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

Boris Haspot (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 in \cite{CMP} where $u_{0}\in B^{\NN-1}_{p,\infty}$ with $1\leq p<+\infty$. This improves the analysis of \cite{Da3}, \cite{Da4} and \cite{AP} 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 in \cite{BCD} when the velocity $u$ is not considered Lipschitz.

Domaines: Equations aux dérivées partielles [math.AP]

Boris Haspot. Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity. Annales de l'Institut Fourier, 2012, 62 (5), pp.1717-1763. ⟨hal-00779844⟩