|Date: Tuesday, October 26, 2010
Title: On Dynamic Stability and Non-blowup of 3D Incompressible Euler
and Navier-Stokes equations
Abstract: Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the Seven Millennium Problems. We review some recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. The local geometric regularity of vortex filaments can lead to tremendous cancellation of nonlinear vortex stretching. We also investigate the stabilizing effect of convection in 3D incompressible Euler and Navier-Stokes equations. The convection term is a main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. Our study reveals a surprising nonlinear stabilizing effect that the convection term plays in regularizing the solution. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special structure of the solution and the cancellation between the convection term and the vortex stretching term, we prove nonlinear stability and the global regularity of this class of solutions.
Speaker: Tom Hou