On Dynamic Stability and Global Regularity of the 3D Incompressible Flow
Tom Hou (Caltech)

Whether the 3D incompressible Euler or Navier-Stokes equations can develop a finite time singularity from smooth initial data has been an outstanding open problem. We first review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equations.  We show that the local geometric properties of vortex filaments can lead to dynamic depletion of vortex stretching, thus avoid finite time blowup of the 3D Euler equations.

Further, we perform well-resolved large scale computations of the 3D Euler equations to re-examine the two slightly perturbed anti-parallel vortex tubes which is considered as one of the most promising candidates for finite time blowup of the 3D Euler equations. Our computational results show that there is tremendous dynamic depletion of vortex stretching. The maximum vorticity does not grow faster than double exponential in time.

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 nonlinear structure of the equations, we prove nonlinear stability and the global regularity of a class of solutions of the 3D axisymmetric Navier-Stokes equations with swirl.