This talk is about the dynamics of a patch given by two fluids of different constant densities, evolving by the inhomogeneous Navier-Stokes equations. The main question to address is whether the regularity of the boundary of the initial patch is preserved in time.  Using classical Sobolev spaces for the velocity, we establish the propagation of $C^{1+\gamma}$, $W^{2,\infty} and $C^{2+\gamma}$ regularities with $0