Global regularity of 2D density patches for inhomogeneous Navier-Stokes

TL;DR

This paper addresses Lions' open problem by proving the preservation of initial boundary regularity and curvature in 2D density patches for inhomogeneous Navier-Stokes equations, advancing understanding of free boundary regularity.

Contribution

It establishes the propagation of $C^{1+ ext{gamma}}$ regularity and curvature persistence for density patches in 2D Navier-Stokes, including a proof for $C^{2+ ext{gamma}}$ regularity.

Findings

Propagation of $C^{1+ ext{gamma}}$ regularity for density patches

Persistence of curvature in free boundary

Proof of $C^{2+ ext{gamma}}$ regularity

Abstract

This paper is about Lions' open problem on density patches \cite{LIONS}: whether inhomogeneous incompressible Navier-Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of $C^{1+\gamma}$ regularity with $0<\gamma<1$ in the case of positive density. Furthermore, we go beyond to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for $C^{2+\gamma}$ regularity.