Ill-posedness of incompressible Kelvin-Helmholtz problem with transverse magnetic field

TL;DR

This paper proves that the Kelvin-Helmholtz problem in incompressible ideal MHD with a transverse magnetic field is both linearly and nonlinearly ill-posed, indicating instability unaffected by the magnetic field's transverse orientation.

Contribution

It rigorously establishes the ill-posedness of the Kelvin-Helmholtz problem in this setting, confirming a longstanding hypothesis about the magnetic field's influence.

Findings

Proves linear and nonlinear ill-posedness of the problem

Shows magnetic field does not influence instability development

Confirms theoretical predictions by Chandrasekhar

Abstract

In this paper, we prove the linear and nonlinear ill-posedness of the well-known Kelvin-Helmholtz problem of the incompressible ideal magnetohydrodynamics (MHD) equations with transverse magnetic field. Our proof rigorously verifies that "the development of the Kelvin-Helmholtz instability, in the direction of the streaming, is uninfluenced by the presence of the magnetic field in the transverse direction" which was proposed by S. Chandrasekhar' book named by Hydrodynamic and Hydromagnetic stability.