Global weak solutions of 3D compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data and vacuum

TL;DR

This paper proves the global existence of weak solutions for 3D compressible magnetohydrodynamic equations with large external forces, discontinuous initial data, and vacuum states, using new estimates based on effective viscous flux.

Contribution

It establishes the existence of weak solutions under large external forces and discontinuous initial data, including vacuum, which was previously unresolved.

Findings

Global weak solutions exist under small initial energy.

Vacuum states and large oscillations are allowed in initial data.

New estimates based on effective viscous flux are crucial.

Abstract

We investigate the compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data in a three-dimensional bounded domain under Navier-slip boundary conditions. We show the global existence of weak solutions for such an initial-boundary value problem provided the initial energy is suitably small. In particular, the initial data may contain vacuum states and possibly exhibit large oscillations. To overcome difficulties brought by boundary and large external forces, some new estimates based on the effective viscous flux play crucial roles.