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

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