Advertisement

Archive for Rational Mechanics and Analysis

, Volume 212, Issue 1, pp 219–239 | Cite as

A Regularity Criterion for the Weak Solutions to the Navier–Stokes–Fourier System

  • Eduard Feireisl
  • Antonín Novotný
  • Yongzhong Sun
Article

Abstract

We show that any weak solution to the full Navier–Stokes–Fourier system emanating from the data belonging to the Sobolev space W 3,2 remains regular as long as the velocity gradient is bounded. The proof is based on the weak-strong uniqueness property and parabolic a priori estimates for the local strong solutions.

Keywords

Weak Solution Strong Solution Relative Entropy Bulk Viscosity Regularity Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992). Teubner-Texte Math., vol. 133, pp. 9–126. Teubner, Stuttgart, 1993Google Scholar
  2. 2.
    Amann H.: Linear and quasilinear parabolic problems, I. Birkhäuser Verlag, Basel (1995)CrossRefGoogle Scholar
  3. 3.
    Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94(1), 61–66 (1984)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bresch D., Desjardins B.: Stabilité de solutions faibles globales pour les équations de Navier-Stokes compressibles avec température. C.R. Acad. Sci. Paris 343, 219–224 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bresch D., Desjardins B.: On the existence of global weak solutions to the Navier-Ntokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. 87, 57–90 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Constantin P., Fefferman C.: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42(3), 775–789 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ericksen, J.L.: Introduction to the thermodynamics of solids, revised ed. Appl. Math. Sci. 131, Springer, New York, 1998Google Scholar
  8. 8.
    Fan J., Jiang S., Ou Y.: A blow-up criterion for compressible viscous heat-conductive flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(1), 337–350 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Feireisl E.: Stability of flows of real monoatomic gases. Commun. Partial Differ. Equ. 31, 325–348 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Feireisl E.: Relative entropies in thermodynamics of complete fluid systems. Discr. Cont. Dyn. Syst. Ser. A 32, 3059–3080 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Feireisl E., Novotný A.: Singular limits in thermodynamics of viscous fluids. Birkhäuser-Verlag, Basel (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Feireisl E., Novotný A.: Weak-strong uniqueness property for the full Navier-Ntokes-Fourier system. Arch. Rational Mech. Anal. 204, 683–706 (2012)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Hoff D.: Dynamics of singularity surfaces for compressible viscous flows in two space dimensions. Commun. Pure Appl. Math. 55, 1365–1407 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hoff D., Jenssen H.K.: Symmetric nonbarotropic flows with large data and forces. Arch. Rational Mech. Anal. 173, 297–343 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hoff D., Santos M.M.: Lagrangean structure and propagation of singularities in multidimensional compressible flow. Arch. Ration. Mech. Anal. 188(3), 509–543 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Krylov N.V.: Parabolic equations with VMO coefficients in Nobolev spaces with mixed norms. J. Funct. Anal. 250(2), 521–558 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Krylov N.V., Safonov M.V: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Izvestija 16(2), 151–164 (1981)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Ladyzhenskaya, O.A., Solonnikov, V.A, Uralceva, N.N.: Linear and qusilinear equations of parabolic type. AMS Trans. Math. Monograph 23, Providence, 1968Google Scholar
  19. 19.
    Lions P.-L.: Mathematical Topics in Fluid Dynamics, Incompressible Models, vol 1. Oxford Science Publication, Oxford (1996)Google Scholar
  20. 20.
    Matsumura A.: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first order dissipation. Publ. RIMS Kyoto Univ. 13, 349–379 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Matsumura A., Nishida T.: The initial value problem for the equations of motion of compressible and heat conductive fluids. Comm. Math. Phys. 89, 445–464 (1983)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Prodi G.: Un teorema di unicità per le equazioni di Navier-Ntokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Serrin J.: On the interior regularity of weak solutions of the Navier-Ntokes equations. Arch. Rational Mech. Anal. 9, 187–195 (1962)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Sun Y., Wang C., Zhang Z.: A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navies-Ntokes equations. Arch. Rational Mech. Anal. 201, 727–742 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Tani A.: On the first initial-boundary value problem of compressible viscous fluid motion. Publ. RIMS Kyoto Univ. 13, 193–253 (1977)CrossRefzbMATHGoogle Scholar
  27. 27.
    Valli, A.: A correction to the paper: An existence theorem for compressible viscous fluids [Ann. Mat. Pura Appl. (4) 130, 197–213 (1982) MR 83h:35112]. Ann. Mat. Pura Appl. (4), 132, 399–400 (1983), (1982)Google Scholar
  28. 28.
    Valli A.: An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. 130(4), 197–213 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Valli A., Zajaczkowski M.: Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eduard Feireisl
    • 1
  • Antonín Novotný
    • 2
  • Yongzhong Sun
    • 3
  1. 1.Faculty of Mathematics and Physics, Mathematical InstituteCharles University in PraguePrague 8Czech Republic
  2. 2.IMATH Université du Sud Toulon-VarLa GardeFrance
  3. 3.Department of MathematicsNanjing UniversityNanjingChina

Personalised recommendations