### Local and global well-posedness results for flows of inhomogeneous viscous fluids

R. Danchin

#### Abstract

This paper is devoted to the the study of density-dependent, incompressible Navier-Stokes equations with periodic boundary conditions, or in the whole space. We aim at stating well-posedness in functional spaces as close as possible to the ones imposed by the scaling of the equations. Preliminary results have been obtained in [5] under the assumption that the density is close to a constant. Getting rid of this assumption (by allowing smoother data if necessary) is the main motivation of the present paper. Local well-posedness is stated for data $(\rho_0,u_0)$ such that $(\rho_0-cste)\in H^{{{{\frac N2}}}+\alpha}$ and $\inf\rho_0>0$, and $u_0\in H^{{{{\frac N2}}}-1+\beta}$. The indices $\alpha,\beta>0$ may be taken arbitrarily small. We further derive a blow-up criterion which entails global well-posedness in dimension $N=2$ if there is no vacuum initially.

#### Article information

Source
Adv. Differential Equations, Volume 9, Number 3-4 (2004), 353-386.

Dates
First available in Project Euclid: 18 December 2012