Skip to main content
SpringerLink
Log in

On the Long-Time Behavior of Dissipative Solutions to Models of Non-Newtonian Compressible Fluids

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We identify a class maximal dissipative solutions to models of compressible viscous fluids that maximize the energy dissipation rate. Then we show that any maximal dissipative solution approaches an equilibrium state for large times.

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., Feireisl, E., Novotný, A.: Generalized solutions to models of compressible viscous fluids. Discrete Contin. Dyn. Syst. 41(1), 1–28, 2021

    Article  MathSciNet  Google Scholar 

  2. 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 

  3. Breit, D., Feireisl, E., Hofmanová, M.: Solution semiflow to the isentropic Euler system. Arch. Ration. Mech. Anal. 235(1), 167–194, 2020

    Article  MathSciNet  Google Scholar 

  4. Brenier, Y.: The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem. Commun. Math. Phys. 364(2), 579–605, 2018

    Article  ADS  MathSciNet  Google Scholar 

  5. Dafermos, C.M.: Maximal dissipation in equations of evolution. J. Differ. Equ. 252(1), 567–587, 2012

    Article  ADS  MathSciNet  Google Scholar 

  6. Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179, 1979

    Article  MathSciNet  Google Scholar 

  7. DiPerna, R.J.: Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88, 223–270, 1985

    Article  MathSciNet  Google Scholar 

  8. 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 

  9. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics, 2nd edn. Birkhäuser/Springer, Cham 2017

    Book  Google Scholar 

  10. Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3, 358–392, 2001

    Article  ADS  MathSciNet  Google Scholar 

  11. Feireisl, E., Petzeltová, H.: On the zero-velocity-limit solutions to the Navier–Stokes equations of compressible flow. Manuscr. Math. 97, 109–116, 1998

    Article  MathSciNet  Google Scholar 

  12. Feireisl, E., Petzeltová, H.: Large-time behaviour of solutions to the Navier–Stokes equations of compressible flow. Arch. Ration. Mech. Anal. 150, 77–96, 1999

    Article  MathSciNet  Google Scholar 

  13. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations I. Springer, New York 1994

    Book  Google Scholar 

  14. Geißert, M., Heck, H., Hieber, M.: On the equation \({\rm div}\,u=g\) and Bogovskiĭ’s operator in Sobolev spaces of negative order. In: Partial Differential Equations and Functional Analysis, volume 168 of Oper. Theory Adv. Appl., pp. 113–121. Birkhäuser, Basel (2006)

  15. Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Incompressible Models, vol. 1. Oxford Science Publication, Oxford 1996

    MATH  Google Scholar 

  16. Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Compressible Models, vol. 2. Oxford Science Publication, Oxford 1998

    MATH  Google Scholar 

  17. Málek, J., Nečas, J., Rokyta, M., R\(\mathring{\rm u}\)žička, M.: Weak and Measure-Valued Solutions to Evolutionary PDE’s. Chapman and Hall, London (1996)

  18. Mamontov, A.E.: Global solvability of the multidimensional Navier–Stokes equations of a compressible fluid with nonlinear viscosity. I. Sib. Math. J. 40(2), 351–362, 1999

    Article  Google Scholar 

  19. Mamontov, A.E.: Global solvability of the multidimensional Navier–Stokes equations of a compressible fluid with nonlinear viscosity. II. Sib. Math. J. 40(3), 541–555, 1999

    Article  Google Scholar 

Download references

Acknowledgements

The paper was written when E.F. was visiting the Dong-A University in Busan. He gladly acknowledges the hospitality and support provided.

Author information

Authors and Affiliations

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

    Eduard Feireisl

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

    Eduard Feireisl

  3. Department of Mathematics, Dong-A University, Busan, 49315, Republic of Korea

    Young-Sam Kwon

  4. IMATH, EA 2134, Université de Toulon, BP 20132, 83957, La Garde, France

    Antonín Novotný

Authors
  1. Eduard Feireisl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Young-Sam Kwon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonín Novotný
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduard Feireisl.

Additional information

Communicated by C. Dafermos

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 18-05974S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. The work of Y.-S. K. was partially supported by the National Research Foundation of Korea (NRF-2019H1D3A2A01101128 and NRF2020R1F1A1A01049805). The work of A.N. was partially supported by Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea (NRF-2019H1D3A2A01101128)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feireisl, E., Kwon, YS. & Novotný, A. On the Long-Time Behavior of Dissipative Solutions to Models of Non-Newtonian Compressible Fluids. Arch Rational Mech Anal 241, 475–495 (2021). https://doi.org/10.1007/s00205-021-01657-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-021-01657-0

Search

Navigation