Linear Stability of the Couette Flow in the 3D Isentropic Compressible Navier--Stokes Equations

Abstract

Consider the linear stability of the three dimensional (3D) isentropic compressible Navier--Stokes equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We prove the enhanced dissipation phenomenon for the linearized isentropic compressible Navier--Stokes equations around the Couette flow $(y, 0, 0)^\top$. Moreover, the lift-up phenomenon is also shown in this paper. In particular, due to the loss of the incompressibility condition, the upper bounds for the $x$-average of the perturbed streamwise velocity $u^1_0$ in this paper are worse than those in the corresponding 3D incompressible Navier--Stokes equations [J. Bedrossian, P. Germain, and N. Masmoudi, Ann. Math., 185 (2017), pp. 541--608].

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