Stability of moving N\'eel walls in ferromagnetic thin films

TL;DR

This paper proves the spectral and nonlinear stability of moving 180-degree Ne9el walls in ferromagnetic thin films under weak external magnetic fields, extending previous static stability results.

Contribution

It introduces a spectral analysis framework for moving Ne9el walls and establishes their nonlinear stability under small perturbations in a weak magnetic field.

Findings

Spectral stability of moving Ne9el walls is demonstrated.

Uniform resolvent estimates are established for the linearized operator.

Nonlinear stability of the moving walls is proven under small external fields.

Abstract

This paper studies moving 180-degree N\'eel walls in ferromagnetic thin films under the reduced model for the in-plane magnetization proposed by Capella, Melcher and Otto [5], in the case when a sufficiently weak external magnetic field is applied. It is shown that the linearization around the moving N\'eel wall's phase determines a spectral problem that is a relatively bounded perturbation of the linearization around the static N\'eel wall, which is the solution when the external magnetic field is set to zero and which is spectrally stable. Uniform resolvent-type estimates for the linearized operator around the static wall are established in order to prove the spectral stability of the moving wall upon application of perturbation theory for linear operators. The spectral analysis is the basis to prove, in turn, both the decaying properties of the generated semigroup and the nonlinear stability of the moving N\'eel wall under small perturbations, in the case of a sufficiently weak external magnetic field. The stability of the static N\'eel wall, which was established in a companion paper [4], plays a key role to obtain the main result.