Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on S2

Penghong Zhong

doi:10.1155/2018/5370349

Abstract

The singularity solution for the inhomogeneous Landau-Lifshitz (ILL) equation without damping term in n-dimensional space was investigated. The implicit singularity solution was obtained for the case where the target space is on S2. This solution can be classified into four types that cover the global and local solutions. An estimation of the energy density of one of these types indicates its exact decay rate, which allows a global solution with finite initial energy under n>3. Analysis of the four aperiodic solutions indicates that energy gaps that are first contributions to the literature of ILL will occur for particular coefficient settings, and these are shown graphically.

Highlights

The nonlinear ferromagnetic chain model (FCM) has attracted the attention of physicists and mathematicians
Quantitative models depicting FCM were proposed by Landau and Lifshitz in 1935 [1], and the Landau-Lifshitz equation (LLE) was proposed to the dynamics of the nonequilibrium magnetism system: St = αS ∧ (△S + H) − βS ∧ (S ∧ (△S + H)), (→󳨀x, t) ∈ R3 × R, (1)
Otherwise, when β = 0, the Gilbert damping vanishes, and when β = 0 and H = 0, the LLE degenerates into the Schrodinger map heat flow (SMF), which is an important equation of differential geometry

Summary

Introduction

The nonlinear ferromagnetic chain model (FCM) has attracted the attention of physicists and mathematicians. Ding et al [29] proved that n = 3 or n = 4 dimensional LLE with the Gilbert term will lead to a finite time blowup under specific initial boundary conditions. We studied the blowup and energy gap for n dimensional aperiodic ILL on S2 target and investigated what happens as t tends to infinity when the initial data is smooth and sufficiently large, in particular whether the solution develops distinct behaviors (finite or global time singularity) under the large data. Global smooth (or blowup) theory for ILL was not established, but we discuss some special solutions that form a singularity in finite time and classify these solutions and analyses based on their energy density.

Blowup Solution on a Sphere

Multiple Branches of the Blowup Solutions

Multibranch elliptic case II

Blowup Solution and Energy Gap

Conclusions