Advertisement

Archive for Rational Mechanics and Analysis

, Volume 222, Issue 2, pp 895–926 | Cite as

Incompressible Limit for Compressible Fluids with Stochastic Forcing

  • Dominic Breit
  • Eduard Feireisl
  • Martina Hofmanová
Article

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 a 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.

Mathematics Subject Classification

60H15 35R60 76N10 35Q30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angot P., Bruneau Ch.-H., Fabrie P.: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81(4), 497–520 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Breit, D., Feireisl, E., Hofmanová, M.: Compressible fluids driven by stochastic forcing: the relative energy inequality and applications. Preprint at arXiv:1510.09001v1
  3. 3.
    Breit, D., Hofmanová, M.: Stochastic Navier-Stokes equations for compressible fluids, to appear in Indiana Univ. Math. J. Preprint at arXiv:1409.2706
  4. 4.
    Brzeźniak Z.: Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4, 1–45 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Capiński M.: A note on uniqueness of stochastic Navier-Stokes equations. Univ. Iagell. Acta Math. 30, 219–228 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Capiński M., Cutland N.J.: Stochastic Navier-Stokes equations. Acta Applicandae Mathematicae 25, 59–85 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Capiński M., Ga¸tarek D.: Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension. J. Funct. Anal. 126(1), 26–35 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge (1992)Google Scholar
  9. 9.
    Desjardins B., Grenier E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1986), 2271–2279 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Desjardins, B., Grenier, E., Lions, P.-L., Masmoudi, N.: Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. (9) 78(5), 461–471 (1999)Google Scholar
  11. 11.
    Feireisl, E., Maslowski, B., Novotný, A.: Compressible fluid flows driven by stochastic forcing. J. Differential Equations 254, 1342–1358 (2013)Google Scholar
  12. 12.
    Feireisl E., Novotný A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid. Mech. 3, 358–392 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Flandoli F., Gaa¸tarek D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gyöngy I., Krylov N.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 105(2), 143–158 (1996)CrossRefzbMATHGoogle Scholar
  15. 15.
    Jakubowski, A.: The almost sure Skorokhod representation for subsequences in nonmetric spaces. Teor. Veroyatnost. i Primenen 42 (1997), no. 1, 209–216; translation in Theory Probab. Appl. 42 (1997), no. 1, 167–174 (1998).Google Scholar
  16. 16.
    Klainerman S., Majda A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klein, R., Botta, N., Schneider, T., Munz, C.D., Roller, S., Meister, A., Hoffmann, L., Sonar, T.: Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Eng. Math. 39, 261–343 (2001)Google Scholar
  18. 18.
    Lighthill J.: On sound generated aerodynamically I. General theory. Proc. R. Soc. Lond. A 211, 564–587 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lighthill J.: On sound generated aerodynamically II. General theory. Proc. R. Soc. Lond. A 222, 1–32 (1954)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford lecture series in mathematics and its applications, 10. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1998)Google Scholar
  21. 21.
    Lions P.-L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl., 71, 585–621 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lions P.-L., Masmoudi N.: Une approche locale de la limite incompressible. (French) [A local approach to the incompressible limit] C. R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    van Neerven, J.M.A.M., Veraar, M.C., Weis, L.: Stochastic integration in UMD Banach spaces. Annals Probab. 35, 1438–1478 (2007)Google Scholar
  24. 24.
    Ondreját M.: Stochastic nonlinear wave equations in local Sobolev spaces. Electron. J. Probab. 15(33), 1041–1091 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ondreját M.: Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Mathematicae 426, 1–63 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Prévôt, C., Röckner, M.: A concise course on stochastic partial differential equations, vol. 1905 of Lecture Notes in Math., Springer, Berlin (2007)Google Scholar
  27. 27.
    Tornatore E.: Global solution of bi-dimensional stochastic equation for a viscous gas. NoDEA Nonlinear Differ. Equ. Appl. 7(4), 343–360 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Dominic Breit
    • 1
  • Eduard Feireisl
    • 2
  • Martina Hofmanová
    • 3
  1. 1.Department of MathematicsHeriot-Watt UniversityRiccarton EdinburghUK
  2. 2.Institute of Mathematics AS CR Žitná 25 CZPraha 1Czech Republic
  3. 3.Technical University Berlin, Institute of MathematicsBerlinGermany

Personalised recommendations