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Probability Theory and Related Fields

, Volume 174, Issue 3–4, pp 981–1032 | Cite as

Stationary solutions to the compressible Navier–Stokes system driven by stochastic forces

  • Dominic Breit
  • Eduard Feireisl
  • Martina HofmanováEmail author
  • Bohdan Maslowski
Article
  • 208 Downloads

Abstract

We study the long-time behavior of solutions to a stochastically driven Navier–Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue–Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. An essential tool in order to obtain the global-in-time estimate is the stationarity of solutions on each approximation level, which provides a certain regularizing effect. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory, due to the underlying martingale structure of the noise.

Keywords

Navier–Stokes system Compressible fluid Stochastic perturbation Stationary solution 

Mathematics Subject Classification

60H15 60H30 35Q30 76M35 76N10 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Dominic Breit
    • 1
  • Eduard Feireisl
    • 2
  • Martina Hofmanová
    • 3
    Email author
  • Bohdan Maslowski
    • 4
  1. 1.Department of MathematicsHeriot-Watt UniversityRiccarton, EdinburghUK
  2. 2.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic
  3. 3.Institute of MathematicsTechnical University BerlinBerlinGermany
  4. 4.Faculty of Mathematics and PhysicsCharles University in PraguePraha 8Czech Republic

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