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This paper is dedicated to the study of both viscous compressible barotropicfluids and Navier-Stokes equation with dependent density, when the viscositycoefficients are variable, in dimension $d\geq2$. We aim at proving the localand global well-posedness for respectively {\it large} and \textit{small}initial data having critical Besov regularity and more precisely we areinterested in extending the class of initial data velocity when we consider theshallow water system, improving the results in \cite{CMZ1,H2} and \cite{arma}.Our result relies on the fact that the velocity $u$ can be written as the sumof the solution $u_{L}$ of the associated linear system and a remaindervelocity term $\bar{u}$; then in the specific case of the shallow-water systemthe remainder term $\bar{u}$ is more regular than $u_{L}$ by taking intoaccount the regularizing effects induced on the bilinear convection term. Inparticular we are able to deal with initial velocity in $\dot{H}^{\N-1}$ asFujita and Kato for the incompressible Navier-Stokes equations (see \cite{FK})with an additional condition of type $u_{0}\in B^{-1}_{\infty,1}$. We wouldlike to point out that this type of result is of particular interest when wewant to deal with the problem of the convergence of the solution ofcompressible system to the incompressible system when the Mach number goes to0.

Existence of strong solutions in a larger space for the shallow-water system