Appears in collection : Harmonic analysis of elliptic and parabolic partial differential equations / Analyse harmonique des équations aux dérivées partielles elliptiques et paraboliques

The inhomogeneous incompressible Navier-Stokes equations that govern the evolution of viscous incompressible flows with non-constant density have received a lot of attention lately. In this talk, we shall mainly focus on the singular situation where the density is discontinuous, which is in particular relevant for describing the evolution of a mixture of two incompressible and non reacting fluids with constant density, or of a drop of liquid in vacuum. We shall highlight the places where tools in harmonic analysis play a key role, and present a few open problems.