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

Peng Gao

doi:10.3934/dcds.2018247

Article Contents

2018, Volume 38, Issue 11: 5649-5684. Doi: 10.3934/dcds.2018247

This issue Previous Article A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion Next Article Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations

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

Peng Gao^,

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

* Corresponding author: Peng Gao
* Corresponding author: Peng Gao

Received: November 30, 2017

Revised: April 30, 2018

Published: August 2018

The first author is supported by NSFC Grant (11601073)

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Abstract

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.

Keywords:

Stochastic Kuramoto-Sivashinsky equation,
stochastic averaging principle,
strong convergence,
fast-slow SPDEs.

Mathematics Subject Classification: Primary: 60H15; Secondary: 70K65, 70K70.

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