### Blow-up or no Blow-up? The Interplay Between Analysis and Computation in the Millennium Problem on Navier-Stokes Equations

- Series
- Stelson Lecture Series
- Time
- Tuesday, October 27, 2009 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Thomas Y. Hou – California Institute of Technology, Applied and Computational Mathematics

**Please Note:** This lecture will be more for the mathematical audience

Whether the 3D incompressible Navier-Stokes equations can develop
a finite time singularity from smooth initial data 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. The local geometric regularity of vortex filaments
can lead to tremendous cancellation of nonlinear vortex stretching.
This is also confirmed by our large scale computations for some of
the most well-known blow-up candidates. We also
investigate the stabilizing effect of convection in 3D incompressible
Euler and Navier-Stokes equations. The convection term is the main source
of nonlinearity for these equations. It is often considered destabilizing
although it conserves energy due to the incompressibility condition. Here
we reveal 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.