of the *BD viscosity coefficients*. In this case our result is very close of the energy initial data with the optimal condition for the scaling $(q_{0},u_{0})\in B^{0}_{\infty,1}\times B^{0}_{N,1})$. In particular it concerns the shallow-water system.

Moreover our method is more flexible than the proofs of D. Hoff in [@5H2], [@5H3], [@5H4] as these works are based crucially on the notion of effective pressure and on a gain of integrability on the velocity which are right only in the case of constant viscosity coefficients.

Our paper is structured as follows. In section \[section2\], we give a few notation and briefly introduce the basic Fourier analysis techniques needed to prove our result. In section \[section3\], we prove estimate on the transport equation. In section \[section4\], we prove the theorem \[theo1\].In section \[section5\] we prove the corollaries \[corollaire1\] and \[corollaire2\]. Two inescapable technical commutator estimates and the proof of paraproduct in hybrid Besov spacesare postponed in appendix.

Littlewood-Paley theory and Besov spaces {#section2}
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Throughout the paper, $C$ stands for a constant whose exact meaning depends on the context. The notation $A\lesssim B$ means that $A\leq CB$. For all Banach space $X$, we denote by $C([0,T],X)$ the set of continuous functions on $[0,T]$ 