\lesssim  N^{2(1-s)} \left( \| w_{0} \|^{2}_{H^{s}} + \| w_{1} \|^{2}_{H^{s-1}} + \| w_{0} \|^{p+1}_{H^{s}} \right)
\end{array}
\label{Eqn:BoundInitMolNrj}$$

In particular, if $w$ is a solution of (\[Eqn:NlkgWdat\]) with data $(w_{0},w_{1})$ satisfying (\[Eqn:BoundA\]) with $A \gtrsim 1$, then there exists a constant $C_{E}$ such that

$$\begin{array}{ll}
E(Iw_{0}) & \leq C_{E} N^{2(1-s)} A^{p+1}
\end{array}$$

\[Prop:EstInitMolNrj\]

[**“Almost Conservation Law ”** [@triroyradnlkg]]{} Assume that $w$ satisfies (\[Eqn:NlkgWdat\]). Let $J=[a,b]$ be a time interval. Let $3 \leq p < 5$ and $s \geq \frac{3p-5}{2p}$. Then

$$\begin{array}{ll}
\left| \sup_{t \in J} E(Iw(t)) - E(Iw(a)) \right| & \lesssim \left| \int_{J} \int_{\mathbb{R}^{3}} | \Re (\overline{\partial_{t} Iw}
(IF(w)-F(Iw)) ) | \, dx
\, dt \right| \\
& \\
& \lesssim \frac{Z^{p+1}(J,w)}{N^{\frac{5-p}{2}-}}
\end{array}
\label{Eqn:Acl}$$

\[Prop:Acl\]

In fact, if one looks carefully at the proof of Proposition \[Prop:Acl\], on notices that

$$\begin{array}{ll}
\left| \sup_{t \in J} E(Iw(t)) - E(Iw(a)) \right| & \lesssim \left| \int_{J} \int_{\mathbb{R}^{3}} | \Re (\overline{\partial_{t} Iw}
(IF(w)-F(Iw)) ) | \, dx
\, dt \right| \\
& \\
& \lesssim \frac{\bar{Z}^{p+1}(J,w)}{N^{\frac{5-p}{2}-}}
\end{array}
\label{Eqn:AclRem}$$

with $\bar{Z}(J,w):= \sup_{m \in \{ 0,1,\frac{p-3}{2},\frac{3p-5}{2p} \}} Z_{m,s}(J,w)$.

\[rem:VarMolNrj\]

[**“Estimate of integrals”** [@triroyradnlkg]]{} Let $J$ be a time interval. Let $w$ be a smooth function. Then for $i=1,2$ we have

$$\begin{array}{ll}
| R_{i}(J,w) | & \lesssim \frac{Z^{p+1}(J,w)}{N^{\frac{5-p}{2}-}}
