Local well-posedness of the Hall-MHD system in $H^s(\mathbb {R}^n)$ with $s>\frac n2$

TL;DR

This paper proves local well-posedness for the Hall-MHD system in Sobolev spaces with lower regularity than previously known, specifically for $s > n/2$, improving earlier results requiring $s > n/2 + 1$.

Contribution

It establishes local well-posedness of the Hall-MHD system in $H^s(R^n)$ for $s > n/2$, reducing the regularity requirement compared to prior work.

Findings

Well-posedness in $H^s$ for $s > n/2$

Improved regularity threshold over previous results

Enhanced understanding of the system's mathematical properties

Abstract

We establish local well-posedness of the Hall-magneto-hydrodynamics (Hall-MHD) system in the Sobolev space $\left(H^s(\mathbb{R}^n)\right)^2$ with $s>\frac n2$. The previously known local well-posedness space was $\left(H^s(\mathbb{R}^n)\right)^2$ with $s>\frac n2+1$. Thus the result presented here is an improvement.