{
  "abstract": "This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in three dimensions. For any prescribed finite set of points $\\{z_j\\}_{j=1}^N \\subset \\mathbb{R}^3$ and an arbitrarily large parameter $M \\gg 1$, we explicitly construct real-analytic initial data generated by Herglotz-type wave fields. We prove that the corresponding classical solution $$ (p,v)\\in C^1([0,T]\\times\\mathbb{R}^3;\\mathbb{R}\\times\\mathbb{R}^3), $$ with $$ T = C_0 \\, M^{-\\frac{3γ+1}{γ-1}}, $$ where $γ> 1$ is the adiabatic index and $C_0 > 0$ is a constant independent of $M$. Throughout the time interval $[0, T]$, the velocity field, its spatial gradient, and the pressure gradient exhibit simultaneous localized amplification at each prescribed point $z_j$. More precisely, each component of the velocity field is bounded below by $M$, each component of the pressure gradient is of size at least $M^{\\frac{2γ}{γ-1}}$, and the velocity gradient exhibits a prescribed anisotropic sign pattern with entries of size at least $M$ at every $z_j$. All implicit constants in these estimates are independent of $M$ and depend solely on the fixed physical parameters and the locations $\\{z_j\\}_{j=1}^N$. This mechanism generates highly localized, shock-like gradient concentrations and implosion-like spatial profiles within the smooth regime. The underlying mechanism relies on the construction of the Herglotz kernel via Helmholtz far-field patterns, together with delicate quantitative energy estimates for the quasilinear hyperbolic system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.29454",
  "author_total": 2,
  "authors": [
    {
      "name": "Xieling Fan"
    },
    {
      "name": "Hongyu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fg080yzn7n52c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.29454",
  "fulltext_url": null,
  "id": "fg080yzn7n52c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.29454",
  "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": "W7166692059",
  "pdf_url": "https://arxiv.org/pdf/2606.29454",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fg080yzn7n52c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2606.29454",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.29454"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Formation of quasi-singularities of shock and implosion type for compressible Euler flows",
  "title_text": "Formation of quasi-singularities of shock and implosion type for compressible Euler flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/fg080yzn7n52c",
  "venue": "arXiv",
  "versions": []
}