  This paper is dedicated to the study of both viscous compressible barotropic
fluids and Navier-Stokes equation with dependent density, when the viscosity
coefficients are variable, in dimension $d\geq2$. We aim at proving the local
and global well-posedness for respectively {\it large} and \textit{small}
initial data having critical Besov regularity and more precisely we are
interested in extending the class of initial data velocity when we consider the
shallow 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 sum
of the solution $u_{L}$ of the associated linear system and a remainder
velocity term $\bar{u}$; then in the specific case of the shallow-water system
the remainder term $\bar{u}$ is more regular than $u_{L}$ by taking into
account the regularizing effects induced on the bilinear convection term. In
particular we are able to deal with initial velocity in $\dot{H}^{\N-1}$ as
Fujita 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 would
like to point out that this type of result is of particular interest when we
want to deal with the problem of the convergence of the solution of
compressible system to the incompressible system when the Mach number goes to
0.
