Advertisement

Communications in Mathematical Physics

, Volume 297, Issue 2, pp 371–400 | Cite as

Incompressible Limit of the Compressible Magnetohydrodynamic Equations with Periodic Boundary Conditions

  • Song Jiang
  • Qiangchang Ju
  • Fucai Li
Article

Abstract

This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the case of the well-prepared initial data.

Keywords

Weak Solution Strong Solution Energy Inequality Global Weak Solution Magnetic Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brenier Y.: Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. PDE 25, 737–754 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Danchin R.: Zero Mach number limit for compressible flows with periodic boundary conditions. Amer. J. Math. 124, 1153–1219 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Desjardins B., Grenier E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455, 2271–2279 (1999)zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Duvaut G., Lions J.L.: Inéquations en thermoélasticité et magnéto-hydrodynamique. Arch. Rat. Mech. Anal. 46, 241–279 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fan J., Yu W.: Strong solution to the compressible MHD equations with vacuum. Nonlinear Anal. Real World Appl. 10, 392–409 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fan J., Yu W.: Global variational solutions to the compressible magnetohydrodynamic equations. Nonlinear Anal. 69, 3637–3660 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gallagher I.: A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations. J. Math. Kyoto Univ. 40, 525–540 (2001)MathSciNetGoogle Scholar
  8. 8.
    Hoff D.: The zero-Mach limit of compressible flows. Commun. Math. Phys. 192, 543–554 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Hu, X., Wang, D.: Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Rat. Mech. Anal., to appear, http://arXiv.org/abs/0904.3587v1[math.AP], 2009
  10. 10.
    Hu X., Wang D.: Global solutions to the three-dimensional full compressible magnetohydrodynamic flows. Commun. Math. Phys. 283, 255–284 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Hu X., Wang D.: Low Mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J. Math. Anal. 41, 1272–1294 (2009)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Jiang, S., Ju, Q.C., Li, F.C.: Combined incompressible and inviscid limit of the compressible magnetohydrodynamic equations in the whole space. http://arXiv.org/abs/0905.3937v1[math.AP], 2009
  13. 13.
    Kawashima S.: Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics. Japan J. Appl. Math. 1, 207–222 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kulikovskiy A.G., Lyubimov G.A.: Magnetohydrodynamics. Addison-Wesley, Reading, MA (1965)Google Scholar
  15. 15.
    Laudau L.D., Lifshitz E.M.: Electrodynamics of continuous media. 2nd ed. Pergamon, New York (1984)Google Scholar
  16. 16.
    Lei Z., Zhou Y.: BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin. Dyn. Syst. 25, 575–583 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Li, F.C., Yu, H.J.: Optimal Decay Rate of Classical Solution to the Compressible Magnetohydrodynamic Equations. Preprint, 2009Google Scholar
  18. 18.
    Lions P.L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Masmoudi N.: Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 199–224 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Masmoudi, N.: Examples of singular limits in hydrodynamics, In: Evolutionary Equations, Vol. III, Handbook of Differential Equations. Amsterdam: North-Holland, 2007, pp. 275–375Google Scholar
  21. 21.
    Polovin R.V., Demutskii V.P.: Fundamentals of Magnetohydrodynamics. Consultants, Bureau, New York (1990)Google Scholar
  22. 22.
    Schochet S.: Fast singular limits of hyperbolic PDEs. J. Diff. Eqns. 114, 476–512 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635–664 (1983)zbMATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Vol’pert A.I., Khudiaev S.I.: On the Cauchy problem for composite systems of nonlinear equations. Mat. Sbornik 87, 504–528 (1972)Google Scholar
  25. 25.
    Zhang J.W., Jiang S., Xie F.: Global weak solutions of an initial boundary value problem for screw pinches in plasma physics. Math. Models Methods Appl. Sci. 19, 833–875 (2009)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.LCP, Institute of Applied Physics and Computational MathematicsBeijingP.R. China
  2. 2.Institute of Applied Physics and Computational MathematicsBeijingP.R. China
  3. 3.Department of MathematicsNanjing UniversityNanjingP.R. China

Personalised recommendations