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 & 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, 18 (1992), 1071.  doi: 10.1016/0362-546X(92)90196-L.  Google Scholar

[2]

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

[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.  doi: 10.1007/BF02505962.  Google Scholar

[4]

W. F. Brown, Micromagnetics,, Interscience, (1963).   Google Scholar

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

[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.  doi: 10.1016/j.na.2006.12.023.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1002/cpa.3028.  Google Scholar

[13]

A. DeSimone, R. V. Kohn, S. Müller and F. Otto, Recent analytical developments in micromagnetics,, in The Science of Hysteresis, 2 (2005), 269.   Google Scholar

[14]

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

[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.  doi: 10.1007/BF01191298.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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.  doi: 10.1137/S0036141097329949.  Google Scholar

[21]

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

[22]

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

[23]

L. Landau, E. Lifshitz and L. Pitaevskii, Electrodynamics of Continuous Media,, Pergamon Press, (1984).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).   Google Scholar

[29]

L. Tartar, The compensated compactness method applied to systems of conservation laws,, in Systems of Nonlinear Partial Differential Equations, (1983), 263.   Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

show all references

References:
[1]

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

[2]

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

[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.  doi: 10.1007/BF02505962.  Google Scholar

[4]

W. F. Brown, Micromagnetics,, Interscience, (1963).   Google Scholar

[5]

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

[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.  doi: 10.1016/j.na.2006.12.023.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1002/cpa.3028.  Google Scholar

[13]

A. DeSimone, R. V. Kohn, S. Müller and F. Otto, Recent analytical developments in micromagnetics,, in The Science of Hysteresis, 2 (2005), 269.   Google Scholar

[14]

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

[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.  doi: 10.1007/BF01191298.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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.  doi: 10.1137/S0036141097329949.  Google Scholar

[21]

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

[22]

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

[23]

L. Landau, E. Lifshitz and L. Pitaevskii, Electrodynamics of Continuous Media,, Pergamon Press, (1984).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).   Google Scholar

[29]

L. Tartar, The compensated compactness method applied to systems of conservation laws,, in Systems of Nonlinear Partial Differential Equations, (1983), 263.   Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

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