Skip to main content
SpringerLink
Log in

Regularity Criteria for the 3D Axisymmetric Non-resistive MHD System in the Swirl Component of the Vorticity

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper, the regularity criteria of the 3D axisymmetric non-resistive MHD system with nonzero swirl are studied. It is proved that if the swirl component of the vorticity belongs to some anisotropic Lorentz spaces or weighted Lebesgue spaces, then strong solutions of this system can be smoothly extended beyond the possible blow-up time T. In particular, no a priori assumptions on the magnetic field are imposed, and we also obtain a new result for the axisymmetric Navier–Stokes system.

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

References

  1. Ai, X., Li, Z.: Global smooth solutions to the 3D non-resistive MHD equations with low regularity axisymmetric data. Commun. Math. Sci. 20, 1979–1994 (2022)

    Article  MathSciNet  Google Scholar 

  2. Benbernou, S., Terbeche, M., Ragusa, M.A., Zhang, Z.: A note on the regularity criterion for 3D MHD equations in \(\dot{B}^{-1}_{\infty, \infty }\) space. Appl. Math. Comput. 238, 245–249 (2014)

    MathSciNet  Google Scholar 

  3. Blozinski, A.P.: Multivariate rearrangements and Banach function spaces with mixed norms. Trans. Am. Math. Soc. 263, 149–167 (1981)

    Article  MathSciNet  Google Scholar 

  4. Chae, D., Lee, J.: On the regularity of axisymmetric solutions of the Navier–Stokes equations. Math. Z. 239, 645–671 (2002)

    Article  MathSciNet  Google Scholar 

  5. 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  Google Scholar 

  6. Chen, H., Fang, D., Zhang, T.: Regularity of 3D axisymmetric Navier–Stokes equations. Discrete Contin. Dyn. Syst. 37, 1923–1939 (2017)

    Article  MathSciNet  Google Scholar 

  7. Chikami, N.: On Gagliardo–Nirenberg type inequalities in Fourier–Herz spaces. J. Funct. Anal. 275, 1138–1172 (2018)

    Article  MathSciNet  Google Scholar 

  8. Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

  10. Fefferman, C., McCormick, D., Robinson, J., Rodrigo, J.: Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Arch. Ration. Mech. Anal. 223, 677–691 (2017)

    Article  MathSciNet  Google Scholar 

  11. Fernandez, D.L.: Lorentz spaces, with mixed norms. J. Funct. Anal. 25, 128–146 (1977)

    Article  MathSciNet  Google Scholar 

  12. Hajaiej, H., Molinet, L., Ozawa, T., Wang, B.: Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized Boson equations, In: Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kokyuroku Bessatsu, vol.B26, Res. Inst. Math. Sci. (RIMS), Kyoto, pp. 159-175 (2011)

  13. Jiu, Q., Liu, J.: Regularity criteria to the axisymmetric incompressible Magneto-hydrodynamics equations. Dyn. Partial Differ. Equ. 15, 109–126 (2018)

    Article  MathSciNet  Google Scholar 

  14. Jiu, Q., Liu, J.: Global regularity for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. Discrete Contin. Dyn. Syst. 35, 301–322 (2015)

    Article  MathSciNet  Google Scholar 

  15. Jiu, Q., Yu, H., Zheng, X.: Global well-posedness for axisymmetric MHD system with only vertical viscosity. J. Differ. Equ. 263, 2954–2990 (2017)

    Article  MathSciNet  Google Scholar 

  16. Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)

    Article  MathSciNet  Google Scholar 

  17. Khai, D.Q., Tri, N.M.: Solutions in mixed-norm Sobolev–Lorentz spaces to the initial value problem for the Navier–Stokes equations. J. Math. Anal. Appl. 417, 819–833 (2014)

    Article  MathSciNet  Google Scholar 

  18. Lei, Z.: On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Differ. Equ. 259, 3202–3215 (2015)

    Article  MathSciNet  Google Scholar 

  19. Li, J., Tan, W., Yin, Z.: Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces. Adv. Math. 317, 786–798 (2017)

    Article  MathSciNet  Google Scholar 

  20. Li, Z.: Critical conditions on \(\omega ^\theta \) imply the regularity of axially symmetric MHD-Boussinesq systems. J. Math. Anal. Appl. 505, 125451 (2022)

    Article  MathSciNet  Google Scholar 

  21. Li, Z., Pan, X.: One component regularity criteria for the axially symmetric MHD-Boussinesq system. Discrete Contin. Dyn. Syst. 42, 2333–2353 (2022)

    Article  MathSciNet  Google Scholar 

  22. Li, Z.: Regularity criteria for the 3D axisymmetric non-resistive MHD system. Commun. Nonlinear Sci. Numer. Simul. 125, 107367 (2023)

    Article  MathSciNet  Google Scholar 

  23. Li, Z., Liu, W.: Regularity criteria for the 3D axisymmetric non-resistive MHD system in Lorentz spaces. Results Math. 78, 8 (2023). https://doi.org/10.1007/s00025-023-01863-0

    Article  MathSciNet  Google Scholar 

  24. Liu, Y.: Global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field. Acta Appl. Math. 155, 21–39 (2018)

    Article  MathSciNet  Google Scholar 

  25. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Univ. Press, Cambridge (2002)

    Google Scholar 

  26. Politano, H., Pouquet, A., Sulem, P.L.: Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 2, 2931–2939 (1995)

    Article  MathSciNet  Google Scholar 

  27. Priest, E., Forbes, T.: Magnetic Reconnection. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  28. Sunthrayuth, P., Alderremy, A., Ghani, F., Tchalla, A., Aly, S., Elmasry, Y.: Unsteady MHD flow for fractional Casson channel fluid in a porous medium: an application of the Caputo-Fabrizio time-fractional derivative. J. Funct. Spaces 22, 2765924 (2022)

    MathSciNet  Google Scholar 

  29. Wang, P., Guo, Z.: Global axisymmetric solutions to the 3D MHD equations with nonzero swirl. J. Geom. Anal. 32, 258 (2022)

    Article  MathSciNet  Google Scholar 

  30. Wang, P., Guo, Z.: Global well-posedness for axisymmetric MHD equations with vertical dissipation and vertical magnetic diffusion. Nonlinearity 35, 2147–2174 (2022)

    Article  MathSciNet  Google Scholar 

  31. Wei, W., Wang, Y., Ye, Y.: Calderón-Zygmund theory in Lorentz mixed-norm spaces and its application to compressible fluids. Math. Nachr. 2, 1–17 (2023). https://doi.org/10.1002/mana.202200364

    Article  Google Scholar 

  32. Wu, F.: Global energy conservation for distributional solutions to incompressible Hall-MHD equations without resistivity. Filomat 37, 9741–9751 (2023)

    MathSciNet  Google Scholar 

  33. Yuan, B., Li, F.: Regularity criteria of axisymmetric weak solutions to the 3D magneto-hydrodynamic equations. Acta Math. Appl. Sin. Eng. Ser. 29, 289–302 (2013)

    Article  Google Scholar 

  34. Zhang, Z.: Remarks on regularity criteria for the Navier–Stokes equations with axisymmetric data. Ann. Polon. Math. 117, 1–16 (2016)

    Article  MathSciNet  Google Scholar 

  35. Zhang, Z., Rao, J.: Global well-posedness of 3D axisymmetric MHD system with large swirl magnetic field. J. Math. Anal. Appl. 516, 126483 (2022)

    Article  MathSciNet  Google Scholar 

  36. Zhang, Z., Yao, Z.: 3D axisymmetric MHD system with regularity in the swirl component of the vorticity. Comput. Math. Appl. 73, 2573–2580 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Li is partially supported by the National Natural Science Foundation of China (Program No. 12201491), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 22JK0475), and Young Talent Fund of Association for Science and Technology in Shaanxi, China (Program No. 20230525). Liu is partially supported by the National Natural Science Foundation of China (Program No. 12301292), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 23JK0762), and the Scientific Research Foundation of Yulin University (Program No. 2023GK14).

Author information

Authors and Affiliations

  1. School of Sciences, Xi’an University of Technology, Xi’an, 710054, China

    Zhouyu Li

  2. School of Mathematics and Statistics, Yulin University, Yulin, 719000, China

    Pan Liu

Authors
  1. Zhouyu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhouyu Li.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper. Our manuscript has no associated data.

Additional information

Publisher's Note

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

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

Li, Z., Liu, P. Regularity Criteria for the 3D Axisymmetric Non-resistive MHD System in the Swirl Component of the Vorticity. J Geom Anal 34, 37 (2024). https://doi.org/10.1007/s12220-023-01478-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-023-01478-5

Keywords

Mathematics Subject Classification

Search

Navigation