Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces

• DOI: 10.3934/dcds.2019025
• Publication Date: 2018-10-31T22:02:46Z
• Authors: Ze Li; ; Lifeng Zhao
• Keywords: Harmonic map; Landau–Lifshitz–Gilbert equation; Harmonic; Linearization; Heat flow

In this paper, we prove that the solution of Landau-Lifshitz flow $u(t, x)$ from $\mathbb{H}^2$ to converges some harmonic map as $t\to∞$. The main idea is construct Tao's caloric gauge in case where nontrivial maps exist and use it convergence maps. On one side, since our stationary solutions are asymptotically stable under heat flow, Tao provides a natural geometric linearization. other although there infinite numbers $\Bbb H^2$ H^2$, initiated for any given $t>0$ same $u(0, x)$. two observations enable us reduce decay corresponding tension field. This also works dispersive flows, see succeeding on wave instance.