# American Institute of Mathematical Sciences

November  2018, 38(11): 5649-5684. doi: 10.3934/dcds.2018247

## Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation

 School of Mathematics and Statistics, and Center for Mathematics, and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

* Corresponding author: Peng Gao

Received  December 2017 Revised  May 2018 Published  August 2018

Fund Project: The first author is supported by NSFC Grant (11601073).

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.

Citation: Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
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