Skip to main content
SpringerLink
Log in

Global Well-Posedness for the Heat-conductive Incompressible Viscous Fluids

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

This paper is concerned with the Cauchy problem derived from the non-stationary motion of heat-conducting incompressible viscous fluids in three-dimensional whole space, where the viscosity and heat-conductivity coefficient vary with the temperature. We establish blow-up criteria and existence of global strong solution provided that the initial data is small enough.

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. Feireisl, E.: Dynamics of compressible flow. Oxford Science Publishers. Clarendon Press, Oxford (2003)

    Google Scholar 

  2. Landau, L., Lifschitz, E.: Lehrbuch der theoretischen Physik. Bd. VI: Hydrodynamik. 3 Aufl., Akademie: Berlin (1978)

  3. Müller, I.: Grundzüge der Thermodynsmik, vol. 3. Aufl., Springer, Berlin (2001)

    Book  Google Scholar 

  4. Kagei, Y.: Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function. Hiroshima Math. J. 25, 251–311 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Beneš, M.: Strong solutions to non-stationary channel flows of heat-conducting viscous incompressible fluids wuth dissipative heating. Acta. App. Math. 116, 237–254 (2011)

    Article  MATH  Google Scholar 

  6. Kakizawa, R.: The initial value problem for motion of incompressible viscous and heat-conductive fluids in Banach space. Hiroshima Math. J 40, 371–402 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Inoue, H.: On heat convection equations with dissipative terms in time dependent domains. Nonlinear Anal. 30, 4441–4448 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kagei, Y., Skowron, M.: Nonstationary flows of nonsymmetric fluids with thermal convection. Hiroshima Math. J. 23, 343–363 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Łukaszewicz, G., KrzyŻanowski, P.: On the heat convection equations with dissipation term in regions with moving boundaries. Math. Methods Appl. Sci. 20, 347–368 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, J., Zhou, Y.: A blow-up criterion of strong solutions to the compressible viscous heat-conductive flows with zero heat conductivity. Acta Appl. Math. 116, 317–327 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sun, Y., Wang, C., Zhang, Z.: A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows. Arch. Ration. Mech. Anal. 201, 727–742 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan, J., Jiang, S., Ou, Y.: A blow-up criterion for compressible viscous heat-conductive flows. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, 337–350 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Shilkin, T.: Classical solvability of the coupled system modelling a heat-convergent poiseuille-type flow. J. Math. Fluid Mech. 7, 72–84 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Consiglieri, L.: Partial Regularity for the Navier-Stokes-Fourier system. Acta Math. Sci. Ser. B Engl. Ed. 31, 1653–1670 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Naumann, J.: On the existence of weak solutions to the equations of non-stationary motion of heat-conducting in compressible viscous fluids. Math. Meth. Appl. Sci. 29, 1883–1906 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Naumann, J., Wolf, J.: Existence of weak solutions to the equations of natural convection with dissipative heating. Advances in Mathehmatical Fluid Mechanics, pp 367–384. Springer-Verlag Berlin Heidelberg (2010)

  17. Bulíček, M., Feireisl, E., Málek, J.: A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlinear Anal. Real World Appl. 10, 992–1015 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Majda, A.: Compressible Fluid Flow and System of Conservation Laws in Several Space Variables. Applied Mathematical Sciences, vol. 53. Springer-Verlag, New York (1984)

    Book  Google Scholar 

  19. Bulíček, M., Kaplický, P., Málek, J.: An L 2-maximal regularity result for the evolutionary Stokes-Fourier system. Appl. Anal. 90, 31–45 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bulíček, M., Málek, J., Shilkin, T.N.: On the regularity of two-dimensional unsteady flows of heat-conducting generalized newtonian fluids. Nonlinear Anal. 19, 89–104 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chae, M., Kang, K., Lee, J.: Existence of smooth solutions to Coupled Chemotaxis-Fluid equations. Discrete Cont. Dyn. Syst. A 33(6), 2271–2297 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equaitons. Arch. Rational Mech. Anal. 9, 187–195 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Majda, A.: Compressible Fluid Flow and System of Conservation Laws in Several Space Variables. Applied Mathematical Sciences, vol. 53. Springer-Verlag, New York (1984)

    Book  Google Scholar 

  24. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, PR, China

    Xia Ye

  2. Department of Mathematics, Jiaxing University, Jiaxing, 314001, PR, China

    Mingxuan Zhu

Authors
  1. Xia Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingxuan Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingxuan Zhu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, X., Zhu, M. Global Well-Posedness for the Heat-conductive Incompressible Viscous Fluids. Math Phys Anal Geom 19, 19 (2016). https://doi.org/10.1007/s11040-016-9224-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-016-9224-y

Keywords

Search

Navigation