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Swirl ill-posedness for incompressible fluids in (sub)critical spaces

  • Tobias Barker
  • Ruilin Hu
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Abstract

For the 3D Navier-Stokes equations with sufficiently smooth initial data having zero swirl u0θ≡ 0, Ladyzhenskaya \cite{L1968} and Ukhovskii and Yudovich \cite{UY1968} established global regularity. It has since remained open as to whether well-posedness still holds for smooth axisymmetric initial data, with the initial swirl being smaller than an \textit{absolute} constant in a critical space. In this remark, we prove \textit{norm inflation} occurs for the Navier-Stokes scalar swirl component, with initial smallness in the critical space Ḣ1/2 or even subcritical Sobolev spaces. We also obtain norm inflation for the 3D Euler swirl \textit{vector} in the Euler subcritical space Ḣ3. Our results are achieved by introducing an anisotropic scale in Luo's supercritical norm inflation construction \cite{Luo24}, as well as performing an anisotropic bootstrap argument to get a sufficiently small swirl remainder term.

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Cite

BibTeXRIS
BibTeX
@misc{barker2026swirl,
  title = {{Swirl ill-posedness for incompressible fluids in (sub)critical spaces}},
  author = {Tobias Barker and Ruilin Hu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33973},
  eprint = {2610.33973},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33973},
}
RIS
TY  - UNPB
TI  - Swirl ill-posedness for incompressible fluids in (sub)critical spaces
AU  - Tobias Barker
AU  - Ruilin Hu
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.33973
UR  - https://arxiv.org/abs/2610.33973
AB  - For the 3D Navier-Stokes equations with sufficiently smooth initial data having zero swirl u_0θ≡ 0, Ladyzhenskaya \cite{L1968} and Ukhovskii and Yudovich \cite{UY1968} established global regularity. It has since remained open as to whether well-posedness still holds for smooth axisymmetric initial data, with the initial swirl being smaller than an \textit{absolute} constant in a critical space. In this remark, we prove \textit{norm inflation} occurs for the Navier-Stokes scalar swirl component, with initial smallness in the critical space Ḣ^1/2 or even subcritical Sobolev spaces. We also obtain norm inflation for the 3D Euler swirl \textit{vector} in the Euler subcritical space Ḣ^3. Our results are achieved by introducing an anisotropic scale in Luo's supercritical norm inflation construction \cite{Luo24}, as well as performing an anisotropic bootstrap argument to get a sufficiently small swirl remainder term.
ER  -