| Abstract | The study of stability and instability of shear flow is an active field. The most important mechanism that stabilizes the viscous fluid is enhanced dissipation. The enhanced dissipation is closely related to the regularity and size of the perturbations for the nonlinear system, so the stability threshold problem is raised. In this talk, I will first review some previous results about nonlinear enhanced dissipation for 2D Couette flow. Then I will present two recent joint works with Li and Masmoudi. The first is regarding optimality of threshold, where we develop a new energy method to study the lower bounds of semigroups generated by time-dependent linearization operators. The second result is the nonlinear enhanced dissipation below the threshold. I will show that to obtain nonlinear enhanced dissipation when the initial perturbation is not small enough, there needs to be some dependence on the regularity and size of the initial perturbation. |
