Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T09:37:35.916Z Has data issue: false hasContentIssue false
  • English
  • Français

On stability and asymptotic behaviours for a degenerate Landau–Lifshitz equation

Published online by Cambridge University Press:  03 June 2015

Baisheng Yan
Baisheng Yan*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China, and Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA, ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the problem concerning stability and asymptotic behaviours of solutions for a degenerate Landau–Lifshitz equation in micromagnetics involving only the non-local magnetostatic energy. Due to the lack of derivative estimates, we do not have the compactness needed for strong convergence and the natural convergence is weak* convergence. By formulating the problem in a new framework of differential inclusions, we show that the Cauchy problems for such an equation are not stable under the weak* convergence of initial data. For the asymptotic behaviours of weak solutions, we establish an estimate on the weak* ω-limit sets that is valid for all initial data satisfying the saturation condition.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 657 - 668
Copyright
Copyright © Royal Society of Edinburgh 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Adams, R. and Fournier, J.. Sobolev spaces, 2nd edn (Academic Press, 2003).Google Scholar
2 Alouges, F. and Soyeur, A.. On global weak solutions for Landau–Lifshitz equations: existence and nonuniqueness. Nonlin. Analysis 18 (1992), 1071–1084.Google Scholar
3 Brown, W. Jr. Micromagnetics (Wiley Interscience, 1963).Google Scholar
4 Carbou, G. and Fabrie, P.. Time average in micromagnetics. J. Diff. Eqns 147 (1998), 383–409.Google Scholar
5 Carbou, G. and Fabrie, P.. Regular solutions for Landau–Lifshitz equation in a bounded domain. Diff. Integ. Eqns 14 (2001), 213–229.Google Scholar
6 Dacorogna, B.. Direct methods in the calculus of variations, 2nd edn (Springer, 2008).Google Scholar
7 Dacorogna, B. and Marcellini, P.. Implicit partial differential equations (Birkhäuser, 1999).Google Scholar
8 Simone, A. De. Energy minimizers for large ferromagnetic bodies. Arch. Ration. Mech. Analysis 125 (1993), 99–143.Google Scholar
9 Deng, W. and Yan, B.. Quasi-stationary limit and a degenerate Landau–Lifshitz equation of ferromagnetism. Appl. Math. Res. eXpress 2013 (2013), 277–296.Google Scholar
10 Gilbarg, D. and Trudinger, N.. Elliptic partial differential equations of second order, 2nd edn (Springer, 1983).Google Scholar
11 James, R. and Kinderlehrer, D.. Frustration in ferromagnetic materials. Continuum Mech. Thermodyn. 2 (1990), 215–239.Google Scholar
12 Jochmann, F.. Existence of solutions and a quasi-stationary limit for a hyperbolic system describing ferromagnetism. SIAM J. Math. Analysis 34 (2002), 315–340.Google Scholar
13 Jochmann, F.. Aysmptotic behavior of the electromagnetic field for a micromagnetism equation without exchange energy. SIAM J. Math. Analysis 37 (2005), 276–290.Google Scholar
14 Joly, P., Komech, A. and Vacus, O.. On transitions to stationary states in a Maxwell–Landau–Lifshitz–Gilbert system. SIAM J. Math. Analysis 31 (1999), 346–374.Google Scholar
15 Joly, J., Metivier, G. and Rauch, J.. Global solutions to Maxwell equations in a ferromagnetic medium. Annales H. Poincaré 1 (2000), 307–340.Google Scholar
16 Kruzík, M. and Prohl, A.. Recent developments in the modeling, analysis, and numerics of ferromagnetism. SIAM Rev. 48 (2006), 439–483.Google Scholar
17 Landau, L. and Lifshitz, E.. On the theory of the dispersion of magnetic permeability of ferromagnetic bodies. Phys. Z. Sowj. 8 (1935), 153–169.Google Scholar
18 Morrey, C. Jr. Quasiconvexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952), 25–53.Google Scholar
19 Pedregal, P. and Yan, B.. A duality method for micromagnetics. SIAM J. Math. Analysis 41 (2010), 2431–2452.Google Scholar
20 Visintin, A.. On Landau–Lifshitz equation for ferromagnetism. Jpn. J. Appl. Math. 2 (1985), 69–84.Google Scholar
21 Yan, B.. On the equilibrium set of magnetostatic energy by differential inclusion. Calc. Var. 47 (2013), 547–565.Google Scholar