UAH > Math > Colloquia > 2/28/2005
I will discuss randomly forced Navier-Stokes system in 2D and 3D. The problem of existence and uniqueness of statistically stationary solutions for randomly forced 2D Navier-Stokes system was intensively studied in the past decade. One of the successful approaches was the one due to E, Mattingly and Sinai. It involved a reduction to a finite-dimensional system with memory and analysis of this finite-dimensional non-Markov ("Gibbsian") system. I will give a new general result (joint with Jonathan Mattingly) for stochastic systems with memory and show that it is applicable to 2D Navier-Stokes. As for the 3D situation, not so much is known so far. I will give a new existence-uniqueness theorem for stationary solutions in 3D under some smallness conditions on the random forcing. The stationary solution will be constructed and studied with the help of a beautiful stochastic cascade construction due to Le Jan and Sznitman.