium.
The remaining of this paper is organized as follows. Some preliminaries are presented in Section 2, the local existence and global existence are presented, respectively, in Sections 3 and 4. In Section 5, we obtain the existence and uniqueness of the invariant measure as well as strong law of large numbers, and convergence to equilibrium. As usual, constants may change from one line to the next; we denote by a constant which depends on some parameter .
2. Preliminaries on the Burgers Equation
Let be a row vector valued function on . And it denotes the following: Let be infinitely differentiable 2-dimensional vector field on , and let be infinitely differentiable 2-dimensional vector field with compact support strictly contained in . We denote by the closure of in , whose norms are denoted by , when . Let be the closure of in and whose norms are denoted by and , respectively. Without confusion, set as the inner product in or . For , let be the norm of vector filed in Lebesgue spaces . represents the norm in the usual sobolev spaces for real valued functions on and ; stands for the norm in the usual Lebesgue spaces for real valued functions on . Denote ; then and . Since coincides with , we can endow with the norm . The operator is positive self-adjoint with compact resolvent; we denote by the eigenvalues of , and by the eigenvectors which is a corresponding complete orthonormal system in satisfying for some positive constant C. We remark that . We define the bilinear operator as for all . Then, (2) is equivalent to the following abstract equation: is the Wiener process having the following representative: in which and are a sequence of mutually independent 1-dimensional Brownian motions in a fixed probability space adapted to a filtration .
It can be derived from  that the solution to the linear problem corresponding to (2) with the following initial condition: is unique, and when , it has the form of Let then is a solution to (2) if and only if it solves the following evolution equation: So, we see that when is fixed, this equation is in fact a deterministic equation. From now on, we will study the equation of the form (11) to get the existence and uniqueness of the solution a.s. .
3. Local Existence in Time
Lemma 2. For any , if , then has a version which is -Hölder continuous with respect to , with any .
Proof. Let and ; then Then, we have So, by the estimate of and , we arrive at For , , we get Therefore, As is a Gaussian random variable, we obtain for By Kolmogorov’ test theorem, we get the conclusion.
Remark 3. An example of the noise satisfying condition of Lemma 2 is where is a sequence of independent 1-dimensional Brownian motion, and satisfies It is so because the eigenvalues of the operator , in 2-dimensional space, behave like .
Remark 4. Another example of stochastic noise satisfying Lemma 2 is where , is an isomorphism in , and
Proof. Choose a in , and set
For the second term on the right hand side of (25),
In the following, we will estimate , respectively, . Since is contraction on , it is known that
for all , , , , and only depends on , , and . Before calculating each , we outline the Sobolev embedding principle in fractional Sobolev spaces as follows:
where is the dimension of the spatial. Let , , , satisfying (29) such that
For , by (27) and Theorem A.8 in , we get where satisfying The last inequality follows by (30). For the other term added to , we have So, by (31)–(34), we have Similarly, we get for that For , by Theorem A.8 in , we get where
For , we have For the first term on the right hand side of (37), by (27), we have For the second term on the right hand side of (37), by (27), we obtain From (37) to (41), we get for that Analogously, for , we get By (26), (35), (36), (42), and (43), we have As , by (44), for , we have Since by Lemma 2, For the last term on the right hand side of (25), we have Therefore, So by (25), (45), and (48), when is small enough, For each , set , . To simplify the notation in the following calculation, we denote , . Then, So, In order to simplify the notation, we set where Then, we estimate , , respectively. For , we have We first consider For the other term added to , By (54)–(56), Analogously, for , For , by (53), we have For the first term on the right hand side of (59), we have For , For the first term on the right hand side of (60), we arrive at For the second term on the right hand side of (60), we obtain By (59)–(63), we get for that Similarly, we get for that By (52), (53), (57), (58), (64), and (65), we have For the second term on the right hand side of (51), we have where Then, Similarly, we can get the same estimate for . So, we have By (51), (66), and (70), we have By (49), (71), and fixed point principle, we get the conclusion.
Remark 6. By making some minor modifications in the proof of Lemma 5, we can see that the conclusion in Lemma 5 is also true for (1). Our original aim is to get the global well-posedness of (1), but we find that the dissipative term cannot dominate the nonlinear term . So, we introduce the dissipative term which will also play an important role in obtaining the ergodicity.
4. Global Existence
Proof. Let be a sequence of vectors which satisfies and , , such that in sense of . Let be a sequence of regular process, such that in when or . For , , , where . Then, by (74), we have If satisfies then, is regular, such that Taking inner product with respect to in (78), we have For simplicity, we calculate the third term on the left hand side of (79) first as follows: where . For , we have In the following, we estimate the four terms for , respectively. For the first term, For the second term, by (75), we have similarly, for the third term, For the last term, by (75) and (76), By (81)–(85), it follows that Similarly, For , For the first term on the right hand side of (88), we deduce that where . For the second term on the right hand side of (88), we have Analogously, for the third term on the right hand side of (88), we see that For the last term, by (75) and (76), we have By (88)–(92), we get Analogously, for , it follows that By (80) and the estimates of , and , see (86), (87), (93), and (94), we have For the last term on the left hand side of (79), we have By (79), (95), and (96), we get Rearranging the above inequality, we deduce that Let , and be small enough, such that So, we integrate with respect to on both sides of (98) to obtain where , by Gronwall’s inequality, we arrive at By (100) and (101), we have Multiplying on both sides of (78), and integrating with respect to , we have which is equivalent to We first estimate the second term on the right hand side of (104) as follows: For , we have For , we have By interpolation inequality, there exists some , such that Then, where the last inequality follows from (101). For , we deduce that For , we arrive at For , we obtain By (106) and (109)–(112), Similarly, for , we infer that For , we have By interpolation inequality and (101), we deduce that For , we have Similarly, for , As for , we get By (115)-(119), we arrive at Analogously to , we have By (105) and the estimates of , see (113), (114), (120), and (121), we get that For the first term on the right hand side of (104), we have By (104), (122), and (123), By the Gronwall inequality, we get Let , by Fatou Lemma,
5. Invariant Measures
In this section, we will establish the existence of invariant measure for (2). Analogously to , we extend the Wiener process to by setting where is another -valued Wiener process satisfying conditions in Lemma 2 and being independent of . For any , we consider the following equation: By Theorem 7, we know that there exists unique solution. In order to obtain the invariant measure, we should show that the family of laws is tight. Since is compact, for any , we only need to show that is bounded in probability in . As we know, is the mild solution of (8) with the following initial condition: Making the classical change of variable , (128) is equivalent to with initial condition In order to get the invariant measure of (131), it is enough to show that is bounded in probability in , for some . That is what we have to do in Theorem 8 below.