奇异Navier-Stokes流的相容性-实现框架：容许族与非刚性核心力学
A compatibility--realization framework for singular Navier--Stokes flows: admissible families and non-rigid core mechanics
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中文总结 AI 辅助
本文提出一个相容性-实现框架，构造受迫三维不可压Navier-Stokes方程的奇异解族，证明可观测纤维内存在非刚性，轴运动学不唯一决定压力-粘度分配。
中文摘要 AI 辅助
我们引入了一个相容性-实现框架，用于构造受迫三维不可压缩Navier-Stokes方程的奇异解。该框架将精确平衡和匹配约束（在选定的几何表示中表达）与特定完备化方案的充分条件分离开来。完备化的实现按可观测纤维组织，区分了预定的核心运动学与周围的力和输运机制。基于本文中指定的依赖于源的完备化输入，我们获得了具有零初始速度、光滑紧支撑强迫、有界动能和有限时间无界速度的解族。这些解族包括：在固定轴运动和第一速度梯度下前导粘性功的反转；在一个固定正则压力迹下通过静止奇异中心的符号转变；以及跨越零前导直接粘性涡量供给阈值的连通拉伸族。在临界轮廓附近，可实现迹数据包含一个无限维解析轴向速度和压力扰动空间中的邻域，而中心第一速度梯度和临界涡量平衡保持固定。我们还确定了所选实现类中联合角向输运算子的完整核和显式右逆。结果确立了可观测纤维内的非刚性，并表明轴运动学本身并不能决定局部的压力-粘度分配。直接轮廓选择和动态应力生成被视为构造的耦合组成部分，而非整个设计空间的定义。
英文摘要
We introduce a compatibility--realization framework for constructing singular solutions to the forced three-dimensional incompressible Navier--Stokes equations. The framework separates exact balance and matching constraints, expressed in a chosen geometric representation, from the sufficient conditions of a particular completion scheme. Completed realizations are organized by observable fibers, distinguishing prescribed core kinematics from the surrounding force and transport mechanisms. Conditional on the source-dependent completion input specified in this paper, we obtain families with zero initial velocity, smooth compactly supported forcing, bounded kinetic energy and finite-time unbounded velocity. These include a reversal of leading viscous work at fixed axis motion and first velocity gradient, a signed transition through a stationary singular center under one fixed canonical pressure trace, and a connected stretching family crossing the threshold of zero leading direct viscous vorticity supply. Near a critical profile, the realizable trace data contain a neighborhood in an infinite-dimensional space of analytic axial-velocity and pressure perturbations, while the central first velocity gradient and the critical vorticity balance remain fixed. We also identify the full kernel and an explicit right inverse of the joint angular-transport operator in the chosen realization class. The results establish non-rigidity within observable fibers and show that axis kinematics alone do not determine the local pressure--viscosity partition. Direct profile selection and dynamical stress generation are treated as coupled components of construction, rather than as definitions of the entire design space.
