正缺陷问题：无外力Navier-Stokes爆破程序化搜索的目标与可接受性准则

中文摘要 AI 辅助

2026年9月7日至8日，程序化搜索产生了奇点：在每个固定粘度下出现受迫Navier-Stokes奇点，即Clay问题的陈述(C)和(D)，以及两个Euler奇点。无外力问题，即陈述(A)和(B)，仍未解决，对其搜索需要一个目标。本文确定了一个目标：正缺陷问题，即来自周期立方体上光滑数据的Leray-Hopf解在有限时间窗口内，在固定粘度下，损失的能量超过粘度所移除的能量。正缺陷意味着爆破，从而对陈述(B)给出否定答案；反之则未知。本文证明该目标等价于通过Littlewood-Paley壳层的能量通量（Onsager湍流理论粗粒化通量的Fourier侧形式）在窗口上的平均值存在一个下限；陈述了候选解的必要条件：速度中的Type II奇点时间、能量集中在零长度集合上、压力不在L^2中、速度不在Onsager临界类L^3_t B^{1/3}_{3,c_0}中、对坍缩到固定稳态Euler轮廓的阻碍；并陈述了不能证明正缺陷的条件：没有有限计算能证明Galerkin一致上限，选择和强迫将问题回归到正缺陷。在128^3和256^3分辨率下的伪谱搜索显示了约化中哪个条件起约束作用：细壳层通量下限在三分之一周转时间内保持在上限的1%至6%以内，并在Kolmogorov波数处尺度上失效，因此候选解必须在级联深度上不同于一般湍流，而非在时间上。每个未另行标记的蕴含式都是Lean 4中Mathlib上的定理；该库不包含Navier-Stokes对象，方程仅通过假设进入。

英文摘要

On 7 and 8 September 2026 programmatic search produced singularities: a forced Navier-Stokes singularity at every fixed viscosity, statements (C) and (D) of the Clay problem, and two Euler singularities. The unforced problem, statements (A) and (B), stands open, and a search for it needs a target. This paper fixes one: the positive defect problem, that a Leray-Hopf solution from smooth data on the periodic cube loses energy on a finite window, at fixed viscosity, beyond what viscosity removes. A positive defect implies blowup and so a negative answer to statement (B); the converse is not known. The paper proves the target equivalent to a floor on the energy flux through the Littlewood-Paley shells, the Fourier-side form of the coarse-grained flux of the Onsager theory of turbulence, averaged over the window; states necessary conditions on a candidate: a singular time of Type II in velocity, energy concentrating on a set of zero length, a pressure outside L^2, a velocity outside the Onsager-critical class L^3_t B^{1/3}_{3,c_0}, an obstruction to collapse onto a fixed steady Euler profile; and states what cannot certify one: no finite computation witnesses a Galerkin-uniform ceiling, and selection and forcing return the question to a positive defect. A pseudo-spectral search at 128^3 and 256^3 shows which condition of the reduction binds: the fine-shell flux floor holds to within 1 to 6 percent of the ceiling for a third of a turnover time, and fails in scale at the Kolmogorov wavenumber, so a candidate must differ from generic turbulence in the depth of its cascade, not in its timing. Every implication not marked otherwise is a theorem in Lean 4 over Mathlib; the library contains no Navier-Stokes object, and the equation enters only through hypotheses.

发表机构

• Stony Brook University（石溪大学）
• GlimmAnalytics LLC（Glimm分析有限责任公司）

机构由 AI 辅助整理，请以论文原文为准。