Vanishing Mach Number Limit of Stochastic Compressible Flows

Abstract

We study the vanishing Mach number limit for the stochastic Navier-Stokes equations with $\gamma$-type pressure laws, with focus on the one-dimensional case. We prove that, if the stochastic term vanishes with respect to the Mach number sufficiently fast, the deviation from the incompressible state of the solutions (for $\gamma \geq 1$) and the invariant measures (for $\gamma = 1$) is governed by a linear stochastic acoustic system in the limit. In particular, the critically sufficient decay rate for the stochastic term is slower than the corresponding results with deterministic external forcing due to the martingale structure of the noise term, and the blow-up of the noise term for the fluctuation system can be allowed.