Local well-posedness results for density-dependent incompressible fluids

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in $\R^{N}$ with $N\geq2$. We address the question of well-posedness for {\it 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 where $u_{0}\in B^{\NN-1}_{p,\infty}$ with $1\leq p<+\infty$. This improves the analysis of H. Abidi, R. Danchin and M. Paicu where $u_{0}$ is considered belonging to $B^{\NN-1}_{p,1}$ with $1\leq p<2N$. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin when the velocity $u$ is not considered Lipschitz.