Global regularity of 2D density patches for inhomogeneous Navier-Stokes

This paper is about Lions' open problem on density patches \cite{LIONS}: whether inhomogeneous incompressible Navier-Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of $C^{1+\gamma}$ regularity with $0<\gamma<1$ in the case of positive density. Furthermore, we go beyond to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for $C^{2+\gamma}$ regularity.