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

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 $u_0\in L^2(D\times \mathrm{\Omega })$ this problem has a unique global solution $u$ in the space $L^2(\mathrm{\Omega },C([0,T],L^2(D)))$ for any $T>0$, and the solution map $u_0\mapsto u$ is Lipschitz continuous.