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

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 $(1-\eta ){\mathbf{1}}_{{\text{Om}}_0}+{\mathbf{1}}_{{\text{Om}}_0^c}$ for some small enough constant $\eta$ and some $W^{k+2,p}$ domain ${\text{Om}}_0,$ 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 ${\text{Om}}_0$ is preserved by time evolution.