Local well-posedness for the Landau-Lifshitz equation with helicity term

TL;DR

This paper establishes local well-posedness results for the Landau-Lifshitz equation with helicity term, using analysis based on the modified Schrödinger map equation, for initial data in specific Sobolev spaces.

Contribution

It proves local well-posedness for the Landau-Lifshitz equation with helicity term in certain Sobolev spaces, extending previous results to include the chiral interaction term.

Findings

Well-posedness in $ar{k} + H^s$ for $s \\ge 3$ with $s \\in \\mathbb{Z}$

Well-posedness in $ar{k} + H^s$ for $s > 2$ with $s \\in \\mathbb{R}$ under homotopy assumptions

Analysis based on the modified Schrödinger map equation

Abstract

We consider the initial value problem for the Landau-Lifshitz equation with helicity term (chiral interaction term), which arises from the Dzyaloshinskii-Moriya interaction. We prove that it is well-posed locally-in-time in the space $\bar{k} +H^s$ for $s\ge 3$ with $s\in \mathbb{Z}$ and $\bar{k}={}^t(0,0,1)$. We also show that if we further assume that the solution is homotopic to constant maps, then local well-posedness holds in the space $\bar{k} + H^s$ for $s>2$ with $s\in \mathbb{R}$. Our proof is base on the analysis via the modified Schr\"odinger map equation.