Skip to main content
SpringerLink
Log in

Compressible Fluid Motion With Uncertain Data

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

We propose a suitable analytical framework to perform numerical analysis of problems arising in compressible fluid models with uncertain data. We discuss both weak and strong stochastic approach, where the former is based on the knowledge of the mere distribution (law) of the random data typical for the Monte-Carlo and related methods, while the latter assumes the data to be known as a random variable on a given probability space aiming at obtaining the associated solution in the same form. As an example of the strong approach, we discuss the stochastic collocation method based on a piecewise constant approximation of the random data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Breit, D., Feireisl, E., Hofmanová, M.: Local strong solutions to the stochastic compressible Navier-Stokes system. Comm. Partial Differ. Equ. 43(2), 313–345 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckmaster, T., Cao-Labora, G., Gómez-Serrano, J.: Smooth imploding solutions for 3D compressible fluid. Preprint (2022)

  3. Buckmaster, T., Vicol, V.: Nonuniqueness of weak solutions to the Navier-Stokes equation. Ann. of Math. (2) 189(1), 101–144 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Constantin, P., Wu, J.: Statistical solutions of the Navier-Stokes equations on the phase space of vorticity and the inviscid limits. J. Math. Phys. 38(6), 3031–3045 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Cuesta, J.A., Matrán, C.: Strong laws of large numbers in abstract spaces via Skorohod’s representation theorem. Sankhyā Ser. A 48(1), 98–103 (1986)

    MathSciNet  Google Scholar 

  6. Fanelli, F., Feireisl, E.: Statistical solutions to the barotropic Navier-Stokes system. J. Stat. Phys. 181(1), 212–245 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Feireisl, E., Hošek, R., Maltese, D., Novotný, A.: Error estimates for a numerical method for the compressible Navier-Stokes system on sufficiently smooth domains. ESAIM Math. Model. Numeri. Anal. 51, 279–319 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feireisl, E., Hošek, R., Maltese, D., Novotný, A.: Unconditional convergence and error estimates for bounded numerical solutions of the barotropic Navier-Stokes system. Numer. Methods Partial Differ. Equ. 33(4), 1208–1223 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feireisl, E., Lukáčová-Medvid’ová, M., Mizerová, H., She, B.: Numer. Anal. Compressible Fluid Flows. Springer-Verlag, Cham (2022)

    MATH  Google Scholar 

  10. Feireisl, E., Lukáčová-Medvi’ ová, M.: Convergence of a stochastic collocation finite volume method for the compressible Navier–Stokes system. Archive Preprint Series, 2021. arxiv preprint No. 07435

  11. Foias, C., Mondaini, C.F., Titi, E.S.: A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics. SIAM J. Appl. Dyn. Syst. 15(4), 2109–2142 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Foias, C., Rosa, R.M.S., Temam, R.M.: Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations. J. Dynam. Differ. Equ. 31(3), 1689–1741 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gallagher, I.: A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations. J. Math. Kyoto Univ. 40(3), 525–540 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Gallavotti, G.: Stat. Mech. Short Treatise. Springer-Verlag, Heidelberg (1999)

    Google Scholar 

  15. Koley, U., Risebro, N.H., Schwab, C., Weber, F.: A multilevel Monte Carlo finite difference method for random scalar degenerate convection-diffusion equations. J. Hyperbolic Differ. Equ. 14(3), 415–454 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuo, F.Y., Scheichl, R., Schwab, C., Sloan, I.H., Ullmann, E.: Multilevel quasi-Monte Carlo methods for lognormal diffusion problems. Math. Comp. 86(308), 2827–2860 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kwon, Y.-S., Novotný, A.: Construction of weak solutions to compressible Navier-Stokes equations with general inflow/outflow boundary conditions via a numerical approximation. Numer. Math. 149(4), 717–778 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kwon, Y.-S., Novotný, A.: Consistency, convergence and error estimates for a mixed finite element-finite volume scheme to compressible Navier-Stokes equations with general inflow/outflow boundary data. IMA J. Numer. Anal. 42(1), 107–164 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leonardi, F., Mishra, S., Schwab, C.: Numerical approximation of statistical solutions of planar, incompressible flows. Math. Models Methods Appl. Sci. 26(13), 2471–2523 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible and heat conductive fluids. Comm. Math. Phys. 89, 445–464 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Merle, F., Raphael, P., Rodnianski, I., Szeftel, J.: On the implosion of a three dimensional compressible fluid. Arxive Preprint Series, arxiv preprint No. 11009, 2019 (1912)

  22. Mishra, S., Schwab, Ch.: Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comp. 81(280), 1979–2018 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun, Y., Wang, C., Zhang, Z.: A Beale-Kato-Majda criterion for the 3-D compressible Navier-Stokes equations. J. Math. Pures Appl. 95(1), 36–47 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sun, Y., Wang, C., Zhang, Z.: A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows. Arch. Ration. Mech. Anal. 201(2), 727–742 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tani, A.: On the first initial-boundary value problem of compressible viscous fluid motion. Publ. RIMS Kyoto Univ. 13, 193–253 (1977)

    Article  MATH  Google Scholar 

  26. Valli, A., Zajaczkowski, M.: Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Vishik, M. J., Fursikov, A. V.: Mathematical problems of statistical hydromechanics, volume 9 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht. Translated from the 1980 Russian original [ MR0591678] by D. A. Leites (1988)

Download references

Acknowledgements

A part of this work was done during author’s visit to the Basque Center for Applied Mathematics in Bilbao. The support and friendly atmosphere are acknowledged with thanks.

Author information

Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67, Praha 1, Czech Republic

    Eduard Feireisl

Authors
  1. Eduard Feireisl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduard Feireisl.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Communicated by T. Ozawa.

Dedicated to the memory of my friend Antonín Novotný

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of E.F. was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 21–02411S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.

This article is part of the topical collection “In memory of Antonin Novotny” edited by Eduard Feireisl, Paolo Galdi, and Milan Pokorny.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feireisl, E. Compressible Fluid Motion With Uncertain Data. J. Math. Fluid Mech. 24, 96 (2022). https://doi.org/10.1007/s00021-022-00727-x

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-022-00727-x

Keywords

Search

Navigation