In this talk we focus on the incompressible Navier–Stokes equations with variable
density. The aim is to prove existence and uniqueness results in the case of a discontinuous
initial density (typically we are interested in discontinuity along an interface).
In the first part of the talk, by making use of Fourier analysis techniques, we establish the existence of global-in-time unique solutions in a critical
functional framework, under some smallness condition over the initial data,
In the second part, we use another approach to avoid the smallness condition over the nonhomogeneity : as a matter of fact, one may consider any density bounded
and bounded away from zero and still get a unique solution. The velocity is required to have subcritical regularity, though.
In all the talk, the Lagrangian formulation for describing the flow plays a key role in the analysis.