Incompressible flows with piecewise constant density

TL;DR

This paper proves local and global existence and uniqueness results for incompressible Navier-Stokes equations with discontinuous, piecewise constant density, using Lagrangian methods, in multiple dimensions under various initial conditions.

Contribution

It establishes the first rigorous results for incompressible flows with piecewise constant density, including cases with large jumps and near-constant densities.

Findings

Local-in-time existence of unique solutions in 2D and 3D.

Global existence in 2D for densities close to constant.

Uniqueness in any dimension for broader velocity classes.

Abstract

We investigate the incompressible Navier-Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous ini- tial density. In dimension n = 2, 3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. Let us emphasize that all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n-dimension if, in addition, the initial velocity is small. The Lagrangian formula- tion for describing the flow plays a key role in the analysis that is proposed in the present paper.