Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T07:53:51.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Landau-Lifshitz equation of ferromagnetism with external magnetic field

Part of: Applications to specific types of physical systems Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  09 April 2009

P. Y. H. Pang ,
J. Xiao  and
F. Zhou
P. Y. H. Pang
Affiliation:
Department of Mathematics, National University of singapore, 2 Science Drive 2, Republic of Singapore 117543 e-mail: [email protected]
J. Xiao
Affiliation:
Department of Mathematics, National University of singapore, 2 Science Drive 2, Republic of Singapore 117543
F. Zhou
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200062, People's Republic of China, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article, we prove the existence and uniqueness of solution for the Cauchy problem of the Landau-Lifshitz equation of ferromagnetism with external magnetic field. We also show that the solution is globally regular with the exception of at most finitely many blow-up points. An energy identity at blow-up points is presented.

Keywords

Landau-Lifshitz equationregularityenergy identity

MSC classification

Secondary: 35Q60: PDEs in connection with optics and electromagnetic theory 35B65: Smoothness and regularity of solutions 82D40: Magnetic materials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 299 - 316
Copyright
Copyright © Australian Mathematical Society 2002

References

[1] Chang, K. C. and Ding, W. Y., ‘A result on the global existence for the heat flows of harmonic maps from D 2 into S 2’, in: Nemantics: mathematical and physical aspects (eds. Coron, J.-M., Ghidaglia, J.-M. and Helein, F.) (Kluwer, Dordrecht, 1990) pp. 37ߝ48.Google Scholar
[2] Chang, K. C., Ding, W. Y. and Ye, R., ‘Finite time blow-up of the heat flow of harmonic maps from surfaces’, J. Differential Geom. 36 (1992), 507ߝ515.CrossRefGoogle Scholar
[3] Chen, Y. M., ‘Blow-up analysis for heat flow of harmonic maps’, in: Nemantics: mathematical and physical aspects (eds. Coron, J.-M., Ghidaglia, J.-M. and Helein, F.) (Kluwer, Dordrecht, 1990) pp. 49ߝ64.Google Scholar
[4] Chen, Y. M. and Guo, B., ‘Two dimensional Landau-Lifshitz equation’, J. Partial Differential Equations 9 (1996), 313ߝ322.Google Scholar
[5] Ding, W. Y. and Tian, G., ‘Energy identity of a class of approximate harmonic maps from surfaces’, Comm. Anal. Geom. 4 (1995), 543ߝ554.CrossRefGoogle Scholar
[6] Guo, B. and Hong, M. C., ‘The Landau-Lifshitz equations of the ferromagnetic chain and harmonic maps’, Calc. Var. Partial Differential Equations 1 (1993), 311ߝ334.CrossRefGoogle Scholar
[7] Hamilton, R. S., Harmonic maps of manifolds with boundary, Lecture Notes in Math. 471 (Springer, Berlin, 1975).CrossRefGoogle Scholar
[8] Hong, M. C., ‘The Landau-Lifshitz equations with the external field—a new extension for the harmonic maps with values in S 2 ’, Math. Z. 220 (1995), 171ߝ188.CrossRefGoogle Scholar
[9] Hong, M. C. and Lemaire, L., ‘Multiple solutions of the static Landau-Lifshitz equation from B 2 to S 2 ’, Math. Z. 220 (1995), 295ߝ306.CrossRefGoogle Scholar
[10] Landau, L. D. and Lifshitz, E. M., ‘On the theory of the dispersion of magnetic permeability in ferromagnetic bodies’, in Phys. Z. Sowj. 8 (1935), 153;Google Scholar
reproduced in Collected works of L. D. Landau (Pergamon Press, New York, 1965) pp. 101–114.Google Scholar
[11] Lin, F. H. and Wang, C. Y., ‘Energy identity of harmonic maps from surfaces at finite singular time’, Calc. Var. Partial Differential Equations 6 (1998), 369ߝ380.Google Scholar
[12] Qing, J., ‘On singularities of the heat flow for harmonic maps from surfaces into spheres’, Comm. Anal. Geom. 2 (1995), 297ߝ315.CrossRefGoogle Scholar
[13] Qing, J. and Tian, G., ‘Bubbling of the heat flows for harmonic maps from surfaces’, Comm. Pure Appl. Math. 50 (1997), 295ߝ310.3.0.CO;2-5>CrossRefGoogle Scholar
[14] Shen, C. L. and Zhou, Q., ‘The Landau-Lifshitz equations with ferro-magnetic chain’, Chinese Ann Math. Ser. A 18 (1997), 253ߝ262.Google Scholar
[15] Struwe, M., ‘On the evolution of harmonic maps of Riemannian surfaces’, Comm. Math. Helv. 60 (1985), 558ߝ581.CrossRefGoogle Scholar
[16] Topping, P. M., ‘Rigidity in the harmonic map heat flow’, J. Differential Geom. 45 (1997), 593ߝ610.CrossRefGoogle Scholar
[17] Wang, Y. D., ‘Heisenberg chain systems from compact manifolds into S 2 ’, J. Math. Phys. 39 (1998), 363ߝ371.CrossRefGoogle Scholar