Long-time instability of planar Poiseuille-type flow in compressible fluid

It is well-known that at the high Reynolds number, the linearized Navier-Stokes equations around the inviscid stable shear profile admit growing mode solutions due to the destabilizing effect of the viscosity. This phenomenon, called Tollmien-Schlichting instability, has been rigorously justified by Grenier-Guo-Nguyen [Adv. Math. 292 (2016); Duke J. Math. 165 (2016)] for Poiseuille flows and boundary layers in the incompressible fluid. To reveal this intrinsic instability mechanism in the compressible setting, in this paper, we study the long-time instability of the Poiseuille flow in a channel. Note that this instability arises in a low-frequency regime instead of a high-frequency regime for the Prandtl boundary layer. The proof is based on the quasi-compressible-Stokes iteration introduced by Yang-Zhang in [50] and subtle analysis of the dispersion relation for the instability. Note that we do not require symmetric conditions on the background shear flow or perturbations.