Effect of weak elasticity on Kelvin-Helmholtz instability

In this paper, we present an analysis of 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, Ulow and Uᵤₚₚ, where Ulow and Uᵤₚₚ represent the lower and upper critical velocities, respectively. We demonstrate that if the rectilinear solutions satisfy Ulow+cε₀< v̇⁺₁ < Uᵤₚₚ, the linear and nonlinear ill-posedness of Kelvin-Helmholtz problem for two-dimensional ideal compressible elastic fluids is established.