• 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  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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.   Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.   Google Scholar

[1]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[2]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[3]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[4]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[5]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[6]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021009

[7]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[8]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[9]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321

[10]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[11]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[12]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[13]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[14]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[15]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[16]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[17]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[18]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[19]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[20]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (64)
  • HTML views (44)
  • Cited by (0)

Other articles
by authors

[Back to Top]