Renormalization for the Laplacian and global well-possness of the Landau\u2013Lifshitz\u2013Gilbert equation in dimensions n geq3

Penghong Zhong,Fengong Wu,Shengxiang Tang

doi:10.1186/s13661-020-01377-6

Abstract

The global solution of the n geq3 Landau–Lifshitz–Gilbert equation on mathbb{S}^{2} is studied under the cylindrical symmetric coordinates. An equivalent complex-valued equation in cylindrical symmetric coordinates is obtained by the Hasimoto transformation. A renormalization for the Laplacian is used to transform this equivalent system to a Ginzberg–Landau type system in which the Strichartz estimate can be applied. The global H^{2} well-posedness of the Cauchy problem for the Landau–Lifshitz–Gilbert equation is established.

Highlights

IntroductionIn this paper we consider the Cauchy problem for the Landau–Lifshitz–Gilbert (LLG) equation [14]:
In this paper we consider the Cauchy problem for the Landau–Lifshitz–Gilbert (LLG) equation [14]:∂ S = αS× S – βS×(S× S), (1)∂t where S = (S1, S2, S3), α, β ∈ [0, 1], the α-term denotes the exchange interaction, while the β-term is the Gilbert damping term
There are a number of different definitions that are all equivalent to the Schrödinger map equation and the harmonic map heat flow under the smooth setting

Summary

Introduction

In this paper we consider the Cauchy problem for the Landau–Lifshitz–Gilbert (LLG) equation [14]:. If β = 0, Theorem 1.1 indicates the solution of the n ≥ 3 Schrödinger map equation is global under the smallness and embedded conditions. Z L∞ t L2x + Z Lqt Lpx φ L2x + ψ Lqt Lpx. As is well known, the dispersive inequality, Hardy–Littlewood–Sobolev inequality and space-time Strichartz estimate are very useful tools to prove the regularity of the Schrödinger equation. The dispersive inequality, Hardy–Littlewood–Sobolev inequality and space-time Strichartz estimate are very useful tools to prove the regularity of the Schrödinger equation Similar to this situation, Lemmas 2.1–2.5 (especially the spacetime estimate Lemma 2.5) will be essential for solving the nonlinear Ginzburg–Landau equations. The functions described by (33) satisfy the general Legendre differential equation with the indicated values of the parameters as follows:. By this inequality and the Hardy inequality, the space norm of R2 can be bounded by

L2r to control the norm

L2Rn dτ