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

In this article, we show that the Navier-Stokes system with variable density and viscosity is locally well-posed in the Besov space Ḃ N p p 1(R N ) × ( Ḃ N p −1 p 1 (R N ) )N , for 1 < p ≤ 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 [8]). Moreover, we prove existence and uniqueness in the Sobolev space H N 2 (R ) × ( H N 2 −1+α(RN ) )N for α > 0, generalizing R. Danchin’s result for the case where viscosity is constant (see [7]).