Skip to main content
SpringerLink
Log in

Existence of Stationary Weak Solutions for the Heat Conducting Flows

  • Living reference work entry
  • First Online:
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Abstract

The steady compressible Navier–Stokes–Fourier system is considered, with either Dirichlet or Navier boundary conditions for the velocity and the heat flux on the boundary proportional to the difference of the temperature inside and outside. In dependence on several parameters, i.e., the adiabatic constant γ appearing in the pressure law p(ϱ, ϑ) ∼ ϱ γ +ϱ ϑ and the growth exponent in the heat conductivity, i.e., κ(ϑ) ∼ (1 +ϑ m), and without any restriction on the size of the data, the main ideas of the construction of weak and variational entropy solutions for the three-dimensional flows with temperature-dependent viscosity coefficients are explained. Further, the case when it is possible to prove existence of solutions with bounded density is reviewed. The main changes in the construction of solutions for the two-dimensional flows are mentioned, and finally, results for more complex systems are reviewed, where the steady compressible Navier–Stokes–Fourier equations play an important role.

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

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. Š. Axmann, M. Pokorný, 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  MATH  Google Scholar 

  2. J. Březina, A. Novotný, On weak solutions of steady Navier–Stokes equations for monoatomoc gas. Comment. Math. Univ. Carol. 49, 611–632 (2008)

    MATH  Google Scholar 

  3. R. Danchin, P.B. Mucha, The divergence equation in rough spaces. J. Math. Anal. Appl. 386(1), 9–31 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Dou, F. Jiang, S. Jiang, Y.-F. Yang, Existence of strong solutions to the steady Navier-Stokes equations for a compressible heat-conductive fluid with large forces. J. Math. Pures Appl. 103(5), 1163–1197 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Elizier, A. Ghatak, H. Hora, An Introduction to Equations of States, Theory and Applications (Cambridge University Press, Cambridge, 1996)

    Google Scholar 

  6. R. Erban, On the existence of solutions to the Navier–Styokes equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 26, 489–517 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)

    Google Scholar 

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

    MATH  Google Scholar 

  9. E. Feireisl, P.B. Mucha, A. Novotný, M. Pokorný, Time-periodic solutions to the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204(3), 745–786 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Feireisl, A. Novotný, A Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2009)

    Google Scholar 

  11. E. Feireisl, D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics (AIMS, Springfield, 2010)

    MATH  Google Scholar 

  12. J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for steady viscous compressible flow in 3-D. J. Math. Pures Appl. 97, 85–97 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Giovangigli, M. Pokorný, E. Zatorska, On the steady flow of reactive gaseous mixture. Analysis 35(4), 319–341 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Jesslé, A. Novotný, Existence of renormalized weak solutions to the steady equations describing compressible fluids in barotropic regime. J. Math. Pures Appl. 99, 280–296 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Jesslé, A. Novotný, M. Pokorný, Steady Navier–Stokes–Fourier system with slip boundary conditions. Math. Models Methods Appl. Sci. 24, 751–781 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Jiang, C. Zhou, Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(4), 485–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. O. Kreml, Š. Nečasová, M. Pokorný, On the steady equations for compressible radiative gas. Z. für angewandte Math. und Phys. 64, 539–571 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Kufner, O. John, S. Fučík, Function Spaces (Academia, Praha, 1977)

    MATH  Google Scholar 

  19. P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol.2: Compressible Models (Oxford Science Publication, Oxford, 1998)

    Google Scholar 

  20. L. Maligranda, Orlicz Spaces and Interpolation (Campinas SP, Brasil, 1989)

    MATH  Google Scholar 

  21. P.B. Mucha, On cylindrical symmetric flows through pipe-like domains. J. Differ. Eqn. 201, 304–323 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. P.B. Mucha, M. Pokorný, On a new approach to the issue of existence and regularity for the steady compressible Navier–Stokes equations. Nonlinearity 19, 1747–1768 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. P.B. Mucha, M. Pokorný, On the steady compressible Navier–Stokes–Fourier system. Comm. Math. Phys. 288, 349–377 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. P.B. Mucha, M. Pokorný, Weak solutions to equations of steady compressible heat conducting fluids. Math. Models Methods Appl. Sci. 20, 1–29 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. P.B. Mucha, M. Pokorný, The rot-div system in exterior domains. J. Math. Fluid Mech. 16(4), 701–720 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. P.B. Mucha, R. Rautmann, Convergence of Rothe’s scheme for the Navier-Stokes equations with slip conditions in 2D domains. ZAMM Z. Angew. Math. Mech. 86(9), 691–701 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Novotný, M. Pokorný, Steady compressible Navier–Stokes–Fourier system for monoatomic gas and its generalizations. J. Differ. Eqn. 251, 270–315 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Novotný, M. Pokorný, Weak and variational solutions to steady equations for compressible heat conducting fluids. SIAM J. Math. Anal. 43, 270–315 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Novotný, M. Pokorný, Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions. Appl. Math. 56, 137–160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow (Oxford University Press, Oxford, 2004)

    MATH  Google Scholar 

  31. G. Panasenko, K. Pileckas, Divergence equation in thin-tube structures. Appl. Anal. 94(7), 1450–1459 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Pecharová, M. Pokorný, Steady compressible Navier-Stokes-Fourier system in two space dimensions. Comment. Math. Univ. Carolin. 51, 653–679 (2010)

    MathSciNet  MATH  Google Scholar 

  33. P.I. Plotnikov, J. Sokolowski, On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier–Stokes equations. J. Math. Fluid Mech. 7, 529–573 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. P.I. Plotnikov, W. Weigant, Steady 3D viscous compressible flows with adiabatic exponent γ ∈ (1, ∞). J. Math. Pures Appl. 104, 58–82 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Pokorný, On the steady solutions to a model of compressible heat conducting fluid in two space dimensions. J. Part. Differ. Eqn. 24(4), 334–350 (2011)

    MathSciNet  MATH  Google Scholar 

  36. M. Pokorný, P.B. Mucha, 3D steady compressible Navier–Stokes equations. Cont. Discret. Dyn. Syst. S 1, 151–163 (2008)

    MathSciNet  MATH  Google Scholar 

  37. V.A. Solonnikov, Overdetermined elliptic boundary value problems. Zap. Nauch. Sem. LOMI 21, 112–158 (1971)

    MathSciNet  MATH  Google Scholar 

  38. R. Vodák, The problem div v = f and singular integrals in Orlicz spaces. Acta Univ. Olomuc. Fac. Rerum Nat. Math. 41, 161–173 (2002)

    MATH  Google Scholar 

  39. E. Zatorska, On the steady flow of a multicomponent, compressible, chemically reacting gas. Nonlinearity 24, 3267–3278 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. E. Zatorska, Analysis of semidiscretization of the compressible Navier-Stokes equations. J. Math. Anal. Appl. 386(2), 559–580 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. X. Zhong, Weak solutions to the three-dimensional steady full compressible Navier-Stokes system. Nonlinear Anal. 127, 71–93 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. W.P. Ziemer, Weakly Differentiable Functions (Springer, New York, 1989)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The work of P.B. Mucha has been partly supported by Polish NCN grant No 2014/13/B/ST1/03094. The work of M. Pokorný has been partially supported by the Czech Science Foundation, grant No. 16-03230S.

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland

    Piotr B. Mucha

  2. Faculty of Mathematics and Physics, Mathematical Institute of Charles University, Charles University in Prague, Praha 8, Czech Republic

    Milan Pokorný

  3. Department of Mathematics, Imperial College London, London, UK

    Ewelina Zatorska

Authors
  1. Piotr B. Mucha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milan Pokorný
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ewelina Zatorska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Piotr B. Mucha .

Editor information

Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this entry

Cite this entry

Mucha, P.B., Pokorný, M., Zatorska, E. (2016). Existence of Stationary Weak Solutions for the Heat Conducting Flows. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_64-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10151-4_64-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Online ISBN: 978-3-319-10151-4

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation