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Local and global well-posedness results for flows of inhomogeneous viscous fluids. (English) Zbl 1103.35085
Summary: This paper is devoted to 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 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 $$\rho_0^{-1}\in H^{\frac N2+\alpha}$$ 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.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 42B25 Maximal functions, Littlewood-Paley theory 46N20 Applications of functional analysis to differential and integral equations