SMOOTH SOLUTIONS OF THE LANDAU-LIFSHITZ-BLOCH EQUATION

Qiaoxin Li; Boling Guo; Ming Zeng

doi:10.11948/20200376

2021 Volume 11 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(6): 2713-2721

Next Article Previous Article

Qiaoxin Li, Boling Guo, Ming Zeng. SMOOTH SOLUTIONS OF THE LANDAU-LIFSHITZ-BLOCH EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2713-2721. doi: 10.11948/20200376

Citation:

Qiaoxin Li, Boling Guo, Ming Zeng. SMOOTH SOLUTIONS OF THE LANDAU-LIFSHITZ-BLOCH EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2713-2721. doi: 10.11948/20200376

SMOOTH SOLUTIONS OF THE LANDAU-LIFSHITZ-BLOCH EQUATION

1.
Department of Mathematics and Physics, North China Electric Power University, 102206, Beijing, China
2.
Institute of Applied Physics and Computational Mathematics, P.O.Box 8009-28, 100088, Beijing, China
3.
College of Applied Sciences, Beijing University of Technology, PingLeYuan 100, Chaoyang District, 100124, Beijing, China

Corresponding author: Email: liqiaoxin@126.com(Q. Li)
Fund Project: The authors were supported by National Natural Science Foundation of China (NSFC-11801552)

Abstract

Abstract

Landau-Lifshitz-Bloch equation is often used to model micromagnetic phenomenon under high temperature. This article proves the existence of smooth solutions of the equation in $ \mathbb{R}^2$ and $ \mathbb{R}^3$, and a small initial value condition should be added in the latter case. These results can also be generalized to periodical boundary value case.
- Landau-Lifshitz-Bloch equation /
- a priori estimate /
- Gagliardo-Nir-enberg inequality
MSC: 35B35, 35Q55

FullText(HTML)

References

[1]	I. Bejenaru, A. D. Ionescu, C. E. Kenig and D. Tartaru. Global Schrodinger maps in dimensions d ≤ 2: small data in the critical Sobolev spaces, Annals Math., 2011, 173, 1443-1506. doi: 10.4007/annals.2011.173.3.5 CrossRef Google Scholar
[2]	A. Berti and C. Giorgi, Derivation of the Landau-Lifshitz-Bloch equation from continuum thermodynamics, Phys. B., 2016, 500, 142-153. doi: 10.1016/j.physb.2016.07.035 CrossRef Google Scholar
[3]	T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Annals of Mathematics, 2019, 189, 101-144. doi: 10.4007/annals.2019.189.1.3 CrossRef Google Scholar
[4]	N. Chang, J. Shatah and K. Uhlanbeck, Schrodinger maps, Comm. Pure Appl. Math., 2000, 53(5), 590-602. doi: 10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R CrossRef Google Scholar
[5]	D. A. Garamin, Generalized equation of motion for a ferromaganet, Phys. A., 1991, 172, 470-491. doi: 10.1016/0378-4371(91)90395-S CrossRef Google Scholar
[6]	D. A. Garamin, Dynamices of elliptiv domain wall, Phys, A., 1991, 178, 467-492. doi: 10.1016/0378-4371(91)90033-9 CrossRef Google Scholar
[7]	D. A. Garamin, Forker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Phys. Rev. B., 1997, 55, 3060-3057. Google Scholar
[8]	B. Guo and Y. Han, Global smooth solution of Hydrodynamical equation for the Heisenberg paremagnet, Math. Meth. Appl. Sci., 2004, 181-191. Google Scholar
[9]	B. Guo and S. Ding, Landau-Lifshitz equations, World Scientific Publishing, Hackensack, NJ., 2008. Google Scholar
[10]	K. Ngan Le, Weak solutions of the Landau-Lifshitz-Bloch equation, Journal of Differential Equation, 2016, 261, 6699-6717. doi: 10.1016/j.jde.2016.09.002 CrossRef Google Scholar
[11]	B. Li and M. Han, Exact peakon solutions given by the generalized hyperbolic functions for some nonlinear wave equations, Journal of Applied Analysis and Computation, 2020, 10(4), 1708-1719. doi: 10.11948/20200139 CrossRef Google Scholar
[12]	Z. Li, Q. Li, P. He, J. Liang, W. Liu and G. Fu, Domain-wall solutions of spinor Bose-Einstein condensates in an optical lattice, Physical Review A., 2010, 81, 015602. doi: 10.1103/PhysRevA.81.015602 CrossRef Google Scholar
[13]	J. Liang and J. Li, Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity, Journal of Applied Analysis and Com- putation, 2018, 8, 1194-1210. Google Scholar
[14]	R. Liu, H. Liu and J. Xin, Attractor for the non-autonomous long wave-short wave resonance interaction equation with damping, Journal of Applied Analysis and Computation, 2020, 10(3), 1149-1169. doi: 10.11948/20190246 CrossRef Google Scholar
[15]	J. Wang and L. Tian, Boundary controllability for the time-fractional nonlinear korteweg-de vries (KDV) equation, Journal of Applied Analysis and Computation, 2020, 10(2), 411-426. doi: 10.11948/20180018 CrossRef Google Scholar
[16]	S. Wu, Laplace inversion for the solution of an abstract heat equation without the forward transform of the source term, Journal of Numerical Mathematics, 2017, 25(3), 185-198. Google Scholar
[17]	T. Tao and L. Zhang, On the continous periodic weak solutions of Boussinesq equations, SIAM J. Math. Anal., 2018, 50(1), 1120-1162. doi: 10.1137/17M1115526 CrossRef Google Scholar