Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in R with N ≥ 2. We address the question of well-posedness for large data having critical Besov regularity and we aim at stating well-posedness in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon in [7] where u0 ∈ B N p −1 p,∞ with 1 ≤ p < +∞. This improves the analysis of [13], [14] and [2] where u0 is considered belonging to B N p −1 p,1 with 1 ≤ p < 2N . Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin in [5] when the velocity u is not considered Lipschitz.