Global weak solutions to the density-dependent Hall-magnetohydrodynamics system

TL;DR

This paper proves the global existence of finite energy weak solutions for a 3D density-dependent Hall-MHD system with boundary conditions, addressing challenges posed by the Hall-effect term's degeneracy.

Contribution

It establishes the existence of weak solutions for the complex density-dependent Hall-MHD system with boundary conditions, handling degeneracy issues.

Findings

Existence of global weak solutions proven.

Density bounds are preserved over time.

Compactness arguments for magnetic field established.

Abstract

We are concerned with the global existence of finite energy weak solutions to 3D density-dependent magnetohydrodynamics (MHD) system with Hall-effect set in a general smooth bounded domain. The perfectly conducting wall boundary condition is imposed on the magnetic field. Due to the degeneracy of Hall-effect term (a tri-linear term) in vacuum, we assumed initial density lies in the bounded function space and having a positive lower bound. Particularly, these bounds are preserved by the density transport equation which helps yield a satisfying compactness argument of the magnetic field.