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

Susana Gutiérrez (1) , André de Laire (2, 3)

1 University of Birmingham [Birmingham]
2 MEPHYSTO-POST - Méthodes quantitatives pour les modèles aléatoires de la physique
3 LPP - Laboratoire Paul Painlevé - UMR 8524

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere, at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

Mots clés: blow up; heat flow for harmonic maps; quasi-harmonic sphere; Landau-Lifshitz-Gilbert equation; backward self-similar solutions; ferromagnetic spin chain; self-similar expanders; asymptotics

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]

Susana Gutiérrez, André de Laire. Self-similar shrinkers of the one-dimensional Landau-Lifshitz-Gilbert equation. 2020. ⟨hal-02480999v1⟩