Abstract (EN)

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 result improves the analysis of R. Danchin in \cite{DW}, of Chen et al in \cite{C2} and of the author in \cite{H1, H2} inasmuch as we may take initial density 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 unknown called \textit{effective velocity} to weaken one the coupling between the density and the velocity. In particular for the first time we obtain uniqueness without imposing hypothesis on the gradient of the density.