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

Susana Gutiérrez; André De Laire

doi:10.1088/1361-6544/ab1296

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

Susana Gutiérrez, André De Laire

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

150 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 language	English
Pages (from-to)	2522-2563
Number of pages	42
Journal	Nonlinearity
Volume	32
Issue number	7
Early online date	18 Jun 2019
DOIs	https://doi.org/10.1088/1361-6544/ab1296
Publication status	Published - 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
General Physics and Astronomy
Applied Mathematics

Access to Document

10.1088/1361-6544/ab1296Licence: None: All rights reserved

Gutierrez_and_de_Laire_The_Cauchy_problem_Nonlinearity_2019
This is an author-created, un-copyedited version of an article published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1361-6544/ab1296. Susana Gutiérrez and André de Laire, 2019, Nonlinearity, 32, 2522
Accepted author manuscript, 716 KBLicence: Other (please specify with Rights Statement)

https://iopscience.iop.org/article/10.1088/1361-6544/ab1296Licence: None: All rights reserved

Singular vortex dynamics and nonlinear Schrodinger equations
Gutierrez, S.
Engineering & Physical Science Research Council
30/09/12 → 29/09/14
Project: Research Councils

Cite this

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

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 Schr{\"o}dinger equation obtained via the stereographic projection and techniques introduced by Koch and Tataru.",

keywords = "Complex Ginzburg-Landau equation, Discontinuous initial data, Dissipative Schr{\"o}dinger equation, Ferromagnetic spin chain, Heat-flow for harmonic maps, Landau-Lifshitz-Gilbert equation, Self-similar solutions, Stability",

author = "Susana Guti{\'e}rrez and {De Laire}, Andr{\'e}",

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: {\textcopyright} 2019 IOP Publishing Ltd.",

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N2 - 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.

AB - 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.

KW - Complex Ginzburg-Landau equation

KW - Discontinuous initial data

KW - Dissipative Schrödinger equation

KW - Ferromagnetic spin chain

KW - Heat-flow for harmonic maps

KW - Landau-Lifshitz-Gilbert equation

KW - Self-similar solutions

KW - Stability

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UR - https://arxiv.org/abs/1701.03083

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SP - 2522

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JO - Nonlinearity

JF - Nonlinearity

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ER -

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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Projects

Singular vortex dynamics and nonlinear Schrodinger equations

Cite this