Non-uniqueness of entropy-conserving solutions to the ideal compressible MHD equations

TL;DR

This paper demonstrates that for certain initial conditions, the ideal compressible MHD equations admit infinitely many entropy-conserving weak solutions, highlighting non-uniqueness in these equations.

Contribution

The paper applies convex integration to show non-uniqueness of entropy-conserving solutions for 2D ideal compressible MHD equations, including the isentropic case.

Findings

Existence of infinitely many solutions for specific initial data.

Non-uniqueness persists in the isentropic case.

Convex integration effectively constructs multiple solutions.

Abstract

In this note we consider the ideal compressible magneto-hydrodynamics (MHD) equations in a special two dimensional setting. We show that there exist particular initial data for which one obtains infinitely many entropy-conserving weak solutions by using the convex integration technique. Our result is also true for the isentropic case.