Kinetic model for the many-particle PCC model in the normalized case

Let $M$ be the state manifold, let $R>0$, let $v$ denote the velocity variable, and let ${\mathcal K}$ be the interaction kernel. Let $\mu_t^N$ be the empirical measure of the normalized many-particle PCC model, and suppose that the initial empirical measures satisfy the same assumptions as in the non-normalized case: $\mu_0^N$ converge in probability as $N\to\infty$ to a deterministic absolutely continuous probability measure $f_0\,|dx\wedge\operatorname{Vol}_M|$. Assume also that the denominator in the normalized interaction law does not vanish. Kinetic model for the many-particle PCC model in the normalized case. For every $t\in[0,\infty)$, $\mu_t^N$ converges towards a deterministic absolutely continuous probability measure $f_t\,|dx\wedge\operatorname{Vol}_M|$, where $f(x,\alpha,t)\equiv f_t(x,\alpha)$ satisfies the kinetic equation for the single-particle model with ${\mathcal J}={\mathcal J}_f(x,t)$, where

$${\mathcal J}_f(x,t)=\frac{\displaystyle\int_{{\mathbb R}^n\times M}{\mathcal K}\left(\frac{|x-y|}{R}\right)\,v\,f(y,\alpha,t)\,|dx\wedge\operatorname{Vol}_M|(y,\alpha)}{\displaystyle\left|\int_{{\mathbb R}^n\times M}{\mathcal K}\left(\frac{|x-y|}{R}\right)\,v\,f(y,\alpha,t)\,|dx\wedge\operatorname{Vol}_M|(y,\alpha)\right|}.$$

The normalized interaction is the expected mean-field expression, but proving convergence is substantially harder because of the denominator's possible singularity; the source notes that this issue remains unresolved even for the Vicsek model except when the normalization singularity is regularized.