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Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions
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 |
We follow the idea of Wang [
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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