This talk concerns the evolution of two incompressible, immiscible fluids in two dimensions, governed by the inhomogeneous Navier-Stokes equations. We present new global-in-time well-posedness results, showing the preservation of the natural \( C^{1+\gamma} \) Hölder regularity of the moving interface, for \( 0<\gamma<1 \). This is the first approach that allows for nonnegative density driven by a low-regularity initial velocity, while also remaining valid in the presence of a small viscosity jump.