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

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

1 University of Birmingham [Birmingham]
2 Paradyse - Systèmes de particules et systèmes dynamiques

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

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. Journal of Evolution Equations, 2020, ⟨10.1007/s00028-020-00589-8⟩. ⟨hal-02480999v2⟩