Recent results for the Landau–Lifshitz equation

Abstract

We give a survey on some recent results concerning the Landau–Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. We revisit the Cauchy problem for the anisotropic LL equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. We also examine two approximations of the LL equation given by of the Sine–Gordon equation and cubic Schrödinger equations, arising in certain singular limits of strong easy-plane and easy-axis anisotropy, respectively. Concerning localized solutions, we review the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical frameworkFinally, we survey results concerning the existence, uniqueness and stability of self-similar solutions (expanders and shrinkers) for the isotropic LL equation with Gilbert term. Since expanders are associated with a singular initial condition with a jump discontinuity, we also review their well-posedness in spaces linked to the BMO space.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    Actually, in [97] they do not study of the difference between two solutions. It is only asserted that uniqueness followed from regularity, which it is not clear in this case; see also [65].

References

  1. 1.

    Alouges, F., Soyeur, A.: On global weak solutions for Landau–Lifshitz equations: existence and nonuniqueness. Nonlinear Anal. 18(11), 1071–1084 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bahri, Y.: Asymptotic stability in the energy space for dark solitons of the Landau–Lifshitz equation. Anal. PDE 9(3), 645–697 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Bahri, Y.: On the asymptotic stability in the energy space for multi-solitons of the Landau–Lifshitz equation. Trans. Am. Math. Soc. 370(7), 4683–4707 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Banica, V., Vega, L.: On the Dirac delta as initial condition for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(4), 697–711 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

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

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Banica, V., Vega, L.: Scattering for 1D cubic NLS and singular vortex dynamics. J. Eur. Math. Soc. (JEMS) 14(1), 209–253 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Banica, V., Vega, L.: Stability of the self-similar dynamics of a vortex filament. Arch. Ration. Mech. Anal. 210(3), 673–712 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Banica, V., Vega, L.: 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)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Bejenaru, I., Ionescu, A., Kenig, C., Tataru, D.: Global Schrödinger maps in dimensions \(d \ge 2\): Small data in the critical Sobolev spaces. Ann. Math. 173(3), 1443–1506 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Béthuel, F., Danchin, R., Smets, D.: On the linear wave regime of the Gross–Pitaevskii equation. J. Anal. Math. 110(1), 297–338 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Béthuel, F., Gravejat, P., Saut, J.-C.: Existence and properties of travelling waves for the Gross-Pitaevskii equation. In: Farina , A., Saut, J.-C. (eds.), Stationary and time dependent Gross-Pitaevskii equations, volume 473 of Contemp. Math., pp. 55–104. Amer. Math. Soc., Providence, RI (2008)

  12. 12.

    Béthuel, F., Gravejat, P., Saut, J.-C., Smets, D.: Orbital stability of the black soliton for the Gross–Pitaevskii equation. Indiana Univ. Math. J 57(6), 2611–2642 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Béthuel, F., Gravejat, P., Saut, J.-C., Smets, D.: On the Korteweg–de Vries long-wave approximation of the Gross–Pitaevskii equation II. Commun. Partial Differ. Equ. 35(1), 113–164 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Béthuel, F., Gravejat, P., Smets, D.: Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation. Ann. Inst. Fourier 64(1) (2014)

  15. 15.

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

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

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

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Bikbaev, R., Bobenko, A., Its, A.: Landau–Lifshitz equation, uniaxial anisotropy case: theory of exact solutions. Theor. Math. Phys. 178(2), 143–193 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Bishop, A., Long, K.: Nonlinear excitations in classical ferromagnetic chains. J. Phys. A 12(8), 1325–1339 (1979)

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Bresch, D., Gisclon, M., Lacroix-Violet, I.: On Navier–Stokes–Korteweg and Euler–Korteweg systems: application to quantum fluids models. Arch. Ration. Mech. Anal. 233(3), 975–1025 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

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

    Article  Google Scholar 

  22. 22.

    Buckingham, R., Miller, P.: Exact solutions of semiclassical non-characteristic Cauchy problems for the Sine-Gordon equation. Phys. D 237(18), 2296–2341 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Carles, R., Danchin, R., Saut, J.-C.: Madelung, Gross-Pitaevskii and Korteweg. Nonlinearity 25(10), 2843–2873 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol. 10. Amer. Math. Soc, Providence (2003)

  25. 25.

    Chang, N.-H., Shatah, J., Uhlenbeck, K.: Schrödinger maps. Commun. Pure Appl. Math. 53(5), 590–602 (2000)

    MATH  Article  Google Scholar 

  26. 26.

    Chiron, D.: Error bounds for the (KdV)/(KP-I) and the (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations. Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 31(6), 1175–1230 (2014)

  27. 27.

    Chiron, D., Rousset, F.: The KdV/KP-I limit of the nonlinear Schrödinger equation. SIAM J. Math. Anal. 42(1), 64–96 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Chousionis, V., Erdogan, M.B., Tzirakis, N.: Fractal solutions of linear and nonlinear dispersive partial differential equations. Proc. Lond. Math. Soc (3) 110(3), 543–564 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Cimrák, I.: A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism. Arch. Comput. Methods Eng. 15(3), 277–309 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Coron, J.-M.: Nonuniqueness for the heat flow of harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(4), 335–344 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Daniel, M., Lakshmanan, M.: Perturbation of solitons in the classical continuum isotropic Heisenberg spin system. Phys. A 120(1), 125–152 (1983)

    Article  Google Scholar 

  32. 32.

    de Laire, A.: Global well-posedness for a nonlocal Gross–Pitaevskii equation with non-zero condition at infinity. Commun. Partial Differ. Equ. 35(11), 2021–2058 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

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

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

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

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    de Laire, A., Gravejat, P.: 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)

  36. 36.

    de Laire, A., Gravejat, P.: The cubic Schrödinger regime of the Landau-Lifshitz equation with a strong easy-axis anisotropy. Rev. Mat. Iberoam. 37(1), 95–128 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    de Laire, A., Mennuni, P.: Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity. Discrete Contin. Dyn. Syst. 40(1), 635–682 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Deruelle, A., Lamm, T.: Existence of expanders of the harmonic map flow. arXiv:1801.08012

  39. 39.

    Ding, S., Wang, C.: Finite time singularity of the Landau-Lifshitz-Gilbert equation. Int. Math. Res. Not. IMRN (4):Art. ID rnm012, 25 (2007)

  40. 40.

    Ding, W., Wang, Y.: Schrödinger flow of maps into symplectic manifolds. Sci. Chin. Ser. A 41(7), 746–755 (1998)

    MATH  Article  Google Scholar 

  41. 41.

    Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20(5), 385–524 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

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

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Faddeev, L., Takhtajan, L.: Hamiltonian methods in the theory of solitons. Classics in Mathematics. Springer, Berlin (2007). Translated by A.G. Reyman

  44. 44.

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

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

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

  46. 46.

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

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Gérard, P., Zhang, Z.: Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation. J. Math. Pures Appl. 91(2), 178–210 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

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

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Germain, P., Rousset, F.: Long wave limit for Schrödinger maps. J. Eur. Math. Soc. 21(8), 2517–2602 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

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

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

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

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Giga, M.-H., Giga, Y., Saal, J.: 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

  53. 53.

    Gilbert, T.L.: A lagrangian formulation of the gyromagnetic equation of the magnetization field. Phys. Rev. 100, 1243 (1955)

    Google Scholar 

  54. 54.

    Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry I. J. Funct. Anal. 74(1), 160–197 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Grünrock, A.: Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS. Int. Math. Res. Not. 41, 2525–2558 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Guan, M., Gustafson, S., Kang, K., Tsai, T.-P.: Global questions for map evolution equations. In: Singularities in PDE and the calculus of variations, volume 44 of CRM Proc. Lecture Notes, pp. 61–74. Amer. Math. Soc, Providence, RI (2008)

  57. 57.

    Guo, B., Ding, S.: Landau-Lifshitz equations. Frontiers of Research with the Chinese Academy of Sciences, vol. 1. World Scientific, Hackensack (2008)

  58. 58.

    Gustafson, S., Shatah, J.: The stability of localized solutions of Landau-Lifshitz equations. Commun. Pure Appl. Math. 55(9), 1136–1159 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

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

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

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

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Gutiérrez, S., de Laire, A.: Self-similar shrinkers of the one-dimensional Landau-Lifshitz-Gilbert equation. J. Evol. Equ. 21(1), 473–501 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  62. 62.

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

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

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

    MathSciNet  MATH  Article  Google Scholar 

  64. 64.

    Hasimoto, H.: A soliton on a vortex filament. J. Fluid Mech. 51, 477–485 (1972)

    MathSciNet  MATH  Article  Google Scholar 

  65. 65.

    Jerrard, R., Smets, D.: On Schrödinger maps from \({\mathbb{T}}^1\) to \({\mathbb{S}}^2\). Ann. Sci. Éc. Norm. Sup. 45(4), 635–678 (2012)

    MATH  Google Scholar 

  66. 66.

    Jia, H., Sverak, V., Tsai, T.-P.: Self-similar solutions to the nonstationary Navier–Stokes equations. In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 461–507. Springer, New York (2018)

  67. 67.

    Kenig, C.E., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106(3), 617–633 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  68. 68.

    Koch, H., Lamm, T.: Geometric flows with rough initial data. Asian J. Math. 16(2), 209–235 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  69. 69.

    Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  70. 70.

    Kohn, R., DeSimone, A., Otto, F., Mueller, S.: Recent analytical developments in micromagnetics. In: G. Bertotti and I. D. Mayergoyz, editors, The science of hysteresis. Vol. II: Physical modeling, micromagnetics, and magnetization dynamics, pp. 269–381. Elsevier, Amsterdam (2006)

  71. 71.

    Lakshmanan, M.: 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)

  72. 72.

    Lakshmanan, M., Nakamura, K.: Landau-Lifshitz equation of ferromagnetism: exact treatment of the Gilbert damping. Phys. Rev. Lett. 53, 2497–2499 (1984)

    Article  Google Scholar 

  73. 73.

    Lakshmanan, M., Ruijgrok, T., Thompson, C.: On the dynamics of a continuum spin system. Phys. A 84(3), 577–590 (1976)

    MathSciNet  Article  Google Scholar 

  74. 74.

    Lakshmanan, M., Ruijgrok, T.W., Thompson, C.: On the dynamics of a continuum spin system. Phys. A 84(3), 577–590 (1976)

    MathSciNet  Article  Google Scholar 

  75. 75.

    Lamb, G.L. Jr.: Elements of Soliton Theory. Wiley, New York (1980). Pure and Applied Mathematics, A Wiley-Interscience Publication

  76. 76.

    Landau, L., Lifshitz, E.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Zeitsch. der Sow. 8, 153–169 (1935)

    MATH  Google Scholar 

  77. 77.

    Lemarié-Rieusset, P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall/CRC Research Notes in Mathematics, vol. 431. Chapman & Hall/CRC, Boca Raton, FL (2002)

  78. 78.

    Lin, F., Wang, C.: The Analysis of Harmonic Maps and Their Heat Flows. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008)

  79. 79.

    Lin, J., Lai, B., Wang, C.: Global well-posedness of the Landau-Lifshitz-Gilbert equation for initial data in Morrey spaces. Calc. Var. Partial Differ. Equ. 54(1), 665–692 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  80. 80.

    Madelung, E.: Quantumtheorie in Hydrodynamische form. Zts. f. Phys. 40, 322–326 (1926)

    MATH  Article  Google Scholar 

  81. 81.

    Martel, Y., Merle, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18(1), 55–80 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  82. 82.

    Martel, Y., Merle, F.: Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Math. Ann. 341(2), 391–427 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  83. 83.

    Martel, Y., Merle, F.: Stability of two soliton collision for nonintegrable gKdV equations. Commun. Math. Phys. 286(1), 39–79 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  84. 84.

    Martel, Y., Merle, F.: Inelastic interaction of nearly equal solitons for the quartic gKdV equation. Invent. Math. 183(3), 563–648 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  85. 85.

    Martel, Y., Merle, F., Tsai, T.-P.: Stability and asymptotic stability in the energy space of the sum of \(N\) solitons for subcritical gKdV equations. Commun. Math. Phys. 231(2), 347–373 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  86. 86.

    Martel, Y., Merle, F., Tsai, T.-P.: Stability in \(H^1\) of the sum of \(K\) solitary waves for some nonlinear Schrödinger equations. Duke Math. J. 133(3), 405–466 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  87. 87.

    McGahagan, H.: An approximation scheme for Schrödinger maps. Commun. Partial Differ. Equ. 32(3), 375–400 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  88. 88.

    Melcher, C.: Global solvability of the Cauchy problem for the Landau–Lifshitz–Gilbert equation in higher dimensions. Indiana Univ. Math. J. 61(3), 1175–1200 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  89. 89.

    Moser, J.: A rapidly convergent iteration method and non-linear differential equations. II. Ann. Scuola Norm. Sup. Pisa (3) 20(3), 499–535 (1966)

    MathSciNet  MATH  Google Scholar 

  90. 90.

    Nahmod, A., Shatah, J., Vega, L., Zeng, C.: Schrödinger maps and their associated frame systems. Int. Math. Res. Not. 1–29, 2007 (2007)

    MATH  Google Scholar 

  91. 91.

    Rodnianski, I., Rubinstein, Y.A., Staffilani, G.: On the global well-posedness of the one-dimensional Schrödinger map flow. Anal. PDE 2(2), 187–209 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  92. 92.

    Shatah, J., Zeng, C.: Schrödinger maps and anti-ferromagnetic chains. Commun. Math. Phys. 262(2), 299–315 (2006)

    MATH  Article  Google Scholar 

  93. 93.

    Sklyanin, E.: On complete integrability of the Landau-Lifshitz equation. Technical Report E-3-79, Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences (1979)

  94. 94.

    Song, C., Wang, Y.: Uniqueness of Schrödinger flow on manifolds. Commun. Anal. Geom. 26(1), 217–235 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  95. 95.

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

  96. 96.

    Struwe, M.: Geometric evolution problems. In Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), volume 2 of IAS/Park City Math. Ser., pages 257–339. Amer. Math. Soc., Providence, RI (1996)

  97. 97.

    Sulem, P.-L., Sulem, C., Bardos, C.: On the continuous limit for a system of classical spins. Commun. Math. Phys. 107(3), 431–454 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  98. 98.

    Taylor, M.: Partial Differential Equations III. Applied Mathematical Sciences, vol. 117, Second edition. Springer, New York (2011)

  99. 99.

    Tjon, J., Wright, J.: Solitons in the continuous Heisenberg spin chain. Phys. Rev. B 15(7), 3470–3476 (1977)

    Article  Google Scholar 

  100. 100.

    Vargas, A., Vega, L.: Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite \(L^2\) norm. J. Math. Pures Appl (9) 80(10), 1029–1044 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  101. 101.

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

    Article  Google Scholar 

  102. 102.

    Wang, C.: Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal. 200(1), 1–19 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  103. 103.

    Wei, D.: Micromagnetics and Recording Materials. Springer Briefs in Applied Sciences and Technology. Springer, Berlin (2012)

    Book  Google Scholar 

  104. 104.

    Zhou, Y., Guo, B.: Existence of weak solution for boundary problems of systems of ferro-magnetic chain. Sci. Chin. Ser. A 27(8), 799–811 (1984)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to André de Laire.

Additional information

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

Verify currency and authenticity via CrossMark

Cite this article

de Laire, A. Recent results for the Landau–Lifshitz equation. SeMA (2021). https://doi.org/10.1007/s40324-021-00254-1

Download citation

Keywords

  • Landau-Lifshitz-Gilbert equation
  • Ferromagnetic spin chain
  • Nonlinear Schrödinger equation
  • Asymptotic regimes
  • Solitons
  • Self-similar solutions

Mathematics Subject Classification

  • 82D40
  • 35Q55
  • 35C06
  • 35C20
  • 58E20
  • 35C07
  • 35B35
  • 37K05
  • 35C08
  • 35R05