point $x$ and starting time $s$, and one may define an invertible flow map $\Phi_{s,t}: \R^{d} \to \R^{d}$ by $\Phi_{s,t}(x)=X^{s,x}_{t}$. Denoting $\Psi_{s,t}=\Phi_{s,t}^{-1}$, the unique solution $\rho$ to is given explicitly via the formula $$\label{eq:ContEqRepFormula}
\rho(t,x)=\rho_{0}(\Psi_{0,t}(x))\text{exp}\left (\int_{0}^{t}\Div u(s,\Psi_{s,t}(x))\ds \right).$$ However, in many applications in PDE, the conditions on the velocity field $u$ required for this method of solving are not known to hold a priori. For instance, the continuity equation appears prominently in the study of compressible fluid dynamics (see for instance [@MR1637634]), where the velocity field $u$ is coupled to $\rho$ through Newton’s laws for transport of momentum. There, one has limited knowledge about the regularity of the velocity field $u$, and in particular no a priori control of $\Div u$ in $L^{1}_{t}(L^{\infty}_{x})$. The most classical result studying with a rough velocity field $u$ is the work of Di-Perna and Lions [@MR1022305], which proves the existence and uniqueness of bounded weak solutions to starting from any bounded initial $\rho_{0}$ provided $u \in L^{1}_{t}(W^{1,1}_{x})$ and $\Div u \in L^{1}_{t}(L^{\infty}_{x})$. Generally, the condition on $\Div u$ cannot be improved in the deterministic setting. A simple counterexample appears in dimension one by taking $u(x)=\text{sign}(x)|x|^{\alpha}$ for $\alpha \in (0,1)$. In this case, the corresponding 