Weak-strong uniqueness for the Landau-Lifshitz-Gilbert equation in micromagnetics

TL;DR

This paper proves weak-strong uniqueness for the Landau-Lifshitz-Gilbert equation in micromagnetics, including complex physical terms, providing a framework applicable to coupled nonlinear PDE systems.

Contribution

It extends weak-strong uniqueness results to include physically relevant lower-order terms and complex interactions in the LLG equation.

Findings

Weak solutions coincide with strong solutions when the latter exist.

Includes physically relevant terms like Zeeman, anisotropy, stray field, and Dzyaloshinskii-Moriya.

Provides a template for analyzing coupled nonlinear PDE systems.

Abstract

We consider the time-dependent Landau-Lifshitz-Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii-Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weak-strong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.