Ergodicity of the 2D stochastic Navier-Stokes equations
by
Martin Hairer
The University of Warwick
Coauthors: Jonathan C. Mattingly

The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in the entire phase space. Unlike in previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property and an approximate integration by parts formula.

For the same class of noises, we also exhibit a space of test functions in which the semigroup generated by the stochastic Navier-Stokes equations possesses a spectral gap. This result is based on the construction of a new metric on the space of velocity fields which captures the geometry of the probabilistic mixing.

Date received: July 12, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-39.