AI 中文总结
本文针对二维Navier-Stokes方程，提出随机批次涡 blob 方法并分析其平均场误差，将计算复杂度降至O(N)，证明了该方法的误差界及粒子边际收敛性。
AI 中文摘要
我们针对全平面上涡量形式的二维Navier-Stokes方程，提出并分析了随机批次涡 blob 方法。涡 blob 方法基于含N个粒子的相互作用粒子系统，计算复杂度为O(N²)，而随机批次方法可将其降至O(N)，相关研究见文献[JinLiLiu2020]。我们的主要结果是定量的律级平均场误差估计，其对blob半径的依赖仍为代数形式。我们分别处理两类主要误差机制：通过局部耦合辅助划分、对称律比较及Fisher信息耗散控制随机批次误差；利用Biot-Savart核的奇性与无散结构估计平均场涨落。对于光滑、严格正的初始涡量，我们在每个有限时间区间上证明了归一化相对熵界，阶为O(ε⁻⁴τ² + N⁻¹)，其中常数与N、τ、ε无关。此处τ为批次刷新间隔，ε为blob半径。由此可得，当N→∞且ε⁻²τ→0时，固定粒子边际在L¹中强收敛到正则化涡量解的张量积。
英文摘要
We propose and analyze the random batch vortex blob method for the 2D Navier--Stokes equation in vorticity form on the whole plane. The vortex blob method is based on an interacting particle system of $N$ particles with computational complexity of $O(N^2)$, which is reduced to $O(N)$ by the random batch method \cite{JinLiLiu2020}. Our main result is a quantitative law level mean field error estimate whose dependence on the blob radius remains algebraic. We treat the two main error mechanisms separately. The random batch error is controlled through a locally coupled auxiliary partition, a symmetric law comparison, and Fisher-information dissipation. The mean field fluctuation is estimated by exploiting the oddness and divergence-free structure of the Biot--Savart kernel. For smooth, strictly positive initial vorticity, we prove on every finite time interval a normalized relative-entropy bound of order $ O\!\left(\varepsilon^{-4}τ^2+N^{-1} \right), $ with constants independent of $N$, $τ$, and $varepsilon$. Here $τ$ is the batch refreshing interval and $\varepsilon$ is the blob radius. As a consequence, the fixed-particle marginals converge strongly in $L^1$ to tensor products of the regularized vorticity solution when $N\to\infty$ and $\varepsilon^{-2}τ\to 0$.
