\int_{0}^{t}\int_{\R^{d}}1_{\rho \geq \overline{\rho}}\rho^{2} \dx\ds \nonumber \leq \, M \,\int_{0}^{t}E(s)\ds \nonumber,\end{aligned}$$ where we used the definition of Pucci’s maximal operator, followed by , and finally the lower bound . In the regions where $\rho$ is below $\overline{\rho}$, we appeal to the regularity estimate and the upper bound to find $$\begin{aligned}
&\E \int_{0}^{t}\int_{\R^{d}}1_{\rho<\overline{\rho}}\rho^{2}\left [\partial_{t}\varphi + u \cdot \nabla \varphi-\Div u\varphi \right ]+\frac{1}{2}\Psi(\rho)\Delta \varphi \dx\ds \nonumber \\
&\leq (\overline{\rho})^{2}\big (\|\partial_{t}\varphi\|_{\LL{1}{1}}+\|u\|_{L^{\infty}_{t,x}}\|\nabla \varphi \|_{L^{1}_{t,x}}+\|\Div u\|_{L^{1}_{t,x}}\|\varphi \|_{L^{\infty}_{t,x}}+\frac{1}{2}\sup_{\rho \in [0,\overline{\rho}]}\Psi(\rho)\|\Delta \varphi \|_{\LL{1}{1}} \big)\nonumber\\
&\leq M(\overline{\rho})^{2}\left (1+\|u\|_{\LL{\infty}{\infty}}+\|\Div u \|_{\LL{1}{1}}+\Psi(\overline{\rho}) \right) \nonumber.\end{aligned}$$ In summary, we find the inequality $$E(t)-E(0)\leq M\left ( \int_{0}^{t}E(s)\ds+(\overline{\rho})^{2}\left (1+\|u\|_{\LL{\infty}{\infty}}+\|\Div u \|_{\LL{1}{1}}+\Psi(\overline{\rho}) \right)\right ).$$ By Gronwall’s inequality, we obtain .

Existence of generalized kinetic solutions {#sec:exist}
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In order to solve , we start by analyzing the more regular SPDE. Namely, we will smooth the irregular vector field $u$ and use a BGK type approximation of the kinetic measure in the spirit of [@MR2064166]. $$\label{eq:BGK}
\begin{cases}
\partial_{t}\fe + \Div_{x,v}(a_{\epsilon}f_{\epsilon}) +\Div_{x}(bf_{\epsilon} \circ\dot{W})=\epsilon^{-1} \left (\chi(\rho_{\epsilon})-f_{\epsilon} \right )\\
f_{\epsilon} \mid_{t=0}=\chi(\rho_{0}), 
\end{cases}$$ where $$\label{eq:DefOfRho}
\rho_{\epsilon}:=\int_{\R}f_{\epsilon}\dv.$$ The regularized drift coefficient $a_{\epsilon}$ is defined as follows. Denoting by $\eta_{\epsilon}$ the standard mollifier in $\R^{d}$, we define $u_{\epsilon}: \R_{+} \times \R^{d} \to \R^{d}$ and $a_{\epsilon}: \R_{+} \times \R^{d+1} \to \R^{d+1}$ by $$\label{eq:regularizedVelocity}
u_{\epsilon}(t,x):=\frac{1}{2\epsilon}\int_{(t-\epsilon)\wedge 