On 2D Navier-Stokes free boundary: nonnegative density and small viscosity contrast

TL;DR

This paper proves global well-posedness and regularity preservation for a two-fluid Navier-Stokes free boundary problem with nonnegative density and small viscosity contrast, advancing understanding of fluid interface dynamics.

Contribution

It establishes the first global-in-time well-posedness result allowing nonnegative density and low-regularity initial velocity with small viscosity jump.

Findings

Preservation of $C^{1+eta}$ regularity of the free boundary.

Global-in-time existence of solutions.

Applicability to low-regularity initial data.

Abstract

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 $C^{1+\gamma}$ H\"older regularity of the free boundary, for $0<\gamma<1$. 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.