| Abstract: |
| We consider the physically relevant fully compressible setting of the Rayleigh--B\enard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to the gravitational force. 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. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure -- a stationary statistical solution sitting on the attractor. In addition, the Birkhoff--Khinchin ergodic theorem
provides convergence of ergodic averages of solutions belonging to the attractor a.s. with respect to the invariant measure. |
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