(2022) Larios, Adam; Rahman, Mohammad Mahabubur (TTU); Yamazaki, Kazuo (TTU)

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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.

We consider the three-dimensional magnetohydrodynamics system forced by noise that is white in both time and space. Its complexity due to four non-linear terms makes its analysis very intricate. Nevertheless, taking advantage of its structure and adapting the theory of paracontrolled distributions from [30], we prove its local well-posedness. A first challenge is to find an appropriate paracontrolled ansatz which must consist of both the velocity and the magnetic fields. Second challenge is that for some non-linear terms, renormalizations cannot be achieved individually; we overcome this obstacle by strategically coupling certain terms together rather than separately. Our proof is also inspired by the work of [70].