Equation de Navier-Stokes avec densité et viscosité variables dans l’espace critique

Abstract

In this article, we show that the Navier-Stokes system with variable density and viscosity is locally well-posed in the Besov space

for $1<p\le N$ when the initial density approaches a strictly positive constant. This result generalizes the work by R. Danchin for the case where the viscosity is constant and $p=2$ (see [Danchin, R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1311-1334.]). Moreover, we prove existence and uniqueness in the Sobolev space\arriba{2}

for $\alpha >0,$ generalizing R. Danchin's result for the case where viscosity is constant (see [Danchin, R.: Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353-386.]).