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

Abstract

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 $O\left({\epsilon }^{-4}{\tau }^2+N^{-1}\right),$ with constants independent of $N$, $\tau$, and $varepsilon$. Here $\tau$ is the batch refreshing interval and $\epsilon$ 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 ${\epsilon }^{-2}\tau \to 0$.