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Weak and strong solutions to Landau-Lifshitz-Bloch-Maxwell equations with polarization. (English) Zbl 1467.35306

Summary: The Landau-Lifshitz-Bloch-Maxwell equations with polarization describe the evolution of the mean fields in continuous ferromagnetics. In this paper, we firstly use the energy method to prove the existence of weak solution to the Landau-Lifshitz-Bloch-Maxwell equations with polarization for the viscosity problem in two dimensions. Then we prove that the estimates are uniformed in \(\varepsilon\) for solutions to viscosity problem, and letting \(\epsilon \to 0\) we obtain the global weak solution for the Landau-Lifshitz-Bloch-Maxwell equations with polarization. Finally combining the a priori estimates, we obtain the existence of smooth solutions for the Landau-Lifshitz-Bloch-Maxwell equations with polarization in two dimensions.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
82D40 Statistical mechanics of magnetic materials
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
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[1] Aygelman, S., Parabolic System (1994)
[2] Bejenaru, I.; Ionescu, A. D.; Kenig, C. E.; Tartaru, D., Global Schrödinger maps in dimensions \(d \geqslant 2\): small data in the critical Sobolev spaces, Ann. Math., 173, 1443-1506 (2011) · Zbl 1233.35112
[3] Bejenaru, I.; Tartaru, D., Near Soliton Evolution for Equivariant Schrödinger Maps in Two Spatial Dimensions, Memoirs Amer. Math. Soc., vol. 228 (2014), No. 1069 · Zbl 1303.58009
[4] Berti, A.; Giorgi, C., Derivation of the Landau-Lifshitz-Bloch equation from continuum thermodynamics, Physica B, 500, 142-153 (2016)
[5] Berti, V.; Fabrizio, M.; Giorgi, C., A three-dimensional phase transition model in ferromagnetism: existence and uniqueness, J. Math. Anal. Appl., 355, 661-674 (2009) · Zbl 1173.35301
[6] Böttcher, C., Theory of Electric Polarization (1952), Elsevier · Zbl 0049.26703
[7] Chang, N.; Shatah, J.; Uhlanbeck, K., Schrödinger maps, Commun. Pure Appl. Math., 53, 5, 590-602 (2000) · Zbl 1028.35134
[8] DeBye, P., Polar Molecules (1945), Dover · JFM 55.1179.04
[9] Ding, S.; Wang, C., Finite time singularity of the Landau-Lifshitz-Gilbert equation, Int. Math. Res. Not. (2007) · Zbl 1130.35304
[10] Fröhlich, H., Theory of Dielectrics (1949), Oxford University Press
[11] Garanin, D., Generalized equation of motion for a ferromagnet, Physica A, 172, 470-491 (1991)
[12] Garanin, D., Dynamics of elliptic domain walls, Physica A, 178, 467-492 (1991)
[13] Garanin, D., Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Phys. Rev. B, 55, 3040-3057 (1997)
[14] Garanin, D.; Ishchenko, V.; Panina, L., Dynamics of an ensemble of single-domain magnetic particles, Teor. Mat. Fiz., 82, 242 (1990)
[15] Greenberg, J.; MacCamy, R.; Coffman, C., On the long-time behavior of ferroelectric systems, Physica D, 134, 362-383 (1999) · Zbl 0980.78008
[16] Guo, B.; Ding, S., Landau-Lifshitz Equations, Frontiers of Research with the Chinese Academy of Sciences, vol. 1 (2008), World Scientific Publishing Co. Pty. Ltd.: World Scientific Publishing Co. Pty. Ltd. Hackensack, NJ · Zbl 1158.35096
[17] Landau, L.; Lifshitz, E., Electrodynamics of Continuous Media (1960), Pergamon Press · Zbl 0122.45002
[18] Le, K., Weak solutions of the Landau-Lifshitz-Bloch equation, J. Differ. Equ., 261, 6699-6717 (2016) · Zbl 1351.82114
[19] Li, Z.; He, P.; Li, L.; Liang, J.; Liu, W., Magnetic soliton and soliton collisions of spinor Bose-Einstein condensates in an optical lattice, Phys. Rev. A, 71, Article 053611 pp. (2005)
[20] Li, D.; Li, Qi.; He, Pe.; Liang, J.; Liu, W.; Fu, G., Domain-wall solutions of spinor Bose-Einstein condensates in an optical lattice, Phys. Rev. A, 81, Article 015602 pp. (2010)
[21] Zhong, W., Physics of Ferroelectric (2000), Science Press, (in Chinese)
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