  We prove existence and uniqueness of global weak solutions of the
Navier-Stokes equations that govern the motion of a compressible viscous fluid
with density-dependent viscosity in two space dimensions. The initial velocity
belongs to the Sobolev space $H^1(\mathbb{R}^2)$ and the initial fluid density
is H\"older continuous on both sides of a $\mathcal{C}^{1+\alpha}$ interface
with some geometrical assumption. We show that this configuration persists over
time: the initial interface is transported by the flow to an interface that is
as regular as the initial one. Our result generalizes previous known of Hoff
[21], Hoff and Santos [22] concerning the propagation of regularity for
discontinuity surfaces by allowing more general nonlinear pressure law and
density-dependent viscosity. Moreover, it supplements the work by Danchin,
Fanelli and Paicu [7] with global well-posedness, even for density-dependent
viscosity and we obtain uniqueness in a large space.
