B$ and $\|
        I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([T_{0},T^{'}])} \geq B $. There are two options:

        -   $$\begin{array}{ll}
            B & \geq e^{(C_{1} M e^{C_{1} (N^{1-s} A )^{\alpha}})^{C_{1} (N^{1-s} A)^{\alpha}}}
            \end{array}
            \label{Eqn:BoundB1}$$

            with $C_{1}$ defined in Proposition \[Prop:SepLocMol\]. Then, in this case, applying Proposition \[Prop:SepLocMol\] to $J=[t_{l},T_{0}]$, we see that there exists $T_{l} \in J$ such that

            $$\begin{array}{ll}
            E(Iw_{l}-Iv_{l}, T_{l}) & \leq \left( C_{E} N^{2(1-s)} A^{p+1} - l(1-\epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right)(1+ \beta) - (N^{s-1} A^{-1})^{\alpha} \\
            & \leq C_{E} N^{2(1-s)} A^{p+1} - (l+1)(1- \epsilon_{0})(N^{s-1} \eta_{1} A^{-1})^{\alpha}
            \end{array}
            \label{Eqn:MakeDecr}$$

            by our choice of $\beta$ (see (\[Eqn:DefBeta\])) and

            $$\begin{array}{ll}
            \| I v_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ( T_{l},T^{'} ) } & \leq \frac{1}{C_{2}^{C_{2} L(l+1) +1 }}
            \end{array}$$

            and letting $w_{l+1}$ be the solution of (\[Eqn:NlkgWdat\]) with data $w_{l+1}(T_{l}):=(w_{l}-v_{l})(T_{l})$, we see that, by using $\mathcal{P}_{C,Q}(l+1)$ and Proposition \[prop:PerturbArg\]

            $$\begin{array}{ll}
            B & \leq \| I w_{l} \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ( [T_{0},T^{'}] ) } \\
            & \leq \| I w_{l} \|_{ L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ( [T_{l},T^{'}] ) } \\ \
            & \lesssim L(l+1)  \\
            \end{array}$$

            Therefore, in this case,

            $$\begin{array}{ll}
            \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}])} & \lesssim L(l+1)
            \end{array}$$

        -   There exists a constant $\tilde{C}$ large enough such that

            $$\begin{array}{l}
            B \leq e^{(C_{1} M e^{C_{1} (N^{1-s} A )^{\alpha}})^{C_{1} 