AI 中文总结
本文研究紧致受迫纳维-斯托克斯爆破产生的奇异数据分布，证明在$H^s$拓扑中稠密性当且仅当$s<1/2$，通过精确插入构造光滑力，并给出完整理论分析。
AI 中文摘要
从OpenAI构造的紧致、光滑受迫纳维-斯托克斯爆破解出发，我们研究其产生的奇异数据的分布。在三维环面上，固定粘度和时间范围，从静止出发在该时间内产生经典破裂的光滑力在继承的时间积分空间$H^s$拓扑中是稠密的，当且仅当$s<1/2$（范数定义见第\ref{sec:intro}节）。正面结果源于对每条正则参考轨迹的精确插入。一个局部向量势移除集中奇异包周围的背景，使得所有非线性交叉项消失，且修改后的力在奇异时刻保持光滑。我们给出了完整的支撑构造、导数估计、分数阶Sobolev标度和经典寿命论证。插入的轨迹在能量和耗散范数下强收敛。一个独立的临界力自举，随后进行$H^1$估计和高Sobolev延拓，提供了在$s\geq 1/2$时非稠密所需的正则开集。进一步的结果描述了扩展数据投影、每个固定的光滑初速度切片、混合力范数、无限维变体和内部无滑移实现。该构造改变力；它不对单一指定力分类奇异初速度。
英文摘要
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 $\mathbb{T}^3$ and $\mathbb{R}^3$ exactly when $s<1/2$.
Comments20 Pages. The Lean4 formalization project of this article can be found in https://github.com/mathzhuonichi/blowup_density. V3 is a simplified combination of arxiv: 2609.10262 v1 and arXiv: 2609.10269 v1 with better writing.V4 updated abstract metadata to match the manuscript; manuscript unchanged. Comments are welcomed
