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This paper is dedicated to the study of viscous compressible barotropicfluids in dimension $N\geq2$. We address the question of the global existenceof strong solutions for initial data close from a constant state havingcritical Besov regularity. In a first time, this article show the recentresults of \cite{CD} and \cite{CMZ} with a new proof. Our result relies on anew a priori estimate for the velocity, where we introduce a new structure to\textit{kill} the coupling between the density and the velocity as in\cite{H2}. We study so a new variable that we call effective velocity. In asecond time we improve the results of \cite{CD} and \cite{CMZ} by adding someregularity on the initial data in particular $\rho_{0}$ is in $H^{1}$. In thiscase we obtain global strong solutions for a class of large initial data on thedensity and the velocity which in particular improve the results of D. Hoff in\cite{5H4}. We conclude by generalizing these results for general viscositycoefficients.

Existence of global strong solutions in critical spaces for barotropic viscous fluids