Local well-posedness for the Hall-MHD system in optimal Sobolev Spaces

Abstract

We show that the viscous resistive magneto-hydrodynamics system with Hall effect is locally well-posed in $H^s({\mathbb{R}}^n)\times H^{s+1-\epsilon }({\mathbb{R}}^n)$ with $s>\frac{n}{2}-1$ and any small enough $\epsilon >0$ such that $s+1-\epsilon >\frac{n}{2}$. This space is to date the largest local well-posedness space in the class of Sobolev spaces for the system. It is also optimal according to the predominant scalings of the two equations in the system.

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