On the global regularity of 2-D density patch for inhomogeneous incompressible viscous flow

TL;DR

This paper proves the global regularity and existence of solutions for 2-D inhomogeneous incompressible Navier-Stokes equations with density patches, confirming the preservation of domain regularity over time.

Contribution

It establishes the global existence and regularity propagation for density patches in 2-D inhomogeneous Navier-Stokes flows, addressing an open question by P.-L. Lions.

Findings

Global existence of solutions with density patches

Preservation of domain regularity over time

Solutions with initial vorticity in L^1 ∩ L^p

Abstract

Toward P.-L. Lions' open question in \cite{Lions96} concerning the propagation of regularity for density patch, we establish the global existence of solutions to the 2-D inhomogeneous incompressible Navier-Stokes system with initial density given by $(1-\eta){\bf 1}_{\Om_0}+{\bf 1}_{\Om_0^c}$ for some small enough constant $\eta$ and some $W^{k+2,p}$ domain $\Om_0,$ and with initial vorticity belonging to $L^1\cap L^p$ and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain $\Om_0$ is preserved by time evolution.