Abstract

The Navier-Stokes equations describe how the velocity field of a viscous fluid evolves in time. This is a non-linear PDE, for which many fundamental questions remain unanswered. Despite the deterministic nature of these equations, stochastic methods are extremely useful (and are sometimes necessary) in their study. I will survey a few such well known methods, and then focus on a recent approach by which the Navier-Stokes equations can be exactly expressed as an expected value over noisy particle trajectories.

This formulation allows us to construct particle systems for the Navier-Stokes equations. Curiously the particle systems constructed in this manner do not completely dissipate their energy after long time. For this reason, a similar particle system for the Burgers' equations (a simplified model of Navier-Stokes) develops singularities (shocks) in finite time (which is unexpected, as the analogous particle system for 2D Navier-Stokes has no singularities). The main result is that stopping and restarting the Burgers' particle system often enough smooths out the singularities. This is surprising, since stopping and restarting corresponds to an operator with norm 1, and no regularizing properties!

FRIDAY, December 5, 2008
Time: 4:30 P.M.
Location: Wean Hall 7500

Refreshments at 4:00, Wean 6220.