Any references would be much appreciated. I'm looking for some well-posedness results for flow problems of the following type (note my issue is that I'm considering a singular non-local term).
Let $\mathcal{P}(\mathbb{R}^d)$ be the space of probability measures on $\mathbb{R}^d$. Let $\mu_0$ be some absolutely continuous (w.r.t Lebesgue) that is in $\mathcal{P}(\mathbb{R}^d)$, lets $\textit{also}$ denote its density by $\mu_0$.
Consider the following flow $X:\mathbb{R}^+\times \mathbb{R}^d\to\mathbb{R}^d$ of the vector field $b$
\begin{equation} \begin{cases} &\partial_t X = b[\mu_0]\circ X, \\ & X(0,\cdot)=\text{id}. \end{cases} \end{equation}
where $b:\mathcal{P}(\mathbb{R}^d)\times \mathbb{R}^d \to \mathbb{R}^d$ is a non-local vector field $b[\mu](x)=\int B(x-y)\mu(dy) $, in the form of a convolution.
At least formally the push-forward measure $\mu(t,\cdot)=\big(X(t,\cdot)\big)_{\#}\mu_0$ solves the continuity equation \begin{equation} \begin{cases} \partial_t \mu(t,\cdot) + \text{div} \big( \mu(t,\cdot) b[\mu_0]\big)=0 \\ \mu(t,\cdot)|_{t=0}=\mu_0. \end{cases} \end{equation}
I'm interested well-posedness of the above two equations and their relation via the push-forward. DiPerna-Lions theory ("Ordinary differential equations, transport theory and Sobolev spaces") has some stuff but im interested in the case of singular kernals $B$, for example
\begin{equation*} B(r):= C \begin{cases} {\|r\|^{-(d-2)}} & \text{for}~d>2 \\ \log\|r\| &\text{for}~d=2 \end{cases}, \end{equation*} for some $C\in \mathbb{R}$.