Skip to main content
Log in

Liouville-Type Theorems for the 3D Stationary MHD Equations

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the Liouville-type theorems for the 3D stationary incompressible MHD equations. Using the Caccioppoli type estimate, we proved the smooth solutions (ub) are identically equal to zero when \((u,b)\in L^{p}({\mathbb {R}}^{3}),\ p\in (\frac{3}{2},3).\) In addition, under an additional assumption in the setting of the Sobolev space of negative order \(\dot{H}^{-1}({\mathbb {R}}^{3}),\) we can extend the index \(p\in (3,+\infty ).\) In fact, our results combine with the result of Yuan and Xiao (J Math Anal Appl 491(2):124343, 2020) that \(p\in [2,\frac{9}{2}],\) which implies a very intriguing and novel result for the 3D stationary MHD equations with \( p\in (\frac{3}{2},+\infty ).\)

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

  1. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Springer Monographs in Mathematics, 2nd edn. Springer, New York (2011)

    Google Scholar 

  2. Chae, D.: Liouville-type theorems for the forced Euler equations and the Navier–Stokes equations. Commun. Math. Phys. 326, 37–48 (2014)

    Article  MathSciNet  Google Scholar 

  3. Chae, D., Wolf, J.: On Liouville type theorems for the steady Navier–Stokes equations in \({\mathbb{R}} ^{3},\). J. Differ. Equ. 261(10), 5541–5560 (2016)

    Article  Google Scholar 

  4. Seregin, G.: Liouville type theorem for stationary Navier–Stokes equations. Nonlinearity 29(8), 2191–2195 (2016)

    Article  MathSciNet  Google Scholar 

  5. Seregin, G.: Remarks on Liouville type theorems for steady-state Navier–Stokes equations. Algebra Anal. 30(2), 238–248 (2018)

    MathSciNet  Google Scholar 

  6. Chae, D.: Anisotropic Liouville type theorem for the stationary Navier–Stokes equations in \({\mathbb{R}}^{3}\). Appl. Math. Lett. 142, 108655 (2023)

    Article  Google Scholar 

  7. JarrÍn, O.: A short note on the Liouville problem for the steady-state Navier-Stokes equations. Archiv. der. Mathematik. 121, 303–315 (2023)

    Article  MathSciNet  Google Scholar 

  8. Chae, D.: Note on the Liouville type problem for the stationary Navier–Stokes equations in \({\mathbb{R}}^{3}\). J. Differ. Equ. 268(3), 1043–1049 (2020)

    Article  Google Scholar 

  9. Chamorro, D., Jarrín, O., Lemarié-Rieusset, P.-G.: Some Liouville theorems for stationary Navier–Stokes equations in Lebesgue and Morrey spaces. Ann. Inst. H. Poincaré C Anal. Non Linéaire 38(3), 689–710 (2021)

    Article  MathSciNet  Google Scholar 

  10. Yuan, B., Xiao, Y.: Liouville-type theorems for the 3D stationary Navier–Stokes, MHD and Hall-MHD equations. J. Math. Anal. Appl. 491(2), 124343 (2020)

    Article  MathSciNet  Google Scholar 

  11. Chae, D., Degond, P., Liu, J.-G.: Well-posedness for Hall-magnetohydrodynamics. Ann. Inst. H. Poincaré C Anal. Non Linéaire 31(3), 555–565 (2014)

    Article  MathSciNet  Google Scholar 

  12. Zhang, Z., Yang, X., Qiu, S.: Remarks on Liouville type result for the 3D Hall-MHD system. J. Part. Differ. Equ. 28(3), 286–290 (2015)

    MathSciNet  Google Scholar 

  13. Chae, D., Wolf, J.: On Liouville type theorems for the stationary MHD and Hall-MHD systems. J. Differ. Equ. 295, 233–248 (2021)

    Article  MathSciNet  Google Scholar 

  14. Chae, D., Kim, J., Wolf, J.: On Liouville-type theorems for the stationary MHD and the Hall-MHD systems in \({\mathbb{R}}^{3}\). Zeitschrift für angewandte Mathematik und Physik 73(2), 66 (2022)

    Article  Google Scholar 

  15. Fan, H., Wang, M.: The Liouville type theorem for the stationary magnetohydrodynamic equations. J. Math. Phys. 62(5), 031503 (2021)

    Article  MathSciNet  Google Scholar 

  16. Wang, W., Wang, Y.: Liouville-type theorems for the stationary MHD equations in 2D. Nonlinearity 32(11), 4483 (2019)

    Article  MathSciNet  Google Scholar 

  17. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Non-linear Elliptic Systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)

Download references

Acknowledgements

This work was supported by Anhui Education Bureau under Grant No. KJ2019A0556 and Scientific Research Project for the Excellent Youth Scholars of Higher Education of Anhui Province under Grant No. 2023AH030073.

Author information

Authors and Affiliations

Authors

Contributions

Zhang Provides the main idea and write part of the content; Zu wrote part of the manuscript and correct some errors in article. All authors reviewed the manuscript.

Corresponding author

Correspondence to Hui Zhang.

Ethics declarations

Conflict of Interest

The authors declared that they have no conflict of interest.

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

Zhang, H., Zu, Q. Liouville-Type Theorems for the 3D Stationary MHD Equations. Mediterr. J. Math. 21, 132 (2024). https://doi.org/10.1007/s00009-024-02675-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-024-02675-4

Keywords

Mathematics Subject Classification

Navigation