• Previous Article
    Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound
  • CPAA Home
  • This Issue
  • Next Article
    Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions
January  2021, 20(1): 319-338. doi: 10.3934/cpaa.2020268

Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

2. 

School of Mathematics and Information Sciences, Guangzhou University

3. 

Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, AMSS, and School of Mathematical Sciences, UCAS, Beijing 100190, China

* Corresponding author

Received  January 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: The authors are supported partially by NSFC grant (No.11731001). The author Y. Wang is supported partially by NSFC grant (No.11971400) and Guangdong Basic and Applied Basic Research Foundation Grant (No. 2020A1515011019)

We follow the idea of Wang [21] to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a $ n $-dimensional Euclidean domain $ \Omega $ or a $ n $-dimensional closed Riemannian manifold $ M $ into a 2-dimensional unit sphere $ \mathbb{S}^{2} $. Our conclusions extend a series of related results obtained in the previous literature.

Citation: Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure and Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
References:
[1]

F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness, Nonlinear Anal., 18 (1992), 1071-1084.  doi: 10.1016/0362-546X(92)90196-L.

[2]

L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B, 54 (1996), 9353.

[3]

F. Bethuel, J. M. Coron, J. M. Ghidaglia and A. Soyeur, Nonlinear Diffusion Equations and Their Equilibrium States, 3, Gregynog, Birkhaüser, 1989. doi: 10.1007/978-1-4612-0393-3_7.

[4]

G. Bonithon, Landau-Lifschitz-Gilbert equation with applied electric current, Discrete Contin. Dyn. Syst., (2007), 138-144.

[5]

G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differ. Integral Equ., 14 (2001), 213-229. 

[6]

G. Carbou and R. Jizzini, Very regular solutions for the Landau-Lifschitz equation with electric current, Chin. Ann. Math. Ser. B, 39 (2018), 889-916.  doi: 10.1007/s11401-018-0103-7.

[7]

B. Chen and Y. D. Wang, Finite-time blow up for heat flow of self-induced harmonic maps, preprint. doi: 10.1512/iumj.2015.64.5499.

[8]

Y. M. Chen, The weak solutions to the Evolution problems of harmonic maps, Math. Z., 201 (1989), 69-74.  doi: 10.1007/BF01161995.

[9]

Y. M. ChenM. C. Hong and N. Hungerbühler, Heat flow of p-harmonic maps with values into sphere, Math. Z., 215 (1994), 25-35.  doi: 10.1007/BF02571698.

[10]

W. Y. Ding and Y. D. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A, 44 (2001), 1446-1464.  doi: 10.1007/BF02877074.

[11]

T. L. Gilbert, A Lagrangian formulation of gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243-1255. 

[12]

Z. L. Jia and Y. D. Wang, Global weak solutions to Landau-Lifshitz equations into compact Lie algebras, Front. Math. China, 14 (2019), 1163-1196.  doi: 10.1007/s11464-019-0803-7.

[13]

H. KohnoG. TataraJ. Shibata and Y. Suzuki, Microscopic calculation of spin torques and forces, J. Magn. Magn. Mater., 310 (2006), 2020-2022. 

[14]

L. D. Landau and E. M. Lifshitz, On the theory of dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Soviet., 8 (1935), 153-169. 

[15]

F. H. Lin, Nonlinear theory of defexts in nematic liquid crystals; phase transition and flow phenomena, Commun. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.

[16]

J. Simon, Compact sets in the space $L^p([0, T];B)$, Ann. Mat. Pura. Appl., 4 (1987), 65-96.  doi: 10.1007/BF01762360.

[17]

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

[18]

J. C. Slonczewski, Current-driven excitation of magnetic multilayer, J. Magn. Magn. Mater., 159 (1996), 635-642. 

[19]

M. Tilioua, Current-induced magnetization dynamics. Global existence of weak solutions, J. Math. Anal. Appl., 373 (2011), 635-642.  doi: 10.1016/j.jmaa.2010.08.024.

[20]

A. Visintin, On Landau-Lifshitz' equations for ferromagnetism, Japan J. Appl. Math., 2 (1985), 69-84.  doi: 10.1007/BF03167039.

[21]

Y. D. Wang, Heisenberg chain systems from compact manifolds into $ \mathbb{S}^2$, J. Math. Phy., 39 (1998), 363-371.  doi: 10.1063/1.532335.

[22]

K. Wehrheim, Uhlenbeck Compactness, EMS Publishing House, USA, 2004. doi: 10.4171/004.

[23]

Y. ZhouB. Guo and S. B. Tan, Existence and uniqueness of smooth solution for system of ferro-magnetic chain, Sci. China Ser. A, 34 (1991), 257-266. 

show all references

References:
[1]

F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness, Nonlinear Anal., 18 (1992), 1071-1084.  doi: 10.1016/0362-546X(92)90196-L.

[2]

L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B, 54 (1996), 9353.

[3]

F. Bethuel, J. M. Coron, J. M. Ghidaglia and A. Soyeur, Nonlinear Diffusion Equations and Their Equilibrium States, 3, Gregynog, Birkhaüser, 1989. doi: 10.1007/978-1-4612-0393-3_7.

[4]

G. Bonithon, Landau-Lifschitz-Gilbert equation with applied electric current, Discrete Contin. Dyn. Syst., (2007), 138-144.

[5]

G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differ. Integral Equ., 14 (2001), 213-229. 

[6]

G. Carbou and R. Jizzini, Very regular solutions for the Landau-Lifschitz equation with electric current, Chin. Ann. Math. Ser. B, 39 (2018), 889-916.  doi: 10.1007/s11401-018-0103-7.

[7]

B. Chen and Y. D. Wang, Finite-time blow up for heat flow of self-induced harmonic maps, preprint. doi: 10.1512/iumj.2015.64.5499.

[8]

Y. M. Chen, The weak solutions to the Evolution problems of harmonic maps, Math. Z., 201 (1989), 69-74.  doi: 10.1007/BF01161995.

[9]

Y. M. ChenM. C. Hong and N. Hungerbühler, Heat flow of p-harmonic maps with values into sphere, Math. Z., 215 (1994), 25-35.  doi: 10.1007/BF02571698.

[10]

W. Y. Ding and Y. D. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A, 44 (2001), 1446-1464.  doi: 10.1007/BF02877074.

[11]

T. L. Gilbert, A Lagrangian formulation of gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243-1255. 

[12]

Z. L. Jia and Y. D. Wang, Global weak solutions to Landau-Lifshitz equations into compact Lie algebras, Front. Math. China, 14 (2019), 1163-1196.  doi: 10.1007/s11464-019-0803-7.

[13]

H. KohnoG. TataraJ. Shibata and Y. Suzuki, Microscopic calculation of spin torques and forces, J. Magn. Magn. Mater., 310 (2006), 2020-2022. 

[14]

L. D. Landau and E. M. Lifshitz, On the theory of dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Soviet., 8 (1935), 153-169. 

[15]

F. H. Lin, Nonlinear theory of defexts in nematic liquid crystals; phase transition and flow phenomena, Commun. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.

[16]

J. Simon, Compact sets in the space $L^p([0, T];B)$, Ann. Mat. Pura. Appl., 4 (1987), 65-96.  doi: 10.1007/BF01762360.

[17]

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

[18]

J. C. Slonczewski, Current-driven excitation of magnetic multilayer, J. Magn. Magn. Mater., 159 (1996), 635-642. 

[19]

M. Tilioua, Current-induced magnetization dynamics. Global existence of weak solutions, J. Math. Anal. Appl., 373 (2011), 635-642.  doi: 10.1016/j.jmaa.2010.08.024.

[20]

A. Visintin, On Landau-Lifshitz' equations for ferromagnetism, Japan J. Appl. Math., 2 (1985), 69-84.  doi: 10.1007/BF03167039.

[21]

Y. D. Wang, Heisenberg chain systems from compact manifolds into $ \mathbb{S}^2$, J. Math. Phy., 39 (1998), 363-371.  doi: 10.1063/1.532335.

[22]

K. Wehrheim, Uhlenbeck Compactness, EMS Publishing House, USA, 2004. doi: 10.4171/004.

[23]

Y. ZhouB. Guo and S. B. Tan, Existence and uniqueness of smooth solution for system of ferro-magnetic chain, Sci. China Ser. A, 34 (1991), 257-266. 

[1]

Zonglin Jia, Youde Wang. Global weak solutions to Landau-Lifshtiz systems with spin-polarized transport. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1903-1935. doi: 10.3934/dcds.2020099

[2]

Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87

[3]

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

[4]

Carlos J. García-Cervera, Xiao-Ping Wang. A note on 'Spin-polarized transport: Existence of weak solutions'. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2761-2763. doi: 10.3934/dcdsb.2015.20.2761

[5]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[6]

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

[7]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133

[8]

Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199

[9]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867

[10]

Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic and Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1

[11]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643

[12]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[13]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116

[14]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357

[15]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[16]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211

[17]

Catherine Choquet, Mohammed Moumni, Mouhcine Tilioua. Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 35-57. doi: 10.3934/dcdss.2018003

[18]

Gaël Bonithon. Landau-Lifschitz-Gilbert equation with applied eletric current. Conference Publications, 2007, 2007 (Special) : 138-144. doi: 10.3934/proc.2007.2007.138

[19]

Yasir Ali, Arshad Alam Khan. Exact solution of magnetohydrodynamic slip flow and heat transfer over an oscillating and translating porous plate. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 595-606. doi: 10.3934/dcdss.2018034

[20]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks and Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (217)
  • HTML views (50)
  • Cited by (0)

Other articles
by authors

[Back to Top]