December  2013, 2(4): 599-620. doi: 10.3934/eect.2013.2.599

On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics

1. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States

2. 

Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, United States

Received  September 2012 Revised  February 2013 Published  November 2013

In this paper, we study Landau-Lifshitz equations of ferromagnetism with a total energy that does not include a so-called exchange energy. Many problems, including existence, stability, regularity and asymptotic behaviors, have been extensively studied for such equations of models with the exchange energy. The problems turn out quite different and challenging for Landau-Lifshitz equations of no-exchange energy models because the usual methods based on certain compactness do not apply. We present a new method for the existence of global weak solution to the Landau-Lifshitz equation of no-exchange energy models based on the existence of regular solutions for smooth data and certain stability of the solutions. We also study higher time regularity, energy identity and asymptotic behaviors in some special cases for weak solutions.
Citation: Wei Deng, Baisheng Yan. On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics. Evolution Equations and Control Theory, 2013, 2 (4) : 599-620. doi: 10.3934/eect.2013.2.599
References:
[1]

F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and Nonuniqueness, Nonlinear Analysis, TMA, 18 (1992), 1071-1084. doi: 10.1016/0362-546X(92)90196-L.

[2]

J. M. Ball, A. Taheri and M. Winter, Local minimizers in micromagnetics and related problems, Calc. Var., 14 (2002), 1-27. doi: 10.1007/s005260100085.

[3]

M. Bertsch, P. Podio-Guidugli and V. Valente, On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pura Appl., 179 (2001), 331-360. doi: 10.1007/BF02505962.

[4]

W. F. Brown, Micromagnetics, Interscience, New York, 1963.

[5]

A. Capella, C. Melcher and F. Otto, Effective dynamic in ferromagnetic thin films and the motion of Neel walls, Nonlinearity, 20 (2007), 2519-2537. doi: 10.1088/0951-7715/20/11/004.

[6]

I. Cimrak and R. V. Keer, Higher order regularity results in 3D for the Landau-Lifshitz equation with an exchange field, Nonlinear Analysis, 68 (2008), 1316-1331. doi: 10.1016/j.na.2006.12.023.

[7]

G. Carbou and P. Fabrie, Time average in micromagnetics, J. Diff. Equations, 147 (1998), 383-409. doi: 10.1006/jdeq.1998.3444.

[8]

G. Carbou and P. Fabrie, Regular solutions for Landau-Lifshitz equation in a bounded domain, Diff. Int. Equations, 14 (2001), 213-229.

[9]

B. Dacorogna and I. Fonseca, A-B quasiconvexity and implicit partial differential equations, Calc. Var. Partial Differential Equations, 14 (2002), 115-149. doi: 10.1007/s005260100092.

[10]

W. Deng and B. Yan, Quasi-stationary limit and a degenerate Landau-Lifshitz equation of ferromagnetism, Applied Mathematics Research Express, 2013(2) (2013), 277-296. doi: 10.1093/amrx/abs019.

[11]

A. DeSimone, Energy minimizers for large ferromagnetic bodies, Arch. Rational Mech. Anal., 125 (1993), 99-143. doi: 10.1007/BF00376811.

[12]

A. DeSimone, R. V. Kohn, S. Müller and F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math., 55 (2002), 1408-1460. doi: 10.1002/cpa.3028.

[13]

A. DeSimone, R. V. Kohn, S. Müller and F. Otto, Recent analytical developments in micromagnetics, in The Science of Hysteresis, (eds G Bertotti and I Mayergoyz), Elsevier Academic Press, New York, 2 (2005), 269-381.

[14]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions, Int. J. Engineering Science, 47 (2009), 821-839. doi: 10.1016/j.ijengsci.2009.05.010.

[15]

B. Guo and M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations, 1 (1993), 311-334. doi: 10.1007/BF01191298.

[16]

R. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Cont. Mech. Thermodyn., 2 (1990), 215-239. doi: 10.1007/BF01129598.

[17]

F. Jochmann, Existence of solutions and a quasi-stationary limit for a hyperbolic system describing ferromagnetism, SIAM J. Math. Anal., 34 (2002), 315-340. doi: 10.1137/S0036141001392293.

[18]

F. Jochmann, Aysmptotic behavior of the electromagnetic field for a micromagnetism equation without exchange energy, SIAM J. Math. Anal., 37 (2005), 276-290. doi: 10.1137/S0036141004443324.

[19]

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

[20]

P. Joly, A. Komech and O. Vacus, On transitions to stationary states in a Maxwell-Landau-Lifshitz-Gilbert system, SIAM J. Math. Anal., 31 (1999), 346-374. doi: 10.1137/S0036141097329949.

[21]

M. Kruzík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Review, 48 (2006), 439-483. doi: 10.1137/S0036144504446187.

[22]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability of ferromagnetic bodies, Phys. Z. Sowj., 8 (1935), 153-169.

[23]

L. Landau, E. Lifshitz and L. Pitaevskii, Electrodynamics of Continuous Media, Pergamon Press, New York, 1984.

[24]

C. Melcher, Thin-film limits for Landau-Lifshitz-Gilbert equations, SIAM J. Math. Anal., 42 (2010) , 519-537. doi: 10.1137/090762646.

[25]

R. Moser, Boundary vortices for thin ferromagnetic films, Arch. Rational Mech. Anal., 174 (2004), 267-300. doi: 10.1007/s00205-004-0329-2.

[26]

P. Pedregal and B. Yan, On two-dimensional ferromagnetism, Proc. R. Soc. Edinburgh, 139A (2009), 575-594. doi: 10.1017/S0308210507000662.

[27]

P. Pedregal and B. Yan, A duality method for micromagnetics, SIAM J. Math. Anal., 41 (2010), 2431-2452. doi: 10.1137/080738179.

[28]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp.

[29]

L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, (J. M. Ball ed.), NATO ASI Series, Vol. CIII, D. Reidel, (1983), 263-285.

[30]

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

[31]

B. Yan, Characterization of energy minimizers in micromagnetics, J. Math. Anal. Appl., 374 (2011), 230-243. doi: 10.1016/j.jmaa.2010.08.045.

[32]

B. Yan, On the equilibrium set of magnetostatic energy by differential inclusion, Calc. Var. Partial Differential Equations, 47 (2013), 547-565. doi: 10.1007/s00526-012-0527-y.

[33]

B. Yan, On stability and asymptotic behaviors for a degenerate Landau-Lifshitz equation,, Preprint submitted., (). 

show all references

References:
[1]

F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and Nonuniqueness, Nonlinear Analysis, TMA, 18 (1992), 1071-1084. doi: 10.1016/0362-546X(92)90196-L.

[2]

J. M. Ball, A. Taheri and M. Winter, Local minimizers in micromagnetics and related problems, Calc. Var., 14 (2002), 1-27. doi: 10.1007/s005260100085.

[3]

M. Bertsch, P. Podio-Guidugli and V. Valente, On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pura Appl., 179 (2001), 331-360. doi: 10.1007/BF02505962.

[4]

W. F. Brown, Micromagnetics, Interscience, New York, 1963.

[5]

A. Capella, C. Melcher and F. Otto, Effective dynamic in ferromagnetic thin films and the motion of Neel walls, Nonlinearity, 20 (2007), 2519-2537. doi: 10.1088/0951-7715/20/11/004.

[6]

I. Cimrak and R. V. Keer, Higher order regularity results in 3D for the Landau-Lifshitz equation with an exchange field, Nonlinear Analysis, 68 (2008), 1316-1331. doi: 10.1016/j.na.2006.12.023.

[7]

G. Carbou and P. Fabrie, Time average in micromagnetics, J. Diff. Equations, 147 (1998), 383-409. doi: 10.1006/jdeq.1998.3444.

[8]

G. Carbou and P. Fabrie, Regular solutions for Landau-Lifshitz equation in a bounded domain, Diff. Int. Equations, 14 (2001), 213-229.

[9]

B. Dacorogna and I. Fonseca, A-B quasiconvexity and implicit partial differential equations, Calc. Var. Partial Differential Equations, 14 (2002), 115-149. doi: 10.1007/s005260100092.

[10]

W. Deng and B. Yan, Quasi-stationary limit and a degenerate Landau-Lifshitz equation of ferromagnetism, Applied Mathematics Research Express, 2013(2) (2013), 277-296. doi: 10.1093/amrx/abs019.

[11]

A. DeSimone, Energy minimizers for large ferromagnetic bodies, Arch. Rational Mech. Anal., 125 (1993), 99-143. doi: 10.1007/BF00376811.

[12]

A. DeSimone, R. V. Kohn, S. Müller and F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math., 55 (2002), 1408-1460. doi: 10.1002/cpa.3028.

[13]

A. DeSimone, R. V. Kohn, S. Müller and F. Otto, Recent analytical developments in micromagnetics, in The Science of Hysteresis, (eds G Bertotti and I Mayergoyz), Elsevier Academic Press, New York, 2 (2005), 269-381.

[14]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions, Int. J. Engineering Science, 47 (2009), 821-839. doi: 10.1016/j.ijengsci.2009.05.010.

[15]

B. Guo and M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations, 1 (1993), 311-334. doi: 10.1007/BF01191298.

[16]

R. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Cont. Mech. Thermodyn., 2 (1990), 215-239. doi: 10.1007/BF01129598.

[17]

F. Jochmann, Existence of solutions and a quasi-stationary limit for a hyperbolic system describing ferromagnetism, SIAM J. Math. Anal., 34 (2002), 315-340. doi: 10.1137/S0036141001392293.

[18]

F. Jochmann, Aysmptotic behavior of the electromagnetic field for a micromagnetism equation without exchange energy, SIAM J. Math. Anal., 37 (2005), 276-290. doi: 10.1137/S0036141004443324.

[19]

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

[20]

P. Joly, A. Komech and O. Vacus, On transitions to stationary states in a Maxwell-Landau-Lifshitz-Gilbert system, SIAM J. Math. Anal., 31 (1999), 346-374. doi: 10.1137/S0036141097329949.

[21]

M. Kruzík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Review, 48 (2006), 439-483. doi: 10.1137/S0036144504446187.

[22]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability of ferromagnetic bodies, Phys. Z. Sowj., 8 (1935), 153-169.

[23]

L. Landau, E. Lifshitz and L. Pitaevskii, Electrodynamics of Continuous Media, Pergamon Press, New York, 1984.

[24]

C. Melcher, Thin-film limits for Landau-Lifshitz-Gilbert equations, SIAM J. Math. Anal., 42 (2010) , 519-537. doi: 10.1137/090762646.

[25]

R. Moser, Boundary vortices for thin ferromagnetic films, Arch. Rational Mech. Anal., 174 (2004), 267-300. doi: 10.1007/s00205-004-0329-2.

[26]

P. Pedregal and B. Yan, On two-dimensional ferromagnetism, Proc. R. Soc. Edinburgh, 139A (2009), 575-594. doi: 10.1017/S0308210507000662.

[27]

P. Pedregal and B. Yan, A duality method for micromagnetics, SIAM J. Math. Anal., 41 (2010), 2431-2452. doi: 10.1137/080738179.

[28]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp.

[29]

L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, (J. M. Ball ed.), NATO ASI Series, Vol. CIII, D. Reidel, (1983), 263-285.

[30]

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

[31]

B. Yan, Characterization of energy minimizers in micromagnetics, J. Math. Anal. Appl., 374 (2011), 230-243. doi: 10.1016/j.jmaa.2010.08.045.

[32]

B. Yan, On the equilibrium set of magnetostatic energy by differential inclusion, Calc. Var. Partial Differential Equations, 47 (2013), 547-565. doi: 10.1007/s00526-012-0527-y.

[33]

B. Yan, On stability and asymptotic behaviors for a degenerate Landau-Lifshitz equation,, Preprint submitted., (). 

[1]

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

[2]

Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199

[3]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644

[4]

Ze Li, Lifeng Zhao. Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 607-638. doi: 10.3934/dcds.2019025

[5]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

[6]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867

[7]

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133

[8]

Changyou Wang, Shenzhou Zheng. Energy identity for a class of approximate biharmonic maps into sphere in dimension four. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 861-878. doi: 10.3934/dcds.2013.33.861

[9]

Catherine Choquet, Mohammed Moumni, Mouhcine Tilioua. Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 35-57. doi: 10.3934/dcdss.2018003

[10]

Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463

[11]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056

[12]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[13]

Boling Guo, Fangfang Li. Global smooth solution for the Sipn-Polarized transport equation with Landau-Lifshitz-Bloch equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2825-2840. doi: 10.3934/dcdsb.2020034

[14]

Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic and Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333

[15]

Tram Thi Ngoc Nguyen, Anne Wald. On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging. Inverse Problems and Imaging, 2022, 16 (1) : 89-117. doi: 10.3934/ipi.2021042

[16]

Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices. Networks and Heterogeneous Media, 2012, 7 (1) : 179-196. doi: 10.3934/nhm.2012.7.179

[17]

Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129

[18]

Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. Remark on a semirelativistic equation in the energy space. Conference Publications, 2015, 2015 (special) : 473-478. doi: 10.3934/proc.2015.0473

[19]

Joseph A. Biello, Peter R. Kramer, Yuri Lvov. Stages of energy transfer in the FPU model. Conference Publications, 2003, 2003 (Special) : 113-122. doi: 10.3934/proc.2003.2003.113

[20]

Immanuel Ben Porat. Local conditional regularity for the Landau equation with Coulomb potential. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022010

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]