Stability of the density patches problem with vacuum for incompressible inhomogeneous viscous flows

TL;DR

This paper proves the stability of solutions to the inhomogeneous incompressible Navier-Stokes equations with vacuum, including classical density patches, by establishing stability estimates without initial density weighting.

Contribution

It demonstrates stability of solutions with vacuum in the inhomogeneous Navier-Stokes system, extending previous results by removing the need for weighted estimates in the presence of vacuum.

Findings

Solutions exhibit exponential decay.

Stability holds in L2(H1) for velocity and negative Sobolev spaces for density.

Applicable to classical density patches with characteristic functions.

Abstract

We consider the inhomogeneous incompressible Navier-Stokes system in a smooth two or three dimensional bounded domain, in the case where the initial density is only bounded. Existence and uniqueness for such initial data was shown recently in [10], but the stability issue was left open. After observing that the solutions constructed in [10] have exponential decay, a result of independent interest, we prove the stability with respect to initial data, first in Lagrangian coordinates, and then in the Eulerian frame. We actually obtain stability in $L_2({\mathbb R}_+;H^1(\Omega))$ for the velocity and in a negative Sobolev space for the density. Let us underline that, as opposed to prior works, in case of vacuum, our stability estimates are not weighted by the initial densities. Hence, our result applies in particular to the classical density patches problem, where the density is a characteristic function.