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

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