Sala Seminari .
We consider the physically relevant fully compressible setting of the
Rayleigh-Benard problem of
a fluid confined between two parallel plates, heated from the bottom,
and subjected to the 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. The standard technique of Krylov and Bogolyubov
then yields the existence of an invariant measure – a stationary
statistical solution sitting on the global 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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