Global Well-posedness of the Stochastic Kuramoto-Sivashinsky Equation with Multiplicative Noise

TL;DR

This paper proves the global existence and uniqueness of solutions for the stochastic Kuramoto-Sivashinsky equation with multiplicative noise in bounded domains, under certain conditions, ensuring well-posedness in a probabilistic setting.

Contribution

It establishes the global well-posedness and Lipschitz continuity of the solution map for the stochastic Kuramoto-Sivashinsky equation with multiplicative noise, extending previous results to this stochastic context.

Findings

Unique global solutions exist under certain conditions.

Solution map is Lipschitz continuous.

Results hold for any initial data in L^2(D×Ω).

Abstract

Global well-posedness of the initial-boundary value problem for the stochastic Kuramoto-Sivashinsky equation in a bounded domain $D$ with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data $u_0\in L^2(D\times \Omega)$ this problem has a unique global solution $u$ in the space $L^2(\Omega,C([0,T],L^2({D})))$ for any $T>0$, and the solution map $u_0\mapsto u$ is Lipschitz continuous.