In this talk I’ll describe an overview of results on the Navier-Stokes equations for viscous compressible flows both in the deterministic and stochastic frame- work. A contrast will be drawn between the one-dimensional flows and the multidimensional case.

In the case of 1d isentropic compressible flow, the existence of invariant mea- sures will be established by extending the classical Krylov-Bogolyubov method and by providing suitable polynomial and exponential moment bounds on the solution together with path-wise estimates.

In the case, of 3d compressible heat conducting flows, the global existence of finite-energy weak martingale solutions to the stochastic Navier-Stokes for large initial data is established. The notion of solution and corresponding com- pactness analysis can be viewed as a stochastic counterpart to the notion of variational solutions introduced by E. Feireisl (2004).