Title :

Recent results for the Landau-Lifshitz equation

Author(s) :

De Laire, André [Auteur]
Systèmes de particules et systèmes dynamiques [Paradyse]

Journal title :

SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada

Pages :

253-295

Publisher :

Springer

Publication date :

2022

ISSN :

2254-3902

English keyword(s) :

Landau-Lifshitz-Gilbert equation
ferromagnetic spin chain
nonlinear Schrödinger equation
asymptotic regimes
solitons
self-similar solutions

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

We give a survey on some recent results concerning the Landau-Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. We ...
Show more >We give a survey on some recent results concerning the Landau-Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. We revisit the Cauchy problem for the anisotropic Landau-Lifshitz equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. We also examine two approximations of the Landau-Lifshitz equation given by of the Sine-Gordon equation and cubic Schrödinger equations, arising in certain singular limits of strong easy-plane and easy-axis anisotropy, respectively. Concerning localized solutions, we review the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical framework. Finally, we survey results concerning the existence, uniqueness and stability of self-similar solutions (expanders and shrinkers) for the isotropic LL equation with Gilbert term. Since expanders are associated with a singular initial condition with a jump discontinuity, we also review their well-posedness in spaces linked to the BMO space.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions
Ondes déterministes et aléatoires

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL