# American Institute of Mathematical Sciences

April  2020, 25(4): 1345-1360. doi: 10.3934/dcdsb.2019230

## Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions

 1 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, 410114, China 2 Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China 3 The Graduate School of China Academy of Engineering Physics, Beijing, 100088, China

* Corresponding author: Jing Li

Received  May 2018 Revised  June 2019 Published  April 2020 Early access  November 2019

Fund Project: The first author is supported by NSFC grant 11301040, Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ3519) and Scientific Research Project of Hunan Provincial Office of Education (Grant no. 17B003).

In this paper, by using the Gal${\rm\ddot{e}}$rkin method and energy estimates, the global weak solution and the smooth solution to the generalized Landau-Lifshitz-Bloch (GLLB) equation in high dimensions are obtained.

Citation: Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230
##### References:
 [1] A. Berti and C. Giorgi, Derivation of the Landau-Lifshitz-Bloch equation from continuum thermodynamics, Physica B: Condensed Matter, 500 (2016), 142-153.  doi: 10.1016/j.physb.2016.07.035. [2] V. Berti, M. Fabrizio and C. Giorgi, A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness, Journal of Mathematical Analysis and Applications, 355 (2009), 661-674.  doi: 10.1016/j.jmaa.2009.01.065. [3] D. A. Garanin, Generalized equation of motion for a ferromagnet, Physica A: Statistical Mechanics and its Applications, 172 (1991), 470-491.  doi: 10.1016/0378-4371(91)90395-S. [4] D. A. Garanin, V. V. lshtchenko and L. V. Panina, Dynamics of an ensemble of single-domain magnetic particles, Theoretical and Mathematical Physics, 82 (1990), 169-179.  doi: 10.1007/BF01079045. [5] B. L. Guo and S. J. Ding, Landau-Lifshitz Equations, 1$^{nd}$ edition, World Scientific Publishing Co. Pty. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812778765. [6] K. N. Le, Weak solutions of the Landau-Lifshitz-Bloch equation, Journal of Differential Equations, 261 (2016), 6699-6717.  doi: 10.1016/j.jde.2016.09.002.

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##### References:
 [1] A. Berti and C. Giorgi, Derivation of the Landau-Lifshitz-Bloch equation from continuum thermodynamics, Physica B: Condensed Matter, 500 (2016), 142-153.  doi: 10.1016/j.physb.2016.07.035. [2] V. Berti, M. Fabrizio and C. Giorgi, A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness, Journal of Mathematical Analysis and Applications, 355 (2009), 661-674.  doi: 10.1016/j.jmaa.2009.01.065. [3] D. A. Garanin, Generalized equation of motion for a ferromagnet, Physica A: Statistical Mechanics and its Applications, 172 (1991), 470-491.  doi: 10.1016/0378-4371(91)90395-S. [4] D. A. Garanin, V. V. lshtchenko and L. V. Panina, Dynamics of an ensemble of single-domain magnetic particles, Theoretical and Mathematical Physics, 82 (1990), 169-179.  doi: 10.1007/BF01079045. [5] B. L. Guo and S. J. Ding, Landau-Lifshitz Equations, 1$^{nd}$ edition, World Scientific Publishing Co. Pty. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812778765. [6] K. N. Le, Weak solutions of the Landau-Lifshitz-Bloch equation, Journal of Differential Equations, 261 (2016), 6699-6717.  doi: 10.1016/j.jde.2016.09.002.
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