AVERAGING PRINCIPLE FOR STOCHASTIC KURAMOTO-SIVASHINSKY EQUATION WITH A FAST OSCILLATION

. This work concerns the problem associated with averaging principle for a stochastic Kuramoto-Sivashinsky equation with slow and fast time- scales. This model can be translated into a multiscale stochastic partial diﬀerential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with diﬀerent time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic Kuramoto-Sivashinsky equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single stochastic Kuramoto-Sivashinsky equation with a modiﬁed coeﬃcient.


Introduction. The Kuramoto-Sivashinsky equation
u t + γu xxxx + βu xx + uu x = 0 is a one-dimensional model for turbulence and wave propagation in reaction-diffusion systems, where γ and β are coefficients accounting for the long-wave instability and the short-wave dissipation respectively, was derived in various physical contexts.
In order to consider a more realistic model our problem, it is sensible to consider some kind of stochastic perturbation represented by a noise term in the equations. Stochastic Kuramoto-Sivashinsky equation is a important equation, a large amount of work has been devoted to the study of the stochastic Kuramoto-Sivashinsky equation: [3,4,13,14,22,42,43].
Throughout the paper, we will take for the sake of simplicity. All the results can be extended without difficulty to any constants γ > 0, β.
1.1. Motivation and problem. In this paper, we will be concerned with the following stochastic Kuramoto-Sivashinsky equation with slow and fast time-scales where T > 0, I = (0, 1), Q = I ×(0, T ), u ε , v ε are two real functions and {W 1 (t)} t≥0 and {W 2 (t)} t≥0 are L 2 (I)−valued mutually independent Q 1 and Q 2 Wiener processes, the terms f (u, v) and g(u, v) are external forces depending on u and v.
The first motivation of considering (1) is that (1) can be seen as a stochastic Kuramoto-Sivashinsky equation with a fast oscillating perturbation ( * ) where v ε (t) is governed by the stochastic reaction-diffusion equation System ( * ) is a model for phase turbulence in reaction-diffusion systems(see [27,28,29]) in a random environment with stochastic perturbation v ε which denotes the dramatically varying temperature, it also can be used as a model for plane flame propagation(see [37]) with stochastic perturbation in a random environment, describing the combined influence of diffusion and thermal conduction of the gas on stability of a plane flame front in a random environment. More generally, this model also describes incipient instabilities in a variety of physical and chemical systems(see [10,25,30]) in a random environment. This model also arises in the modeling of the flow of a thin film of viscous liquid falling down on an inclined plane subject to an applied electric field(see [24]) in a random environment.
The second motivation of considering (1) is that the nonlinear coupled stochastic Kuramoto-Sivashinsky-heat equations (1) with fast and slow time scales may describe the surface waves on multilayered liquid films in a random environment. Indeed, in order to combine dissipative and dispersive features, and to simultaneously support stable solitary-pulse, a model consisting of a Kuramoto-Sivashinsky equation, linearly coupled to an extra linear dissipative equation, is proposed in [32] under the name of the stabilized Kuramoto-Sivashinsky system. The model applies to a description of surface waves on multilayered liquid films. Almost all physical systems have a certain hierarchy in which not all components evolve at the same rate, i.e., some of components vary very rapidly, while others change very slowly, see [36], so we consider multiscale stochastic partial differential equations (1).
The nonlinear coupled stochastic Kuramoto-Sivashinsky-heat equations (1) with fast and slow time scales may describe the surface waves on multilayered liquid films in a random environment, two real wave fields u, v evolve at the different rates. Then (1) is a multiscale stochastic partial differential equations. Multiscale stochastic partial differential equations arise as models for various complex systems, such model arises from describing multiscale phenomena in, for example, nonlinear oscillations, material sciences, automatic control, fluids dynamics, chemical kinetics and in other areas leading to mathematical description involving "slow" and "fast" phase variables.
According to tha above motivations, the asymptotic study of the behavior ε → 0 of (1) is of great interest. Very often, one needs to simulate or predict the time evolution of the slow component of (1) without solving the full system of equations, then a reduced system which governs the slow motion over a long time scale is highly desirable. In this respect, the question of how the physical effects at large time scales influence the dynamics of (1) is arisen. This mathematical question arises naturally which are important from the point of view of dynamical systems from both physical and mathematical standpoints. We focus on this question and by using averaging principle, we show that, under some dissipative conditions on fast variable equation, the complexities effects at large time scales to the asymptotic behavior of the slow component can be omitted or neglected in some sense. More precisely, the slow process u ε converges toū in the strong way whereū is the solution of the following reduced problem Thus, with the help of averaging principle, • we can establish an effective approximation for slow process u ε with respect to the limit ε → 0, this can predicting the time evolution of the slow component u ε . • we can extract effective dynamic system (2) from complex system (1), it provides an effective tool to analyze qualitative behaviors of (1). It makes the interaction between nonlinearity, uncertainty and multiple scales of (1) more clear. • it enormously reduce the computational load of (1) although computer technology is highly efficient nowadays. It follows from the above facts that averaging principle can help us understand and investigate the physical phenomenon described by (1). On the one hand, the averaging principle for (1) shows that the dramatically varying temperature v ε does not affect the phase turbulence and plane flame propagation. On the other hand, it can be seen that when the propagation speed of the interface waves is high enough, the surface waves are not effected by the interface waves.
The theory of averaging principle has a long and rich history, which has been applied in many fields, such as, celestial mechanics, wireless communication, signal processing, oscillation theory and radio physics. The averaging principle in the stochastic ordinary differential equations setup was first considered by Khasminskii [26] which proved that an averaging principle holds in weak sense, and has been an active research field on which there is a great deal of literature. In recent years, there are many interesting results for stochastic system in infinite dimensional space: [1,6,7,8,9,12,18,16,20,17,19,34,38,41,39,40].

1.2.
Mathematical setting and assumptions. We introduce the following mathematical setting: Throughout the paper, the letter C denotes positive constants whose value may change in different occasions. We will write the dependence of constant on parameters explicitly if it is essential.
Let (Ω, F, {F t } t≥0 , P ) be a complete filtered probability space. Let Y be a Banach space, and let C([0, T ]; Y ) be the Banach space of all Y −valued strongly continuous functions defined on [0, T ]. We denote by L p < ∞. All the above spaces are endowed with the canonical norm.
We denote by L 2 (I) the space of all Lebesgue square integrable functions on I. The inner product on L 2 (I) is for any u, v ∈ L 2 (I). The norm on L 2 (I) is for any u ∈ L 2 (I).
L 2 (I), H s (I)(s ≥ 0) are the classical Sobolev spaces of functions on I. The definition of H s (I) can be found in [31], the norm on H s (I) is · H s .
Through this paper, we make the following assumption (H): 1. There exist constants L f , L g such that f and g satisfy for any u, v ∈ R. f and g are Lipschitz continuous, that is We assume that α λ − 2L g > 0, where λ > 0 is the smallest constant such that the following inequality holds where u ∈ H 1 0 (I) or I udx = 0.
1.3. Main result. Now, we are in a position to present the main result in this paper.
Theorem 1.1. Suppose that the hypothesis (H) holds and u 0 ∈ H 2 (I) ∩ H 1 0 (I), v 0 ∈ L 2 (I), (u ε , v ε ) is the solution of (1) andū is the solution of the effective dynamics equation (2), then for any T > 0, any p > 0, we have and µ u is an invariant measure for the fast motion with frozen slow component Moreover, if p > 5 4 , there exists a positive constant C(p) such that if 0 < p ≤ 5 4 , for any κ > 0, there exists a positive constant C(p, κ) such that The rest of this paper is structured as follows. In Section 2, we gather all the necessary tools. Section 3 is devoted to the proof of Theorem 3.2. Section 4 is devoted to the proof of Theorem 1.1.

2.
Preliminaries. This section is devoted to some preliminaries for the proof of Theorem 1.1.
Proposition 2. For any p, T > 0, we have Proof. By the same method as in [12,Lemma 4.2], we can prove this proposition.
2.3. The fast motion equation. We first consider the frozen equation associate to fast motion for fixed slow component We denote v u,X the solution to (5), we now discuss the asymptotic behavior of the fast equation (5).
By the same method as in [2], we can obtain the existence of the invariant measure for (5), namely, we have Proposition 3. For u, X, Y ∈ L 2 (I), let v u,X be the solution of in I × (0, +∞) in (0, +∞) in I.
1) There exists a positive constant C such that v u,X satisfies: for t ≥ 0.
2) There is unique invariant measure µ u for the Markov semigroup P u t associated with the system (6) in L 2 (I). Moreover, we have 3) There exists a positive constant C such that v u,X satisfies: for t ≥ 0.
Proof. 1) • By applying the generalized Itô formula with 1 2 v u,X 2 , we can obtain that Taking mathematical expectation from both sides of above equation, we have It follows from Young's inequality that Hence, by applying Lemma 2.3 with E v u,X (t) 2 , we have

PENG GAO
• It is easy to see thus, it follows from the energy method that Thus, we have Thus, we have 2) (7) implies that for any u ∈ L 2 (I) that there is unique invariant measure µ u for the Markov semigroup P u t associated with the system (6) in L 2 (I) such that for any ϕ ∈ B b (L 2 (I)) the space of bounded functions on L 2 (I).
Then by repeating the standard argument as in [8,Proposition 4.2] and [9, Lemma 3.4], the invariant measure satisfies

AVERAGING PRINCIPLE FOR SKS EQUATION 5657
3) According to the invariant property of µ u , (2) and (7), we have 3. Well-posedness and some a priori estimates of (1). Let us explain what we mean by a solution of (1) where C is a positive constant depending on ε, T, Q 1 , Q 2 . 3.1. Local existence. In this section, we will take ε = 1 for the sake of simplicity. All the results can be extended without difficulty to the general case.

PENG GAO
The truncated equation corresponding to (1) is the following stochastic partial differential equations: It is easy to see that for any Indeed, we set F (u) = uu x . Noting the fact for We have without any loss of generality, assume that By Finally, collecting the above estimates (9)-(11), we get This implies that • The estimate of (12) and (13) that For a sufficiently small T 0 , Φ R (u, v) is a contraction mapping on X T0 . Hence, by applying the Banach contraction principle, Φ R (u, v) has a unique fixed point in X T0 , which is the unique local solution to (8) on the interval [0, T 0 ]. Since T 0 does not depend on the initial value (u 0 , v 0 ), this solution may be extended to the whole interval [0, T ].
We denote by (u R , v R ) this unique mild solution and let with the usual convention that inf ∅ = ∞.
We define a local solution to (1) as follows Indeed, for any t ∈ [0, τ ] Proceeding as in the proof of (14), we can obtain where C(t) is a monotonically increasing function and C(0) = 0. If we take t sufficiently small, we can obtain Repeating the same argument for the interval [t, 2t] and so on yields for the whole interval [0, τ ]. According to this, we can know the above definition of local solution to (1) is well defined. If τ ∞ < +∞, the definition of (u, v) yields P−a.s. 3.2. Some a priori estimates of (u ε , v ε ).
Proof. For simplicity, we will omit the index ε. It is also suffice to prove this inequality holds when p is large enough. We apply the Itô formula (see [11,35]) with v ε 2p (p ≥ 2) and obtain that by taking mathematical expectation from both sides of above equation, we have Then, it follows from the property of g that then, by using the Young inequality, we have hence, by comparison theorem We apply the Itô formula (see [11,35]) with u ε 2p and obtain that Plug this inequality into (15), we have is the unique solution to (1), then for any p > 0, there exist constants C 1 , C 2 such that the solution (u ε , v ε ) satisfies where C 1 is dependent of p, T, u 0 , v 0 and C 2 is dependent of ε, p, T, u 0 , v 0 .
Proof. It is also suffice to prove these inequalities hold when p is large enough. 1) Noting Indeed, Indeed, this can be obtained from Proposition 2.
• Estimate Indeed, it follows from the Agmon's inequality, Gagliardo-Nirenberg inequality that

thus, it follows from this that
With the help of the above estimates, we arrive at by applying Gronwall inequality and Young inequality, we have With the help of the above estimates, we arrive at This completes the proof of Proposition 6.

4.2.
Well-posedness for the averaged equation (2). By the same method as in Theorem 3.2 and Proposition 6, we can obtain the following proposition.

4.3.
Hölder continuity of time variable for u ε . Next, we are going to provide a Hölder continuity of time variable for u ε .
Proof. Let us write here Id denotes the identity operator. * Due to [33], there is a C such that for all x ∈ H 2 (I), and then, according to the above estimate and Proposition 7, we have ≤Ch 2p . * In view of the Burkholder-Davis-Gundy inequality and Hölder inequality, it yields 2p ≤ Ch p . * With the help of the above estimates, we arrive at (16).
Thus (û ε ,v ε ) satisfies By the same method as in Proposition 5, Proposition 6 and Proposition 7, it holds that is the unique solution to (17), then for any p > 0, there exists a constant C such that the solution (û ε ,v ε ) satisfies where C is dependent of p, T, u 0 , v 0 but independent of ε ∈ (0, 1).

4.5.
The errors of u ε −û ε and v ε −v ε . Proposition 11. There exists a constant C such that Proof. • We prove the first inequality. Indeed, we have it is easy to see that v ε −v ε satisfies the following SPDE

AVERAGING PRINCIPLE FOR SKS EQUATION 5671
For t ∈ [0, T ] with t ∈ [kδ, (k + 1)δ), applying Itô formula to (18) By taking mathematical expectation from both sides of above equation, we have Due to Proposition 9, it holds that
• We prove the second inequality. Indeed, we have It follows from Proposition 1 that thus, it follows from Proposition 7 and Proposition 9 that Thus, it follows from the Gronwall inequality that This completes the proof of Proposition 11. 4.6. The error ofû ε −ū. Next we prove strong convergence of the auxiliary procesŝ u ε to the averaging solution processū.
Proof. It is easy to see it follows from Proposition 8 that the averaged equation In mild sense, we introduce the following decomposition We define the stopping time τ ε n = inf{t > 0 : û ε (t) H 1 + ū(t) H 1 > n} for any n ≥ 1, and ε > 0.
• For J 1 , • For J 2 , where m t = [ t δ ]. * For J 21 , by a time shift transformation, we can obtain that for any fixed p and where W * 2 (t) is the shift version of W 2 (t) and hence they have the same distribution. LetW (t) be a Wiener process defined on the same stochastic basis and independent of W 1 (t) and W 2 (t). Construct a process v u ε (pδ),v ε (pδ) (t) ∈ L 2 (I) by means hereW (t) is the scaled version ofW (t). By comparing the above two equations, we see that where ∼ denotes a coincidence in distribution sense. Thus, we have It follows from (22) and the property of semigroup {S 1 (t)} t≥0 that J k (s, r) In view of the above inequality, Proposition 3 and the method in [18,16,20,17,41,39,40], there holds J k (s, r) ≤ Ce −µ(s−r) , where µ > 0. Thus if choose δ = δ(ε) such that δ ε sufficiently large, we have  ( t mtδ f (u ε (m t δ),v ε (s)) −f (u ε (s)) 2p ds)] ≤Cδ 2p−1 . * For J 24 , using the contractive property of semigroup, Lipschitz continuity off and Proposition 11, we have If p > 5 4 , we have If 0 < p ≤ 5 4 , for any κ > 0, it holds that