Science China Mathematics

, Volume 62, Issue 11, pp 2229–2248 | Cite as

Small Alfvén number limit for incompressible magneto-hydrodynamics in a domain with boundaries

  • Song JiangEmail author
  • Qiangchang Ju
  • Xin Xu


for any fixed Alfvén number, the local well-posedness is proved for the equations of three-dimensional ideal incompressible magneto-hydrodynamics in a domain with boundaries. Under appropriate conditions, a smooth solution is shown to exist in a time interval independent of the Alfvén number, and the solutions of the original system tend to the solutions of a two-dimensional Euler flow coupled with a linear transport equation as the Alfvén number goes to zero.


MHD equations vanishing Alfvén number limit small Mach number uniform estimates 


34D15 35A01 



This work was supported by the Israel Science Foundation-National Natural Science Foundation of China Joint Research Program (Grant No. 11761141008). The first author was supported by the National Basic Research Program of China (Grant No. 2014CB745002) and National Natural Science Foundation of China (Grant No. 11631008). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11571046, 11471028 and 11671225) and Beijing Natural Science Foundation (Grant No. 1182004). The authors thank Professor Steve Schochet from Tel-Aviv University for helpful discussions.


  1. 1.
    Alazard T. Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions. Adv Differential Equations, 2005, 10: 19–44MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alekseev G V. Solvability of a homogeneous initial-boundary value problem for equations of magneto-hydrodynamics of an ideal fluid (in Russian). Dinamika Sploshn Sredy, 1982, 57: 3–20Google Scholar
  3. 3.
    Browning G, Kreiss H O. Problems with different time scales for nonlinear partial differential equations. SIAM J Appl Math, 1982, 42: 704–718MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen S. On the initial-boundary value problems for quasilinear symmetric hyperbolic system with characteristic boundary. Chin Ann Math Ser B, 1982, 3: 223–232MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cheng B, Ju Q, Schochet S. Three-scale singular limits of evolutionary PDEs. Arch Ration Mech Anal, 2018, 229: 601–625MathSciNetCrossRefGoogle Scholar
  6. 6.
    Goto S. Singular limits of the incompressible ideal magneto-fluid motion with respect to the Alfvén number. Hokkaido Math J, 1990, 19: 175–187MathSciNetCrossRefGoogle Scholar
  7. 7.
    He F, Fan J, Zhou Y. Local existence and blow-up criterion of the ideal density-dependent flows. Bound Value Probl, 2016, 2016: 101MathSciNetCrossRefGoogle Scholar
  8. 8.
    Isozaki H. Singular limits for the compressible Euler equation in an exterior domain. J Reine Angew Math, 1987, 381: 1–36MathSciNetzbMATHGoogle Scholar
  9. 9.
    Jiang S, Ju Q, Li F. Incompressible limit of the nonisentropic ideal magnetohydrodynamic equations. SIAM J Math Anal, 2006, 48: 302–319MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jiang S, Ju Q, Li F. Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Comm Math Phys, 2010, 297: 371–400MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jiang S, Ju Q, Li F. Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients. SIAM J Math Anal, 2010, 42: 2539–2553MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kawashima S, Yanagisawa T, Shizuta Y. Mixed problems for quasi-linear symmetric hyperbolic systems. Proc Japan Acad Ser A Math Sci, 1987, 63: 243–246MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klainerman S, Majda A. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm Pure Appl Math, 1981, 34: 481–524MathSciNetCrossRefGoogle Scholar
  14. 14.
    Klainerman S, Majda A. Compressible and incompressible fluids. Commu Pure Appl Math, 1982, 35: 629–651MathSciNetCrossRefGoogle Scholar
  15. 15.
    Klein R, Botta N, Schneider T, et al. Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J Engrg Math, 2001, 39: 261–343MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kukucka P. Singular limits of the equations of magnetohydrodynamics. J Math Fluid Mech, 2011, 13: 173–189MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kwon Y S, Trivisa K. On the incompressible limits for the full magnetohydrodynamics flows. J Differential Equations, 2011, 251: 1990–2023MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lax P D, Phillips R S. Local boundary conditions for dissipative symmetric linear differential operators. Comm Pure Appl Math, 1960, 13: 427–455MathSciNetCrossRefGoogle Scholar
  19. 19.
    Majda A, Osher S. Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary. Comm Pure Appl Math, 1975, 28: 607–675MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ohkubo T. Well posedness for quasi-linear hyperbolic mixed problems with characteristic boundary. Hokkaido Math J, 1989, 18: 79–123MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schmidt P G. On a magnetohydrodynamic problem of Euler type. J Differential Equations, 1988, 74: 318–335MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schochet S. The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit. Comm Math Phys, 1986, 104: 49–75MathSciNetCrossRefGoogle Scholar
  23. 23.
    Schochet S. Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation. J Differential Equations, 1986, 68: 400–428MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schochet S. Asymptotics for symmetric hyperbolic systems with a large parameter. J Differential Equations, 1988, 75: 1–27MathSciNetCrossRefGoogle Scholar
  25. 25.
    Secchi P. On the equations of ideal incompressible magnetohydrodynamics. Rend Semin Mat Univ Padova, 1993, 90: 103–119MathSciNetzbMATHGoogle Scholar
  26. 26.
    Secchi P. Well-Posedness of characteristic symmetric hyperbolic systems. Arch Ration Mech Anal, 1996, 134: 155–197MathSciNetCrossRefGoogle Scholar
  27. 27.
    Takayama M. Initial boundary value problem for the equations of ideal magneto-hydrodynamics in a half space. In: Mathematical Analysis in Fluid and Gas Dynamics, vol. 1322. Kyoto: Research Institute for Mathematical Sciences, Kyoto University, 2003, 79–84Google Scholar
  28. 28.
    Yanagisawa T. The initial boundary value problem for the equations of ideal magnetohydrodynamics. Hokkaido Math J, 1987, 16: 295–314MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yanagisawa T, Matsumura A. The fixed boundary value problems for the equations of ideal magneto-hydrodynamics with a perfectly conducting wall condition. Comm Math Phys, 1991, 136: 119–140MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina

Personalised recommendations