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

Susana Gutierrez; 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 Gutierrez, André de Laire

Mathematics

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4 Citations (Scopus)

121 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
Article number	2522
Pages (from-to)	2522-2563
Number of pages	42
Journal	Nonlinearity
Volume	32
Issue number	7
DOIs	https://doi.org/10.1088/1361-6544/ab1296
Publication status	Published - 18 Jun 2019

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

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 Gutierrez and {de Laire}, Andr{\'e}",

year = "2019",

month = jun,

day = "18",

doi = "10.1088/1361-6544/ab1296",

language = "English",

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pages = "2522--2563",

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TY - JOUR

T1 - The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions

AU - Gutierrez, Susana

AU - de Laire, André

PY - 2019/6/18

Y1 - 2019/6/18

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

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

U2 - 10.1088/1361-6544/ab1296

DO - 10.1088/1361-6544/ab1296

M3 - Article

VL - 32

SP - 2522

EP - 2563

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 7

M1 - 2522

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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