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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])
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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 

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