Abstract

In this talk, we consider the dynamical Landau-Lifshitz equation (LL) without Gilbert damping, which is a PDE arising in micromagnetism. From PDE point of view, LL can be considered as geometric version of the Schrödinger equation, and has been studied in the community of nonlinear dispersive PDEs, referred to as Schrödinger maps. Starting with a short review on known results, I will introduce my result on the local well-posedness of initial value problem for LL with helicity term (Dzyaloshinskii-Moriya interaction). The difficulty is that the helicity term has quadratic derivative nonlinearity. To handle this, we exploit the skew-adjoint structure of helicity term to cancel bad part of the nonlinearity in the energy method.