Stability of shock solutions to piston problem for the magnetohydrodynamics

Abstract

This article is devoted to studying one-dimensional piston problem for the compressible Magnetohydrodynamics when the piston is pushed forward relatively. If the initial data and the piston speed are both given some small \(\mathrm {BV}\) perturbations, then global stability of large shock waves to this problem has been established, and the long time behavior has been also considered.

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References

  1. 1.

    Amadori, D.: Initial boundary value problem for nonlinear systems of conservation laws. NoDEA 4, 1–42 (1997)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bressan, A.: Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford University Press, New York (2000)

    Google Scholar 

  3. 3.

    Cabannes, H.: Theoretical Magnetofluid Dynamics. Applied Math and Mechanics series, vol. 13. Academic Press, New York (1970)

    Google Scholar 

  4. 4.

    Chen, G.Q., Chen, S., Wang, D., Wang, Z.: A multidimensional piston problem for the Euler equations for compressible flow. Dis. Cont. Dyn. Sys. 12, 361–383 (2005)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chen, G.-Q., Wang, Y.-G.: Existence and stability of compressible current-vortex sheets in three-dimensional Magnetohydrodynamics. Arch. Ration. Mech. Anal. 187, 369–408 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chen, S.: A singular multidimensional piston problem in compressible flow. J. Differ. Equ. 189, 292–317 (2003)

    Article  Google Scholar 

  7. 7.

    Chen, S., Wang, Z., Zhang, Y.: Global existence of shock front solutions for the axially symmetric piston problem for compressible fluids. J. Hyper. Diff. Eqns. 1, 51–84 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chen, S., Wang, Z., Zhang, Y.: Global existence of shock front solution to axially symmetric piston problem in compressible flow. Z. Angew. Math. Phys. 59, 434C456 (2008)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Applied Mathematical Sciences, vol. 21. Springer, New York (1976)

    Google Scholar 

  10. 10.

    Cowling, T.G.: Universe Magnetogasdynamics. Science Press, Beijing (1987)

    Google Scholar 

  11. 11.

    Ding, M.: Existence and stability of rarefaction wave to 1-D piston problem for the relativistic full Euler equations. J. Differ. Eqs. 262, 6068–6108 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ding, M., Li, Y.: Global existence and non-relativistic global limits of entropy solutions to the 1-D piston problem for the isentropic relativistic Euler equations. J. Math. Phys. 54(031506), 28 (2013)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Ding, M., Li, Y.: An overview of piston problems in fluid dynamics, Hyperbolic Conservation Laws and Related Analysis with Applications. In: Springer Proc. Math. Stat., 49, Springer, Heidelberg, pp. 161–191 (2014)

  14. 14.

    Ding, M., Li, Y.: Stability and Non-relativistic limits of rarefaction wave to the 1-D piston problem for the relativistic Euler equations. Z. Angew. Math. Phys. 68, Art. 43, 32. (2017)

  15. 15.

    Ding, M., Yuan, H.R.: Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Dis. Contin. Dyn. Syst. 38, 2911C2943 (2018)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Ding, M., Kuang, J., Zhang, Y.: Global stability of rarefaction wave to the 1-D piston problem for the compressible full Euler equations. J. Math. Anal. Appl. 448, 1228–1264 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Glimm, J., Lax, P.D.: Decay of Solutions of Systems of Hyperbolic Conservation Laws. Memoirs of the American Mathematical Society, 101. American Mathematical Society, Providence, RI (1970)

  18. 18.

    Hu, Y., Sheng, W.: The Riemann problem of conservation laws in Magnetogasdynamics. Commun. Pure Appl. Anal. 2, 755–769 (2013)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    KuLikovskiy, A.G., Lyubimov, G.A.: Magnetohydrodynamics. Addision-Wesley, Reading, NY (1965)

    Google Scholar 

  20. 20.

    Lax, P.: Hyperbolic system of conservation laws, II. Comm. Pure. Appl. Math. 10, 537–566 (1957)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Li, T.-T., Qin, T.: Physics and Parital Differential Equations, Translated by Yachun Li, Volume II, Higher Education Press, Beijing, (2014)

  22. 22.

    Liu, T.P.: Large time behavior of initial and initial-boundary value problems of a general system of hyperbolic conservation laws. Commun. Math. Phys. 55, 163–177 (1977)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Liu, T.P.: The free piston problem for gas dynamics. J. Differ. Eqns. 30, 175–191 (1978)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Liu, Y., Sun, W.: Riemann problem and wave interactions in Magnetogasdynamics. J. Nath. Anal. Appl. 397, 454–466 (2013)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Sekhar, T.R., Sharma, V.D.: Riemann problem and elementary wave interactions in isentropic magnetogasdynamics. Nonlinear Anal. Real World Appl. 11, 619–639 (2010)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Smoller, J.: Shock Waves Reaction-Diffusion Equations. Spring, New York (1983)

    Google Scholar 

  27. 27.

    Wang, Z.: Global existence of shock front solution to 1-dimensional piston problem. Chin. Ann. Math. Ser. A 26, 549–560 (2005). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Wang, Z.: Local existence of shock front solution to the axi-symmetrical piston problem in compressible flow. Acta. Math. Sini., Engl. Ser. 20, 589–604 (2004)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The research of Min Ding was supported in part by the National Natural Science Foundation of China under Grant Nos. 11701435 and 11626176, and in part by “the Fundamental Research Funds for the Central Universities(WUT: 2018IVB013 and 2018IB015)”.

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Correspondence to Min Ding.

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Ding, M. Stability of shock solutions to piston problem for the magnetohydrodynamics. Z. Angew. Math. Phys. 72, 59 (2021). https://doi.org/10.1007/s00033-021-01490-1

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Keywords

  • Shock waves
  • Magnetohydrodynamics
  • Wave interaction
  • Long time behavior

Mathematics Subject Classification

  • 35B35
  • 35L67
  • 76L05
  • 76W05
  • 35B40