Stochastic and Multiscale Modeling and Computation Seminar by Jiahong Wu: Stabilizing Phenomenon for Incompressible Fluids
Speaker: Jiahong Wu, professor of mathematics, University of Notre Dame
Title: Stabilizing Phenomenon for Incompressible Fluids
Abstract: This talk presents recent stability results on several PDE systems modeling fluid flows. These results reflect a seemingly universal stabilizing phenomenon exhibited in quite different fluids. The 3D incompressible Euler equation can blow up in a finite time. Even small data would not help. But when the 3D Euler is coupled with the non-Newtonian stress tensor in the Oldroyd-B model, small smooth data always lead to global and stable solutions. Solutions of the 2D Navier-Stokes in R^2 with dissipation in only one direction are not known to be stable, but the Boussinesq system involving this Navier-Stokes is always stable near the hydrostatic equilibrium. The buoyancy forcing helps stabilize the fluid. The 3D incompressible Navier-Stokes equation with dissipation in only one direction is not known to always have global solutions even when the initial data are small. However, when this Navier-Stokes is coupled with the magnetic field in the magneto-hydrodynamic system, solutions near a background magnetic field are shown to be always global in time. In all these examples the systems governing the perturbations can be converted to damped wave equations, which reveal the smoothing and stabilizing effect.