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The Bénard problem with random perturbations: Dissipativity and invariant measures. (English) Zbl 0876.35082
The so-called Bénard problem is considered. A viscous fluid, in a rectangular container, is heated from below and the top surface temperature is constant. Heating the fluid, its density changes and the gradient of density causes a motion of particles from the bottom to the top of the container. (The direction of motion is considered to be vertical.) The density is supposed to be constant (incompressibility condition) and the motion is modelled as the one generated by a force which is proportional to gravitational force and temperature gradient. The noise represents all the perturbations acting on the system, which is not isolated. So the system under consideration consists of the Navier-Stokes equations perturbated by white noise, coupled with the heat equation. The author proves an existence and uniqueness result for the solution and the existence of invariant measures for the associated semigroup.
MSC:
35Q30Stokes and Navier-Stokes equations
35R60PDEs with randomness, stochastic PDE
76D05Navier-Stokes equations (fluid dynamics)