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Applications of Mathematics

, 56:137 | Cite as

Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions

  • Antonín NovotnýEmail author
  • Milan Pokorný
Article

Abstract

We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law p(ϱ, ϑ) ∼ ϱ γ + ϱϑ if γ > 1 and p(ϱ, ϑ) ∼ ϱ ln α (1 + ϱ) + ϱϑ if γ = 1, α > 0, depending on the model for the heat flux.

Keywords

steady compressible Navier-Stokes-Fourier system weak solution entropy inequality Orlicz spaces compensated compactness renormalized solution 

MSC 2010

76N10 35Q30 

References

  1. [1]
    R. Erban: On the existence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 26 (2003), 489–517.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    E. Feireisl, A. Novotný: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel, 2009.Google Scholar
  3. [3]
    J. Frehse, M. Steinhauer, W. Weigant: The Dirichlet Problem for Steady Viscous Compressible Flow in 3-D. Preprint University of Bonn, SFB 611, No. 347 (2007), http://www.iam.uni-bonn.de/sfb611/.
  4. [4]
    J. Frehse, M. Steinhauer, W. Weigant: The Dirichlet problem for viscous compressible isothermal Navier-Stokes equations in two-dimensions. Arch. Ration. Mech. Anal. 198 (2010), 1–12.MathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Kufner, O. John, S. Fučík: Function spaces. Academia, Praha, 1977.zbMATHGoogle Scholar
  6. [6]
    P.-L. Lions: Mathematical Topics in Fluid Dynamics, Vol. 2: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications, Vol. 10. Clarendon Press, Oxford, 1998.Google Scholar
  7. [7]
    L. Maligranda: Orlicz Spaces and Interpolation. Univ. Estadual de Campinas, Campinas, 1989.zbMATHGoogle Scholar
  8. [8]
    P.B. Mucha, M. Pokorný: On the steady compressible Navier-Stokes-Fourier system. Commun. Math. Phys. 288 (2009), 349–377.zbMATHCrossRefGoogle Scholar
  9. [9]
    P.B. Mucha, M. Pokorný: Weak solutions to equations of steady compressible heat conducting fluids. Math. Models Methods Appl. Sci. 20 (2010), 785–813.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. Novotný, M. Pokorný: Steady compressible Navier-Stokes-Fourier system for monoatomic gas and its generalizations. J. Differ. Equations. Accepted. See also Preprint Series of Nečas Center for Mathematical Modeling, http://www.karlin.mff.cuni.cz/ncmm/research/Preprints/servirPrintsYY.php?y=2010. Preprint No. 2010-021.
  11. [11]
    A. Novotný, M. Pokorný: Weak and variational solutions to steady equations for compressible heat conducting fluids. Submitted. See also Preprint Series of Nečas Center for Mathematical Modeling, http://www.karlin.mff.cuni.cz/ncmm/research/Preprints/servirPrintsYY.php?y=2010. Preprint No. 2010-023.
  12. [12]
    A. Novotný, I. Straškraba: Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press, Oxford, 2004.zbMATHGoogle Scholar
  13. [13]
    P. Pecharová, M. Pokorný: Steady compressible Navier-Stokes-Fourier system in two space dimensions. Commentat. Math. Univ. Carol. To appear.Google Scholar
  14. [14]
    P. I. Plotnikov, J. Sokołowski: On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations. J. Math. Fluid Mech. 7 (2005), 529–573.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    P. I. Plotnikov, J. Sokołowski: Concentrations of stationary solutions to compressible Navier-Stokes equations. Commun. Math. Phys. 258 (2005), 567–608.zbMATHCrossRefGoogle Scholar
  16. [16]
    P. I. Plotnikov, J. Sokołowski: Stationary solutions of Navier-Stokes equations for diatomic gases. Russ. Math. Surv. 62 (2007), 561–593.zbMATHCrossRefGoogle Scholar
  17. [17]
    R. Vodák: The problem div v = f and singular integrals on Orlicz spaces. Acta Univ. Palacki. Olomuc, Fac. Rerum Nat. Math. 41 (2002), 161–173.zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2011

Authors and Affiliations

  1. 1.IMATHUniversité du Sud Toulon-VarLa GardeFrance
  2. 2.Mathematical Institute of Charles UniversityPraha 8Czech Republic

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