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

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow $(1-y^2,0)$ 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 $e^{-c\sqrt{\nu}t}$. As an application, we prove that if the initial perturbation of vortiticy satisfies $$\|\omega_0\|_{L^2}\leq c_0\nu^{\frac{3}{4}},$$ for some small constant $c_0>0$ independent of the viscosity $\nu$, then the solution dose not transition away from the Poiseuille flow for any time.