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

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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

Bo Chen ^1, and Youde Wang ^2,3,,

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.

Keywords: Landau Lifshitz flow, spin-polarized current, heat flow, global weak solution, auxiliary approximation equation.

Mathematics Subject Classification: Primary: 35Q60, 78A25; Secondary: 58J35.

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. Chen, M. 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. Kohno, G. Tatara, J. 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. Sulem, C. 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. Zhou, B. 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. Chen, M. 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. Kohno, G. Tatara, J. 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. Sulem, C. 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. Zhou, B. 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.

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