Weak and strong solutions of equations of compressible magnetohydrodynamics

TL;DR

This paper reviews the mathematical analysis of compressible magnetohydrodynamics, focusing on weak and strong solutions, existence results, and special one-dimensional cases with simplified models.

Contribution

It provides a comprehensive overview of existence results for weak and strong solutions in compressible MHD, including methods and special cases.

Findings

Existence of weak solutions using renormalized solutions.

Existence of strong solutions near equilibrium states.

Simplified one-dimensional models solved by standard techniques.

Abstract

This article proposes a review of the analysis of the system of magnetohydrodynamics (MHD). First, we give an account of the modelling asumptions. Then, the results of existence of weak solutions, using the notion of renormalized solutions. Then, existence of strong solutions in the neighbourhood of equilibrium states is reviewed, in particular with the method of Kawashima and Shizuta. Finally, the special case of dimension one is highlighted : the use of Lagrangian coordinates gives a simpler system, which is solved by standard techniques.