Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

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 ρ0=η1boldnormal1Ω 0+η2boldnormal1Ω0c. Here (η1,η2) is any pair of positive constants and Ω0 is a bounded, simply connected W3,p(ℝ2) domain. We then prove that for any positive time t, the density ρ(t)=η1boldnormal1Ω(t)+η2boldnormal1Ω(t)c, with the domain Ω(t) preserving the W3,p‐boundary regularity. © 2018 Wiley Periodicals, Inc.

A well-posedness result for viscous compressible fluids with only bounded density

“…In the particular case where ρ 0 is given by (0.6) then, once the Lipschitz control of the transport field is available, it is possible to propagate the Lipschitz regularity of the domain D, as it is just advected by the (Lipschitz continuous) flow of the velocity field. Based on that observation, further developments and more accurate informations on the evolution of the boundary of D have been obtained very recently by X. Liao and P. Zhang in [30,31], and by the first author with X. Zhang [17] and with P. B. Mucha [15]. In most of those works, a key ingredient is the propagation of tangential regularity, in the spirit of the seminal work by J.-Y Chemin in [4,5] dedicated to the vortex patch problem for the incompressible Euler equations.…”

The Incompressible Navier‐Stokes Equations in Vacuum

“…The main issue is to establish that the smoothness of D 0 is preserved through the time evolution (see [10,13,21,22]).…”

Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system

“…We remark that similar idea was first used by Hmidi and Keraani [14] for two dimensional incompressible Euler system, which also works (without change) for the transport diffusion equation. In the inhomogeneous context, similar idea was used by Liao and the author [23] in order to propagate fractional Besov regularities for the velocity field of the two dimensional incompressible inhomogeneous Navier-Stokes system. Remark 2.2.…”

“…To avoid the difficulty caused by vacuum, Liao and the author [22] investigated the case when the system (1.1) is supplemented with the initial density, ρ 0 (x) = η 1 1 Ω 0 + η 2 1 Ω c 0 , for a pair of positive constants (η 1 , η 2 ) with |η 1 − η 2 | being sufficiently small, and where Ω 0 is a bounded, simply connected 2D domain with W k+2,p -boundary regularity for k ∈ N . This smallness assumption for the difference between η 1 and η 2 was removed by the authors in [23]. Danchin and Zhang [13], Gancedo and Garcia-Juarez [18] proved the propagation of C k+γ regularity of the density patch to (1.1).…”