Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

This paper presents some progress toward an open question proposed by P.-L. Lions [26] concerning the propagation of regularities of density patches for viscous inhomogeneous incompressible flow. We first establish the global-in-time well-posedness of the two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density rho 0=eta(1)1 Omega(0)+eta(2)1 Omega(c)(0). Here (eta(1),eta(2)) is any pair of positive constants and Omega(0) is a bounded, simply connected W-3,W-p(R-2) domain. We then prove that for any positive time t, the density rho(t)=eta(1)1(Omega(t))+eta(2)1(Omega(t)c), with the domain Omega(t) preserving the W-3,W-p-boundary regularity. (c) 2018 Wiley Periodicals, Inc.