 ( \sqrt{\int_0 ^t \mathcal{E} } + \e \sqrt{\kappa} \sqrt{\mathcal{D}(t) } \right ) \sqrt{\mathcal{E}(t) } \right. \\
&\left. + \left (\sqrt{\int_0 ^t \mathcal{E} } +\e^{\frac{1}{2}}\left (\sqrt{\mathcal{D}(t)} + \sqrt{\int_0 ^t \mathcal{E} } \right ) \right ) \sqrt{\mathcal{E}(t) } \right ] \\
& \lesssim (\| \nabla_x u^\beta \|_{L^\infty_{tx} } + V(\beta) \sqrt{\kappa} ) \sqrt{\mathcal{E}(t) } \left ( \int_0 ^ t \mathcal{E} + \e \mathcal{D}(t) \right ) \lesssim \sqrt{\mathcal{E} (t) } \left ( \| \nabla_x u^\beta \|_{L^\infty_{tx} } \int_0 ^t \mathcal{E} + \mathcal{D}(t) \right )
\end{split}\notag
\end{equation}
where the first factor $\| \nabla_x u^\beta \|_{L^\infty_{tx} }$ comes from the case $s_1 = 1$ and the second factor $V(\beta) \sqrt{\kappa} $ comes from the case case $s_2 = 2, s_2 = 0$. Also we have used \eqref{ekappabeta} to bury the contribution of $\|\nabla_x u^\beta \|_{L^\infty_{t,x} }$ in $\mathcal{D}(t) $. 

Therefore, we have
\begin{equation}  \label{nonlinearestimate}
\eqref{nonlinearity} \lesssim \left ( (\| \nabla_x u^\beta \|_{L^\infty_{tx} }+1) \int_0 ^t \mathcal{E} + \mathcal{D}(t) \right )  \sqrt{\mathcal{E} (t) }
\end{equation}

Summing up \eqref{sourceestimate}, \eqref{linearestimate}, \eqref{Lcommutatorestimate}, \eqref{momestimate}, \eqref{nonlinearestimate}, we have
\begin{equation}  \label{Energy}
\begin{split}
\mathcal{E}(t) + \mathcal{D}(t)  &\lesssim  \mathcal{E}(0)+ (\|\nabla_x u^\beta \|_{L^\infty_{t,x} }^2 + 1 + \sqrt{\mathcal{E}(t) } ) \int_0 ^t \mathcal{E} + \kappa  + \sqrt{\mathcal{E}(t) } \mathcal{D}(t)  \\
&+ e^{-\frac{1}{(\e \sqrt{\kappa})^{o(1) } } } \sum_{s \le 2} \| w_s \partial^s f \|_{L^\infty ((0, t); L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) ) } ^2.
\end{split}
\end{equation}

\subsubsection{$L^\infty$ control}

From Proposition \ref{prop_infty} and \eqref{Remainder} we obtain the following:
\begin{equation}  \label{LinfinityRem}
\begin{split}
&\| w_0 f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )}
\\
 \lesssim& \  \| w_0 f_{R0} \|_{L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 )} + \frac{1}{\e} \sqrt{\mathcal{E}(t)} + \e \| w_0 f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )}  ^2 \\
&+ \e \kappa V(\beta) \| w_0 f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} + \e^3 \kappa V(\beta) + \e \kappa^2 V(\beta)
\end{split}
\end{equation}
Here we have used Lemma \ref{Gammabound} to bound the right-hand side of \eqref{Remainder}. Proceeding similar argument to \eqref{pRemainder}, for $1 \le s \le 2$, we obtain
\begin{equation} \notag
\begin{split}
\| w_s \partial^s f_R \|&_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \lesssim  \| w_s \partial^s f_{R0} \|_{L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 )} + \frac{1}{\e\kappa^{\frac{s}{2}}} \mathcal{E}(t)\\
& + \e \| w_s \partial^s f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \| w_0 f_R \| _{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \\
& + \e \kappa V(\beta)  \| w_s \partial^s f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \\
& + \e V(\beta) \sum_{s' < s} \| w_{s'} \partial^{s'} f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \\
& + \e V(\beta) \sum_{s_1+s_2 \le s, s_1, s_2 < s} \| w_{s_1} \partial^{s_1} f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )}\| w_{s_2} \partial^{s_2} f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \\
&+ \e^3 \kappa V(\beta) + \e \kappa^2 V(\beta)
\end{split}
\end{equation}
Here we have used a pointwise bound $w_0 > \nu^2 w_1 > \nu^4 w_2$ for the third line. Therefore, we have
\begin{equation}  \label{Linfestimate}
\begin{split}
\mathcal{F}(t) &\lesssim \mathcal{F}(0) + \e^2 + \sqrt{\mathcal{E}(t)} + \mathcal{F}(t)^2, \\
\| w_1 \partial^1 f_R \|&_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \lesssim  \| w_1 \partial^1 f_{R0} \|_{L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 )} + \mathcal{F}(t) \| w_1 \partial^1 f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )}  \\
&+ \frac{1}{\e \sqrt{\kappa} } \sqrt{\mathcal{E}(t)} + \e V(\beta) \left ( 1 + \frac{\mathcal{F}(t)}{\e} \right )^2, \\
\| w_2 \partial^2 f_R \|&_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )} \lesssim  \| w_2 \partial^2 f_{R0} \|_{L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 )} + \mathcal{F}(t) \| w_2 \partial^2 f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )}  \\
&+ \frac{1}{\e \kappa} \sqrt{\mathcal{E}(t)} + \e V(\beta) \left ( 1 + \frac{\mathcal{F}(t)}{\e} +\| w_1 \partial^1 f_R \|_{L^\infty ((0, t) ; L^\infty(\mathbb{T}^2 \times \mathbb{R}^3 ) )}   \right )^2
\end{split}
\end{equation}
In particular,