Skip to main content
SpringerLink
Log in

Time-periodic solutions for 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

The 2D magnetohydrodynamics equations with horizontal dissipation and horizontal magnetic diffusion are considered. The classical solution in \(H^k\) \((k\ge 2)\) has been obtained; due to partial dissipation and strong nonlinearity, the global well-posedness of weak solution in \(H^1\) is still unknown. In this paper, by combining classic Galerkin’s method with Brouwer’s fixed point theorem, existence of time-periodic solution in \(H^1\) with small initial values is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Cai, H., Tan, Z.: Time periodic solutions to the three dimensional equations of compressible magnetohydrodynamic flows. Discrete Contin. Dyn. Syst. 36, 1847–1868 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cai, H., Tan, Z.: Periodic solutions to the compressible magnetohydrodynamic equations in a periodic domain. J. Math. Anal. Appl. 426, 172–193 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cao, C., Regmi, D., Wu, J.: The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J. Differ. Equ. 254, 2661–2681 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, C., Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226, 1803–1822 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, C., Wu, J., Yuan, B.: The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion. SIAM J. Math. Anal. 46, 588–602 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chemin, J., McCormick, D., Robinson, J., Rodrigo, J.: Local existence for the non-resistive MHD equations in Besov spaces. Adv. Math. 286, 1–31 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dong, B., Jia, Y., Li, J., Wu, J.: Global regularity and time decay for the 2D magnetohydrodynamic equations with fractional dissipation and partial magnetic diffusion. J. Math. Fluid Mech. 20, 1541–1565 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dragomir, S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers Inc, Hauppauge (2003)

    MATH  Google Scholar 

  9. Du, L., Zhou, D.: Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion. SIAM J. Math. Anal. 47, 1562–1589 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duvaut, G., Lions, J.L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)

    Article  MATH  Google Scholar 

  11. Fan, J., Ozawa, T.: Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. Kinet. Relat. Models 7, 45–56 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Galdi, P., Hieber, M., Kashiwabara, T.: Strong time-periodic solutions to the 3D primitive equations subject to arbitrary large forces. Nonlinearity 30, 3979–3992 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, C., Liu, M., Guo, X.: Global smooth solutions of the 2D MHD equations for a class of large data without magnetic diffusion. Appl. Math. Lett. 78, 59–64 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hussein, A., Saal, M., Wrona, M.: Primitive equations with horizontal viscosity: the initial value and the time-periodic problem for physical boundary conditions. Discrete Contin. Dyn. Syst. 41, 3063–3092 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiu, Q., Zhao, J.: Global regularity of 2D generalized MHD equations with magnetic diffusion. Z. Angew. Math. Phys. 66, 677–687 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, Y.: Global well-posedness for the three-dimensional full compressible viscous non-resistive MHD system. J. Math. Fluid Mech. 24, 1–24 (2022)

    Article  MathSciNet  Google Scholar 

  18. Morimoto, H.: Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete Contin. Dyn. Syst. Ser. S 5, 631–639 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Paicu, M., Zhu, N.: Global regularity for the 2D MHD and tropical climate model with horizontal dissipation. J. Nonlinear Sci. 31, 1–39 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Serrin, J.: A note on the existence of periodic solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 3, 120–122 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tan, Z., Wang, H.: Time periodic solutions of compressible magnetohydrodynamic equations. Nonlinear Anal. 76, 153–164 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, J., Zhu, Y.: Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium. Adv. Math. 377, 107466 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wan, R.: On the uniqueness for the 2D MHD equations without magnetic diffusion. Nonlinear Anal. Real World Appl. 30, 32–40 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xiao, Y., Yuan, B.: Global existence and large time behavior of solutions to 3D MHD system near equilibrium. Result Math. 76, 1–14 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ye, Z.: Note on global regularity of 2D regularized MHD equations with zero magnetic diffusion. J. Math. Anal. Appl. 468, 1090–1097 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, L., Li, S.: A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion. Nonlinear Anal. Real World Appl. 21, 197–206 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhou, Y., Fan, J.: A regularity criterion for the 2D MHD system with zero magnetic diffusivity. J. Math. Anal. Appl. 378, 169–172 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the editors and referees for their very valuable suggestions and comments.

Funding

This study was supported partially by a China NSF Grant Nos. 11701269, 11801270, 11901342, 12271293.

Author information

Authors and Affiliations

  1. School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, People’s Republic of China

    Chengfeng Sun & Fang Zhang

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Hui Liu

  3. College of Sciences, Nanjing Agricultural University, Nanjing, 210095, People’s Republic of China

    Qingkun Xiao

Authors
  1. Chengfeng Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qingkun Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

CS contributed to conceptualization, methodology, and writing—original draft; FZ contributed to investigation; HL contributed to investigation and writing—review and editing; QX contributed to investigation and validation.

Corresponding author

Correspondence to Hui Liu.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

The authors have approved the manuscript submission and acknowledged that the submitted work is original and the content of the manuscript has not been published or submitted for publication elsewhere.

Competing interests

The authors declare no competing interests and Funding.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported partially by a China NSF Grant Nos. 11701269, 11801270, 11901342, 12271293.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, C., Zhang, F., Liu, H. et al. Time-periodic solutions for 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. Z. Angew. Math. Phys. 74, 42 (2023). https://doi.org/10.1007/s00033-022-01927-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01927-1

Keywords

Mathematics Subject Classification

Search

Navigation