July  2020, 25(7): 2825-2840. doi: 10.3934/dcdsb.2020034

Global smooth solution for the Sipn-Polarized transport equation with Landau-Lifshitz-Bloch equation

1. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

2. 

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: FangFang Li

Received  June 2019 Published  July 2020 Early access  April 2020

Fund Project: The authors are supported by the National Natural Science Foundation of China (Grant No. 11801067)

The Landau-Lifshitz-Bloch equation is often used to describe micromagnetic phenomenon under high temperature. In this paper, we establish the existence and uniqueness of global smooth solution for the initial problem of the spin-polarized transport equation with Landau-Lifshitz-Bloch equation in dimension two.

Citation: Boling Guo, Fangfang Li. Global smooth solution for the Sipn-Polarized transport equation with Landau-Lifshitz-Bloch equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2825-2840. doi: 10.3934/dcdsb.2020034
References:
[1]

V. BertiM. Fabrizio and C. Giorgi, A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness, J. Math. Anal. Appl., 355 (2009), 661-674.  doi: 10.1016/j.jmaa.2009.01.065.

[2]

C. Garcia-Cervera and X. P. Wang, Spin-polarized currents in ferromagnetic multilayers, J. Comput. Phys., 224 (2007), 699-711.  doi: 10.1016/j.jcp.2006.10.029.

[3]

D. A. GaraninV. V. lshtchenko and L. V. Panina, Dynamics of an ensemble of single-domain magnetic particles, Theoretical Math. Phys., 82 (l990), 169-179.  doi: 10.1007/BF01079045.

[4]

D. A. Garanin, Generalized equation of motion for a ferromagnet, Phys. A: Statistical Mech. Appl., 172 (1991), 470-491.  doi: 10.1016/0378-4371(91)90395-S.

[5]

B. L. Guo and S. J. Ding, Landau–Lifshitz Equations, Frontiers of Research with the Chinese Academy of Sciences, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/6658.

[6]

B. L. Guo and X. K. Pu, Global smooth solutions of the spin polarized transport equation, Electron. J. Differential Equations, (2008), 15pp. doi: 10.1080/14689360802423530.

[7]

L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, in Collected Papers of L.D. Landau, 1965, Pergamon Press Ltd., 101–114. doi: 10.1016/B978-0-08-010586-4.50023-7.

[8]

K. N. Le, Weak solutions of the Landau–Lifshitz–Bloch equation, J. Differential Equations, 261 (2016), 6699-6717.  doi: 10.1016/j.jde.2016.09.002.

[9]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

show all references

References:
[1]

V. BertiM. Fabrizio and C. Giorgi, A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness, J. Math. Anal. Appl., 355 (2009), 661-674.  doi: 10.1016/j.jmaa.2009.01.065.

[2]

C. Garcia-Cervera and X. P. Wang, Spin-polarized currents in ferromagnetic multilayers, J. Comput. Phys., 224 (2007), 699-711.  doi: 10.1016/j.jcp.2006.10.029.

[3]

D. A. GaraninV. V. lshtchenko and L. V. Panina, Dynamics of an ensemble of single-domain magnetic particles, Theoretical Math. Phys., 82 (l990), 169-179.  doi: 10.1007/BF01079045.

[4]

D. A. Garanin, Generalized equation of motion for a ferromagnet, Phys. A: Statistical Mech. Appl., 172 (1991), 470-491.  doi: 10.1016/0378-4371(91)90395-S.

[5]

B. L. Guo and S. J. Ding, Landau–Lifshitz Equations, Frontiers of Research with the Chinese Academy of Sciences, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/6658.

[6]

B. L. Guo and X. K. Pu, Global smooth solutions of the spin polarized transport equation, Electron. J. Differential Equations, (2008), 15pp. doi: 10.1080/14689360802423530.

[7]

L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, in Collected Papers of L.D. Landau, 1965, Pergamon Press Ltd., 101–114. doi: 10.1016/B978-0-08-010586-4.50023-7.

[8]

K. N. Le, Weak solutions of the Landau–Lifshitz–Bloch equation, J. Differential Equations, 261 (2016), 6699-6717.  doi: 10.1016/j.jde.2016.09.002.

[9]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

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