Skip to main content
Log in

A Regularity Criterion for the 2D Full Compressible MHD Equations with Zero Heat Conductivity

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper we establish a regularity criterion for the 2D full compressible MHD equations with zero heat conductivity and initial vacuum in a bounded domain.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Brézis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Partial Differential Equations 5, 773–789 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Duan, Q.: Blowup of smooth solution for non-isentropic magnetohydrodynamic equations without heat conductivity. Math. Methods Appl. Sci. 40(6), 1865–1879 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ducomet, B., Feireisl, E.: The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars. Comm. Math. Phys. 266, 595–629 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fan, J., Yu, W.: Global variational solutions to the compressible magnetohydrodynamic equations. Nonlinear Anal. 69, 3637–3660 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fan, J., Yu, W.: Strong solution to the compressible magnetohydrodynamic equations with vacuum. Nonlinear Anal. RWA 10, 392–409 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan, J., Li, F., Nakamura, G., Tan, Z.: Regularity criteria for the three-dimensional magnetohydrodynamic equations. J. Differential Equations 256(8), 2858–2875 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, J., Li, F., Nakamura, G.: A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity. Discrete Contin. Dyn. Syst. Ser. B 23(4), 1757–1766 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Gao, Z., Tan, Z., Wu, G.: Global existence and convergence rates of smooth solutions for the 3-D compressible magnetohydrodynamic equations without heat conductivity. Acta Math. Sci. Ser. B (Engl. Ed.) 34(1), 93–106 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Hu, X., Wang, D.: Global solutions to the three-dimensional full compressible magnetohydrodynamic flows. Comm. Math. Phys. 283, 255–284 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, X., Li, J.: Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows. Commun. Math. Phys. 324, 147–171 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, X., Wang, Y.: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differential Equations 254, 511–527 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, X., Wang, Y.: \(L^{\infty }\) continuation principle to the non-baratropic non-resistive magnetohydrodynamic equations without heat conductivity. Math. Methods Appl. Sci. 39(14), 4234–4245 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Huang, T., Wang, C., Wen, H.: Strong solutions of the compressible nematic liquid crystal flow. J. Differential Equations 252, 2222–2265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiang, F.: Nonlinear thermal instability in compressible viscous flows without heat conductivity. J. Math. Fluid Mech. 20(4), 1509–1539 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, S., Ju, Q., Li, F.: Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations. Nonlinearity 25, 1351–1365 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jiang, S., Ju, Q., Li, F., Xin, Z.: Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data. Adv. Math. 259, 384–420 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  18. Liu, T.-P., Zeng, Y.: Compressible Navier-Stokes equations with zero heat conductivity. J. Differ. Equ. 153, 225–291 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pu, X., Guo, B.: Global existence and convergence rates of smooth solutions for the full compressible MHD equations. Z. Angew. Math. Phys. 64, 519–538 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sun, Y., Wang, C., Zhang, Z.: A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. J. Math. Pures Appl. 95, 36–47 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tan, Z., Xu, Q., Wang, H.: Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete Contin. Dyn. Syst. 35(10), 5083–5105 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vincenti, W., Kruger, C.: Introduction to Physical Gas Dynamics. Wiley, New York (1965)

    Google Scholar 

  23. Von Wahl, W.: Estimating \(\nabla u\) by \(\mathrm {div}\,u\) and \(\mathrm {curl}\,u\). Math. Methods Appl. Sci. 15, 123–143 (1992)

    MathSciNet  MATH  Google Scholar 

  24. Xi, X., Guo, B., Xie, B., Fang, S.: Nonlinear thermal instability in the magnetohydrodynamics problem without heat conductivity. J. Differ. Equ. 263(10), 6635–6683 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhong, X.: Singularity formation to the two-dimensional full compressible Navier-Stokes equations with zero heat conduction in a bounded domain (2018). arXiv:1810.01265v1

Download references

Acknowledgements

The authors are very grateful to the referees for their helpful suggestions, which improved the earlier version of this paper. This work is supported by NSFC (Grant No. 11671193) and the Fundamental Research Funds for the Central Universities (Grant No. NS2012122).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiuhui Yang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, X. A Regularity Criterion for the 2D Full Compressible MHD Equations with Zero Heat Conductivity. Acta Appl Math 169, 523–531 (2020). https://doi.org/10.1007/s10440-020-00309-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-020-00309-x

Keywords

Mathematics Subject Classification (2000)

Navigation