Global well-posedness for the two-dimensional incompressible and inhomogeneous Navier-Stokes equations with rough density via dynamic interpolation

The incompressible and inhomogeneous Navier-Stokes system (INS) governs the evolution of fluids which, although incompressible, have non constant density. This is a coupling between a transport equation for the density, and an evolution equation similar to the "classical" Navier-Stokes equation for the velocity. It is known since Kazhikhov's seminal paper in 1974 that any initial data with finite energy velocity and strictly positive bounded density generates a global weak solution of finite energy for (INS). But, except in the constant density case and in dimension two, it is not known whether these solutions are unique.