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abstract: '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 [@CD] and [@CMZ] with a new proof. Our result relies on a new a priori estimate for the velocity, where we introduce a new structure to *kill* the coupling between the density and the velocity as in [@H2]. We study so a new variable that we call effective velocity. In a second time we improve the results of [@CD] and [@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 [@5H4]. We conclude by generalizing these results for general viscosity coefficients.'
author:
- 'Boris Haspot [^1]'
title: Existence of global strong solutions in critical spaces for barotropic viscous fluids
---

Introduction
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The motion of a general barotropic compressible fluid is described by 