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

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 $(1-\eta){\bf 1}<em>{\Om_0}+{\bf 1}</em>{\Om_0<sup>c}$ for some small enough constant $\eta$ and some $W<sup>{k+2,p}$ domain $\Om_0,$ and with initial vorticity belonging to $L^1\cap L<sup>p$ and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain $\Om_0$ is preserved by time evolution.