Math Seminar: "Ergodicity for the stochastic partial differential equations of fluid mechanics"
4:10pm Neill 5W
I will try to discuss a particular topic of research direction in the stochastic partial differential equations (SPDE) of fluid mechanics such as the Navier-Stokes equations with noise, namely ergodicity, although it is applicable in many other fields. More specifically I will discuss the question of the existence of a unique invariant, and consequently ergodic, measure for such SPDE. If proven, informally speaking, such a measure describes the statistical equilibrium to which the system of SPDE approaches. The proof of its existence requires an application of the classical Krylov-Bogoliubov theorem while the proof of its uniqueness requires, due to the classical Doobs theorem, verifying the irreducible property of the transition function and the strong Feller property of the associated Markovian semigroup. A more advanced research topic in this direction, which has caught much attention in the last decade, is the study of the exponential convergence of the Markovian transition semigroup toward this unique invariant measure via a coupling method.