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Self-similar shrinkers of the one-dimensional Landau–Lifshitz–Gilbert equation

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Abstract

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere \(\mathbb {S}^2\), at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

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References

  1. M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.

  2. H. Amann. Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Rational Mech. Anal., 92(2):153–192, 1986.

    Article  MathSciNet  Google Scholar 

  3. V. Banica and L. Vega. On the stability of a singular vortex dynamics. Comm. Math. Phys., 286(2):593–627, 2009.

    Article  MathSciNet  Google Scholar 

  4. V. Banica and L. Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on \(\mathbb{S}^2\). Commun. Pure Appl. Anal., 17(4):1317–1329, 2018.

    Article  MathSciNet  Google Scholar 

  5. M. F. Bidaut-Véron. Self-similar solutions of the \(p\)-Laplace heat equation: the case when \(p>2\). Proc. Roy. Soc. Edinburgh Sect. A, 139(1):1–43, 2009.

    Article  MathSciNet  Google Scholar 

  6. P. Biernat and P. Bizoń. Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres. Nonlinearity, 24(8):2211–2228, 2011.

    Article  MathSciNet  Google Scholar 

  7. P. Biernat and R. Donninger. Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow. Nonlinearity, 31(8):3543, 2018.

    Article  MathSciNet  Google Scholar 

  8. P. Bizoń and A. Wasserman. Nonexistence of shrinkers for the harmonic map flow in higher dimensions. Int. Math. Res. Not. IMRN, (17):7757–7762, 2015.

    Article  MathSciNet  Google Scholar 

  9. G. Broggi, P. F. Meier, R. Stoop, and R. Badii. Nonlinear dynamics of a model for parallel pumping in ferromagnets. Phys. Rev. A, 35:365–368, 1987.

    Article  Google Scholar 

  10. T. F. Buttke. A numerical study of superfluid turbulence in the self-induction approximation. Journal of Computational Physics, 76(2):301–326, 1988.

    Article  MathSciNet  Google Scholar 

  11. G. Darboux. Leçons sur la théorie générale des surfaces. I, II. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Sceaux, 1993. Généralités. Coordonnées curvilignes. Surfaces minima. [Generalities. Curvilinear coordinates. Minimum surfaces], Les congruences et les équations linéaires aux dérivées partielles. Les lignes tracées sur les surfaces. [Congruences and linear partial differential equations. Lines traced on surfaces], Reprint of the second (1914) edition (I) and the second (1915) edition (II), Cours de Géométrie de la Faculté des Sciences. [Course on Geometry of the Faculty of Science].

  12. A. de Laire. Minimal energy for the traveling waves of the Landau-Lifshitz equation. SIAM J. Math. Anal., 46(1):96–132, 2014.

    Article  MathSciNet  Google Scholar 

  13. A. de Laire and P. Gravejat. Stability in the energy space for chains of solitons of the Landau–Lifshitz equation. J. Differential Equations, 258(1):1–80, 2015.

    Article  MathSciNet  Google Scholar 

  14. A. de Laire and P. Gravejat. The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy. Ann. Inst. Henri Poincaré, Analyse Non Linéaire , 35(7):1885–1945, 2018.

    Article  MathSciNet  Google Scholar 

  15. A. de Laire and P. Gravejat. The cubic Schrödinger regime of the Landau–Lifshitz equation with a strong easy-axis anisotropy. Rev. Mat. Iberoamericana, in press.

  16. F. Demontis, G. Ortenzi, and M. Sommacal. Heisenberg ferromagnetism as an evolution of a spherical indicatrix: localized solutions and elliptic dispersionless reduction. Electron. J. Differential Equations, 106:1–34, 2018.

    MathSciNet  MATH  Google Scholar 

  17. A. Deruelle and T. Lamm. Existence of expanders of the harmonic map flow. Preprint arXiv:1801.08012.

  18. J. Eggers and M. A. Fontelos. The role of self-similarity in singularities of partial differential equations. Nonlinearity, 22(1):1–9, 2009.

    Article  MathSciNet  Google Scholar 

  19. H. Fan. Existence of the self-similar solutions in the heat flow of harmonic maps. Sci. China Ser. A, 42(2):113–132, 1999.

    Article  MathSciNet  Google Scholar 

  20. O. Gamayun and O. Lisovyy. On self-similar solutions of the vortex filament equation. J. Math. Phys., 60(8):083510, 13, 2019.

  21. A. Gastel. Singularities of first kind in the harmonic map and Yang–Mills heat flows. Math. Z., 242(1):47–62, 2002.

    Article  MathSciNet  Google Scholar 

  22. P. Germain, T.-E. Ghoul, and H. Miura. On uniqueness for the harmonic map heat flow in supercritical dimensions. Comm. Pure Appl. Math., 70(12):2247–2299, 2017.

    Article  MathSciNet  Google Scholar 

  23. P. Germain and M. Rupflin. Selfsimilar expanders of the harmonic map flow. Ann. Inst. H. Poincaré Anal. Non Linéaire, 28(5):743–773, 2011.

    Article  MathSciNet  Google Scholar 

  24. P. Germain, J. Shatah, and C. Zeng. Self-similar solutions for the Schrödinger map equation. Math. Z., 264(3):697–707, 2010.

    Article  MathSciNet  Google Scholar 

  25. M. Giaquinta. Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1983.

  26. M.-H. Giga, Y. Giga, and J. Saal. Nonlinear partial differential equations, volume 79 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 2010. Asymptotic behavior of solutions and self-similar solutions.

  27. T. L. Gilbert. A Lagrangian formulation of the gyromagnetic equation of the magnetization field. Phys. Rev., 100:1243, 1955.

    Google Scholar 

  28. R. D. Gordon. Values of Mills’ ratio of area to bounding ordinate and of the normal probability integral for large values of the argument. Ann. Math. Statistics, 12:364–366, 1941.

    Article  MathSciNet  Google Scholar 

  29. B. Guo and S. Ding. Landau–Lifshitz equations, volume 1 of Frontiers of Research with the Chinese Academy of Sciences. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

  30. B. L. Guo and M. C. Hong. The Landau–Lifshitz equation of the ferromagnetic spin chain and harmonic maps. Calc. Var. Partial Differential Equations, 1(3):311–334, 1993.

    Article  MathSciNet  Google Scholar 

  31. S. Gutiérrez. Vortex filaments and 1D cubic Schrödinger equations: singularity formation. Commun. Appl. Anal., 15(2-4):457–474, 2011.

    MathSciNet  MATH  Google Scholar 

  32. S. Gutiérrez and A. de Laire. Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation. Nonlinearity, 28(5):1307–1350, 2015.

    Article  MathSciNet  Google Scholar 

  33. S. Gutiérrez and A. de Laire. The Cauchy problem for the Landau–Lifshitz–Gilbert equation in BMO and self-similar solutions. Nonlinearity, 32(7):2522–2563, 2019.

    Article  MathSciNet  Google Scholar 

  34. S. Gutiérrez, J. Rivas, and L. Vega. Formation of singularities and self-similar vortex motion under the localized induction approximation. Comm. Partial Differential Equations, 28(5-6):927–968, 2003.

    Article  MathSciNet  Google Scholar 

  35. S. Gutiérrez and L. Vega. Self-similar solutions of the localized induction approximation: singularity formation. Nonlinearity, 17:2091–2136, 2004.

    Article  MathSciNet  Google Scholar 

  36. T. Ilmanen. Lectures on mean curvature flow and related equations (lecture notes). In ICTP, Trieste, 1995.

  37. H. Jia, V. Sverák, and T.-P. Tsai. Self-similar solutions to the nonstationary Navier–Stokes equations. In Handbook of mathematical analysis in mechanics of viscous fluids, pages 461–507. Springer, Cham, 2018.

  38. J. Jost. Riemannian geometry and geometric analysis. Universitext. Springer-Verlag, Berlin, fifth edition, 2008.

  39. O. A. Ladyzhenskaya and N. N. Ural’tseva. Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968.

  40. M. Lakshmanan. The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 369(1939):1280–1300, 2011.

    MathSciNet  MATH  Google Scholar 

  41. M. Lakshmanan, T. W. Ruijgrok, and C. Thompson. On the dynamics of a continuum spin system. Physica A: Statistical Mechanics and its Applications, 84(3):577–590, 1976.

    Article  MathSciNet  Google Scholar 

  42. G. L. Lamb, Jr. Elements of soliton theory. John Wiley & Sons Inc., New York, 1980. Pure and Applied Mathematics, A Wiley-Interscience Publication.

  43. L. Landau and E. Lifshitz. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion, 8:153–169, 1935.

    MATH  Google Scholar 

  44. F. Lin and C. Wang. Harmonic and quasi-harmonic spheres. Comm. Anal. Geom., 7(2):397–429, 1999.

    Article  MathSciNet  Google Scholar 

  45. F. Lin and C. Wang. The analysis of harmonic maps and their heat flows. World Scientific Publishing Co. Pte. Ltd. , Hackensack, NJ, 2008.

    Book  Google Scholar 

  46. T. Lipniacki. Shape-preserving solutions for quantum vortex motion under localized induction approximation. Phys. Fluids, 15(6):1381–1395, 2003.

    Article  MathSciNet  Google Scholar 

  47. S. Montiel and A. Ros. Curves and surfaces, volume 69 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, second edition, 2009. Translated from the 1998 Spanish original by Montiel and edited by Donald Babbitt.

  48. P. Quittner and P. Souplet. Superlinear parabolic problems. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states.

  49. F. Schulz. Regularity theory for quasilinear elliptic systems and Monge–Ampère equations in two dimensions, volume 1445 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1990.

  50. D. J. Struik. Lectures on Classical Differential Geometry. Addison-Wesley Press, Inc., Cambridge, Mass., 1950.

    MATH  Google Scholar 

  51. M. Struwe. On the evolution of harmonic maps in higher dimensions. J. Differential Geom., 28(3):485–502, 1988.

    Article  MathSciNet  Google Scholar 

  52. F. Waldner, D. R. Barberis, and H. Yamazaki. Route to chaos by irregular periods: Simulations of parallel pumping in ferromagnets. Phys. Rev. A, 31:420–431, 1985.

    Article  Google Scholar 

  53. D. Wei. Micromagnetics and Recording Materials. SpringerBriefs in Applied Sciences and Technology. Springer Berlin Heidelberg, 2012.

  54. D. Xu and C. Zhou. A remark on the quasi-harmonic spheres. Appl. Math. J. Chinese Univ. Ser. B, 17(2):164–170, 2002.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

S. Gutiérrez was partially supported by ERCEA Advanced Grant 2014 669689—HADE. The Université de Lille also supported S. Gutiérrez’s research visit during July 2018 through their Invited Research Speaker Scheme. A. de Laire was partially supported by the Labex CEMPI (ANR-11-LABX-0007-01), the ANR project “Dispersive and random waves” (ANR-18-CE40-0020-01), and the MATH-AmSud Project EEQUADD-II.

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  1. School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    Susana Gutiérrez

  2. CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, Univ. Lille, 59000, Lille, France

    André de Laire

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Gutiérrez, S., de Laire, A. Self-similar shrinkers of the one-dimensional Landau–Lifshitz–Gilbert equation. J. Evol. Equ. 21, 473–501 (2021). https://doi.org/10.1007/s00028-020-00589-8

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