The Rayleigh Benard 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 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.

Time periodic motion of temperature driven compressible fluids

“…• The concept of weak solution for the Navier-Stokes-Fourier system based on a combination of the entropy inequality and the ballistic energy balance developed in [7]. • Uniform bounds and large time asymptotics of the weak solutions in the spirit of [14].…”

“…The problems with inhomogeneous Dirichlet boundary conditions require an alternative approach developed in [7], where the energy is replaced by the ballistic energy, for which the boundary flux is again controllable. This approach has been used recently in [14], where the existence of bounded absorbing sets and asymptotic compactness of bounded trajectories was established.…”