{
  "abstract": "The data-to-solution map of the incompressible Euler equations is continuous but not uniformly continuous on Sobolev spaces, whereas viscosity makes the Navier--Stokes solution map Lipschitz continuous on small data, uniformly in time. This paper identifies the behavior of the solution map for a hybrid system sitting between these two extremes: the incompressible Oldroyd--B model in $\\mathbb{R}^d$, $d=2,3$, in the high-Reynolds-number, high-Weissenberg-number regime. The velocity equation is a forced Euler equation and the only dissipation is a fractional diffusion $η(-Δ)^β$, $\\frac12\\leβ\\le1$, acting on stress. We prove that, on the ball of small $H^s$ data ($s>1+\\frac d2$) on which global solutions exist, the solution map is Lipschitz continuous with respect to the $H^{r}$-norm, $r=s-(2β-1)$, uniformly for all time, and Lipschitz into the natural dissipation spaces; the constant is explicit and tends to $\\sqrt3$ as the size of the data tends to zero. Consequently the map is Hölder continuous into $C_b([0,\\infty);H^{s'})$ for every $s'<s$, and is Lipschitz in $H^s$ itself when $β=\\frac12$. Thus, although the velocity field is only transported, the solution map behaves like that of the Navier--Stokes equations rather than that of the Euler equations. The proof exploits the damped wave structure generated by the coupling between $u$ and $\\nabla\\cdotτ$ through a Lyapunov functional with a cross term $(δu,\\nabla\\cdotδτ)_{H^{r-β}}$, which transfers the stress dissipation to the velocity difference, together with a low-frequency interpolation argument that keeps all Gronwall coefficients integrable on $(0,\\infty)$. The loss of $2β-1$ derivatives is exactly the gap between the hidden velocity dissipation of order $2(1-β)$ and the dissipation required by the Lipschitz estimate, and we show that it cannot be reduced by the energy method.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34498",
  "author_total": 3,
  "authors": [
    {
      "name": "Gerard Misiołek"
    },
    {
      "name": "Jiahong Wu"
    },
    {
      "name": "Mingyu Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/637f1gjkn4ahb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34498",
  "fulltext_url": null,
  "id": "637f1gjkn4ahb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34498",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7223915458",
  "pdf_url": "https://arxiv.org/pdf/2610.34498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/637f1gjkn4ahb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Regularity of the solution map for the Oldroyd--B model with fractional stress dissipation",
  "title_text": "Regularity of the solution map for the Oldroyd--B model with fractional stress dissipation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/637f1gjkn4ahb",
  "venue": "arXiv",
  "versions": []
}