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2004 Local and global well-posedness results for flows of inhomogeneous viscous fluids
R. Danchin
Adv. Differential Equations 9(3-4): 353-386 (2004).


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.


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R. Danchin. "Local and global well-posedness results for flows of inhomogeneous viscous fluids." Adv. Differential Equations 9 (3-4) 353 - 386, 2004.


Published: 2004
First available in Project Euclid: 18 December 2012

zbMATH: 1103.35085
MathSciNet: MR2100632

Primary: 35Q30
Secondary: 35B30, 76D03, 76D05

Rights: Copyright © 2004 Khayyam Publishing, Inc.


Vol.9 • No. 3-4 • 2004
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