Malliavin calculus and ergodic properties of highly degenerate 2D stochastic Navier--Stokes equation

2004/09/03 by Martin Hairer, Hairer, Martin, Jonathan C. Mattingly +3
Mathematics · Physics and Astronomy · #35R60 #37A60 #37L55 #37N10 #60H07 #60H15 #76F20 #76F55 #7A25 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.AP #math.DS #math.MP #math.PR #msc:35R60 #msc:37A60 #msc:37L55 #msc:37N10 #msc:60H07 #msc:60H15 #msc:76F20 #msc:76F55 #msc:7A25

paper · pdf · doi:10.48550/arxiv.math/0409057

A summary of two recent results for the stochastic Navier--Stokes euqations

arxiv created 2004/09/03 · arxiv updated 2009/12/01

Abstract

The objective of this note is to present the results from the two recent papers. We study the Navier--Stokes equation on the two--dimensional torus when forced by a finite dimensional white Gaussian noise. We give conditions under which both the law of the solution at any time t>0, projected on a finite dimensional subspace, has a smooth density with respect to Lebesgue measure and the solution itself is ergodic. In particular, our results hold for specific choices of four dimensional white Gaussian noise. Under additional assumptions, we show that the preceding density is everywhere strictly positive.

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