Well-Posedness in Critical Spaces for Barotropic Viscous Fluids with Truly Not Constant Density

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 sole additional assumption is that the initial density be bounded away from zero. This improves the analysis of Danchin (Citation2001) where the smallness of for some positive constant was needed. Our result relies on a new a priori estimate for a class of parabolic systems with variable coefficients, which is likely to be useful for the investigation of other models in fluid mechanics. Keywords: Barotropic viscous fluidsBesov spacesCritical spacesLittlewood–Paley theoryScalingWell-posednessMathematics Subject Classification: Primary 76N10Secondary 35B35, 35Q30 Notes 1With our definition of , one can take κ = 9/16. 2Remark the use of the tilde norm.