Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case. Issue 4 (27th August 2018)

• Record Type: Journal Article
• Authors: Liao, Xian; Zhang, Ping
• Is Part Of: Communications on pure and applied mathematics. Volume 72:Issue 4(2019:Apr.)
• Page Start: 835
• Page End: 884
• Publication Date: 2018-08-27
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21782
• Languages: English
• ISSNs: 0010-3640

Abstract

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.