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Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case
Author(s) -
Liao Xian,
Zhang Ping
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21782
Subject(s) - compressibility , mathematics , bounded function , domain (mathematical analysis) , boundary (topology) , flow (mathematics) , mathematical analysis , incompressible flow , open domain , constant (computer programming) , mechanics , geometry , physics , computer science , question answering , natural language processing , programming language
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.