The Incompressible Navier‐Stokes Equations in Vacuum

Abstract

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier‐Stokes equations supplemented with H1 initial velocity and only bounded nonnegative density. In contrast to all the previous works on those topics, we do not require regularity or a positive lower bound for the initial density or compatibility conditions for the initial velocity and still obtain unique solutions. Those solutions are global in the two‐dimensional case for general data, and in the three‐dimensional case if the velocity satisfies a suitable scaling‐invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in his 1996 book Mathematical Topics in Fluid Mechanics, vol. 1, Incompressible Models, concerning the evolution of a drop of incompressible viscous fluid in the vacuum. © 2018 Wiley Periodicals, Inc.

Striated Regularity of 2-D Inhomogeneous Incompressible Navier–Stokes System with Variable Viscosity

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In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier-Stokes equations with constant viscosity, the main difficulty here lies in the fact that the L 1 in time Lipschitz estimate of the velocity field can not be obtained by energy method (see [11,20,21] for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid ([6, 7]), namely, striated regularity can help to get the L ∞ boundedness of the double Riesz transform, we derive the a priori L 1 in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space.

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“…When one assumes that the viscous coefficient is a positive constant, there are tremendous literatures on this topic. One may check [1,10,11,18,19,20,21,23] and the references therein. In general, Lions [22] proved the global existence of weak solutions to (1.1) with finite Date: December 12, 2017.…”

Between homogeneous and inhomogeneous Navier–Stokes systems: The issue of stability

“…Proof of Lemma 2.2. For the proof, we mainly use the spirit of the corresponding part in [7] (see also [8,Lemma 5.2]). For any z ∈ X T , we define Ψ(z) ≡ ∇∆ −1 div (Id − A)z + R .…”

“…Questions concerned with the low regularity of initial density or the possibility of vacuum states are the subjects of current studies of (IHS) system (1.1) (see e.g. [5,6,8,13,19,27]).…”

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

“…Danchin and Zhang [13], Gancedo and Garcia-Juarez [18] proved the propagation of C k+γ regularity of the density patch to (1.1). Lately Danchin and Mucha [11] can allow vacuum.…”