Effect of weak elasticity on Kelvin-Helmholtz instability

Abstract

In this paper, we present an analysis of the Kelvin-Helmholtz instability in two-dimensional ideal compressible elastic flows, providing a rigorous confirmation that weak elasticity has a destabilizing effect on the Kelvin-Helmholtz instability. There are two critical velocities, $U_{\text{low}}$ and $U_{\text{upp}}$, where $U_{\text{low}}$ and $U_{\text{upp}}$ represent the lower and upper critical velocities, respectively. We demonstrate that when the magnitude of the rectilinear solutions satisfies $U_{\text{low}}+c{\epsilon }_0\le |{\dot{v}}_1^{+}|\le U_{\text{upp}}-c{\epsilon }_0$, the linear and nonlinear ill-posedness of the piecewise smooth solutions of the Kelvin-Helmholtz problem for two-dimensional ideal compressible elastic fluids is established uniformly, where $c$ is the sound speed and ${\epsilon }_0$ is some small enough positive constant.

arXiv comment34pages