Ill-posedness of free boundary problem of the incompressible ideal MHD

TL;DR

This paper demonstrates that the free boundary problem for incompressible ideal MHD in two dimensions is ill-posed in Sobolev spaces, regardless of the vacuum permeability value.

Contribution

It establishes the ill-posedness of the free boundary problem for 2D incompressible ideal MHD uniformly for all positive vacuum permeability values.

Findings

The problem is ill-posed in Sobolev spaces.

Ill-posedness holds for any positive vacuum permeability.

The analysis is uniform across all positive bc_0 values.

Abstract

In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $\mu_0$, in Sobolev spaces. The analysis is uniform for any $\mu_0>0$.