The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions

TL;DR

This paper establishes global well-posedness for the Landau-Lifshitz-Gilbert equation with small BMO initial data, constructs self-similar solutions in any dimension, and analyzes their stability and multiplicity, using advanced PDE techniques.

Contribution

It introduces a novel approach to prove well-posedness and self-similar solutions for the Landau-Lifshitz-Gilbert equation in BMO spaces, extending previous results.

Findings

Global well-posedness for small BMO initial data

Existence of self-similar solutions in all dimensions

Multiple solutions possible with strong damping

Abstract

We prove a global well-posedness result for the Landau-Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any dimension. In the one-dimensional case, we characterize the self-similar solutions associated with an initial data given by some ($\mathbb{S}^2$-valued) step function and establish their stability. We also show the existence of multiple solutions if the damping is strong enough. Our arguments rely on the study of a dissipative quasilinear Schr\"odinger obtained via the stereographic projection and techniques introduced by Koch and Tataru.