Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case. Issue 4 (27th August 2018) - Liao, Xian; Zhang, Ping
Abstract : 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 ρ 0 = η 1 1 Ω     0 + η 2 1 Ω 0 c . Here ( η 1, η 2 ) is any pair of positive constants and Ω0 is a bounded, simply connected W 3, p ( ℝ 2 ) domain. We then prove that for any positive time t, the density ρ ( t ) = η 1 1 Ω ( t ) + η 2 1 Ω ( t ) c, with the domain Ω( t ) preserving the W 3, p ‐boundary regularity. © 2018 Wiley Periodicals, Inc.