{
  "abstract": "We study the Kuramoto--Sivashinsky (KS) equation in one, two, and three spatial dimensions subject to a codimension-one constraint. The constraint preserves the \\(L^2\\)-norm of the solution and consequently provides a uniform a priori bound that plays a central role in the higher-order energy estimates. In one spatial dimension, we establish global well-posedness by means of a Fourier--Galerkin approximation and compactness arguments. In two and three dimensions, we first prove the existence and uniqueness of local strong solutions using analytic semigroup theory and the Banach fixed-point theorem. In two dimensions, the preserved \\(L^2\\)-norm, combined with the Agmon inequality and suitable higher-order energy estimates, allows us to control the nonlinear terms and obtain global-in-time a priori bounds. Consequently, we establish the existence and uniqueness of global strong solutions for arbitrary sufficiently regular initial data, without any smallness assumption. This provides a global well-posedness result for the constrained two-dimensional KS equation, in contrast with the corresponding unconstrained problem, for which global well-posedness of strong solutions remains open in general. In three dimensions, although the constraint yields a uniform \\(L^2\\)-bound, the available interpolation and Agmon estimates are insufficient to close the higher-order energy estimate. This identifies a genuine dimension-dependent obstruction to extending the global regularity argument to three dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00248",
  "author_total": 2,
  "authors": [
    {
      "name": "Harsh Tiwari"
    },
    {
      "name": "Manil T. Mohan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ep1krttjmn3wy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00248",
  "fulltext_url": null,
  "id": "ep1krttjmn3wy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00248",
  "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": "W7215993075",
  "pdf_url": "https://arxiv.org/pdf/2610.00248",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ep1krttjmn3wy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00248",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00248"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint",
  "title_text": "Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/ep1krttjmn3wy",
  "venue": "arXiv",
  "versions": []
}