On the Yudovich solutions for the ideal MHD equations

TL;DR

This paper investigates the existence and uniqueness of Yudovich-type weak solutions for 2D inviscid MHD equations, focusing on vortex patches with smooth boundaries under specific geometric constraints.

Contribution

It establishes conditions under which vortex patches with smooth boundaries solve the inviscid MHD equations, introducing a geometric constraint involving magnetic field lines.

Findings

Vortex patches with smooth boundaries can be solutions under certain geometric constraints.

Stationary patches with rectifiable boundaries are characterized as discs for the Euler system.

The paper discusses stationary solutions for both Euler and MHD systems.

Abstract

In this paper, we address the problem of weak solutions of Yudovich type for the inviscid MHD equations in two dimensions. The local-in-time existence and uniqueness of these solutions sound to be hard to achieve due to some terms involving Riesz transforms in the vorticity-current formulation. We shall prove that the vortex patches with smooth boundary offer a suitable class of initial data for which the problem can be solved. However this is only done under a geometric constraint by assuming the boundary of the initial vorticity to be frozen in a magnetic field line. We shall also discuss the stationary patches for the incompressible Euler system $(E)$ and the MHD system. For example, we prove that a stationary simply connected patch with rectifiable boundary for the system $(E)$ is necessarily the characteristic function of a disc.