March  2020, 40(3): 1903-1935. doi: 10.3934/dcds.2020099

Global weak solutions to Landau-Lifshtiz systems with spin-polarized transport

1. 

Institute of Applied Physics and Computational Mathematics, China Academy of Engineering Physics, Beijing 100088, China

2. 

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

* Corresponding author: Youde Wang

Received  July 2019 Revised  September 2019 Published  December 2019

Fund Project: The authors are supported by NSFC grant No.11731001

In this paper, we consider the Landau-Lifshitz-Gilbert systems with spin-polarized transport from a bounded domain in $ \mathbb{R}^3 $ into $ S^2 $ and show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz systems. In particular, we show that the Cauchy problem to Landau-Lifshitz equation without damping but with diffusion process of the spin accumulation admits a global weak solution. The Landau-Lifshtiz system with spin-polarized transport into a compact Lie algebra is also discussed and some similar results are proved. The key ingredients of this article consist of the choices of test functions and approximate equations.

Citation: Zonglin Jia, Youde Wang. Global weak solutions to Landau-Lifshtiz systems with spin-polarized transport. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1903-1935. doi: 10.3934/dcds.2020099
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

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

[3]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998.  Google Scholar

[4]

R. Balakrishnan, On the inhomogeneous Heisenberg chain, Journal of Physics C: Solid State Physics, 15 (1982), 1305-1308.  doi: 10.1088/0022-3719/15/36/007.  Google Scholar

[5]

X. ChenR. Q. Jiang and Y. D. Wang, A class of periodic solutions of one-dimensional Landau-Lifshitz equations, J. Math. Study, 50 (2017), 199-214.  doi: 10.4208/jms.v50n3.17.01.  Google Scholar

[6]

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

[7]

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

[8]

M. DanielK. Porsezian and M. Lakshmanan, On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions, Journal of Mathematical Physics, 35 (1994), 6498-6510.  doi: 10.1063/1.530687.  Google Scholar

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S. J. Ding and B. L. Guo, Existence of partially regular weak solutions to Landau-Lifshitz-Maxwell equations, J. Differential Equations, 244 (2008), 2448-2472.  doi: 10.1016/j.jde.2008.02.029.  Google Scholar

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S. J. DingX. G. Liu and C. Y. Wang, The Landau-Lifshitz-Maxwell equation in dimension three, Pacific J. Math, 243 (2009), 243-276.  doi: 10.2140/pjm.2009.243.243.  Google Scholar

[11]

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

[12]

W. Y. DingH. Y. Wang and Y. D. Wang, Schrödinger flows on compact Hermitian symmetric spaces and related problems, Acta Math. Sin. (Engl. Ser.), 7 (2018), 513-522.  doi: 10.1142/9789813220881_0042.  Google Scholar

[13]

C. J. Garcia-Cervera and X.-P. Wang, Spin-polarized transport: Existence of weak solutions, Discre. Contin. Dynam. Syst. Series B, 7 (2007), 87-100.  doi: 10.3934/dcdsb.2007.7.87.  Google Scholar

[14]

C. J. Garcia-Cervera and X.-P. Wang, A note on 'Spin-polarized transport: Existence of weak solutions', Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2761-2763.  doi: 10.3934/dcdsb.2015.20.2761.  Google Scholar

[15]

K. Gärtner and A. Glitzky, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, SIAM J. Math. Anal., 41 (2009/10), 2489-2513.  doi: 10.1137/080736454.  Google Scholar

[16]

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

[17]

A. Glitzky, Analysis of a spin-polarized drift-diffusion model, Adv. Math. Sci. Appl., 18 (2008), 401-427.   Google Scholar

[18]

R. El Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation, Commun. Math. Sci., 12 (2014), 565-592.  doi: 10.4310/CMS.2014.v12.n3.a9.  Google Scholar

[19]

Z. L. Jia and Y. D. Wang, Local nonautonomous Schrödinger flows on Kähler manifolds, Acta Mathematica Sinica, English Series, 35 (2019), 1251-1299.  doi: 10.1007/s10114-019-8303-y.  Google Scholar

[20]

Z. L. Jia and Y. D. Wang, Global weak solutions to Landau-Lifshitz equations into compact Lie algebras, Acta Mathematica Sinica, English Series, 35 (2019), 1251–1299, arXiv: math/1802.00135. Google Scholar

[21]

J. L. JolyG. Métivier and J. Rauch, Global solutions to Maxwell equations in a ferromagnetic medium, Ann. Henri Poincaré, 1 (2000), 307-340.  doi: 10.1007/PL00001007.  Google Scholar

[22]

O. A. Ladyzhenskaya, The Boundary Value Problem of Mathematical Physics, Applied Mathematical Sciences, 49. Springer-Verlag, New York, 1985. doi: 10.1063/1.3048363.  Google Scholar

[23]

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.  doi: 10.1016/B978-0-08-010523-9.50018-3.  Google Scholar

[24]

C. Melcher, Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions, Indiana Univ. Math. J., 61 (2012), 1175-1200.  doi: 10.1512/iumj.2012.61.4717.  Google Scholar

[25]

S. Possanner and C. Negulescu, Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport, Kinet. Relat. Models, 4 (2011), 1159-1191.  doi: 10.3934/krm.2011.4.1159.  Google Scholar

[26]

X. K. PuM. Wang and W. D. Wang, The Landau-Lifshitz equation of the ferromagnetic spin chain and Oseen-Frank flow, SIAM J. Math. Anal., 49 (2017), 5134-5157.  doi: 10.1137/16M1094907.  Google Scholar

[27]

X. K. Pu and W. D. Wang, Partial regularity to the Landau-Lifshitz equation with spin accumulation, J. Differential Equations, 268 (2020), 707–737, arXiv: math/1808.01798. doi: 10.1016/j.jde.2019.08.034.  Google Scholar

[28]

A. Shpiro, P. M. Levy and S. F. Zhang, Self-consistent treatment of nonequilibrium spin torques in magnetic multilayers, Phys. Rev. B, 67 (2003), 104430. doi: 10.1103/PhysRevB.67.104430.  Google Scholar

[29]

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

[30]

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

[31] M. Wang, Nonlinear Elliptic Equations, Science Press, Beijing, 2010.   Google Scholar
[32]

C. Y. Wang, On Landau-Lifshitz equation in dimensions at most four, Indiana Univ. Math. J., 55 (2006), 1615-1644.  doi: 10.1512/iumj.2006.55.2810.  Google Scholar

[33]

Y.-D. Wang, Heisenberg chain systems from compact manifolds into $S^2$, J. Math. Phys., 39 (1998), 363-371.  doi: 10.1063/1.532335.  Google Scholar

[34] W. XiangZ. Hou and D. Meng, Lectures of Lie Group, Higher Education Press, Beijing, 2014.   Google Scholar
[35]

N. Zamponi, Analysis of a drift-diffusion model with velocity saturation for spin-polarized transport in semiconductors, J. Math. Anal. Appl., 420 (2014), 1167-1181.  doi: 10.1016/j.jmaa.2014.06.065.  Google Scholar

[36]

N. Zamponi and A. Jüngel, Analysis of a coupled spin drift-diffusion Maxwell-Landau-Lifshitz system, J. Differential Equations, 260 (2016), 6828-6854.  doi: 10.1016/j.jde.2016.01.010.  Google Scholar

[37]

S. Zhang, P. M. Levy and A. Fert, Mechanisms of spin-polarized current-driven magnetication switching, Phys. Rev. Lett., 88 (2002), 236601. doi: 10.1103/PhysRevLett.88.236601.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

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

[3]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998.  Google Scholar

[4]

R. Balakrishnan, On the inhomogeneous Heisenberg chain, Journal of Physics C: Solid State Physics, 15 (1982), 1305-1308.  doi: 10.1088/0022-3719/15/36/007.  Google Scholar

[5]

X. ChenR. Q. Jiang and Y. D. Wang, A class of periodic solutions of one-dimensional Landau-Lifshitz equations, J. Math. Study, 50 (2017), 199-214.  doi: 10.4208/jms.v50n3.17.01.  Google Scholar

[6]

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

[7]

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

[8]

M. DanielK. Porsezian and M. Lakshmanan, On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions, Journal of Mathematical Physics, 35 (1994), 6498-6510.  doi: 10.1063/1.530687.  Google Scholar

[9]

S. J. Ding and B. L. Guo, Existence of partially regular weak solutions to Landau-Lifshitz-Maxwell equations, J. Differential Equations, 244 (2008), 2448-2472.  doi: 10.1016/j.jde.2008.02.029.  Google Scholar

[10]

S. J. DingX. G. Liu and C. Y. Wang, The Landau-Lifshitz-Maxwell equation in dimension three, Pacific J. Math, 243 (2009), 243-276.  doi: 10.2140/pjm.2009.243.243.  Google Scholar

[11]

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

[12]

W. Y. DingH. Y. Wang and Y. D. Wang, Schrödinger flows on compact Hermitian symmetric spaces and related problems, Acta Math. Sin. (Engl. Ser.), 7 (2018), 513-522.  doi: 10.1142/9789813220881_0042.  Google Scholar

[13]

C. J. Garcia-Cervera and X.-P. Wang, Spin-polarized transport: Existence of weak solutions, Discre. Contin. Dynam. Syst. Series B, 7 (2007), 87-100.  doi: 10.3934/dcdsb.2007.7.87.  Google Scholar

[14]

C. J. Garcia-Cervera and X.-P. Wang, A note on 'Spin-polarized transport: Existence of weak solutions', Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2761-2763.  doi: 10.3934/dcdsb.2015.20.2761.  Google Scholar

[15]

K. Gärtner and A. Glitzky, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, SIAM J. Math. Anal., 41 (2009/10), 2489-2513.  doi: 10.1137/080736454.  Google Scholar

[16]

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

[17]

A. Glitzky, Analysis of a spin-polarized drift-diffusion model, Adv. Math. Sci. Appl., 18 (2008), 401-427.   Google Scholar

[18]

R. El Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation, Commun. Math. Sci., 12 (2014), 565-592.  doi: 10.4310/CMS.2014.v12.n3.a9.  Google Scholar

[19]

Z. L. Jia and Y. D. Wang, Local nonautonomous Schrödinger flows on Kähler manifolds, Acta Mathematica Sinica, English Series, 35 (2019), 1251-1299.  doi: 10.1007/s10114-019-8303-y.  Google Scholar

[20]

Z. L. Jia and Y. D. Wang, Global weak solutions to Landau-Lifshitz equations into compact Lie algebras, Acta Mathematica Sinica, English Series, 35 (2019), 1251–1299, arXiv: math/1802.00135. Google Scholar

[21]

J. L. JolyG. Métivier and J. Rauch, Global solutions to Maxwell equations in a ferromagnetic medium, Ann. Henri Poincaré, 1 (2000), 307-340.  doi: 10.1007/PL00001007.  Google Scholar

[22]

O. A. Ladyzhenskaya, The Boundary Value Problem of Mathematical Physics, Applied Mathematical Sciences, 49. Springer-Verlag, New York, 1985. doi: 10.1063/1.3048363.  Google Scholar

[23]

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.  doi: 10.1016/B978-0-08-010523-9.50018-3.  Google Scholar

[24]

C. Melcher, Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions, Indiana Univ. Math. J., 61 (2012), 1175-1200.  doi: 10.1512/iumj.2012.61.4717.  Google Scholar

[25]

S. Possanner and C. Negulescu, Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport, Kinet. Relat. Models, 4 (2011), 1159-1191.  doi: 10.3934/krm.2011.4.1159.  Google Scholar

[26]

X. K. PuM. Wang and W. D. Wang, The Landau-Lifshitz equation of the ferromagnetic spin chain and Oseen-Frank flow, SIAM J. Math. Anal., 49 (2017), 5134-5157.  doi: 10.1137/16M1094907.  Google Scholar

[27]

X. K. Pu and W. D. Wang, Partial regularity to the Landau-Lifshitz equation with spin accumulation, J. Differential Equations, 268 (2020), 707–737, arXiv: math/1808.01798. doi: 10.1016/j.jde.2019.08.034.  Google Scholar

[28]

A. Shpiro, P. M. Levy and S. F. Zhang, Self-consistent treatment of nonequilibrium spin torques in magnetic multilayers, Phys. Rev. B, 67 (2003), 104430. doi: 10.1103/PhysRevB.67.104430.  Google Scholar

[29]

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

[30]

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

[31] M. Wang, Nonlinear Elliptic Equations, Science Press, Beijing, 2010.   Google Scholar
[32]

C. Y. Wang, On Landau-Lifshitz equation in dimensions at most four, Indiana Univ. Math. J., 55 (2006), 1615-1644.  doi: 10.1512/iumj.2006.55.2810.  Google Scholar

[33]

Y.-D. Wang, Heisenberg chain systems from compact manifolds into $S^2$, J. Math. Phys., 39 (1998), 363-371.  doi: 10.1063/1.532335.  Google Scholar

[34] W. XiangZ. Hou and D. Meng, Lectures of Lie Group, Higher Education Press, Beijing, 2014.   Google Scholar
[35]

N. Zamponi, Analysis of a drift-diffusion model with velocity saturation for spin-polarized transport in semiconductors, J. Math. Anal. Appl., 420 (2014), 1167-1181.  doi: 10.1016/j.jmaa.2014.06.065.  Google Scholar

[36]

N. Zamponi and A. Jüngel, Analysis of a coupled spin drift-diffusion Maxwell-Landau-Lifshitz system, J. Differential Equations, 260 (2016), 6828-6854.  doi: 10.1016/j.jde.2016.01.010.  Google Scholar

[37]

S. Zhang, P. M. Levy and A. Fert, Mechanisms of spin-polarized current-driven magnetication switching, Phys. Rev. Lett., 88 (2002), 236601. doi: 10.1103/PhysRevLett.88.236601.  Google Scholar

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

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