Existence of strong solutions in critical spaces for barotropic viscous fluids in larger spaces

Abstract

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 Danchin and of the author inasmuch as we may take initial density 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 unknown called effective velocity to weaken one of the couplings between the density and the velocity. In particular, our result is the first in which we obtain uniqueness without imposing hypothesis on the gradient of the density.