Global Solutions to the Three-Dimensional Full Compressible Magnetohydrodynamic Flows

  • Xianpeng Hu
  • Dehua Wang


The equations of the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows are considered in a bounded domain. The viscosity coefficients and heat conductivity can depend on the temperature. A solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global variational weak solution to the three-dimensional full magnetohydrodynamic equations with large data is established.


Global Solution Global Existence Global Weak Solution Magnetic Diffusivity Renormalize Solution 
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  1. 1.
    Admasm, R.A.: Sobolev spaces, Pure and Applied Mathematics, Vol. 65. New York-London: Academic Press, (1975)Google Scholar
  2. 2.
    Cabannes, H.: Theoretical Magnetofluiddynamics. New York: Academic Press, 1970Google Scholar
  3. 3.
    Chen, G.-Q., Wang, D.: Global solution of nonlinear magnetohydrodynamics with large initial data. J. Differ. Eqs. 182, 344–376 (2002)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chen, G.-Q., Wang, D.: Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z. Angew. Math. Phys. 54, 608–632 (2003)zbMATHGoogle Scholar
  5. 5.
    DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Ducomet, B., Feireisl, E.: The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars. Commun. Math. Phys. 226, 595–629 (2006)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Fan, J., Jiang, S., Nakamura, G.: Vanishing shear viscosity limit in the magnetohydrodynamic equations. Commun. Math. Phys. 270, 691–708 (2007)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Feireisl, E.: On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53, 1707–1740 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Feireisl, E.: Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications, 26. Oxford: Oxford University Press, 2004Google Scholar
  10. 10.
    Freistühler, H., Szmolyan, P.: Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves. SIAM J. Math. Anal. 26, 112–128 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gerebeau, J.F., Bris, C.L., Lelievre, T.: Mathematical methods for the magnetohydrodynamics of liquid metals. Oxford: Oxford University Press, 2006Google Scholar
  12. 12.
    Goedbloed, H., Poedts, S.: Principles of magnetohydrodynamics with applications to laboratory and astrophysical plasmas. Cambridge: Cambridge University Press, 2004Google Scholar
  13. 13.
    Hoff, D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rat. Mech. Anal. 132, 1–14 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hoff, D.: Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids. Arch. Rat. Mech. Anal. 139, 303–354 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hoff, D., Tsyganov, E.: Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z. Angew. Math. Phys. 56, 791–804 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hu, X., Wang, D.: Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Submitted for publicationGoogle Scholar
  17. 17.
    Kazhikhov, V., Shelukhin, V.V.: Unique global solution with respect to time of initial-boundary-value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kawashima, S., Okada, M.: Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A Math. Sci. 58, 384–387 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kulikovskiy, A.G., Lyubimov, G.A.: Magnetohydrodynamics. Reading, MA: Addison-Wesley, 1965Google Scholar
  20. 20.
    Laudau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 2nd ed., New York: Pergamon, 1984Google Scholar
  21. 21.
    Lions, P.L.: Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press, 1996Google Scholar
  22. 22.
    Lions, P.L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press, 1998Google Scholar
  23. 23.
    Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Memoirs Amer. Math. Soc. 599, 1997Google Scholar
  24. 24.
    Novotný, A., Straškraba, I.: Introduction to the theory of compressible flow. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  25. 25.
    Wang, D.: Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM J. Appl. Math. 63, 1424–1441 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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