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Ergodicity for the dissipative Boussinesq equations with random forcing. (English) Zbl 1113.35141
For the two-dimensional stochastically forced Boussinesq equation the existence of a unique stationary measure is proved if all the determining modes are forced. The existence of an invariant measure is proved by the standard compactness argument. The proof of the uniqueness is reduced to two parts. In the first part the regularity of the transition probability densities is shown, i.e. the hypoellipticity of the diffusion operator. Then the authors prove the irreducibility of the associated Markov process in the sense starting from any initial position, the dynamics enter any neighborhood of the origin infinitely often. These two results allow to prove the main ergodicity result following to the previous results on the stochastically forced Navier-Stokes equations. The Galerkin truncations of the three-dimensional Boussinesq equations under degenerate stochastic forcing are also studied.
35Q35PDEs in connection with fluid mechanics
35Q60PDEs in connection with optics and electromagnetic theory
60H40White noise theory
76D99Incompressible viscous fluids
76M35Stochastic analysis (fluid mechanics)