Self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation

TL;DR

This paper analytically studies self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation, revealing how damping influences solutions originating from discontinuous, infinite-energy initial data, with implications for related models like Schrödinger maps.

Contribution

It constructs a family of global, smooth solutions with finite energy for all positive times from discontinuous initial data, analyzing their dependence on the damping parameter and connecting to related models.

Findings

Existence of global solutions from infinite-energy initial data.

Solutions are smooth and finite energy for positive times.

Behavior varies with the Gilbert damping parameter.

Abstract

We consider the one-dimensional Landau-Lifshitz-Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated to a discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter. We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schr\"odinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1d-Schr\"odinger map equation.