  In our recent work dedicated to the Boussinesq equations [Danchin and Zhang
2016], we established the persistence of solutions with piecewise constant
temperature along interfaces with H\"older regularity. We here address the same
problem for the inhomogeneous Navier-Stokes equations satisfied by a viscous
incompressibleand inhomogeneous fluid. We establish that, indeed, in the
slightly inhomogeneous case, patches of densities with $\mathcal{C}^{1,
\varepsilon}$ regularity propagate for all time. As in [Danchin and Zhang
2016], our result follows from the conservation of H\"older regularityalong
vector fields moving with the flow. The proof of that latter result is based on
commutator estimates involving para-vector fields, and multiplier spaces. The
overall analysis is more complicated than in [Danchin and Zhang 2016] however,
since the coupling between the mass and velocity equations in the inhomogeneous
Navier-Stokes equations is \emph{quasilinear} while it is linear for the
Boussinesq equations.
