Global Regularity of 2D Navier-Stokes Free Boundary with Small Viscosity Contrast

Abstract

This paper studies the dynamics of two incompressible immiscible fluids in 2D 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 [32] for $H^{5/2}$ Sobolev regularity of the interface. Here we provide a new approach which allows to obtain preservation of the natural $C^{1+\gamma }$ Hölder regularity of the interface for all $0<\gamma <1$. Our proof is direct and allows for low Sobolev regularity of the initial velocity without any extra technicality. It uses new quantitative harmonic analysis bounds for $C^{\gamma }$ norms of even singular integral operators on characteristic functions of $C^{1+\gamma }$ domains [21].