发表机构
Sungkyunkwan University; Linyi University（成均馆大学; 临沂大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明三维不可压缩纳维-斯托克斯流的端点动能测度中的点原子会迫使局部压力集中，通过建立相对压力功恒等式等揭示了相关压力功机制。
AI 中文摘要
我们证明，在平坦环面上的三维不可压缩纳维-斯托克斯流的端点动能测度中的一个点原子，会迫使实际压力出现定量集中。在每个重启时间τ，通过由u驱动的无压力分量平流扩散方程演化u(τ)，并减去得到的被动场z_τ，以此进行同态约束-响应比较。记r_τ = u - z_τ，g_τ = Q z_τ，我们得到一个精确的相对压力功恒等式。若该原子的质量为m，纳什平滑会使z_τ最终非原子，而r_τ保留至少为m的最终原子质量；因此，在每次固定重启后，累积压力功的最终上极限至少为m/2。介观局域化给出了规范不变L2压力振荡的显式尺度下界，以及收缩终端圆柱中压力梯度的局部L2质量的下界。由此，在原子点的每个终端邻域上，压力的时间模函数及其梯度均不满足平方可积性。该相对压力功原理可推广到约束螺线型奥森演化：当τ趋近于T*时，零初始响应在弱尾拓扑中消失，尽管每次固定重启仍保留相同的下界。这确定了一种由不可压缩约束产生的必要压力功机制，该机制将端点原子与局部压力浓度关联起来，无需完全正则性假设。
英文摘要
We prove that a point atom in an endpoint kinetic-energy measure of a three-dimensional incompressible Navier-Stokes flow on the flat torus forces quantitative concentration of the actual pressure. At each restart time tau, a same-state constraint-response comparison evolves u(tau) by the pressure-free componentwise advection-diffusion equation driven by u, and subtracts the resulting passive field z_tau. Writing r_tau = u - z_tau and g_tau = Q z_tau, we obtain an exact relative pressure-work identity. If the atom has mass m, Nash smoothing makes z_tau terminally non-atomic while r_tau retains terminal atomic mass at least m; hence, after every fixed restart, the terminal limsup of the accumulated pressure work is at least m/2. Mesoscopic localization yields scale-explicit lower bounds for gauge-invariant L2 pressure oscillation and for the local L2 mass of the pressure gradient in shrinking terminal cylinders. Consequently, on every terminal neighborhood of the atomic point, the pressure modulo functions of time and its gradient fail to be square integrable. The relative pressure-work principle extends to constrained solenoidal Oseen evolutions: as tau approaches T*, the zero-initial responses vanish in weak-tail topologies, although every fixed restart retains the same lower bound. This identifies a necessary pressure-work mechanism, generated by the incompressibility constraint, that links endpoint atoms to local pressure concentration, without requiring a full regularity hypothesis.
Comments17 pages. Submitted for publication
