For stochastic partial differential equations
(SPDE) in fluid mechanics such as the stochastic Navier-Stokes
equations, magnetohydrodynamics system, Benard and magnetic Benard
problems, micropolar and magneto-micropolar fluid systems, and many
others, besides well-posendess, ergodicity results are also very
important direction of research. In particular, the existence of a
unique invariant, and consequently ergodic, measure for a system
describes informally a statistical equilibrium to which the system
approaches. The proof of its existence in particular relies on the
classical Krylov-Bogoliubov theorem while the uniqueness requires an
application of the classical Doob's theorem and verifying the the
irreducibility of the transition function and the strong Feller
property of its associated Markovian semigroup. A more advanced
research direction is the issue of the exponential convergence of the
trajectories to the unique invariant measure which may be proven via a
coupling method. We discuss these results and proofs.
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