The stochastic compressible Navier-Stokes system on the whole space and some singular limits

Abstract

Firstly, we show the existence of at least one non-trivial solution to the stochastically forced compressible Navier–Stokes system defined on the whole Euclidean space. This solution is deterministically weak in the usual sense of distributions but also weak in the sense of probability, the latter meaning that the underlying probability space, as well as the stochastic driving force, are also unknowns. Secondly, we study various asymptotic results for the above mentioned system when the microscopic time and space variables are rescaled appropriately. Different rescaling leads to various singular versions of this system with coefficients which either blow up or dissipate when they are made small. Subsequently, we are able to show that any family of the solutions constructed above parametrised by the singular coefficients converges to solutions of other fluid dynamic models like the incompressible Navier–Stokes system and the compressible Euler system with corresponding stochastic forcing terms. Crucially, we also consider the case when rotation in the fluid is taken into account.