This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of the global existence of strong solutions for initial data close from a constant state having critical Besov regularity. In a first time, this article show the recent results of \cite{ CD} and \cite{ CMZ} with a new proof. Our result relies on a new 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 a second time we improve the results of \cite{ CD} and \cite{ CMZ} by adding some regularity on the initial data in particular $\rho_{ 0}$ is in $H^{ 1}$. In this case we obtain global strong solutions for a class of large initial data on the density and the velocity which in particular improve the results of D. Hoff in \cite{ 5H4}. We conclude by generalizing these results for general viscosity coefficients.
Categories: math.AP