Regularity Criteria for the Kuramoto–Sivashinsky Equation in Dimensions Two and Three

Abstract

AbstractWe propose and prove several regularity criteria for the 2D and 3D Kuramoto–Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution $$\phi $$ ϕ , the vector solution $$u\triangleq \nabla \phi $$ u ≜ ∇ ϕ , as well as the divergence $$\text {div}(u)=\Delta \phi $$ div ( u ) = Δ ϕ , and each component of u and $$\nabla u$$ ∇ u . We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.