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

Abstract

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 interface. Here we provide a new approach which allows us 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 technicalities. It uses new quantitative harmonic analysis bounds for $C^{\gamma }$ norms of even singular integral operators on characteristic functions of $C^{1+\gamma }$ domains [Gancedo and García-Juárez, J. Funct. Anal. 283 (2022)].