## Advances in Differential Equations

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

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867948

**Mathematical Reviews number (MathSciNet)**

MR2100632

**Zentralblatt MATH identifier**

1103.35085

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]

#### Citation

Danchin, R. Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), no. 3-4, 353--386. https://projecteuclid.org/euclid.ade/1355867948