AI 中文总结
针对趋近有限终端时间的三维纳维-斯托克斯流，证明点原子强制全尾动力学刚性，得到受约束奥森传播子饱和等结果，揭示延迟二阶算子预算在端点附近失效。
AI 中文摘要
考虑平坦环面上光滑无外力作用的三维纳维-斯托克斯流趋近有限终端时间，其动能密度沿整个时间变量收敛到唯一的端点测度。我们证明每个点原子会强制产生同母体的全尾动力学刚性。预先指定的水平穿越目录与嵌套局部霍奇分解生成正交包，从整个包尾中提取单个伴随向后算子，其终端能量集中在该原子处。柯西饱和随后将该伴随算子锁定到所有足够晚的包，得到受约束奥森传播子及其伴随算子的一致双参数饱和，同时一阶耗散消失。因此，每个足够晚的固定根后代具有无限延迟二阶作用与不可积的正涡量产生，等价地，仅由纳维-斯托克斯母体确定的延迟二阶算子预算在任意接近端点处必然失效。
英文摘要
Let a smooth, unforced, three-dimensional Navier--Stokes flow on the flat torus approach a finite terminal time. Its kinetic-energy densities converge, along the full time variable, to a unique endpoint measure. We prove that each point atom forces a same-parent, full-tail dynamical rigidity. A preassigned level-crossing catalogue and nested local Hodge projections produce orthogonal packets. From the entire packet tail, we extract a single backward adjoint whose terminal energy concentrates at the atom. Cauchy saturation then locks this adjoint to every sufficiently late packet and yields uniform two-parameter saturation of both the constrained Oseen propagator and its adjoint, together with vanishing first-order dissipation. Consequently, every sufficiently late fixed-root descendant has infinite delayed second-order action and non-integrable positive enstrophy production. Equivalently, a delayed second-order operator budget determined solely by the Navier--Stokes parent must fail arbitrarily close to the endpoint.
