0.\end{aligned}$$ Note that we used again the maximum principle in the last step. Combining with , we find that $\varphi_{\epsilon}$ satisfies with $$M^{5}_{u,\gamma,p,b}=M_{u,\gamma,p,b}^{4}+\frac{1}{2}\Lambda C_{1}\max \left \{R^{-1},R^{-2}\right \}.$$\
*Step 5: Lower bounds*\
The final step is to verify the further properties of $\varphi_{\epsilon}$. First, we claim that on all of $\R^{d}$, we have the lower bound $$\label{eq:lowerBdonPhi}
\varphi_{\epsilon} \geq \min \left \{1,\frac{1}{2p} \right \}.$$ Indeed, a first observation is that $$\label{eq:lowerboundinBall}
(\varphi_{\epsilon}-\widehat{\varphi})1_{A_{0,2R}} \geq \frac{1}{2p}.$$ This follows from two facts. The first is that, by construction, each $\varphi_{\epsilon,k}$ is bounded from below by $1/(2p)$ on $B_{2R}$. The second is that $\eta_{1}$ takes the value $1$ on $A_{0,r_{2}}$ while each $\eta_{k}$ (for $k \geq 2$) takes the value one on $A_{r_{k},r_{k+1}}$ and the union of $A_{0,r_{2}}$ with $\{ A_{r_{k},r_{k+1}}\}_{k=2}^{N}$ is $A_{0,2R}$. Combining with the lower bound \[eq:trivSubSol\] for $\widehat{\varphi}$ outside $A_{0,2R}$ completes the proof.\
Finally, using the interior estimates and the maximum principle , together with the support properties and bounds for the cutoffs, we find $$\|\varphi_{\epsilon}\|_{L^{\infty}_{t,x}}+\|\{\nabla_{t,x},\nabla^{2}_{x}\}\varphi_{\epsilon}\|_{L^{1}_{t,x}} \leq M_{u,p}^{6}.$$ Finally, setting the constant $M$ to be the maximum of $M_{u,p}^{5}$ and $M_{u,p}^{6}$ we complete the claim for $\epsilon<\epsilon_{0}(p)$ depending on $p$ through the $\gamma$ selected in Step 3.

\[lem:AveragedVelocityDecay\] Assume that $u$ satisfies 