Well-posedness in critical spaces for barotropic viscous fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of well-posedness for {\it large} data having critical Besov regularity. %Our sole additional assumption is that %the initial density be bounded away from zero. This improves the analysis of \cite{DL} where the smallness of $\rho_{0}$ for some positive constant $\bar{\rho}$ was needed. Our result improve the analysis of R. Danchin by the fact that we choose initial density more general in $B^{\NN}_{p,1}$ with $1\leq p<+\infty$. 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. In particular our result is the first where we obtain uniqueness without imposing hypothesis on the gradient of the density.