The Rayleigh–Bénard Problem for Compressible Fluid Flows

Abstract

We consider the physically relevant fully compressible setting of the Rayleigh–Bénard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor mathcal {A}. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure—a stationary statistical solution sitting on mathcal {A}. In addition, the Birkhoff–Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to mathcal {A} a.s. with respect to the invariant measure.