Theodore Y. Wu Lecture in Aerospace

Professor Thomas Yizhao Hou will give a presentation entitled "Blow-up or No Blow-up? Fluid Dynamic Perspective of the Clay Millennium Problem." Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the Seven Millennium Problems posted by the Clay Mathematical Institute. 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. Our study shows that the convection term could have a stabilizing effect for certain flow geometry. This is demonstrated through two reduced models of the 3D incompressible Navier-Stokes equations, which show that local flattening of the vortex structure, and the effect of convection could lead to dynamic depletion of the vortex stretching term. Finally we present a new class of solutions to the 3D Euler and Navier-Stokes equations that could lead to a strong nonlinear alignment in the vortex stretching term and have the potential to develop a finite time vortex sheet singularity. However, the Kelvin-Helmholtz instability of the fluid flow eventually destroys such nonlinear alignment and lead to the development of turbulence instead of forming a finite time singularity.