Global regularity of 2D Navier-Stokes free boundary with small viscosity contrast

This paper studies the dynamics of two incompressible immiscible fluids in two dimensions modeled by the inhomogeneous Navier-Stokes equations. We prove that if initially the viscosity contrast is small then there is global-in-time regularity. This result has been proved recently in Paicu and Zhang [Comm. Math. Phys. 376 (2020)] for H-5/2 Sobolev regularity of the inter-face. Here we provide a new approach which allows us to obtain preservation of the natural C (1+Y) Holder regularity of the interface for all 0 < y < 1. Our proof is direct and allows for low Sobolev regularity of the initial velocity without any extra technicalities. It uses new quantitative harmonic analysis bounds for C-Y norms of even singular integral operators on characteristic functions of C (1+Y) domains [Gancedo and Garcia-Juarez, J. Funct. Anal. 283 (2022)].