facts on Besov spaces.\
In Section $4$, we are going to establish the proof of crucial Lemma \[general lemma for critical element \], which provides the proof of the first part of Theorem \[Big key theorem\] : there exists some sup-critical elements in $\dot{H}^s$. The second part of the proof is postponed in Section $6$, where we build some sup-critical elements not only in $\dot{H}^s$, but also in others spaces, such as $\dot{B}^{\frac{1}{2}}_{2,\infty} $ and $\dot{B}^{s'}_{2,\infty} $, with ${\displaystyle}{s<s'< 2s -\frac{1}{2}\cdotp}$ To carry out this, we need some estimates on the fluctuation part of the solution, which will be provided in Section $5$.\
Then in Section $7$, we give an analogue sup-inf critical criteria. It turns out that among sup-critical solutions, there exists some of them which are sup-inf-critical in the sense of they reach the biggest infimum limit. Section $8$ is devoted to the proof of Lemma \[lemme allure de la solution\], which gives the structure of a Navier-Stokes solution associated with a bounded sequence of data in $\dot{H}^s$. We recall to the reader that such structure result has been partially proved in [@P], except for the orthogonality property of Navier-Stokes solution in $\dot{H}^s$-norm. As a result, we give 