we will case the existence of global strong solutions but with only a condition of smallness for the density on $\|q_{0}\|_{\widetilde{B}^{{\frac{N}{2}}-1,{\frac{N}{2}}}_{2,p,1}\cap \widetilde{B}^{0,1}_{2,r}}$. In particular it improves widely the results of [@CMZ] and [@CD], this result is close from the results of D. Hoff in [@5H4] and [@5H2] except that we are in critical space for the scaling (except concerning the fact that $q_{0}\in \widetilde{B}^{0,1}_{2,r}$).\
In particular as in [@CD], we can take $u_{0}(x)=\phi(x)sin({\varepsilon}^{-1}x\cdot\omega)n$ where $\omega$ and $n$ stand for any unit vectors of ${\mathbb{R}}^{N}$ and $\phi$ for any smooth compactly supported function then we have if $p_{1}>N$: $$\|u_{0}\|_{B^{\frac{N}{p_{1}}-1}_{p_{1},1}}\leq C{\varepsilon}^{1-\frac{N}{p_{1}}},$$ so that the smallness condition is satisfied by $u_{0}$ if ${\varepsilon}$ small enough. However we remark that $u_{0}$ is arbitrarly big in $L^{3}$. On the other hand, $u_{0}$ belongs to Scwartz class ${\cal S}$ hence also to $B^{0}_{N,1}$ so that uniqueness holds true by persistency results.

We can observe that $u_{0}\in B^{0}_{2,r}$ corresponds exactly to the energy space when $r=2$, in this sense this additional regularity on the velocity seems very natural. We will explain in the corollary \[corollaire2\] why it is perfectly adapted in the case of specific viscosity coefficients. Indeed in the general case $q_{0}\in B^{1}_{2,r}$ is not in 