Enhanced dissipation and transition threshold for the 2-D plane Poiseuille flow via resolvent estimate

Abstract

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow (1−y2,0) in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate e−cνt. As an application, we prove that if the initial perturbation of vorticity satisfies‖ω0‖L2≤c0ν34 for some small constant c0>0 independent of the viscosity ν, then the solution does not transition away from the Poiseuille flow for any time.