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).
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References
Lubomír Baňas, Zdzislaw Brzeźniak, Mikhail Neklyudov, and Andreas Prohl. A convergent finite-element-based discretization of the stochastic landau–lifshitz–gilbert equation. IMA Journal of Numerical Analysis, page drt020, 2013.
Florent Berthelin and Julien Vovelle. Stochastic isentropic euler equations. arXiv preprint arXiv:1310.8093, 2013.
Dominic Breit and Martina Hofmanova. Stochastic navier stokes equations for compressible fluids. Indiana University of Mathematics Journal, 65:1183–1250, 2016.
Zdzisław Brzeźniak and Martin Ondreját. Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces. Ann. Probab., 41(3B):1938–1977, 2013.
Giuseppe Da Prato and Jerzy Zabczyk. Stochastic equations in infinite dimensions. Cambridge university press, 2014.
Eduard Feireisl. On compactness of solutions to the compressible isentropic navier-stokes equations when the density is not square integrable. Commentationes Mathematicae Universitatis Carolinae, 42(1):83–98, 2001.
Eduard Feireisl. Dynamics of viscous compressible fluids, volume 26. Oxford University Press Oxford, 2004.
Eduard Feireisl, Bohdan Maslowski, and Antonín Novotnỳ. Compressible fluid flows driven by stochastic forcing. Journal of Differential Equations, 254(3):1342–1358, 2013.
A. Jakubowski. The almost sure Skorokhod representation for subsequences in nonmetric spaces. Teor. Veroyatnost. i Primenen., 42(1):209–216, 1997.
N. V. Krylov. On the Itô-Wentzell formula for distribution-valued processes and related topics. Probab. Theory Related Fields, 150(1-2):295–319, 2011.
Olga Aleksandrovna Ladyzhenskaia, Vsevolod Alekseevich Solonnikov, and Nina N Ural’tseva. Linear and quasi-linear equations of parabolic type, volume 23. American Mathematical Soc., 1988.
Pierre-Louis Lions. Mathematical topics in fluid mechanics. volume 2: Compressible models. 1998.
Antoine Mellet and Alexis Vasseur. A bound from below for the temperature in compressible navier–stokes equations. Monatshefte für Mathematik, 157(2):143–161, 2009.
Scott Smith. Random perturbations of viscous compressible fluids: Global existence of weak solutions. SIAM Journal of Mathematical Analysis, to appear.
Hans Triebel. Theory of function spaces. III, volume 100 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 2006.
Aad W. van der Vaart and Jon A. Wellner. Weak convergence and empirical processes. Springer Series in Statistics. Springer-Verlag, New York, 1996. With applications to statistics.
Dehua Wang, Huaqiao Wang, et al. Global existence of martingale solutions to the three-dimensional stochastic compressible navier-stokes equations. Differential and Integral Equations, 28(11/12):1105–1154, 2015.
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Scott A. Smith acknowledges the support in part by the National Science Foundation under the Awards DMS-1211519, DMS-1614964 and the support by the Ann G. Wylie Dissertation Fellowship.
Konstantina Trivisa gratefully acknowledges the support in part by the National Science Foundation under the Grant DMS-1614964, by the Simons Foundation Grant 346300 and the Polish Government MNiSW 2015-2019 matching fund.
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Smith, S.A., Trivisa, K. The stochastic Navier–Stokes equations for heat-conducting, compressible fluids: global existence of weak solutions. J. Evol. Equ. 18, 411–465 (2018). https://doi.org/10.1007/s00028-017-0407-1
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DOI: https://doi.org/10.1007/s00028-017-0407-1