三维Navier–Stokes方程全局正则性的几何时滞反馈归约:桥梁、残余项与瓶颈
Year: 2026
Type: dissertation
Abstract: 摘 要 本文将三维不可压缩Navier–Stokes方程的全局正则性问题归约为验证一个几何时滞反馈桥梁(C-1503-H)。本文系统分析了传统瞬时吸收范式的四类不可吸收余项——毕奥–萨伐尔非局部耦合余项、高次幂余项、幅值–方向复合余项以及危险区内涡拉伸源积分——指出瞬时吸收范式无法闭合三维情形的范式层面根源:非线性拉伸在物理上具有延迟效应,t时刻产生的拉伸增量并不被t时刻的耗散立即抵消,而是通过涡量方向场的演化在后续时间窗口中引发几何耗散。据此本文提出允许跨时间切片梯次因果的时滞反馈范式,给出该范式核心桥梁C-1503-H的严格数学表述,并建立包含时间尺度内生性、逻辑方向、危险余项逐项处理与常数普适性四个层面的审查协议。通过基于Navier–Stokes原始方程的完整推导,本文识别出该桥梁证明中的三类本质数学障碍——危险区边界输运通量余项、方向场与应变场动态耦合余项以及毕奥–萨伐尔非局部应变输入余项——并证明在当前框架下无法以普适常数闭合这些余项。本文由此建立三维Navier–Stokes方程全局正则性与C-1503-H命题成立性的等价归约关系,为这一千禧年问题提供了明确、可溯源的核心障碍刻画与未来研究方向。 关键词:Navier–Stokes方程;全局正则性;时滞反馈;归约定理;几何泛函;残余项 Abstract This paper reduces the global regularity problem for the three-dimensional incompressible Navier–Stokes equations to the verification of a geometric delay-feedback bridge, denoted by C-1503-H. We systematically analyze four types of non-absorbable remainders inherent to the classical instantaneous-absorption paradigm—the Biot–Savart nonlocal coupling remainder, the high-power remainder, the amplitude–direction composite remainder, and the vortex-stretching source integral within the dangerous region—and identify the paradigm-level origin of its failure: the nonlinear stretching is physically delayed, and the stretching increment produced at time t is not cancelled instantaneously by the dissipation at time t, but triggers geometric dissipation in later time windows through the evolution of the vorticity direction field. We therefore propose a delay-feedback paradigm that permits a cascading causality across time slices, give the precise mathematical statement of its core bridge C-1503-H, and establish a four-level scrutiny protocol involving the endogeneity of the time scale, the logical direction, the term-by-term treatment of dangerous remainders, and the universality of the constants. Based on a complete derivation from the primitive Navier–Stokes equations, we identify three essential mathematical obstructions to any proof of the bridge—the boundary flux remainder across the dangerous region, the dynamical coupling remainder between the direction field and the strain field, and the Biot–Savart nonlocal strain-input remainder—and show that they cannot be closed by universal constants within the current framework. An equivalent reduction between the global regularity of the three-dimensional Navier–Stokes equations and the validity of the bridge C-1503-H is thereby established, providing a precise, traceable characterization of the core obstruction and future research directions for this Millennium Problem. Keywords: Navier–Stokes equations; global regularity; delay feedback; reduction theorem; geometric functionals; remainders
Author: 许, 建军
Language: Chinese
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204436079