Skip to main content
SpringerLink
Log in

On Moffatt’s Magnetic Relaxation Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate the stability properties for a family of equations introduced by Moffatt to model magnetic relaxation. These models preserve the topology of magnetic streamlines, contain a cubic nonlinearity, and yet have a favorable \(L^2\) energy structure. We consider the local and global in time well-posedness of these models and establish a difference between the behavior as \(t\rightarrow \infty \) with respect to weak and strong norms.

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

Notes

  1. Here, we say that \(B_1\) and \(B_0\) are topologically equivalent, if \(B_1(X(\alpha )) = \nabla _\alpha X(\alpha ) B_0(\alpha )\) for a volume preserving diffeomorphism \(\alpha \mapsto X(\alpha )\). In contrast, to say that \(B_1\) is topologically accessible from \(B_0\) means that (see e.g. in [Mof21, Section 8.2.1]) \(B_1 = \lim _{t\rightarrow \infty } B(\cdot ,t)\), where B is a solution of (1.3a) with initial datum \(B_0\) and some solenoidal vector field u, under the additional property that \(\int _0^\infty \left| \int _{{\mathbb {T}}^d} B \cdot (B\cdot \nabla u) dx \right| dt < \infty \).

  2. Note in contrast that the cross-helicity \(\int _{{\mathbb {T}}^d} u\cdot B dx\) is expected to vanish as \(t\rightarrow \infty \) since \(B(\cdot ,t)\) remains uniformly bounded in \(L^2\), while \(u(\cdot ,t)\rightarrow 0\) in \(L^2\).

References

  1. Arnold, V.I., Khesin, B.A.: Topological methods in hydrodynamics. In: Applied Mathematical Sciences, vol. 125. Springer, New York (1998)

  2. Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble), 16(fasc. 1):319–361 (1966)

  3. Arnold, V.I.: The asymptotic Hopf invariant and it applications. In: Proceedings of Summer School in Differential Equations, Armenian Academy of Sciences, English translation, vol. 561 (1974)

  4. Bloch, A., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S.: The Euler–Poincaré equations and double bracket dissipation. Commun. Math. Phys. 175(1), 1–42 (1996)

    Article  ADS  Google Scholar 

  5. Brenier, Y.: Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation. Commun. Math. Phys. 330(2), 757–770 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  6. Buckmaster, T., Vicol, V.: Convex integration constructions in hydrodynamics. Bull. Am. Math. Soc. 58(1), 1–44 (2021)

    Article  MathSciNet  Google Scholar 

  7. Castro, Á., Córdoba, D., Gancedo, F., Orive, R.: Incompressible flow in porous media with fractional diffusion. Nonlinearity 22(8), 1791–1815 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  8. Castro, Á., Córdoba, D., Lear, D.: Global existence of quasi-stratified solutions for the confined IPM equation. Arch. Ration. Mech. Anal. 232(1), 437–471 (2019)

    Article  MathSciNet  Google Scholar 

  9. Constantin, P., La, J., Vicol, V.: Remarks on a paper by Gavrilov: Grad–Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications. Geom. Funct. Anal. 29(6), 1773–1793 (2019)

    Article  MathSciNet  Google Scholar 

  10. Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the \(2\)-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994)

    Article  MathSciNet  Google Scholar 

  11. De Lellis, C., Székelyhidi, Jr. L.: The \(h\)-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. (N.S.) 49(3), 347–375 (2012)

  12. DiPerna, R.J., Majda, A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  13. Elgindi, T.M.: On the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation. Arch. Ration. Mech. Anal. 225(2), 573–599 (2017)

    Article  MathSciNet  Google Scholar 

  14. Ebin, D.G., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 2(92), 102–163 (1970)

    Article  MathSciNet  Google Scholar 

  15. Elgindi, T.M., Masmoudi, N.: \(L^\infty \) ill-posedness for a class of equations arising in hydrodynamics. Arch. Ration. Mech. Anal. 235(3), 1979–2025 (2020)

    Article  MathSciNet  Google Scholar 

  16. Enciso, A., Peralta-Salas, D.: Knots and links in steady solutions of the Euler equation. Ann. Math. (2) 175(1), 345–367 (2012)

    Article  MathSciNet  Google Scholar 

  17. Friedlander, S., Vishik, M.: On stability and instability criteria for magnetohydrodynamics. Chaos 5(2), 416–423 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  18. Gavrilov, A.V.: A steady Euler flow with compact support. Geom. Funct. Anal. 29(1), 190–197 (2019)

    Article  MathSciNet  Google Scholar 

  19. Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123(1–2), 116 (1985)

    MathSciNet  MATH  Google Scholar 

  20. Li, D.: On Kato-Ponce and fractional Leibniz. Rev. Mat. Iberoam. 35(1), 23–100 (2019)

    Article  MathSciNet  Google Scholar 

  21. Lin, F., Xu, L., Zhang, P.: Global small solutions of 2-D incompressible MHD system. J. Differ. Equ. 259(10), 5440–5485 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  22. Moffatt, H.K.: Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals. J. Fluid Mech. 159, 359–378 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  23. Moffatt, H.K.: Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Pt. 2: stability considerations. J. Fluid Mech. 166, 359–78 (1986)

    Article  ADS  Google Scholar 

  24. Moffatt, H.K.: Some topological aspects of fluid dynamics. J. Fluid Mech. 914:Paper No. P1 (2021)

  25. Nishiyama, T.: Pseudo-advection method for the two-dimensional stationary Euler equations. In: Proceedings of the American Mathematical Society, pp. 429–432 (2001)

  26. Nishiyama, T.: Construction of three-dimensional stationary Euler flows from pseudo-advected vorticity equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 459(2038), 2393–2398 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  27. Pasqualotto, F.: Nonlinear Waves in General Relativity and Fluid Dynamics. PhD thesis, Princeton University. Thesis (Ph.D.)–Princeton University (2020)

  28. Ren, X., Wu, J., Xiang, Z., Zhang, Z.: Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J. Funct. Anal. 267(2), 503–541 (2014)

    Article  MathSciNet  Google Scholar 

  29. Sadun, L., Vishik, M.: The spectrum of the second variation of the energy for an ideal incompressible fluid. Phys. Lett. A 182(4–6), 394–398 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  30. Vallis, G.K., Carnevale, G.F., Young, W.R.: Extremal energy properties and construction of stable solutions of the Euler equations. J. Fluid Mech. 207, 133–152 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  31. Yudovich, V.I.: On the loss of smoothness of the solutions of Euler’s equations with time. Dinamika Sploshnoi Sredy (Dynamics of Continuous Media) (Novosibirsk, Russia) 16, 71–78 (1974)

  32. Yudovich, V.I.: On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid. Chaos 10(3), 705–719 (2000)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Acknowledgements

R.B. was supported by the NSF Graduate Fellowship Grant 1839302. S.F. was in part supported by NSF grant DMS 1613135. S.F. thanks IAS for its hospitality when she was a Member in 2020-21. V.V. was in part supported by the NSF grant CAREER DMS 1911413. V.V. thanks B. Texier and S. Shkoller for stimulating discussions.

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Rajendra Beekie & Vlad Vicol

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

    Susan Friedlander

Authors
  1. Rajendra Beekie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Susan Friedlander
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vlad Vicol
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rajendra Beekie.

Additional information

Communicated by A. Ionescu.

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

Beekie, R., Friedlander, S. & Vicol, V. On Moffatt’s Magnetic Relaxation Equations. Commun. Math. Phys. 390, 1311–1339 (2022). https://doi.org/10.1007/s00220-021-04289-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-04289-3

Search

Navigation