Regularity criteria for the Kuramoto-Sivashinsky equation in dimensions two and three

Abstract

We 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 $\varphi$, the vector solution $u\triangleq \nabla \varphi$, as well as the divergence $\text{div}(u)=\Delta \varphi$, and each component of $u$ and $\nabla u$. 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.

References in corpus (4)

• Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor
• Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming
• On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto-Sivashinsky equation
• On the three-dimensional magnetohydrodynamics system in scaling-invariant spaces