the proof of such a property, after reminding the ideas of the complete proof.

Profile theory and Tool Box
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We recall the fundamental result due to P. Gérard : the profile decomposition of a bounded sequence in the Sobolev space $\dot{H}^s$. The original motivation of this theory was the desciption, up to extractions, of the defect of compactness in Sobolev embeddings (see for instance the pionneering works of P.-L. Lions in [@PLL], [@PLL2] and H. Brezis, J.-M. Coron in [@BC]. Here, we will use the theorem of P. Gérard [@PG], which gives, up to extractions, the structure of a bounded sequence of $\dot{H}^s$, with $s$ between $0$ and ${\displaystyle}{\frac{3}{2}} \cdotp$ More precisely, the defect of compactness in the critical Sobolev embedding ${\displaystyle}{\dot{H}^s \subset L^p}$ is described in terms of a sum of rescaled and translated orthogonal profiles, up to a small term in $L^p$. For more details about the history of the profile theory, we refer the reader to the paper [@P].

*[(Profile Theorem [@PG])\
\[theo profiles\] Let $(u_{0,n})_{n \in \N}$ be a bounded sequence in $\dot{H}^s$. Then, up to an extraction:\
- There exists a sequence vectors fields, called profiles $(\varphi^{j})_{j \in \N}$ in $\dot{H}^s$.\
- There exists 