  We investigate the incompressible Navier-Stokes equations with variable
density. The aim is to prove existence and uniqueness results in the case of
discontinuous ini- tial density. In dimension n = 2, 3, assuming only that the
initial density is bounded and bounded away from zero, and that the initial
velocity is smooth enough, we get the local-in-time existence of unique
solutions. Uniqueness holds in any dimension and for a wider class of velocity
fields. Let us emphasize that all those results are true for piecewise constant
densities with arbitrarily large jumps. Global results are established in
dimension two if the density is close enough to a positive constant, and in
n-dimension if, in addition, the initial velocity is small. The Lagrangian
formula- tion for describing the flow plays a key role in the analysis that is
proposed in the present paper.
