Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. (English) Zbl 1054.76020
Summary: We prove that the two dimensional Navier-Stokes equations possess an exponentially attracting invariant measure. This result is in fact the consequence of a more general “Harris-like” ergodic theorem applicable to many dissipative stochastic PDEs and stochastic processes with memory. A simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting. To analyze the iterated map, a general “Doeblin-like” theorem is proven. One of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non-Markovian processes.
MSC:
76D05 | Navier-Stokes equations for incompressible viscous fluids |
35Q30 | Navier-Stokes equations |
37H99 | Random dynamical systems |
37L99 | Infinite-dimensional dissipative dynamical systems |
60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
76M35 | Stochastic analysis applied to problems in fluid mechanics |