Skip to main content
SpringerLink
Log in

On Liouville-type theorems for the stationary MHD and the Hall-MHD systems in \({ \mathbb {R} }^3\)

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

Abstract

In this paper, we prove a Liouville-type theorem for the stationary MHD and the stationary Hall-MHD systems. Assuming suitable growth condition at infinity for the mean oscillations for the potential functions, we show that the solutions are trivial. These results generalize the previous results obtained by two of the current authors in Chae and Wolf (J Differ Equ 295:233–248, 2021). To prove our main theorems, we use a refined iteration argument.

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. Chae, D.: Liouville-type theorems for the forced Euler equations and the Navier–Stokes equations. Commun. Math. Phys. 326(1), 37–48 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Chae, D., Yoneda, T.: On the Liouville theorem for the stationary Navier–Stokes equations in a critical space. J. Math. Anal. Appl. 405(2), 706–710 (2013)

    Article  MathSciNet  Google Scholar 

  4. 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 

  5. Chae, D., Wolf, J.: On Liouville type theorem for the stationary Navier–Stokes equations. Calc. Var. Partial Differ. Equ. 58(3), Paper No. 111, 11 pp (2019)

  6. 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 

  7. Chae, D., Kim, J., Wolf, J.: On Liouville type theorems for the stationary non-Newtonian fluid equations. J. Differ. Equ. 302, 710–727 (2021). arXiv:2107.09867

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

    Article  MathSciNet  Google Scholar 

  9. Galdi, G.: An introduction to the mathematical theory of the Navier–Stokes equations: steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, xiv+1018 pp (2011)

  10. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Stud., 105. Princeton University Press, Princeton, NJ. vii+297 pp (1983)

  11. Gilbarg, D., Weinberger, H.F.: Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(2), 381–404 (1978)

  12. Jarrin, O.: Some Liouville theorems for stationary Navier–Stokes equations in Lebesgue and Morrey spaces. J. Math. Anal. Appl., 486, Article 123871 (2020)

  13. Koch, G., Nadirashvili, N., Seregin, G.A., Šverák, V.: Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203(1), 83–105 (2009)

    Article  MathSciNet  Google Scholar 

  14. Korobkov, M., Pileckas, K., Russo, R.: The Liouville theorem for the steady-state Navier–Stokes problem for axially symmetric 3D solutions in absence of swirl. J. Math. Fluid Mech. 17(2), 287–293 (2015)

    Article  MathSciNet  Google Scholar 

  15. Kozono, H., Terasawa, Y., Wakasugi, Y.: A remark on Liouville-type theorems for the stationary Navier–Stokes equations in three space dimensions. J. Funct. Anal. 272(2), 804–818 (2017)

    Article  MathSciNet  Google Scholar 

  16. Li, Z., Niu, P.: Liouville type theorems for the 3D stationary Hall-MHD equations. ZAMM Z. Angew. Math. Mech. 100(5), e201900200, 9 pp (2020)

  17. Schulz, S.: Liouville type theorem for the stationary equations of magnetohydrodynamics. Acta Math. Sci. 39, 491–497 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Seregin, G.A.: Remarks on Liouville type theorems for steady-state Navier-Stokes equations. St. Petersburg Math. J. 30(2), 321–328 (2019)

    Article  MathSciNet  Google Scholar 

  20. Seregin, G.A., Wang, W.: Sufficient conditions on Liouville type theorems for the 3D steady Navier–Stokes equations. St. Petersburg Math. J. 31(2), 387–393 (2020)

    Article  MathSciNet  Google Scholar 

  21. Sohr, H.: The Navier–Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel. x+367 pp (2001)

  22. 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, 10 pp (2020)

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Chae’s research was partially supported by NRF Grants 2021R1A2C1003234 and by the Chung-Ang University research Grant in 2019. Wolf has been supported by NRF Grants 2017R1E1A1A01074536.

Author information

Authors and Affiliations

  1. Department of Mathematics, Chung-Ang University, Seoul, 06974, Republic of Korea

    Dongho Chae, Junha Kim & Jörg Wolf

Authors
  1. Dongho Chae
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junha Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jörg Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongho Chae.

Ethics declarations

Conflict of interest

The authors declare 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chae, D., Kim, J. & Wolf, J. On Liouville-type theorems for the stationary MHD and the Hall-MHD systems in \({ \mathbb {R} }^3\). Z. Angew. Math. Phys. 73, 66 (2022). https://doi.org/10.1007/s00033-022-01701-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01701-3

Keywords

Mathematics Subject Classification

Search

Navigation