Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to $H^{\frac{3}{2}}(\mathbb{R}^3)$

TL;DR

This paper proves local and global well-posedness for the incompressible Hall-MHD system with initial magnetic field in $H^{3/2}$, improving regularity conditions and establishing decay rates for small initial data.

Contribution

It establishes the well-posedness of the Hall-MHD system with less restrictive initial data regularity and introduces new formulations to extend results.

Findings

Local well-posedness for initial data in $H^{1/2+\sigma}$ and $H^{3/2}$

Global well-posedness for small initial data

Optimal decay rates of solutions

Abstract

In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data $(u_0,B_0)\in H^{\frac{1}{2}+\sigma}(\mathbb{R}^3)\times H^{\frac{3}{2}}(\mathbb{R}^3)$ with $\sigma\in (0,2)$. In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $\sigma$ to $0$ with the aid of the new formulation of the Hall-MHD system observed by Danchin and Tan (Commun Partial Differ Equ 46(1):31-65, 2021). Compared with the previous works, our local well-posedness results improve the regularity condition on the initial data. Moreover, we establish the global well-posedness for small initial data in $H^{\frac{1}{2}+\sigma}(\mathbb{R}^3)\times H^{\frac{3}{2}}(\mathbb{R}^3)$ with $\sigma\in (0,2)$, and get the optimal time-decay rates of solutions.