Global Weak Solutions for the Cauchy Problem to One-Dimensional Heat-Conductive MHD Equations of Viscous Non-resistive Gas

  • Yang LiEmail author
  • Lingyu Jiang


Magnetohydrodynamics is concerned with the mutual interactions between moving, electrically conductive fluids and the magnetic field. It is widely applied in astrophysics, thermonuclear reactions and industry. The Cauchy problem to one-dimensional heat-conductive magnetohydrodynamic equations of viscous non-resistive gas is considered under the framework of Lagrangian coordinates. Based on the crucial upper and lower bounds of the density, we first obtain global well-posedness of strong solutions with regular initial data. Then existence of global weak solutions is established via the compactness analysis and the method of weak convergence. Stability of weak solutions is also verified by making full use of the specific mathematical structure of the equations. All results are obtained without any restriction to the size of the initial data.


Non-resistive MHD equations Global weak solutions Cauchy problem 

Mathematics Subject Classification

35M10 35D30 35K55 



The research of Y. Li is supported by NSF of China under grant number 11571167 and Postgraduate Research and Practice Innovation Program of Jiangsu Province under grant number KYCX 18-0028. The authors thank Professor Y. Sun for helpful discussions. In addition, the authors are indebted to the anonymous reviewers for providing valuable comments on the manuscript.


  1. 1.
    Alfvén, H.: Existence of electromagnetic-hydrodynamic waves. Nature 150, 405–406 (1942) CrossRefGoogle Scholar
  2. 2.
    Amosov, A.A., Zlotnik, A.A.: Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas. Sov. Math. Dokl. 38, 1–5 (1989) zbMATHGoogle Scholar
  3. 3.
    Amosov, A.A., Zlotnik, A.A.: Solvability ‘in the large’ of a system of equations of the one-dimensional motion of an inhomogeneous viscous heat-conducting gas. Math. Notes 52, 753–763 (1992) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. North-Holland, Amsterdam (1990) zbMATHGoogle Scholar
  5. 5.
    Cabannes, H.: Theoretical Magnetofluiddynamics. Academic Press, New York (1970) Google Scholar
  6. 6.
    Chen, G., Wang, D.: Global solutions of nonlinear magnetohydrodynamics with large initial data. J. Differ. Equ. 182, 344–376 (2002) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, G., Wang, D.: Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z. Angew. Math. Phys. 54, 608–632 (2003) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fan, J., Hu, Y.: Global strong solutions to the 1-D compressible magnetohydrodynamic equations with zero resistivity. J. Math. Phys. 56, 023101 (2015) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fan, J., Jiang, S., Nakamura, G.: Stability of weak solutions to the equations of magnetohydrodynamics with Lebesgue initial data. J. Differ. Equ. 251, 2025–2036 (2011) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Iskenderova, D.A., Smagulov, Sh.S.: The Cauchy problem for the equations of a viscous heat-conducting gas with degenerate density. Comput. Math. Phys. 33, 1109–1117 (1993) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jiang, S., Zhang, P.: Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas. Q. Appl. Math. 61, 435–449 (2003) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jiang, S., Zhang, J.: On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics. Nonlinearity 30, 3587–3612 (2017) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jiang, S., Zlotnik, A.: Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data. Proc. R. Soc. Edinb. 134A, 939–960 (2004) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kazhikhov, A.V., Smagulov, Sh.S.: Well-posedness and approximation methods for a model of magnetohydrodynamics. Izv. Akad. Nauk. Kaz. SSR Ser. Fiz. Mat. 5, 17–19 (1986) Google Scholar
  15. 15.
    Li, Y.: Global strong solutions to the one-dimensional heat-conductive model for planar non-resistive magnetohydrodynamics with large data. Z. Angew. Math. Phys. 69, 78 (2018) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li, Y., Sun, Y.: Global weak solutions and long time behavior for 1D compressible MHD equations without resistivity (2017). arXiv:1710.08248
  17. 17.
    Lin, F., Xu, L., Zhang, P.: Global small solutions of 2-D incompressible MHD system. J. Differ. Equ. 259, 5440–5485 (2015) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 2, Compressible Models. Clarendon, Oxford (1998) zbMATHGoogle Scholar
  19. 19.
    Nash, J.: Le probléme de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. Fr. 90, 487–497 (1962) CrossRefGoogle Scholar
  20. 20.
    Smagulov, S.S., Durmagambetov, A.A., Iskenderova, D.A.: The Cauchy problem for the equations of magneto-gas dynamics. Differ. Equ. 29, 337–348 (1993) zbMATHGoogle Scholar
  21. 21.
    Wu, J., Wu, Y.: Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion. Adv. Math. 310, 759–888 (2017) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Xu, L., Zhang, P.: Global small solutions to three-dimensional incompressible magnetohydrodynamical system. SIAM J. Math. Anal. 47, 26–65 (2015) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yu, H.: Global classical large solutions with vacuum to 1D compressible MHD with zero resistivity. Acta Appl. Math. 128, 193–209 (2013) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang, J., Xie, F.: Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics. J. Differ. Equ. 245, 1853–1882 (2008) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhang, J., Zhao, X.: On the global solvability and the non-resistive limit of the one-dimensional compressible heat-conductive MHD equations. J. Math. Phys. 58, 031504 (2017) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zlotnik, A.A., Amosov, A.A.: On stability of generalized solutions to the equations of one-dimensional motion of a viscous heat-conducting gas. J. Sib. Math. 38, 663–684 (1997) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zlotnik, A.A., Amosov, A.A.: Stability of generalized solutions to equations of one-dimensional motion of viscous heat-conducting gas. Math. Notes 63, 736–746 (1998) MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Department of MathematicsCentral University of Finances and EconomicsBeijingChina

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