In this paper, we investigate the global existence and uniqueness of strong
solutions to 2D incompressible inhomogeneous Navier-Stokes equations with
viscous coefficient depending on the density and with initial density being
discontinuous across some smooth interface. Compared with the previous results
for the inhomogeneous Navier-Stokes equations with constant viscosity, the main
difficulty here lies in the fact that the $L^1$ in time Lipschitz estimate of
the velocity field can not be obtained by energy method (see \cite{DM17,LZ1,
LZ2} for instance). Motivated by the key idea of Chemin to solve 2-D vortex
patch of ideal fluid (\cite{Chemin91, Chemin93}), namely, striated regularity
can help to get the $L^\infty$ boundedness of the double Riesz transform, we
derive the {\it a priori} $L^1$ in time Lipschitz estimate of the velocity
field under the assumption that the viscous coefficient is close enough to a
positive constant in the bounded function space. As an application, we shall
prove the propagation of $H^3$ regularity of the interface between fluids with
different densities.