Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23753

Asymptotic stability of a Bloch skyrmion for the Landau-Lifshitz-Gilbert equation

Jordan Berthoumieu·Raphaël Côte
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove the asymptotic stability of isolated skyrmions under the flow of the two-dimensional Landau-Lifshitz-Gilbert equation. We consider a micromagnetic energy functional involving the exchange energy, the Dzyaloshinskii-Moriya interaction, and the Zeeman interaction. We are interested in the stability of chiral Bloch skyrmions, which are local minimizers of this energy. We prove that data in H2 which are near in the skyrmion in a slighlty stronger than H1 sense, converge with exponential rate, in H2, to the skyrmion, up to a change of gauge.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jordan Berthoumieu, Raphaël Côte. 2026-10-16. Asymptotic stability of a Bloch skyrmion for the Landau-Lifshitz-Gilbert equation. https://arxiv.org/abs/2610.23753

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantitative Schauder estimates for hypoelliptic equations

We derive Schauder estimates using ideas from Campanato's approach for a general class of local hypoelliptic operators and non-local kinetic equations. The method covers equations in divergence and non-divergence form. In particular our results are applicable to the inhomogeneous Landau and to the non-cutoff Boltzmann equation. The paper is self-contained.

math.AP↗
arXiv2026

Scaling-Critical Theory for the Boltzmann and Landau Equations

This paper develops a scaling-critical theory for the spatially inhomogeneous non-cutoff Boltzmann and Landau equations with very soft potentials. For sufficiently small initial perturbations in a localized, weighted, anisotropic Riesz-potential norm, we prove global well-posedness near a Maxwellian in the whole space. This critical phase-space norm captures the Boltzmann--Landau scaling, the velocity-dependent anisotropy, and the hypoelliptic transport structure. The proof combines frozen-operator estimates, a critical fixed-point argument, and weighted hypocoercive energy estimates. We also develop a short-time pointwise Green-function theory for variable-coefficient kinetic equations with a nonnegative Hölder background. We first construct the small-jump Green function by freezing coefficients along kinetic characteristics and then recover the full kernel through a convergent parametrix expansion. The resulting bounds capture the fractional Kolmogorov geometry near characteristics and rapid decay away from them. These estimates form the analytic core of the scaling-critical theory and provide a robust pointwise framework for further regularity analysis in the soft-potential regime.

math.AP↗
arXiv2026

Nonlinear effects of incoming time-periodic boundary conditions on shock waves for scalar conservation laws

This paper is concerned with the large-time behavior of inviscid and viscous shocks for scalar conservation laws on the half-line $(-\infty,0)$ under an incoming \emph{time-periodic} boundary condition. This problem arises naturally from classical piston problems in fluid mechanics. Even for arbitrarily small periodic oscillations of the boundary data, the effect of the boundary manifests itself through a persistent time-periodic component near the boundary, which behaves as a \emph{boundary wave}. For inviscid shocks moving away from the boundary ($s<0$), this boundary wave repeatedly interacts with the shock over infinitely many periods. We show that the repeated interactions do not generate secular growth, but instead produce a finite \emph{cumulative} correction to the asymptotic shock location. For the inviscid stationary case $s=0$, the time-periodic boundary \emph{modulates} the background stationary shock into a genuinely time-periodic shock configuration, to which the solution converges. In the viscous case with $s<0$, the boundary wave couples with the viscous shock through a transitional region induced by diffusion, and the solution converges asymptotically to their superposition. Our results show that for scalar conservation laws, the persistent boundary wave modifies the shock without producing resonance.

math.AP↗