AI 中文总结
本研究通过受约束薛定谔-感应系统证明流体奇点导致动能方差在有限时间发散，并揭示端点态失去动能算子正则性，为平均能量控制极限提供基准。
AI 中文摘要
本研究使用受约束的薛定谔-感应系统来研究由流体奇点引起的动能涨落。假设所引用的纳维-斯托克斯构造具有零初始速度及其精确外部轮廓，我们从恒定归一化波函数和零连接构造一个周期轨迹。所施加的体积力在归一化奇异时间 $t=1$ 上允许平滑的时空延拓。在此构造中，概率密度保持恒定，平均物质动能保持有界，而规范不变的动能方差发散。我们通过在一个所有振荡修正都消失的精确外部区域上对完整速度大小四次方进行积分，证明了阶为 $(1-t)^{-1/2-5h}$ 的下界增长，其中 $h$ 由源构造固定。在固定具有恒定波函数的规范后，我们识别出 $L^4$ 之外的唯一弱 $L^2$ 连接极限，并通过相关的磁形式构造其正自伴动能算子。我们进一步证明，归一化端点状态属于形式域但不属于算子域：其第一动能谱矩有限，而第二动能谱矩无限。这些结果将流体集中与动能域正则性的丧失联系起来，并为评估流体动力学波形式中平均能量控制的极限提供了动力学基准。
英文摘要
In this study, we use a constrained Schrödinger--induction system to investigate kinetic-energy fluctuations induced by a fluid singularity. Assuming the cited Navier--Stokes construction with zero initial velocity and its exact exterior profile, we construct a periodic trajectory from a constant normalized wave function and zero connection. The prescribed body force admits a smooth space--time extension across the normalized singular time $t=1$. In this construction, the probability density remains constant and the mean matter kinetic energy stays bounded, whereas the gauge-invariant kinetic-energy variance diverges. We prove a lower growth bound of order $(1-t)^{-1/2-5h}$, with $h$ fixed by the source construction, by integrating the fourth power of the complete velocity's magnitude over an exact exterior region where all oscillatory corrections vanish. After fixing a gauge with constant wave function, we identify a unique weak $L^2$ connection limit outside $L^4$ and construct its positive self-adjoint kinetic operator through the associated magnetic form. We further show that the normalized endpoint state belongs to the form domain but not the operator domain: its first kinetic spectral moment is finite, whereas its second is infinite. These results connect fluid concentration to a loss of kinetic-domain regularity and provide a dynamical benchmark for assessing the limits of mean-energy control in hydrodynamic wave formulations.
