\right\rVert}_2  {\left\lVert \hat{u}_S(0,\xi) {\bf 1}_{l=0,|\xi|\leq 1} \right\rVert}_{2} \notag\\
&\lesssim {\left\lVert |\nabla| M_A \omega_S \right\rVert}_2^{\frac{2+2\delta}{2+\delta}}   {\left\lVert M_A \omega_S \right\rVert}_2^{\frac{4+\delta}{2+\delta}}  + {\left\lVert |\nabla| M_A \omega_S \right\rVert}_2 {\left\lVert M_A \omega_S \right\rVert}_2  {\left\lVert  {u}_S \right\rVert}_{2},
\label{eq:step:a:4}\end{aligned}$$ for any $\delta >0$. In the last inequality we have used the Gagliardo-Nirenberg interpolation for the first term. In view of the bound , estimate yields $$\begin{aligned}
|N_A|
&\leq \frac{C}{\nu} {\left\lVert M_A \omega_S \right\rVert}_2^2 \left(  \frac{1}{\nu^{\delta}} {\left\lVert M_A \omega_S \right\rVert}_2^{2+\delta} +{\left\lVert  {u}_S \right\rVert}_{2}^2 \right) + \frac{\nu}{2}  {\left\lVert |\nabla| M_A \omega_S \right\rVert}_2^2\notag\\
&\leq \frac{C}{\nu^{1+\delta}} {\left\lVert M_A \omega_S \right\rVert}_2^2 (1 +  {\left\lVert M_A \omega_S \right\rVert}_2)^{2+\delta} + \frac{\nu}{2}  {\left\lVert |\nabla| M_A \omega_S \right\rVert}_2^2 
\label{eq:step:a:5}\end{aligned}$$ for some positive constant $C \geq 1$, that is independent of $\nu, \epsilon^\prime \leq 1$, but depends on the choice of $\delta>0$. Combining , , and we arrive at $$\begin{aligned}
 \frac{d}{dt} {\left\lVert M_A \omega_S \right\rVert}_2^2 + \nu {\left\lVert  |\nabla| M_A \omega_S \right\rVert}_2^2 & \leq 2\sqrt{\nu} (1+ t) {\left\lVert  |\nabla|^{1/2} M_A\omega_S \right\rVert}_2^2 + \frac{C}{\nu^{1+\delta}}  {\left\lVert M_A \omega_S \right\rVert}_2^2 (1+ {\left\lVert M_A \omega_S \right\rVert}_2)^{2+\delta} \\ 
& \leq 2\sqrt{\nu} (1+ t){\left\lVert  |\nabla| M_A\omega_S \right\rVert}_2{\left\lVert  M_A\omega_S \right\rVert}_2  + \frac{C}{\nu^{1+\delta}}  {\left\lVert M_A \omega_S \right\rVert}_2^2 (1+ {\left\lVert M_A \omega_S \right\rVert}_2)^{2+\delta} \\ 
& \leq \frac{\nu}{2}{\left\lVert  |\nabla| M_A\omega_S \right\rVert}_2^2 + 4(1+ t)^2{\left\lVert  M_A\omega_S \right\rVert}_2^2 + 