with Hölder continuous derivative. Moreover, the techniques also work for the counterpart of , the stochastic continuity equation $$\label{eq:stochCont}
\partial_{t}\rho+\Div(\rho\circ(u+\sigma\dot{W}))=0 \quad \text{on} \quad [0,T] \times \R^{d},$$ as carried out, under various hypotheses (see for instance [@beck2014stochastic] and [@MR3399177]). Indeed, the stochastic flow analysis of [@MR2593276] is sufficient to rule out concentrations in the density for the velocity field given above.

In nonlinear situations this ceases to be the case. Indeed, in the case of Burgers’ equation $$\partial_{t}\rho+\partial_x\rho \circ (\rho+\sigma\dot{W})=0 \quad \text{on} \quad [0,T] \times \R,$$ it has been observed in [@MR2593276] that the inclusion of transport noise does not yield regularizing effects anymore. In contrast, it has been pointed out in [@MR3666564; @CG17] that noise entering the PDE via the flux function, that is, $$\label{eq:SSCL}
\partial_{t}\rho+\frac{1}{2}\partial_x \rho^2 \circ \dot{W}=0 \quad \text{on} \quad [0,T] \times \R,$$ has a regularizing effect, in the sense that quasi-solutions to are more regular than quasi-solutions to the deterministic Burgers’ equation. This type of stochastic scalar conservation law was motivated in [@MR3327520; @MR3666564; @MR3351442] as a continuum limit of interacting particle systems subject to a common noise.

In the present article, we are interested in the following degenerate, non-linear version of . $$\label{eq:SPDE}
\partial_{t}\rho + \Div(\rho 