Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection

We study the Kuramoto-Sivashinsky equation (KSE) in scalar form on two-dimensional torus with and without advection by an incompressible vector field. prove local existence of mild solutions for arbitrary data L2. then issue global existence. KSE presence data, provided advecting velocity field v satisfies certain conditions that ensure dissipation time associated hyperdiffusion-advection is sufficiently small. In absence advection, can be shown only if linearized operator does not admit any growing mode small initial data.