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 φ, the vector solution u , ∇φ, as well as the divergence div(u) = ∆φ, and each component of u and ∇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.

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