{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03866",
  "author_total": 2,
  "authors": [
    {
      "name": "Carlos Garc\\'ıa-Azpeitia"
    },
    {
      "name": "Thomas Hudson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a9trfy8kv5wz0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03866",
  "fulltext_url": null,
  "id": "a9trfy8kv5wz0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03866",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DS"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03866",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a9trfy8kv5wz0.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03866",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03866"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Curve-Supported Concentrations of Vortices and Screw Dislocations Subject to Confinement",
  "title_text": "Curve-Supported Concentrations of Vortices and Screw Dislocations Subject to Confinement",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/a9trfy8kv5wz0",
  "venue": "arXiv",
  "versions": []
}