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 densityρ 0 = η 11 Ω   0+ η 21Ω 0 c. Here( η 1 , η 2 )is any pair of positive constants and Ω 0 is a bounded, simply connectedW 3 , p( ℝ 2 )domain. We then prove that for any positive time t , the density ρ ( t ) = η 11 Ω ( t )+ η 21 Ω( t ) c, with the domain Ω( t ) preserving theW 3 , p‐boundary regularity. © 2018 Wiley Periodicals, Inc.