Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation
Abstract
We derive whole-space distribution results from the established compact, smoothly forced Navier--Stokes blowup construction of OpenAI. For every fixed positive viscosity and deadline, smooth forces causing classical breakdown from rest on ℝ3 are dense in the relative L1tHsx topology exactly when s<1/2, and in the relative L2tHsx topology exactly when s<−1/2. The positive results preserve every fixed initial velocity in H∞σ(R3). They follow from a compact divergence-free removal of a regular background, followed by exact insertion of a single whole-space singular packet. Separate critical estimates prove non-density; low spatial frequencies are controlled explicitly, without a periodic mean or a Poincaré inequality. The conclusions hold for spacetime-compact forces and for the rapidly decaying data class in the Clay formulation. We also obtain density in the usual completed energy-force spaces, strong energy-and-dissipation approximation of regular trajectories, and exact equality of cell observations on a prescribed finite family of whole-space grids. Initial-state spaces, forcing spaces and numerical error regularity are distinguished throughout. No claim of fixed-force instability, loss of weak existence or new formal verification is made.
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BibTeX
@misc{cao2026singular,
title = {{Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation}},
author = {Shaozhen Cao and Zhuoni Chi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10269},
eprint = {2609.10269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10269},
}
RIS
TY - UNPB
TI - Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation
AU - Shaozhen Cao
AU - Zhuoni Chi
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.10269
UR - https://arxiv.org/abs/2609.10269
AB - We derive whole-space distribution results from the established compact, smoothly forced Navier--Stokes blowup construction of OpenAI. For every fixed positive viscosity and deadline, smooth forces causing classical breakdown from rest on ℝ^3 are dense in the relative L^1_tH^s_x topology exactly when s<1/2, and in the relative L^2_tH^s_x topology exactly when s<−1/2. The positive results preserve every fixed initial velocity in H^∞_σ(R^3). They follow from a compact divergence-free removal of a regular background, followed by exact insertion of a single whole-space singular packet. Separate critical estimates prove non-density; low spatial frequencies are controlled explicitly, without a periodic mean or a Poincaré inequality. The conclusions hold for spacetime-compact forces and for the rapidly decaying data class in the Clay formulation. We also obtain density in the usual completed energy-force spaces, strong energy-and-dissipation approximation of regular trajectories, and exact equality of cell observations on a prescribed finite family of whole-space grids. Initial-state spaces, forcing spaces and numerical error regularity are distinguished throughout. No claim of fixed-force instability, loss of weak existence or new formal verification is made.
ER -