Ill-posedness of the Kelvin-Helmholtz problem for compressible Euler fluids

TL;DR

This paper proves the linear and nonlinear ill-posedness of the Kelvin-Helmholtz problem for compressible Euler fluids when the Mach number is within a specific range, marking a first in demonstrating nonlinear ill-posedness in this context.

Contribution

It is the first to establish nonlinear ill-posedness of the Kelvin-Helmholtz problem for compressible Euler fluids within a certain Mach number range.

Findings

Proves linear ill-posedness for certain Mach numbers.

Establishes nonlinear ill-posedness for the Kelvin-Helmholtz problem.

Identifies a specific Mach number range where ill-posedness occurs.

Abstract

In this paper, when the magnitude of the Mach number is strictly between some fixed small enough constant and $\sqrt{2}$, we can prove the linear and nonlinear ill-posedness of the Kelvin-Helmholtz problem for compressible ideal fluids. To our best knowledge, this is the first reslult that proves the nonlinear ill-posedness to the Kelvin-Helmholtz problem for the compressible Euler fluids.