Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier-Stokes system (Q948861)

The author studies the non-homogeneous incompressible Navier-Stokes system in space. The system describes the evolution of an incompressible viscous fluid whose density is non-constant. The author considers new a priori estimates which enable to deal with low-regularity data and vanishing density. In the paper, new well-posedness results improving previous results of Danchin by considering a less regular initial density are obtained. Furthermore, it is obtained the first uniqueness criterion for weak solutions which is at the scaling of the equation. To do so, the author gives a theorem on the existence of local strong solutions, in which no compatibility condition is required and the density does not have to be bounded from below. In addition, a theorem giving a criterion for the uniqueness of energy class solutions is given.