continuity equation and a more regular, deterministic parabolic equation for the moments $\E[\rho^{q}]$ no longer holds. Nonetheless, we show that a duality method is still possible, provided one replaces the role of the linear, parabolic operator on the right hand side of by a fully non-linear operator. Namely, we solve an auxiliary problem of the form $$\partial_{t}\varphi + \mathcal{M}^{+}(D^{2}\varphi)=-|\Div u|,$$ using Pucci’s extremal operators and show that this is sufficient to implement the duality method above. An informal account of the main ideas leading to a priori bounds is carried out in section \[sec:KES\] below. We now introduce our notation and outline the the article.

Notation
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We will use several abbreviations throughout. For function spaces, we use subscripts to abbreviate the domain. For instance, $L^{1}_{t}(W^{1,1}_{x})$ and $L^{1}_{t,x,v}$ are short hand for $L^{1}([0,T];W^{1,1}(\R^{d}))$ and $L^{1}([0,T] \times \R^{d+1})$ respectively. For spaces of measures we use the symbol $\mathcal{M}$, together with a subscript as above to denote the domain and a possible superscript to indicate spaces of positive measures or spaces of locally finite measures. For instance, $\mathcal{M}^{loc}_{t,x,v}$ denotes the space of locally finite Radon measures on $[0,T] \times \R^{d+1}$, while $\mathcal{M}_{v}^{+}$ denotes the non-negative, finite measures on $\R$. In addition, the 