Skip to main content
Springer Nature Link
Log in

Uniqueness and Non-Uniqueness Results for Forced Dyadic MHD Models

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

We construct non-unique Leray–Hopf solutions for some forced dyadic models for magnetohydrodynamics (MHD) when the intermittency dimension \(\delta \) is less than 1. Conventionally, the interaction of the velocity and magnetic fields is a major challenge in the context of MHD. However, in the dyadic MHD model scenario, we exploit to our benefit certain symmetries in the interactions of the fields to obtain a non-uniqueness result. In contrast, uniqueness of the Leray–Hopf solution to the dyadic MHD models is established in the case of \(\delta \ge 1\). Analogous results on uniqueness and non-uniqueness of Leray–Hopf solution are also obtained for dyadic models of MHD with fractional diffusion.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /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.

Institutional subscriptions

Similar content being viewed by others

Data availability

Our manuscript has no associated data.

References

  • Albritton, D., Brué, E., Colombo, M.: Non-uniqueness of Leray solutions of the forced Navier–Stokes equations. Ann. Math. 196(1), 415–455 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Barbato, D., Flandoli, F., Morandin, F.: Energy dissipation and self-similar solutions for an unforced inviscid dyadic model. Trans. Am. Math. Soc. 363(4), 1925–1946 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Barbato, D., Morandin, F., Romito, M.: Smooth solutions for the dyadic model. Nonlinearity 24(11), 3083–3097 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Beekie, R., Buckmaster, T., Vicol, V.: Weak solutions of ideal MHD which do not conserve magnetic helicity. Ann. PDE (2020). https://doi.org/10.1007/s40818-020-0076-1

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, K., Liu, J.: Weak solutions of the three-dimensional hypoviscous elastodynamics with finite kinetic energy. arXiv: 2104.11872 (2021)

  • Cheskidov, A.: Blow-up in finite time for the dyadic model of the Navier–Stokes equations. Trans. Am. Math. Soc. 360(10), 5101–5120 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheskidov, A., Dai, M.: Kolmogorov’s dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations. Proc. Roy. Soc. Edinburg Sect. A 149(2), 429–446 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheskidov, A., Friedlander, S.: The vanishing viscosity limit for a dyadic model. Physica D 238, 783–787 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheskidov, A., Shvydkoy, R.: A unified approach to regularity problems for the 3D Navier-Stokes and Euler equations: the use of Kolmogorov’s dissipation range. J. Math. Fluid Mech. 16(2), 263–273 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheskidov, A., Friedlander, S., Pavlović, N.: Inviscid dyadic model of turbulence: the fixed point and Onsager’s conjecture. J. Math. Phys. 48(6), 065503, 16 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheskidov, A., Friedlander, S., Pavlović, N.: An inviscid dyadic model of turbulence: the global attractor. Discrete Contin. Dyn. Syst. 26(3), 781–794 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Dai, M., Friedlander, S.: Dyadic models for ideal MHD. J. Math. Fluid Mech. 24, 21 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Dai, M.: Blow-up of a dyadic model with intermittency dependence for the Hall MHD. Physica D 428, 133066 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Desnyanskiy, V.N., Novikov, E.A.: Evolution of turbulence spectra toward a similarity regime. Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 10, 127–136 (1974)

    Google Scholar 

  • Faraco, D., Lindberg, S., Székelyhidi, L.: Magnetic helicity, weak solutions and relaxation of ideal MHD. arXiv: 2109.09106 (2021)

  • Faraco, D., Lindberg, S., Székelyhidi, L.: Bounded solutions of ideal MHD with compact support in space-time. Arch. Ration. Mech. Anal. 239, 51–93 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Filonov, N.: Uniqueness of the Leray–Hopf solution for a dyadic model. Trans. Am. Math. Soc. 369(12), 8663–8684 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Filonov, N., Khodunov, P.: Non-uniqueness of Leray–Hopf solutions for a dyadic model. St. Petersburg Math. J. 32, 371–387 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Friedlander, S., Pavlović, N.: Blowup in a three-dimensional vector model for the Euler equations. Commun. Pure Appl. Math. 57(6), 705–725 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Gledzer, E.B.: System of hydrodynamic type admitting two quadratic integrals of motion. Soviet Phys. Dokl. 18, 216–217 (1973)

    MATH  Google Scholar 

  • Gloaguen, C., Léorat, J., Pouquet, A., Grappin, R.: A scalar model for MHD turbulence. Physica D 17(2), 154–182 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  • Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  • Jeong, I., Li, D.: A blow-up result for dyadic models of the Euler equations. Commun. Math. Phys. 337, 1027–1034 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Katz, N., Pavlović, N.: Finite time blow-up for a dyadic model of the Euler equations. Trans. Am. Math. Soc. 357(2), 695–708 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Kiselev, A., Zlatoš, A.: On discrete models of the Euler equation. Int. Math. Res. Not. 38, 2315–2339 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  • L’vov, V.S., Podivilov, E., Pomyalov, A., Procaccia, I., Vandembroucq, D.: Improved shell model of turbulence. Phys. Rev. E (3) 58, 1811–1822 (1998)

    Article  MathSciNet  Google Scholar 

  • Mailybaev, A.A.: Continuous representation for shell models of turbulence. Nonlinearity 28(7), 2479 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Ohkitani, K., Yamada, M.: Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully-developed model of turbulence. Progr. Theoret. Phys. 81, 329–341 (1989)

    Article  MathSciNet  Google Scholar 

  • Onsager, L.: Statistical hydrodynamics. Nuovo Cimento 9(6), 279–287 (1949)

    Article  MathSciNet  Google Scholar 

  • Plunian, F., Stepanov, R., Frick, P.: Shell models of magnetohydrodynamic turbulence. Phys. Rep. 523, 1–60 (2013)

    Article  MathSciNet  Google Scholar 

  • Vishik, M.: Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I. arXiv: 1805.09426 (2018a)

  • Vishik, M.: Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II. arXiv: 1805.09440 (2018b)

Download references

Acknowledgements

M. Dai is partially supported by the NSF Grants DMS-1815069 and DMS-2009422. S. Friedlander is partially supported by the NSF Grant DMS-1613135. S. Friedlander is grateful to IAS for its hospitality in 2020–2021. M. Dai is also grateful to IAS for its hospitality in 2021–2022. The authors are very grateful to the anonymous referees for their constructive suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Mimi Dai

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Susan Friedlander

Authors
  1. Mimi Dai
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Susan Friedlander
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Mimi Dai.

Additional information

Communicated by Paul Newton.

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

Dai, M., Friedlander, S. Uniqueness and Non-Uniqueness Results for Forced Dyadic MHD Models. J Nonlinear Sci 33, 10 (2023). https://doi.org/10.1007/s00332-022-09868-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-022-09868-9

Keywords

Mathematics Subject Classification