{
  "abstract": "We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. Éc. Norm. Supér. (4) \\textbf{58} (2025), no.~4, 943--970], which treats $m$-fold symmetry solutions with $m\\geq4$. We prove that every nonconstant $W^{1,p}$ solution satisfies $\\|g_θ(t)\\|_{L^p}\\to\\infty$ as $t\\to\\pm\\infty$ for $1<p\\leq\\infty$. For $p=1$, the total variation is conserved, but the $L\\log L$ modular tends to infinity whenever it is initially finite. If $D_θg_0$ is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in $W^{α,r}$ for $αr<1$. For such data, every weak $L^2$ infinite-time limit generates a complete $L^2$-precompact orbit. This structural assumption on $D_θg_0$ is automatic for $C^1$ data. In particular, these results answer the question concerning small-scale creation and compact orbit raised by Drivas and Elgindi [EMS Surv. Math. Sci. \\textbf{10} (2023), no.~1, 1--100, Problem~5] for all nonconstant smooth 3-fold symmetric scale-invariant flows. Together with the known theory for $m\\geq4$, they cover the full well-posed scale-invariant range $m\\geq3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.16755",
  "author_total": 3,
  "authors": [
    {
      "name": "Daomin Cao"
    },
    {
      "name": "Junhong Fan"
    },
    {
      "name": "Guolin Qin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dq545fw1khvyc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.16755",
  "fulltext_url": null,
  "id": "dq545fw1khvyc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.16755",
  "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",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7203677087",
  "pdf_url": "https://arxiv.org/pdf/2608.16755",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dq545fw1khvyc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2608.16755",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.16755"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows",
  "title_text": "Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/dq545fw1khvyc",
  "venue": "arXiv",
  "versions": []
}