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Time decay estimates for localized perturbations around a helical state for the Landau-Lifshitz-Gilbert equation

Ikkei Shimizu

TL;DR

This work proves global stability and time decay for localized perturbations of the helical state ${\mathbf{h}}^1$ in the Landau--Lifshitz--Gilbert equation with Dzyaloshinskii--Moriya interaction. By reformulating the perturbation in a moving frame as a complex-valued function $u$ and applying a Bloch--Floquet spectral analysis to the resulting linear operator $A$, the authors obtain sharp low- and high-frequency decay estimates that mirror the heat equation. The nonlinear analysis leverages a frequency-splitting bootstrap, exploiting a cancellation structure in the nonlinearity and obtaining bounds that close for small initial data in $L^1 \cap H^s$, yielding both global existence and a decay rate $|{\mathbf{n}}(t,x) - {\mathbf{h}}(x)| \lesssim (1+\alpha t)^{-d/2}$. The results extend to generalized DMI strength via rescaling and provide a robust framework for stability analysis of spatially periodic equilibria in PDEs. Overall, the paper delivers a rigorous, decay-driven stability theory for helical states in LLG–DMI, with techniques potentially applicable to other periodic magnetization configurations.

Abstract

We study the dynamics of the Landau--Lifshitz--Gilbert equation with the Dzyaloshinskii--Moriya interaction. The equation admits a family of exact stationary solutions, referred to as helical states, which are periodic in one spatial variable and constant in the others. We investigate the dynamical stability of a helical state with respect to perturbations belonging to suitable Lebesgue and Sobolev spaces. Under a smallness assumption on the initial perturbation, we prove global existence and time decay estimates for solutions, demonstrating that the above helical state is stable. The analysis of the relevant linear operator is carried out via the Bloch--Fourier-wave decomposition, where the eigenvalue problem for the reduced operator is characterized by certain Mathieu equations.

Time decay estimates for localized perturbations around a helical state for the Landau-Lifshitz-Gilbert equation

TL;DR

This work proves global stability and time decay for localized perturbations of the helical state in the Landau--Lifshitz--Gilbert equation with Dzyaloshinskii--Moriya interaction. By reformulating the perturbation in a moving frame as a complex-valued function and applying a Bloch--Floquet spectral analysis to the resulting linear operator , the authors obtain sharp low- and high-frequency decay estimates that mirror the heat equation. The nonlinear analysis leverages a frequency-splitting bootstrap, exploiting a cancellation structure in the nonlinearity and obtaining bounds that close for small initial data in , yielding both global existence and a decay rate . The results extend to generalized DMI strength via rescaling and provide a robust framework for stability analysis of spatially periodic equilibria in PDEs. Overall, the paper delivers a rigorous, decay-driven stability theory for helical states in LLG–DMI, with techniques potentially applicable to other periodic magnetization configurations.

Abstract

We study the dynamics of the Landau--Lifshitz--Gilbert equation with the Dzyaloshinskii--Moriya interaction. The equation admits a family of exact stationary solutions, referred to as helical states, which are periodic in one spatial variable and constant in the others. We investigate the dynamical stability of a helical state with respect to perturbations belonging to suitable Lebesgue and Sobolev spaces. Under a smallness assumption on the initial perturbation, we prove global existence and time decay estimates for solutions, demonstrating that the above helical state is stable. The analysis of the relevant linear operator is carried out via the Bloch--Fourier-wave decomposition, where the eigenvalue problem for the reduced operator is characterized by certain Mathieu equations.
Paper Structure (23 sections, 17 theorems, 241 equations)

This paper contains 23 sections, 17 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Introduction and main result
  3. Outline and the idea of the proof
  4. Related works
  5. Notations
  6. Reduction via moving frame
  7. Spectral analysis
  8. Bloch--Floquet decomposition
  9. Eigenvalue problem for $A_\xi$
  10. Lower bound of the spectrum
  11. Asymptotics near $\xi=0$
  12. Linear estimates
  13. Preliminary lemmas
  14. High-frequency estimate
  15. Low-frequency estimate
  16. ...and 8 more sections

Key Result

Proposition 1

Let $s$ be an integer satisfying $s\ge 1$ for $d=1$, and $s\ge 2$ for $d=2,3$. For ${\mathbf{m}}_0\in H^s(\mathbb{R}^d\colon \mathbb{R}^3)$, E1.7 has a unique solution with ${\mathbf{m}}(0,x)={\mathbf{m}}_0$, where $T_{\max} = T_{\max} ({\mathbf{m}}_0)$ is the maximal existence time in this class. Moreover, the solution map ${\mathbf{m}}_0 \mapsto {\mathbf{m}}$ is continuous in the topology $H^{s

Theorems & Definitions (39)

  • Proposition 1: Local well-posedness
  • Remark 1
  • Theorem 1: Time decay estimates
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1: MR493421MR3075381
  • Lemma 2
  • proof
  • ...and 29 more