Calc. Var. (2017) 56:96
Calculus of Variations
Asymptotic behaviors of Landau–Lifshitz ﬂows from R
to Kähler manifolds
· Lifeng Zhao
Received: 5 January 2017 / Accepted: 4 May 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract In this paper, we study the asymptotic behaviors of ﬁnite energy solutions to the
Landau–Lifshitz ﬂows from R
into Kähler manifolds. First, we prove that the solution with
initial data below the critical energy converges to a constant map in the energy space as
t →∞for the compact Riemannian surface targets. In particular, when the target is a two
dimensional sphere, we prove that the solution to the Landau–Lifshitz–Gilbert equation with
initial data having an energy below 4π converges to some constant map in the energy space.
The proof bases on the method of induction on energy and geometric renormalizations.
Second, for general compact Kähler manifolds and initial data of an arbitrary ﬁnite energy,
we obtain a bubbling theorem analogous to the Struwe’s results on the heat ﬂows.
Mathematics Subject Classiﬁcation 35Q56 · 35K55 · 58E50
In this article, we consider the two dimensional Landau–Lifshitz (LL) equation:
u − β J(∇
u(0) = u
Communicated by M. Struwe.
Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences and Department of
Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, China