# American Institute of Mathematical Sciences

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., ().

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##### 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., ().
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