existence and uniqueness of solutions to and use the noise to prevent concentrations. The intuitive hope is as follows: if the noise is temporarily turned off and the pathological behavior from the deterministic theory begins to form, the start of a concentration re-activates the noise and re-distributes the mass more evenly.

Nonetheless, there are several difficulties in realizing this intuition. The first point is that allowing for degeneracy in the dynamics necessarily forces to be non-linear. As a result, in contrast to the linear SPDE , even if $\rho_{0}$ and $u$ are both very regular, the solution to is expected to develop discontinuities (at random) in finite time. Hence, the Lagrangean techniques discussed above, used in [@MR2593276], do not seem to be a viable approach to analyzing .

Instead, it is more natural to try to understand the regularization effect from a purely Eulerian viewpoint. In the linear setting, this has been carried out in [@beck2014stochastic]. The rough idea is that if $\rho$ is a non-negative solution to and $\beta : \R \to \R$ is a smooth function, then $\E[\beta(\rho)]$ solves a closed parabolic equation and the additional diffusion can compensate for the irregular drift. Note however, this is not 