Global Solvability of the Cauchy Problem for the Landau-Lifshitz-Gilbert Equation in Higher Dimensions

TL;DR

This paper establishes the global existence, uniqueness, and long-term behavior of smooth solutions to the Landau-Lifshitz-Gilbert equation in higher dimensions, under small initial gradient conditions, using advanced analytical techniques.

Contribution

It introduces a novel approach combining moving frames and weighted-in-time norms to analyze the Landau-Lifshitz-Gilbert equation in dimensions three and above.

Findings

Proves global smooth solutions exist under small initial gradient conditions.

Demonstrates uniqueness and asymptotic stability of solutions.

Develops a covariant complex Ginzburg-Landau equation framework.

Abstract

We prove existence, uniqueness and asymptotics of global smooth solutions for the Landau-Lifshitz-Gilbert equation in dimension $n \ge 3$, valid under a smallness condition of initial gradients in the $L^n$ norm. The argument is based on the method of moving frames that produces a covariant complex Ginzburg-Landau equation, and a priori estimates that we obtain by the method of weighted-in-time norms as introduced by Fujita and Kato.