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.
Similar content being viewed by others
References
Brezis, H.: Functional analysis. Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011)
Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)
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
Friedlander, S., Rusin, W., Vicol, V.: On the supercritically diffusive magneto-geostrophic equations. Nonlinearity 25(11), 3071–3097 (2012)
Friedlander, S., Suen, A.: Wellposedness and convergence of solutions to a class offorced non-diffusive equations with applications. arXiv:1902.04366
Friedlander, S., Suen, A.: Existence, uniqueness, regularity and instability results for the viscous magneto-geostrophic equation. Nonlinearity 28(9), 3193–3217 (2015)
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)
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)
Friedlander, S., Vicol, V.: On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations. Nonlinearity 24(11), 3019–3042 (2011)
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)
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)
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)
Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars, Paris (1969)
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)
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)
Seregin, G., Silvestre, L., Šverák, V., Zlatoš, A.: On divergence-free drifts. J. Differ. Equ. 252(1), 505–540 (2012)
Shvydkoy, R.: Convex integration for a class of active scalar equations. J. Am. Math. Soc. 24(4), 1159–1174 (2011)
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)
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)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
The author declares no conflict of interest.
Additional information
Communicated by S Friedlander.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-021-00566-2