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It is well-known that at the high Reynolds number, the linearizedNavier-Stokes equations around the inviscid stable shear profile admit growingmode solutions due to the destabilizing effect of the viscosity. Thisphenomenon, called Tollmien-Schlichting instability, has been rigorouslyjustified 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, inthis paper, we study the long-time instability of the Poiseuille flow in achannel. Note that this instability arises in a low-frequency regime instead ofa high-frequency regime for the Prandtl boundary layer. The proof is based onthe quasi-compressible-Stokes iteration introduced by Yang-Zhang in [50] andsubtle analysis of the dispersion relation for the instability. Note that we donot require symmetric conditions on the background shear flow or perturbations.

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