The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions

Susana Gutiérrez, André De Laire

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)
147 Downloads (Pure)

Abstract

We prove a global well-posedness result for the Landau–Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any dimension. In the one-dimensional case, we characterize the self-similar solutions associated with an initial data given by some (-valued) step function and establish their stability. We also show the existence of multiple solutions if the damping is strong enough.

Our arguments rely on the study of a dissipative quasilinear Schrödinger equation obtained via the stereographic projection and techniques introduced by Koch and Tataru.
Original languageEnglish
Pages (from-to)2522-2563
Number of pages42
JournalNonlinearity
Volume32
Issue number7
Early online date18 Jun 2019
DOIs
Publication statusPublished - Jul 2019

Bibliographical note

Funding Information:
A de Laire was partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the MathAmSud program. S Gutierrez was partially supported by the EPSRC grant EP/J01155X/1 and the ERCEA Advanced Grant 2014 669689-HADE.

Publisher Copyright:
© 2019 IOP Publishing Ltd.

Keywords

  • Complex Ginzburg-Landau equation
  • Discontinuous initial data
  • Dissipative Schrödinger equation
  • Ferromagnetic spin chain
  • Heat-flow for harmonic maps
  • Landau-Lifshitz-Gilbert equation
  • Self-similar solutions
  • Stability

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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