\begin{split}
 u_{0,n}(x) &= \sum_{\stackrel{j \in \mathcal{J}_{1}}{j \leqslant J}} \varphi^{j}(x-x_{n,j})
           + \sum_{\stackrel{j \in \mathcal{J}^{{}{c}}_{1}}{j \leqslant J}} \Lambda^{\frac{3}{p}}_{\lambda_{n,j},x_{n,j}}\varphi^{j}(x) + \psi_{n}^{J}(x)\\
 \end{split}$$\
where for any $j \in \mathcal{J}_{1}$, for any $n \in \N$, $\lambda_{n,j}  \equiv 1$.

Under these notations, we claim we have the following structure Lemma of the Navier-Stokes solutions, which proof will be provided in Section $8$.

*[(Profile decomposition of a sequence of Navier-Stokes solutions)\
\[lemme allure de la solution\] Let $(u_{0,n})_{n \geqslant 0}$ be a bounded sequence of initial data in $\dot{H}^s$ which profile decomposition is given by $$u_{0,n}(x) = \sum_{j=0}^{J} \Lambda^{\frac{3}{p}}_{\lambda_{n,j},x_{n,j}}\varphi^{j}(x) + \psi_{n}^{J}(x).$$ Then, ${\displaystyle}{\liminf_{n \geqslant 0}{T_{*}}(u_{0,n}) \geqslant \widetilde{T} \eqdefa {\displaystyle}{\inf_{j \in \mathcal{J}_{1}}{{T_{*}}(\varphi^{j})}} }$ and for any $t < {T_{*}}(u_{0,n}) $, we have $$\label{decomposition de la solution}
\begin{split}  
NS(u_{0,n})(t,x) &= \sum_{j \in \mathcal{J}_{1}} NS(\varphi^{j})(t,x-x_{n,j})\, +\, e^{t\Delta} \Bigl(\sum_{\stackrel{j \in \mathcal{J}^{{}{c}}_{1}}{j \leqslant J}} \Lambda^{\frac{3}{p}}_{\lambda_{n,j},x_{n,j}}\varphi^{j}(x) + \psi_{n}^{J}(x) \Bigr)\,+\,   R_{n}^{J}(t,x)
\end{split}$$ where the remaining term $R_{n}^{J}$ satisfies for any $T < \tilde{T}$, ${\displaystyle}{\lim_{J \to +\infty}\lim_{n \to +\infty} \| R_{n}^{J} \|_{X^{s}_{T}}=0}$.\
Moreover, we have the orthogonality property on the $\dot{H}^s$-norm for any $t < \tilde{T}$ $$\label{Pythagore}
\begin{split}
 \| NS(u_{0,n})(t)  \|^{2}_{\dot{H}^s} &= \sum_{j \in \mathcal{J}_{1}} \| NS(\varphi^{j})(t) \|^{2}_{\dot{H}^s} \,+ \, \Bigl\|  e^{t\Delta} \Bigl(\sum_{\stackrel{j \in \mathcal{J}^{{}{c}}_{1}}{j \leqslant J}} \Lambda^{\frac{3}{p}}_{\lambda_{n,j},x_{n,j}}\varphi^{j} + \psi_{n}^{J} \Bigr) \Bigr\|^{2}_{\dot{H}^s} +  \gamma_{n}^{J}(t).
\end{split}$$ 