{
  "abstract": "We study the stability of transition fronts for the one-dimensional Cahn-Hilliard equation. More precisely, we prove that for any Hölder initial datum sufficiently close to a front in the $L^2$ norm, the corresponding solution of the Cahn-Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. The result holds for a class of double-well potentials in a $C^4$-neighborhood of the standard quartic potential. For the proof, we develop an $L^2$-stability framework adapted to Cahn-Hilliard fronts. The key ingredient is a nontrivial second-order Poincaré-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an $L^2$-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted $L^2$ topology. The stability analysis requires only the $L^2$-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.15658",
  "author_total": 2,
  "authors": [
    {
      "name": "Bongsuk Kwon"
    },
    {
      "name": "Wanyong Shim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jbn566gd1q1x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.15658",
  "fulltext_url": null,
  "id": "8jbn566gd1q1x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.15658",
  "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"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7164905677",
  "pdf_url": "https://arxiv.org/pdf/2606.15658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jbn566gd1q1x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2606.15658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.15658"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "$L^2$-contraction and asymptotic stability of Cahn-Hilliard fronts",
  "title_text": "L^2-contraction and asymptotic stability of Cahn-Hilliard fronts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/8jbn566gd1q1x",
  "venue": "arXiv",
  "versions": []
}