Incompressible Limit for Compressible Fluids with Stochastic Forcing

Abstract

Abstract. We study the asymptotic behavior of the isentropic Navier-Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier-Stokes system is identified as the limit problem.

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

“…• We studied the incompressible limit of the compressible Navier-Stokes with stochastic forcing in our previous paper [1].…”

Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

“…In that case, it even suffices to consider just smooth test functions which are not necessarily compactly supported. See for example [3][4][5].…”

“…A major problem to overcome is the rapid oscillation of acoustic waves due to the lack of compactness. A stochastic counterpart of this theory has very recently been established in [4]. The limit ε of the system…”

“…ε 2 ]dt = Φ( ε , ε u ε )dW (4) has been analyzed under periodic boundary conditions. Given a sequence of the so-called finite energy weak martingale solution for (4) (see next section for definition) where ε ∈ (0, 1), its limit (as ε → 0) is indeed a weak martingale solution to the following incompressible system: div(u) = 0,…”

“…A major drawback in the approach in [4] is that the noise coefficient Φ( , u) has to be linear in the momentum u. This is due to the aforementioned lack of compactness of momentum when ε passes to zero.…”