Global Carleman estimates for the linear stochastic Kuramoto-Sivashinsky equations and their applications

In this paper, we establish two new global Carleman estimates for the linear stochastic Kuramoto-Sivashinsky equations. The first one is for the backward linear stochastic Kuramoto-Sivashinsky equation. Based on this estimate and the duality argument, we obtain the null controllability of the forward linear stochastic Kuramoto-Sivashinsky equation by three controls, one is an internal control in the diffusion term and the others are boundary controls. The second one is for the forward linear stochastic Kuramoto-Sivashinsky equation. According to this estimate, we obtain a unique continuation property for the forward linear stochastic Kuramoto-Sivashinsky equation.