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

*(English)*Zbl 1401.60121Summary: 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.

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

70K65 | Averaging of perturbations for nonlinear problems in mechanics |

70K70 | Systems with slow and fast motions for nonlinear problems in mechanics |

##### Keywords:

stochastic Kuramoto-Sivashinsky equation; stochastic averaging principle; strong convergence; fast-slow SPDE
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\textit{P. Gao}, Discrete Contin. Dyn. Syst. 38, No. 11, 5649--5684 (2018; Zbl 1401.60121)

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