Skip to main content
SpringerLink
Log in

Compressible Navier–Stokes system with the hard sphere pressure law in an exterior domain

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We consider the motion of compressible Navier–Stokes fluids with the hard sphere pressure law around a rigid obstacle when the velocity and the density at infinity are nonzero. This kind of pressure model is largely employed in various physical and industrial applications. We prove the existence of weak solution to the system in the exterior domain.

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. Berthelin, F., Degond, P., Delitala, M., Rascle, M.: A model for the formation and evolution of traffic jams. Arch. Ration. Mech. Anal. 187, 185–220 (2008)

    Article  MathSciNet  Google Scholar 

  2. Berthelin, F., Degond, P., LeBlanc, V., Moutari, S., Rascle, M., Royer, J.: A traffic-flow model with constraints for the modeling of traffic jams. Math. Models Methods Appl. Sci. 18, 1269–1298 (2008)

    Article  MathSciNet  Google Scholar 

  3. Bresch, D., Nečasová, Š, Perrin, C.: Compression effects in heterogeneous media. J. Éc. Polytech. Math. 6, 433–467 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bresch, D., Perrin, C., Zatorska, E.: Singular limit of a Navier–Stokes system leading to a free/congested zones two-phase model. C. R. Math. 352, 685–690 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bresch, D., Renardy, M.: Development of congestion in compressible flow with singular pressure. Asymptot. Anal. 103, 95–101 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Carnahan, N., Starling, K.: Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635–636 (1969)

    Article  Google Scholar 

  7. Choe, H.J., Novotný, A., Yang, M.: Compressible Navier–Stokes system with hard sphere pressure law and general inflow-outflow boundary conditions. J. Differ. Equ. 266, 3066–3099 (2019)

    Article  MathSciNet  Google Scholar 

  8. Degond, J., Hua, P.: Self-organized hydrodynamics with congestion and path formation in crowds. J. Comput. Phys. 237, 299–319 (2013)

    Article  MathSciNet  Google Scholar 

  9. Degond, P., Hua, J., Navoret, L.: Numerical simulations of the Euler system with congestion constraint. J. Comput. Phys. 230, 8057–8088 (2011)

    Article  MathSciNet  Google Scholar 

  10. DiPerna, R., Lions, P.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–548 (1989)

    Article  MathSciNet  Google Scholar 

  11. Feireisl, E., Lu, Y., Novotný, A.: Weak-strong uniqueness for the compressible Navier–Stokes equations with a hard-sphere pressure law. Sci. China Math. 61, 2003–2016 (2018)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. Feireisl, E., Zhang, P.: Quasi-neutral limit for a model of viscous plasma. Arch. Ration. Mech. Anal. 197, 271–295 (2010)

    Article  MathSciNet  Google Scholar 

  14. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations. Vol. I, vol.38 of Springer Tracts in Natural Philosophy. Springer, New York (1994). Linearized steady problems

  15. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Springer Monographs in Mathematics, 2nd edn. Springer, New York (2011). Steady-state problems

  16. Geissert, M., Heck, H., Hieber, M.: On the equation \({\rm div}\,u=g\) and Bogovskiĭ’s operator in Sobolev spaces of negative order. Partial Differ. Equ. Funct. Anal. Oper. Theory Adv. Appl. 168, 113–121 (2006)

  17. Kastler, A., Vichnievsky, R., Bruhat, G.: Cours de physique générale à l’usage de l’enseignement supérieur scientifique et technique. Thermodynamique, Masson et Cie (1962)

  18. Kolafa, J., Labik, S., Malijevsky, A.: Accurate equation of state of the hard sphere fluid in stable and mestable regions. Phys. Chem. Chem. Phys. 6, 2335–2340 (2004)

    Article  Google Scholar 

  19. Kračmar, S., Nečasová, Š., Novotný, A.: The motion of a compressible viscous fluid around rotating body. Ann. Univ. Ferrara Sez. Sci. Mat. 60, 189–208 (2014)

  20. Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 2, vol.10 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1998). Compressible models, Oxford Science Publications

  21. Liu, H.: Carnahan-Starling type equations of state for stable hard disk and hard sphere fluids. Mol. Phys. 119 (2021)

  22. Maury, B.: Prise en compte de la congestion dans les modeles de mouvements de foules, Actes des colloques Caen (2012)

  23. Novo, S.: Compressible Navier–Stokes model with inflow-outflow boundary conditions. J. Math. Fluid Mech. 7, 485–514 (2005)

    Article  MathSciNet  Google Scholar 

  24. Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible flow. Oxford Lecture Series in Mathematics and Its Applications, vol. 27. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  25. Perrin, C., Zatorska, E.: Free/congested two-phase model from weak solutions to multi-dimensional compressible Navier–Stokes equations. Commun. Partial Differ. Equ. 40, 1558–1589 (2015)

    Article  MathSciNet  Google Scholar 

  26. Song, Y., Mason, E.A., Stratt, R.M.: Why does the Carnahan–Starling equation work so well? J. Phys. Chem. 93, 6916–6919 (1989)

    Article  Google Scholar 

Download references

Acknowledgements

Š.N. and A. R. have been supported by the Czech Science Foundation (GAČR) project GA19-04243S. The Institute of Mathematics, CAS, is supported by RVO:67985840. The work of A.N. was partially supported by the distinguished Eduard Čech visiting program at the Institute of Mathematics of the Academy of Sciences of the Czech Republic.

Author information

Authors and Affiliations

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

    Šárka Nečasová & Arnab Roy

  2. IMATH, EA 2134, Université de Toulon Var, BP20132, 83957, La Garde, France

    Antonin Novotný

Authors
  1. Šárka Nečasová
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonin Novotný
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arnab Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Šárka Nečasová.

Additional information

Publisher's Note

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

The article was finished shortly after death of A. Novotny. We never forget him.

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

Nečasová, Š., Novotný, A. & Roy, A. Compressible Navier–Stokes system with the hard sphere pressure law in an exterior domain. Z. Angew. Math. Phys. 73, 197 (2022). https://doi.org/10.1007/s00033-022-01809-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01809-6

Keywords

Mathematics Subject Classification

Search

Navigation