Well-posedness in critical spaces for the system of Navier-Stokes compressible

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N>=2. We address the question of well-posedness for large data having critical Besov regularity. Our result improve the analysis of R. Danchin in [13] and of B. Haspot in [15], by the fact that we choose initial density more general in B N p p,1 with 1 ≤ p < +∞. 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.ou