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Global well-posedness for the 2D non-resistive MHD equations in two kinds of periodic domains

  • Qionglei Chen
  • Xiaoxia RenEmail author
Article
  • 36 Downloads

Abstract

In this paper, we obtain the global well-posedness of non-resistive 2D MHD problem in two kinds of periodic domains: \(\mathbb T\times \mathbb R\) and \(\mathbb T\times (0,1)\), without any assumption on initial data. What’s more, we find that the periodic domains have similar effects with domains with physical boundaries, although Poincaré’s inequality does not hold in the zero mode.

Keywords

MHD equations Periodic domains Anisotropic estimates 

Mathematics Subject Classification

35A01 35Q35 76W05 

Notes

Acknowledgements

Q. Chen is partly supported by the National Science Foundation of China under Grant No11671045. X. Ren is supported by National Postdoctoral Program for Innovative Talents of China under Grant BX201700039.

References

  1. 1.
    Abidi, H., Zhang, P.: On the global solution of a 3-D MHD system with initial data near equilibrium. Commun. Pure Appl. Math. 70, 1509–1561 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cabannes, H.: Theoretical Magneto-Fluid Dynamics. Academic Press, New York (1970)Google Scholar
  3. 3.
    Califano, F., Chiuderi, C.: Resistivity-independent dissipation of magnetrodydrodynamic waves in an inhomogeneous plasma. Phys. Rev. E 60(part B), 4701–4707 (1999)CrossRefGoogle Scholar
  4. 4.
    Chemin, J., McCormick, D., Robinson, J., Rodrigo, J.: Local existence for the non-resistive MHD equations in Besov spaces. arXiv:1503.01651
  5. 5.
    Deng, W., Zhang, P.: Large time behavior of solutions to 3-D MHD system with initial data near equilibrium. arXiv:1702.05260
  6. 6.
    Fefferman, C., McCormick, D., Robinson, J., Rodrigo, J.: Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267, 1035–1056 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu, X., Lin, F.: Global existence for two dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity. arXiv:1405.0082v1
  8. 8.
    Lei, Z.: On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Differ. Equ. 259, 3202–3215 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lin, F., Xu, L., Zhang, P.: Global small solutions of 2-D incompressible MHD system. J. Differ. Equ. 259, 5440–5485 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lin, F., Zhang, P.: Global small solutions to MHD type system (I): 3-D case. Commun. Pure. Appl. Math. 67, 531–580 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lin, F., Zhang, T.: Global small solutions to a complex fluid model in 3D. Arch. Ration. Mech. Anal. 216, 905–920 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pan, R., Zhou, Y., Zhu, Y.: Global classical solutions of 3D viscous MHD system without magnetic diffusion on periodic boxes. Arch. Ration. Mech. Anal. 227, 637–662 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Robinson, C., Rodrigo, L., Sadowski, W.: The Three-Dimensional Navier–Stokes Equations. Cambridge University Press, United Kingdom (2016)CrossRefGoogle Scholar
  14. 14.
    Ren, X., Wu, J., Xiang, Z., Zhang, Z.: Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J. Funct. Anal. 267, 503–541 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ren, X., Xiang, Z., Zhang, Z.: Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain. Nonlinearity 29, 1257–1291 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ren, X., Xiang, Z., Zhang, Z.: Decay of smooth solution for the 3D MHD-type equations without magnetic diffusion. Sci. China Math. 59, 1949–1974 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sohr, H.: The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhauser, Boston (2001)zbMATHGoogle Scholar
  18. 18.
    Tan, Z., Wang, Y.: Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems. SIAM J. Math. Anal. 50, 1432–1470 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wei, D., Zhang, Z.: Transition threshold for the 3D Couette flow in Sobolev space. arXiv:1803.01359
  20. 20.
    Wan, R.: Decay estimate of solutions to the 2D non-resistive incompressible MHD equations. arXiv:1806.11295
  21. 21.
    Xu, L., Zhang, P.: Global small solutions to three dimensional incompressible MHD system. SIAM J. Math. Anal. 47, 26–65 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, T.: An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system. arXiv:1404.3081
  23. 23.
    Zhang, T.: Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field. J. Differ. Equ. 260, 5450–5480 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhai, C., Zhang, T.: Global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system with non-equilibrium background magnetic field. J. Differ. Equ. 261, 3519–3550 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingPeople’s Republic of China

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