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

TL;DR

This paper proves that two-dimensional incompressible fluids with small viscosity contrast maintain global regularity, using a novel harmonic analysis approach that preserves interface regularity under low Sobolev initial conditions.

Contribution

It introduces a new method to establish global regularity for 2D Navier-Stokes free boundary problems with small viscosity contrast, preserving $C^{1+ ext{gamma}}$ regularity.

Findings

Global-in-time regularity for small viscosity contrast

Preservation of $C^{1+ ext{gamma}}$ interface regularity

Applicable to low Sobolev regularity initial velocities

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 $C^{1+\gamma}$ H\"older 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].