Asymptotic stability of viscous shock profiles for compressible planar magnetohydrodynamics system

Abstract

This paper shows time-asymptotic nonlinear stability and existence of the viscous shock profiles to the Cauchy problem of the one-dimensional compressible planar magnetohydrodynamics system, which describes the motions of a conducting fluid in an electro-magnetohydrodynamics system. We can prove that the solutions to the compressible planar magnetohydrodynamics system tend time-asymptotically to the viscous shock profiles provided that the initial disturbance is small and of integral zero.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Chen, G.Q., Wang, D.H.: Global solutions of nonlinear magnetohydrodynamics with large initial data. J. Differ. Equ. 182, 344–376 (2002)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chen, G.Q., Wang, D.H.: Existence and continuous dependence of large solutions for the magnetohydrodynamics equations. Z. Angew. Math. Phys. 54, 608–632 (2003)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chen, Q., Tan, Z.: Global existence and convergence rates of smooth solutions for the compressible magnetohydronamics equations. Nonlinear Anal. 72, 4438–4451 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Fan, J.S., Jiang, S., Nakamura, G.: Vanishing shear viscosity limit in the magnetohydrodynamics equations. Commun. Math. Phys. 270, 691–708 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Gilbarg, D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73, 256–274 (1951)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Ration. Mech. Anal. 95, 325–344 (1986)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Huang, F.M., Li, J., Matsumura, A.: Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system. Arch. Ration. Mech. Anal. 197, 89–116 (2010)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Huang, F.M., Matsumura, A., Xin, Z.P.: Stability of contact discontinuities for the 1-D compressible Navier–Stokes equations. Arch. Ration. Mech. Anal. 179, 55–77 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Huang, F.M., Xin, Z.P., Yang, T.: Contact discontinuity with general perturbations for gas motions. Adv. Math. 219, 1246–1297 (2008)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kawashima, S., Matsumura, A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. math. Phys. 101, 97–127 (1985)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Li, H.L., Xu, X.Y., Zhang, J.W.: Global classical solutions to three dimensional compressible magnetohydrodynamics equations with large oscillations and vaccum. SIAM J. Math. Anal. 45, 1356–1387 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Liu, T.P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. 56, 1–108 (1985)

    MathSciNet  Google Scholar 

  13. 13.

    Liu, T.P.: Shock waves for compressible Navier–Stokes equations are stable. Commun. Pure Appl. Math. 39, 565–594 (1986)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Liu, T.P., Xin, Z.P.: Nonlinear stability of rarefaction waves for compressible Navier–Stokes equations. Commun. Math. Phys. 118, 451–465 (1988)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Matsumura, A., Nishihara, K.: Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math. 3, 1–13 (1986)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Matsumura, A., Nishihara, K.: On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math. 2, 17–25 (1985)

    Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

    Smoller, J.: Shock Waves and Reaction–Diffusion Equations. Springer, New York (1983)

    Google Scholar 

  19. 19.

    Wang, D.H.: Large solutions to the initial boundary value problem for planar magnetohydrodynamics. SIAM J. Appl. Math. 63, 1424–1441 (2003)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ye, X., Zhang, J.W.: On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit. J. Differ. Equ. 260, 3927–3961 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Yin, H.Y.: The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinet. Relat. Models 10, 1235–1253 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Yin, H.Y.: Stability of stationary solutions for inflow problem on the planar magnetohydrodynamics. J. Math. Phys. 59, 023101 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Yin, H.Y.: Stability of composite wave for inflow problem on the planar magnetohydrodynamics. Nonlinear Anal. Real World Appl. 44, 305–333 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Yin, H.Y.: Converge rates towards stationary solutions for the outflow problem of planar magnetohydrodynamics on a half line. Proc. R. Soc. Edinb. Sect. A. 5, 1291–1322 (2019)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors were supported by the National Natural Science Foundation of China (Grant Nos. 12071163 and 11601165) and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant No. ZQN-PY602).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Haiyan Yin.

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

Verify currency and authenticity via CrossMark

Cite this article

Ding, Q., Yin, H. Asymptotic stability of viscous shock profiles for compressible planar magnetohydrodynamics system. Z. Angew. Math. Phys. 72, 11 (2021). https://doi.org/10.1007/s00033-020-01431-4

Download citation

Keywords

  • Planar magnetohydrodynamics system
  • Shock profiles
  • Asymptotic stability

Mathematics Subject Classification

  • 76W05
  • 35B40
  • 74J40
  • 35B35