Discontinuous solutions for the Navier-Stokes equations with density-dependent viscosity

We prove existence of a unique global-in-time weak solutions of the Navier-Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space $H^1(\mathbb{R}^2)$, and the initial fluid density is $\alpha$-H\"older continuous on both sides of a $\mathscr{C}^{1+\alpha}$-regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity 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 [6] with global-in-time well-posedness, even for density-dependent viscosity and we achieve uniqueness in a large space.