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Density of Forces Producing Navier--Stokes Blowup

  • Shaozhen Cao
  • Zhuoni Chi
  • Ping Nie
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Abstract

We study the density of smooth external forces for which the three-dimensional Navier--Stokes equations lose classical regularity by a prescribed time T>0. Starting from the compact forced blowup solution of OpenAI, we construct a blowup solution near every given smooth solution while preserving its initial velocity. A smooth cutoff of a local vector potential makes the given velocity vanish near the support of a rescaled blowup solution. The two velocities then have no nonlinear interaction, and the resulting force remains smooth through the blowup time. For fixed viscosity and zero initial velocity, such forces are dense in the relative L1tHsx topology on both 𝕋3 and ℝ3 exactly when s<1/2.

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Cite

BibTeXRIS
BibTeX
@misc{cao2026density,
  title = {{Density of Forces Producing Navier--Stokes Blowup}},
  author = {Shaozhen Cao and Zhuoni Chi and Ping Nie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10262},
  eprint = {2609.10262},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10262},
}
RIS
TY  - UNPB
TI  - Density of Forces Producing Navier--Stokes Blowup
AU  - Shaozhen Cao
AU  - Zhuoni Chi
AU  - Ping Nie
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.10262
UR  - https://arxiv.org/abs/2609.10262
AB  - We study the density of smooth external forces for which the three-dimensional Navier--Stokes equations lose classical regularity by a prescribed time T>0. Starting from the compact forced blowup solution of OpenAI, we construct a blowup solution near every given smooth solution while preserving its initial velocity. A smooth cutoff of a local vector potential makes the given velocity vanish near the support of a rescaled blowup solution. The two velocities then have no nonlinear interaction, and the resulting force remains smooth through the blowup time. For fixed viscosity and zero initial velocity, such forces are dense in the relative L^1_tH^s_x topology on both 𝕋^3 and ℝ^3 exactly when s<1/2.
ER  -