Abstract
We investigate the well posedness of the stochastic Navier–Stokes equations for viscous, compressible, non-isentropic fluids. The global existence of finite-energy weak martingale solutions for large initial data within a bounded domain of mathbb {R}^d is established under the condition that the adiabatic exponent gamma > d/2. The flow is driven by a stochastic forcing of multiplicative type, white in time and colored in space. This work extends recent results on the isentropic case, the main contribution being to address the issues which arise from coupling with the temperature equation. The notion of solution and corresponding compactness analysis can be viewed as a stochastic counterpart to the work of Feireisl (Dynamics of viscous compressible fluids, vol 26. Oxford University Press, Oxford, 2004).
Highlights
The proof of Lemma 2.1 uses the method of Mellet/Vasseur [13], but tracking a bit more carefully the dependence of the estimate on |g|Lγ
Applying Lemma 2.1 with β = (3 − σ )/2 and arguing as in the last section, we find that: E
Applying (4.15) and Lemma 4.3, we find that lim sup P
Summary
This paper is devoted to the analysis of the initial boundary value problem for the stochastic Navier–Stokes equations for non-isentropic, compressible fluids. The first result is due to [8], which used a deterministic approach in the case where the coefficients σk are independent of the fluid variables. In this setting, one can make a convenient change of variables which turns the SPDE into a random PDE. Three sets of authors [3,14,17] studied independently the case of more general noise coefficients, establishing the existence of global martingale solutions to (1.1) in the barotropic regime. We turn to a more precise statement of our results
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