The Global Solvability Of The Hall-magnetohydrodynamics System In Critical Sobolev Spaces

TL;DR

This paper establishes the global well-posedness and long-term behavior of solutions to the 3D Hall-MHD system in critical Sobolev spaces, extending classical results and analyzing special flow configurations.

Contribution

It provides an elementary proof of global existence for small data in critical Sobolev spaces and explores long-time asymptotics and special flow cases for the Hall-MHD system.

Findings

Global well-posedness for small data in critical Sobolev spaces

Analysis of long-time asymptotics of solutions

Global existence of solutions for 2.5D flows with small initial magnetic field

Abstract

We are concerned with the 3D incompressible Hall-magnetohydro-dynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato's theorem [10] for the Navier-Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita-Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2 1/2 D flows for the Hall-MHD system, and prove the global existence of strong solutions, assuming only that the initial magnetic field is small.