Abstract. The flow of a compressible fluid contained in a bounded domain is considered, with a prescribed temperature on the boundary. Thew iconic example is the Rayleigh-Benard problem. Asymptotic compactness as well as the existence of a bounded absorbing set is obtained. As corollaries, the following properties are established for global in time solutions:
(1) Existence of a trajectory attractor;
(2) Existence of stationary statistical solutions supported by omega-limit sets;
(3) Convergence of ergodic time averages and a conditional validity of ergodic hypothesis.