Skip to main content
SpringerLink
Log in

On the Motion of a Compressible Viscous Fluid Driven by Time Periodic Inflow/Outflow Boundary Conditions

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We consider the barotropic Navier–Stokes system describing the motion of a compressible viscous fluid confined to a bounded domain driven by time periodic inflow/outflow boundary conditions. We show that the problem admits a time periodic solution in the class of weak solutions satisfying the energy inequality.

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. Abbatiello, A.: Time-periodic weak solutions to incompressible generalized Newtonian fluids, accepted in J. Math. Fluid Mech. (2020)

  2. Abbatiello, A., Maremonti, P.: Existence of regular time-periodic solutions to shear-thinning fluids. J. Math. Fluid Mech. 21(2), 29 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  3. Axmann, Š., Pokorný, M.: Time-periodic solutions to the full Navier–Stokes–Fourier system with radiation on the boundary. J. Math. Anal. Appl. 428(1), 414–444 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bogovskii, M.E.: Solution of some vector analysis problems connected with operators div and grad (in Russian). Trudy Sem. S.L. Sobolev 80(1), 5–40 (1980)

    Google Scholar 

  5. Březina, J., Feireisl, E., Novotný, A.: Globally bounded trajectories for the barotropic Navier–Stokes system with general boundary conditions. Commun. Partial. Differ. Equ. 45(12), 1820–1832 (2020)

    Article  MathSciNet  Google Scholar 

  6. Chang, T., Jin, B.J., Novotný, A.: Compressible Navier–Stokes system with general inflow-outflow boundary data. SIAM J. Math. Anal. 51(2), 1238–1278 (2019)

    Article  MathSciNet  Google Scholar 

  7. Chen, G.-Q., Torres, M., Ziemer, W.P.: Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Commun. Pure Appl. Math. 62(2), 242–304 (2009)

    Article  MathSciNet  Google Scholar 

  8. Ciuperca, I.S., Feireisl, E., Jai, M., Petrov, A.: A rigorous derivation of the stationary compressible Reynolds equation via the Navier–Stokes equations. Math. Models Methods Appl. Sci. 28(4), 697–732 (2018)

    Article  MathSciNet  Google Scholar 

  9. Ciuperca, I.S., Feireisl, E., Jai, M., Petrov, A.: Stationary solutions of the Navier–Stokes–Fourier system in planar domains with impermeable boundary. J. Math. Pures Appl. (9) 140, 110–138 (2020)

    Article  MathSciNet  Google Scholar 

  10. Denk, R., Hieber, M., Prüss, J.: Optimal \(L^p-L^q\)-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 193–224 (2007)

    Article  MathSciNet  Google Scholar 

  11. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  12. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

    Google Scholar 

  13. Feireisl, E., Lu, Y., Málek, J.: On PDE analysis of flows of quasi-incompressible fluids. Z. Angew. Math. Mech. 96(4), 491–508 (2016)

    Article  MathSciNet  Google Scholar 

  14. Feireisl, E., Matuš\(\mathring{\rm u}\)-Nečasová, Š, Petzeltová, H., Straškraba, I.: On the motion of a viscous compressible flow driven by a time-periodic external flow. Arch. Rational Mech. Anal. 149, 69–96 (1999)

  15. Feireisl, E., Mucha, P., Novotný, A., Pokorný, M.: Time periodic solutions to the full Navier–Stokes–Fourier system. Arch. Rational. Mech. Anal. 204, 745–786 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  16. Feireisl, E., Zhang, P.: Quasi-neutral limit for a model of viscous plasma. Arch. Ration. Mech. Anal. 197(1), 271–295 (2010)

    Article  MathSciNet  Google Scholar 

  17. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, 2nd edn. Springer, New York (2003)

    Google Scholar 

  18. Galdi, G.P.: Existence and uniqueness of time-periodic solutions to the Navier–Stokes equations in the whole plane. Discrete Contin. Dyn. Syst. Ser. S 6(5), 1237–1257 (2013)

    MathSciNet  Google Scholar 

  19. Galdi, G.P., Kyed, M.: Time-periodic solutions to the Navier–Stokes equations, Handbook of mathematical analysis in mechanics of viscous fluids, Springer, Cham, 2018, pp. 509–578 (2018)

  20. Geißert, M., Heck, H., Hieber, M.: On the equation \({\rm div}\,u=g\) and Bogovskiĭ’s operator in Sobolev spaces of negative order, Partial differential equations and functional analysis. Oper. Theory Adv. Appl., vol. 168, Birkhäuser, Basel, pp. 113–121 (2006)

  21. Girinon, V.: Navier-Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain. J. Math. Fluid Mech. 13(3), 309–339 (2011)

    Article  ADS  MathSciNet  Google Scholar 

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

  23. 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  CAS  Google Scholar 

  24. Kozono, H., Yanagisawa, T.: Leray’s problem on the stationary Navier–Stokes equations with inhomogeneous boundary data. Math. Z. 262(1), 27–39 (2009)

    Article  MathSciNet  Google Scholar 

  25. Maremonti, P., Padula, M.: Existence, uniqueness and attainability of periodic solutions of the Navier-Stokes equations in exterior domains. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 233 (1996), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 27, 142–182, 257

  26. Novo, S.: Compressible Navier–Stokes model with inflow-outflow boundary conditions. J. Math. Fluid Mech. 7(4), 485–514 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  27. Poul, L.: Existence of weak solutions to the Navier–Stokes–Fourier system on Lipschitz domains, Discrete Contin. Dyn. Syst. (2007), no. Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., pp. 834–843 (2007)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Anna Abbatiello

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

    Eduard Feireisl

Authors
  1. Anna Abbatiello
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduard Feireisl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduard Feireisl.

Additional information

Publisher's Note

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

A. Abbatiello: The research of A.A. is supported by Einstein Foundation, Berlin.

E. Feireisl: The research of E.F. leading to these results has received funding from 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. The stay of E.F. at TU Berlin is supported by Einstein Foundation, Berlin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abbatiello, A., Feireisl, E. On the Motion of a Compressible Viscous Fluid Driven by Time Periodic Inflow/Outflow Boundary Conditions. J Dyn Diff Equat 36 (Suppl 1), 105–126 (2024). https://doi.org/10.1007/s10884-021-10004-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-021-10004-6

Keywords

Search

Navigation