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 $C^{1+γ}$ Hölder regularity of the free boundary, for $0<γ<1$. 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.
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