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

TL;DR

This paper establishes new regularity criteria for solutions to the 2D and 3D Kuramoto-Sivashinsky equation, providing theoretical conditions and computational insights into solution behavior.

Contribution

It introduces novel integrability-based regularity criteria for the Kuramoto-Sivashinsky equation in multiple dimensions and explores their computational validation.

Findings

New regularity criteria for 2D and 3D Kuramoto-Sivashinsky solutions

Computational validation with solution snapshots and energy estimates

Insights into solution regularity based on scalar and vector quantities

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 $\phi$, the vector solution $u\triangleq\nabla\phi$, as well as the divergence $\text{div}(u)=\Delta\phi$, 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.