u) +\nabla B(\rho) \circ \dot{W}=0 \quad \text{on} \quad [0,T] \times \R^{d}.$$ The velocity field $u: [0,T] \times \R^{d} \to \R^{d}$ is deterministic and weakly differentiable. It is assumed to be bounded, but with a potentially unbounded divergence, so that mass concentration is possible in the deterministic setting. The noise $\nabla B(\rho) \circ \dot{W}$ may degenerate for some values of the density in the sense that $b(\rho):=B'(\rho)$ may vanish. Motivated from the developments of [@MR3666564], recalled above, we aim to show that the inclusion of nonlinear noise in , despite its nonlinearity and possible degeneracy, causes well-posedness in the sense that mass concentration is prevented. We further note that the SPDE can be derived as a continuum limit of an interacting particle system with common noise and irregular drift along the same lines as the derivation of in [@MR3327520].

In comparison to the linear equation , the perturbation of the velocity field $u$ has a correlation structure which depends on the density, and may temporarily trivialize, reducing the dynamics to those in the deterministic setting (with a random initial condition). We impose only that the noise acts non-trivally for sufficiently large values of the density. Our main objective is to prove 