Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations

TL;DR

This paper establishes the existence and uniqueness of low-energy weak solutions for the 3D compressible magnetohydrodynamics equations, allowing for certain discontinuities and extending previous results for compressible Navier-Stokes equations.

Contribution

It proves the existence and uniqueness of weak solutions with small L^2-norm for 3D compressible MHD, including solutions with codimension-one discontinuities.

Findings

Existence of weak solutions with small L^2-norm

Weak solutions can have codimension-one discontinuities

Extension of results from compressible Navier-Stokes equations

Abstract

We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small $L^2$-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-Stokes equations.