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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

 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. 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.  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. Kohno, G. Tatara, J. 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. Sulem, C. 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. 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.   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. 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.  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. Kohno, G. Tatara, J. 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. Sulem, C. 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. 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.   Google Scholar
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