On the well-posedness of the Hall-MHD system in a critical setting of Besov-Morrey type

TL;DR

This paper establishes local and global well-posedness for the 3D Hall-MHD system using a novel critical Besov-Morrey space framework, broadening initial data regularity conditions compared to previous Sobolev and Besov space results.

Contribution

It introduces a new critical Besov-Morrey space setting for analyzing the Hall-MHD system, enabling well-posedness results with more general initial data.

Findings

Proved local and global well-posedness in the new framework.

Extended the class of initial data beyond traditional Sobolev and Besov spaces.

Demonstrated the applicability of Besov-Morrey spaces to complex PDE systems.

Abstract

In this paper, we address the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our objective is to provide local and global well-posedness results for initial velocity $u_{0}$, magnetic field $B_{0}$ and the current $J_{0}:=\nabla\times B_{0}$ in a new critical framework, namely critical Besov-Morrey spaces. These spaces combine typical characteristics from both Besov and Morrey spaces allowing a broader framework that encompasses the regularity properties inherent in Besov spaces with the Morrey space structure. Compared to previous works in Sobolev and Besov spaces, our approach accommodates a broader class of initial data, ensuring the construction of a unique solution over time.