On the Spectrum of the Linearized Operator Around Compressible Couette Flows Between Two Concentric Cylinders

The spectrum of the linearized operator around the Couette flow of the compressible Navier-Stokes equations between two concentric rotating cylinders is studied when the Mach number is sufficiently small. It is shown that, for sufficiently small Mach numbers, the linearized operator around the Couette flow has a critical eigenvalue which crosses the origin when the rotating speed of the cylinder increases. The critical eigenvalue and the associated eigenprojection converge to the ones for the incompressible Navier-Stokes equations as the Mach number goes to zero.