I'm looking for a reference about some possible generalization of the well-known Di Perna-Lions theory for transport equations (say, on $[0,T] \times \mathbb{R}^d$) of the form

\begin{align} \partial_t u + b \cdot \nabla_x u+ c u=\Gamma(u). \end{align} Here, $b=b(t,x)$, $c=c(t,x)$ (with $(t,x) \in [0,T] \times \mathbb{R}^d$) are given and may be nonsmooth but with integrability properties for $c$ and Sobolev regularity for $b$, and $\Gamma$ is an integral operator of the form \begin{align} \Gamma(u)(t,x)=\int_{\mathbb{R}} K(x,x^*)u(t,x^*) \, \mathrm{d}x^*, \end{align} where $K$ is smooth and given.

When $\Gamma \equiv 0$, the well-posedness theory for that type of equation is by now well-known with the theory of renormalized solutions which enjoy nice properties of strong stability.

I was wondering if something in this direction is known when we add this type of integral operator in the righ-hand side.

Any help or reference is welcome !

I'm looking for a reference about some possible generalization of the well-known Di Perna-Lions theory for transport equations (say, on $[0,T] \times \mathbb{R}^d$) of the form

\begin{align} \partial_t u + b \cdot \nabla_x u+ c u=\Gamma(u). \end{align} Here, $b=b(t,x)$, $c=c(t,x)$ (with $(t,x) \in [0,T] \times \mathbb{R}^d$) are given and may be nonsmooth but with integrability properties for $c$ and Sobolev regularity for $b$, and $\Gamma$ is an integral operator of the form \begin{align} \Gamma(u)(t,x)=\int_{\mathbb{R}^d} K(x,x^*)u(t,x^*) \, \mathrm{d}x^*, \end{align} where $K$ is smooth and given.

When $\Gamma \equiv 0$, the well-posedness theory for that type of equation is by now well-known with the theory of renormalized solutions which enjoy nice properties of strong stability.

I was wondering if something in this direction is known when we add this type of integral operator in the righ-hand side.

Any help or reference is welcome !