The Rayleigh Benard problem for compressible fluid flows

TL;DR

This paper analyzes the fully compressible Rayleigh-Benard problem, demonstrating the existence of a global attractor, invariant measure, and ergodic properties for the system under certain physical restrictions.

Contribution

It extends the analysis of Rayleigh-Benard convection to fully compressible fluids, establishing dissipativity, attractors, and invariant measures in this setting.

Findings

Existence of a bounded absorbing set for weak solutions

Presence of a global compact trajectory attractor A

Convergence of ergodic averages for solutions on A

Abstract

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 A. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure - a stationary statistical solution sitting on A. In addition, the Birkhoff - Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to A a.s. with respect to the invariant measure.