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On the Non-diffusive Magneto-Geostrophic Equation

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Abstract

Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In Friedlander and Vicol (Nonlinearity 24(11)::3019–3042, 2011), the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the \(H^{5/2^{+}}({{\mathbb {T}}}^3)\) norm of the perturbation.

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References

  1. Brezis, H.: Functional analysis. Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011)

    MATH  Google Scholar 

  2. Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)

    Google Scholar 

  3. Friedlander, S., Rusin, W., Vicol, V.: The Magneto-Geostrophic equations: a survey. In: Proceedings of the St. Petersburg Mathematical Society, Volume XV: Advances in Mathematical Analysis of Partial Differential Equations

  4. Friedlander, S., Rusin, W., Vicol, V.: On the supercritically diffusive magneto-geostrophic equations. Nonlinearity 25(11), 3071–3097 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  5. Friedlander, S., Suen, A.: Wellposedness and convergence of solutions to a class offorced non-diffusive equations with applications. arXiv:1902.04366

  6. Friedlander, S., Suen, A.: Existence, uniqueness, regularity and instability results for the viscous magneto-geostrophic equation. Nonlinearity 28(9), 3193–3217 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  7. Friedlander, S., Suen, A.: Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations. Ann. PDE 4(2), 34 (2018)

    Article  MathSciNet  Google Scholar 

  8. Friedlander, S., Vicol, V.: Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(2), 283–301 (2011)

  9. Friedlander, S., Vicol, V.: On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations. Nonlinearity 24(11), 3019–3042 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  10. Friedlander, S., Vicol, V.: Higher regularity of Hölder continuous solutions of parabolic equations with singular drift velocities. J. Math. Fluid Mech. 14(2), 255–266 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  11. Ghil, M., Childress, S.: Topics in geophysical fluid dynamics: atmospheric dynamics, dynamo theory, and climate dynamics, volume 60 of Applied Mathematical Sciences. Springer, New York (1987)

  12. Kenig, C.E., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4(2), 323–347 (1991)

    Article  MathSciNet  Google Scholar 

  13. Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  14. Moffatt, H.K.: Magnetostrophic turbulence and the geodynamo. In: Kaneda, Y. (ed.) IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, pp. 339–346. Springer, Dordrecht (2008)

    Chapter  Google Scholar 

  15. Moffatt, H.K., Loper, D.E.: The magnetostrophic rise of A buoyant parcel in the earth’s core. Geophys. J. Int. 117, 394–402 (1994)

    Article  ADS  Google Scholar 

  16. Seregin, G., Silvestre, L., Šverák, V., Zlatoš, A.: On divergence-free drifts. J. Differ. Equ. 252(1), 505–540 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  17. Shvydkoy, R.: Convex integration for a class of active scalar equations. J. Am. Math. Soc. 24(4), 1159–1174 (2011)

    Article  MathSciNet  Google Scholar 

  18. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III (1993)

  19. Tao, T.: Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI. Local and global analysis (2006)

  20. Temam, R.: Navier–Stokes equations, volume 2 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, revised edition, , : Theory and numerical analysis. With an appendix by F, Thomasset (1979)

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Acknowledgements

The author thanks Ángel Castro and Diego Córdoba for their valuable comments. The author acknowledges helpful conversations with Susan Friedlander and Roman Shvydkoy.

Funding

The author is partially supported by Spanish National Research Project MTM2017-89976-P and ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-556.

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Correspondence to Daniel Lear.

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Communicated by S Friedlander.

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Lear, D. On the Non-diffusive Magneto-Geostrophic Equation. J. Math. Fluid Mech. 23, 31 (2021). https://doi.org/10.1007/s00021-021-00566-2

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