Stability of plane Couette flow for the compressible Navier–Stokes equations with Navier-slip boundary

Abstract

This paper is devoted to the stability analysis of the plane Couette flow for the 3D compressible Navier–Stokes equations with Navier-slip boundary condition at the bottom boundary. It is shown that the plane Couette flow is asymptotically stable for small perturbation provided that the slip length, Reynolds and Mach numbers satisfy 3(1+ν˜)αγ2(ν+α)γ0≤1 and 2αν(ν+α)≤1 for some constant γ0>0. In particular, the Reynolds number ν−1 can be large if the slip length α is suitably small. This means that the constraint required in [11] on the Reynolds number to guarantee the stability of the plane Couette flow can be relaxed and improved so long as the slip effect at the boundary is involved.