Markov selection for the stochastic compressible Navier.Stokes system

We analyze the Markov property of solutions to the compressible Navier.Stokes system perturbed by a general multiplicative stochastic forcing. We show the existence of an almost sure Markov selection to the associated martingale problem. Our proof is based on the abstract framework introduced in Flandoli and Romito (Probab. Theory Related Fields 40 (2008) 407.458). A major difficulty arises from the fact, different from the incompressible case, that the velocity field is not continuous in time. In addition, it cannot be recovered from the variables whose time evolution is described by the Navier.Stokes system, namely, the density and the momentum. We overcome this issue by introducing an auxiliary variable into the Markov selection procedure.