Enhanced dissipation and transition threshold for the Poiseuille-Couette flow

Abstract

In this paper, we consider the hydrodynamic stability of the 2D Navier-Stokes equations on T×R around Poiseuille-Couette flow V=(y2,0)+a(y,0) with a∈(0,1). By means of the hypocoercivity method with time dependent weights and introducing a new estimate on the energy functional, we establish the enhanced dissipation estimates for the linearized Navier-Stokes equations. More precisely, for a⩽10−3ν14, we obtain a decay rate proportional to ν12, which is consistent with the Poiseuille flow. For a=10−3νs with s∈[0,14], the decay rate is proportional to ν1−2s. As an application of the linear estimates, we obtain the result of nonlinear stability transition threshold.