Statistical stationary solutions to the compressible Rayleigh-Benard convection problem

Stationary statistical solutions (invariant measures) represent a standard tool for describing turbulent phenomena in fluid mechanics. We consider the complete Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat conducting fluid driven by boundary temperature fluctuations (Rayleigh-Benard problem). We show that any omega-limit set associated to a global in time (weak) solution supports an invariant measure - stationary statistical solution. The proof is based on careful analysis of propagation of density oscillations (joint work with A.Swierczewska-Gwiazda (Warsaw)).