On 2D Navier-Stokes free boundary: nonnegative density and small viscosity contrast

This paper is concerned with the evolution of two incompressible, immiscible fluids in two dimensions governed by the inhomogeneous Navier-Stokes equations. We prove global-in-time well-posedness, establishing the preservation of the natural $C^{1+\gamma}$ H\"older regularity of the free boundary, for $0<\gamma<1$. This is the first result that allows for nonnegative density driven by a low-regularity initial velocity, while also remaining valid in the presence of a small viscosity jump.