We study the long-time behavior of solutions to a stochastically driven Navier–Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue–Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. An essential tool in order to obtain the global-in-time estimate is the stationarity of solutions on each approximation level, which provides a certain regularizing effect. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory, due to the underlying martingale structure of the noise.
Breit, D., Feireisl, E., Hofmanová, M., & Maslowski, B. (2018). Stationary solutions to the compressible Navier–Stokes system driven by stochastic forces. Probability Theory and Related Fields, 1-52. https://doi.org/10.1007/s00440-018-0875-4