Mean field error estimate of the random batch method for vortex blob dynamics for the 2D Navier--Stokes Equation

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}\tau^2+N^{-1} \right), $ with constants independent of $N$, $\tau$, and $varepsilon$. Here $\tau$ 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 $\varepsilon^{-2}\tau\to 0$.