Weinan Wang

In this talk, I will discuss some recent well-posedness and stability results for several incompressible fluid equations. More precisely, I will first discuss a global well-posedness result for the 2D Boussinesq equations with fractional dissipation and the long-time behavior of solutions. For the Oldroyd-B model, we show that small smooth data lead to global and stable solutions. When the Navier-Stokes is coupled with the magnetic field in the magneto-hydrodynamics (MHD) system, solutions near a background magnetic field are shown to be always global in time. The magnetic field stabilizes the fluid. In the examples for Oldroyd-B and MHD, the systems governing the perturbations can be converted to damped wave equations, which reveal the smoothing and stabilizing effect. If time permits, I will discuss the stabilization and dissipation enhancement in stochastic PDEs together with some open problems.