Kelvin–Helmholtz Instability for Compressible Euler Fluids

ABSTRACT In this paper, we prove the linear and nonlinear ill‐posedness of the Kelvin–Helmholtz problem for compressible ideal fluids when the Mach number lies in the range between a sufficiently small fixed constant and . Inspired by the approach of Guo and Tice (2011), we establish the failure of local‐in‐time continuous dependence on initial data in high‐order Sobolev spaces for solutions of the Kelvin–Helmholtz problem. To the best of our knowledge, this is the first result that rigorously demonstrates nonlinear instability for the Kelvin–Helmholtz problem for compressible Euler fluids.