On the Vanishing Viscosity Limit for Inhomogeneous Incompressible Navier-Stokes Equations on Bounded Domains

TL;DR

This paper investigates the conditions under which solutions of inhomogeneous Navier-Stokes equations on bounded domains converge to Euler solutions as viscosity vanishes, extending Kato's criterion to inhomogeneous fluids.

Contribution

It extends Kato's criterion for the vanishing viscosity limit from homogeneous to inhomogeneous incompressible fluids using a new relative energy functional.

Findings

Convergence in energy space occurs if energy dissipation vanishes in a boundary layer proportional to viscosity.

The result applies to both two and three-dimensional bounded domains.

A new analytical approach using a relative energy functional is introduced.

Abstract

In this paper we study the vanishing viscosity limit for the inhomogeneous incompressible Navier-Stokes equations on bounded domains with no-slip boundary condition in two or three space dimensions. We show that, under suitable assumptions on the density, we can establish the convergence in energy space of Leray-Hopf type solutions of the Navier-Stokes equation to a smooth solution of the Euler equations if and only if the energy dissipation vanishes on a boundary layer with thickness proportional to the viscosity. This extends Kato's criterion for homogeneous Navier-Stokes equations to the inhomogeneous case. We use a new relative energy functional in our proof.