$\eta_{1}$ takes the value $1$ on $A_{0,r_{2}}$ and vanishes outside of $A_{0,r_{3}}$. For $k>1$, $\eta_{k}$ takes the value one on $A_{r_{k},r_{k+1}}$ and vanishes outside of $A_{r_{k-1},r_{k+2}}$. Note that we may choose a constant $M_{u}^{2}$ depending on $u$ such that $$\label{eq:kthCutoffBounds}
\sup_{k \in \{1 \,,.., \,N\}} \|\nabla^{2}_{x}\eta_{k}\|_{L^{\infty}(\R^{d})} + \|\nabla_{x}\eta_{k}\|_{L^{\infty}(\R^{d})} \leq M_{u}^{2}.$$ As our candidate sub-solution, we define $$\varphi_{\epsilon} =\widehat{\varphi}+\sum_{k=1}^{N}\eta_{k}\varphi_{k,\epsilon}.$$ We now claim that $\varphi_{\epsilon}$ satisfies for an $M_{u}$ which will be defined below. Note that $$\begin{aligned}
\label{eq:sub-SolComp}
&\partial_{t}(\varphi_{\epsilon}-\widehat{\varphi})+u_{\epsilon} \cdot \nabla_{x}(\varphi_{\epsilon}-\widehat{\varphi}) + p (\Div_{x} u_{\epsilon})_{-} (\varphi_{\epsilon}-\widehat{\varphi}) + \frac{1}{2}b^{2}\Delta_{x}(\varphi_{\epsilon}-\widehat{\varphi}) \nonumber \\
&=\sum_{k=1}^{N}\eta_{k}\left (\partial_{t}\varphi_{\epsilon,k}+ \frac{1}{2}b^{2}\Delta_{x} \varphi_{\epsilon,k}+p (\Div_{x} u_{\epsilon})_{-} \varphi_{\epsilon,k} \right)+\sum_{k=1}^{N}E_{k,\epsilon},\nonumber\\\end{aligned}$$ where $$\begin{aligned}
E_{k,\epsilon}&=u_{\epsilon}\cdot \nabla_{x}(\eta_{k}\varphi_{k,\epsilon})+  \frac{1}{2}\varphi_{k,\epsilon}b^{2}\Delta_{x} \eta_{k}+b^{2}\nabla_{x}\eta_{k}\cdot \nabla_{x}\varphi_{\epsilon,k}.\end{aligned}$$ By the support properties of $\eta_{k}$, we find that $$\left \| \sum_{k=1}^{N}E_{k,\epsilon} \right \|_{L^{\infty}_{t,x,v}} \leq C \sup_{k \geq 1}\|E_{k,\epsilon}\|_{L^{\infty}([0,T] \times A_{r_{k-1},r_{k+2}} \times \R)}.$$ Moreover, we find that $$\begin{aligned}
\|E_{k,\epsilon}\|_{L^{\infty}([0,T] \times A_{r_{k-1},r_{k+2}}\times \R)} &\leq \|u\|_{L^{\infty}_{t,x}}\|\nabla_{x}\eta_{k}\|_{L^{\infty}(\R^{d})}\|\varphi_{k,\epsilon}\|_{L^{\infty}([0,T] \times A_{r_{k-1},r_{k+2}})}\\
&+ \|u\|_{L^{\infty}_{t,x}}\|\eta_{k}\|_{L^{\infty}(\R^{d})}\|\nabla_{x}\varphi_{k,\epsilon}\|_{L^{\infty}([0,T] \times A_{r_{k-1},r_{k+2}})}\\
&+\|b\|_{L^{\infty}_{v}}^{2}\|\nabla^{2}_{x}\eta_{k}\|_{L^{\infty}(\R^{d})}\|\varphi_{k,\epsilon}\|_{L^{\infty}([0,T] \times A_{r_{k-1},r_{k+2}})}\\
&+ \|b\|_{L^{\infty}_{v}}^{2}\|\nabla_{x}\eta_{k}\|_{L^{\infty}(\R^{d})}\|\nabla_{x}\varphi_{k,\epsilon}\|_{L^{\infty}([0,T] \times A_{r_{k-1},r_{k+2}})}\\
&\leq M_{u,p,b}^{4}\end{aligned}$$ by the bounds for the cutoff together with the estimates and . Next we observe that for almost all $(t,x) \in [0,T] \times B_{4R}$ and all $|v| \geq |v_{0}|$, $$\begin{aligned}
&\eta_{k}\left (\partial_{t}\varphi_{\epsilon,k}+ \frac{1}{2}b^{2}(v)\Delta_{x} \varphi_{\epsilon,k}+p (\Div_{x} u_{\epsilon})_{-} \varphi_{\epsilon,k} \right)\\
&\leq\eta_{k}\left (\partial_{t}\varphi_{\epsilon,k}+\mathcal{M}^{+}_{\frac{\lambda}{4},\Lambda}(D^{2}\varphi_{\epsilon,k})+1_{A_{r_{k-1},r_{k+2}}}p (\Div_{x} u_{\epsilon})_{-} \varphi_{\epsilon,k} \right)\\
&=\eta_{k}(\Div u_{\epsilon})_{-} \left (p\varphi_{k,\epsilon}-1\right)\leq 