Curve-Supported Concentrations of Vortices and Screw Dislocations Subject to Confinement
Abstract
We study the concentration of logarithmically interacting defects, such as Ginzburg--Landau vortices and screw dislocations, on closed curves in two dimensions, subject to a confining field. We prove that a measure supported on a smooth closed curve solves the mean-field transport equation exactly when, in Lagrangian labels, the curve moves with the negative induced velocity and each label keeps its mass. This contour law is a gradient flow of the mean-field energy. We prove that the gradient of the energy on a curve has a leading term proportional to the operator |D| with Fourier symbol |k|, which prevents every smooth equilibrium curve from being a local minimizer, in a gradient-flow analogue of the Kelvin--Helmholtz instability of vortex sheets. In a radially symmetric case with quadratic confinement, we find an explicit circular equilibrium whose linearization is diagonal in Fourier modes, and we show that, as $N\rightarrow \infty $, the linear spectrum of the regular N-gon converges, mode by mode, to that of the circle. Finally, we compute a critical point on two concentric circles with lower energy than the single circle, indicating a rich family of multi-circle equilibria.
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BibTeX
@misc{garcıaazpeitia2026curvesupported,
title = {{Curve-Supported Concentrations of Vortices and Screw Dislocations Subject to Confinement}},
author = {Carlos Garc\'ıa-Azpeitia and Thomas Hudson},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03866},
eprint = {2611.03866},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03866},
}
RIS
TY - UNPB
TI - Curve-Supported Concentrations of Vortices and Screw Dislocations Subject to Confinement
AU - Carlos Garc\'ıa-Azpeitia
AU - Thomas Hudson
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03866
UR - https://arxiv.org/abs/2611.03866
AB - We study the concentration of logarithmically interacting defects, such as Ginzburg--Landau vortices and screw dislocations, on closed curves in two dimensions, subject to a confining field. We prove that a measure supported on a smooth closed curve solves the mean-field transport equation exactly when, in Lagrangian labels, the curve moves with the negative induced velocity and each label keeps its mass. This contour law is a gradient flow of the mean-field energy. We prove that the gradient of the energy on a curve has a leading term proportional to the operator |D| with Fourier symbol |k|, which prevents every smooth equilibrium curve from being a local minimizer, in a gradient-flow analogue of the Kelvin--Helmholtz instability of vortex sheets. In a radially symmetric case with quadratic confinement, we find an explicit circular equilibrium whose linearization is diagonal in Fourier modes, and we show that, as $N\rightarrow \infty $, the linear spectrum of the regular N-gon converges, mode by mode, to that of the circle. Finally, we compute a critical point on two concentric circles with lower energy than the single circle, indicating a rich family of multi-circle equilibria.
ER -