The two dimensional incompressible Navier Stokes equations with small viscosity (corresponding to large Reynolds numbers) play a fundamental role in modeling large scale behavior of fluid flows, such as atmospheric turbulence. Due to the small (or zero) viscosity and lack of global relaxation mechanism for the limiting Euler equation, the long time behavior of solutions is challenging to study mathematically, even in perturbative regimes. The main difficulties are the presence of continuous spectrum, non-self-adjointness of associated linear operators, singular perturbations and strong nonlinear effects.
In this talk, we will review some recent progress on understanding precise dynamics near coherent solutions, such as shear flows and vortices, including linear and nonlinear inviscid damping results, enhanced dissipation effects and vorticity depletion phenomena. Numerical simulations, major open problems and future directions will also be presented.
