  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.
