标题： 三维纳维-斯托克斯方程平稳鞅解的柯尔莫哥洛夫4/5定律的充分条件 显示英文标题

标题： A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations

摘要： 我们证明了在$\mathbb{T}^3$上受时间白噪声（空间彩色）强迫的3D纳维-斯托克斯方程的统计平稳鞅解，在仅假设动能在$\nu \rightarrow 0$时为$o(\nu^{-1})$的情况下（其中$\nu$是逆雷诺数），满足科莫戈罗夫4/5定律（在平均意义下并在合适的惯性范围内）。这起到了弱异常耗散的作用。没有假设能量平衡或额外的正则性（除了所有鞅解从能量不等式中满足的条件之外）。如果力是统计同性的，那么任何同性鞅解在空间上逐点满足球面平均的4/5定律。惯性范围内近似各向同性的额外假设给出了传统的科莫戈罗夫定律版本。我们通过证明能量平衡和一个额外的定量正则性估计作为$\nu \rightarrow 0$可以得出4/5定律（或任何类似的标度定律）不能成立，从而展示了必要条件。 显示英文摘要

摘要： We prove that statistically stationary martingale solutions of the 3D Navier-Stokes equations on $\mathbb{T}^3$ subjected to white-in-time (colored-in-space) forcing satisfy the Kolmogorov 4/5 law (in an averaged sense and over a suitable inertial range) using only the assumption that the kinetic energy is $o(\nu^{-1})$ as $\nu \rightarrow 0$ (where $\nu$ is the inverse Reynolds number). This plays the role of a weak anomalous dissipation. No energy balance or additional regularity is assumed (aside from that satisfied by all martingale solutions from the energy inequality). If the force is statistically homogeneous, then any homogeneous martingale solution satisfies the spherically averaged 4/5 law pointwise in space. An additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the Kolmogorov law. We demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as $\nu \rightarrow 0$ imply that the 4/5 law (or any similar scaling law) cannot hold.