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

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 improves the analysis of R. Danchin (2007) in [13], , of Q. Chen et al. (2010) in [8] and of B. Haspot (2009, 2010) in [15,16] inasmuch as we may take initial density in with 1⩽p<+∞. Our result relies on a new a priori estimate for the velocity, where we introduce a new unknown called 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.