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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 . Here is any pair of positive constants and Ω 0 is a bounded, simply connected domain. We then prove that for any positive time t , the density , with the domain Ω( t ) preserving the ‐boundary regularity. © 2018 Wiley Periodicals, Inc.
Liao et al. (Mon,) studied this question.