Global existence for the two-dimensional Kuramoto–Sivashinsky equation with a shear flow

Abstract We consider the Kuramoto–Sivashinsky equation (KSE) on two-dimensional torus in presence of advection by a given background shear flow. Under assumption that has finite number critical points and there are linearly growing modes only direction shear, we prove global existence solutions with data $$L^2$$ L 2 , using bootstrap argument. The initial can be taken arbitrarily large.