Stability and Transition Threshold for the 2-D Poiseuille flow in a Channel

In this talk, we introduce our recent result about the quantitative stability problem for the 2-D Poiseuille flow $(1-y^2, 0)$ with Navier-slip boundary conditions in a periodic channel. For the linearized Navier-Stokes equations around the 2-D Poiseuille flow, the enhanced dissipation is obtained by using the careful resolvent estimates. For the nonlinear stability transition threshold, we prove that the solution of the Navier-Stokes equations around the 2-D Poiseuille flow does not transition away from the Poiseuille flow provided that the $H^1$ norm of the initial perturbation is less than the 3/4 power of the viscosity. This talk is based on a joint work with Zhilin Lin.