Long-Wave Spectral Instability of Shear Layers for the Compressible Euler Equations
We study the long-wave spectral instability of the two-dimensional compressible Euler equations around smooth monotone shear layers. We construct decaying half-line solutions of the compressible Rayleigh equation through a long-wave expansion and derive a second-order expansion of the matching Wronskian. For every fixed Mach number $m>0$, we prove the existence of unstable modes for sufficiently small wavenumbers. For $m<\sqrt2$, this holds for a class of profiles, with $c_i$ tending to a positive constant as $\alpha\to0$. At $m=\sqrt{2}$, the profile $U_s(Y)=\tanh Y$ admits an unstable mode with $c\to0$ and $c_i$ of order $\alpha^{1/3}$. For $m>\sqrt2$, the same profile remains unstable, with $c\to c_*(m)\in(0,1)$ and $c_i>0$ of order $\alpha$. The corresponding temporal growth rates are of order $\alpha$, $\alpha^{4/3}$ and $\alpha^2$, respectively, showing a change in the long-wave instability scaling at $m=\sqrt2$. In the zero-thickness limit, the supercritical unstable eigenvalue approaches the real axis, consistently with the stability results for supersonic compressible vortex sheets in \cite{CS1,CS2}.