A rigorous justification of the Euler and Navier-Stokes equations with geometric effects

TL;DR

This paper rigorously derives 1D Euler and Navier-Stokes equations with geometric effects from 3D systems in a narrowing cylinder, using energy inequalities and Korn-Poincare's inequality.

Contribution

It provides a rigorous mathematical justification for the reduction of 3D Navier-Stokes to 1D equations in variable cross-section geometries.

Findings

Derivation of 1D equations as asymptotic limits

Use of relative energy inequality for weak solutions

Development of Korn-Poincare's inequality for thin channels

Abstract

We derive the 1D isentropic Euler and Navier-Stokes equations describing the motion of a gas through a nozzle of variable cross section as the asymptotic limit of the 3D isentropic Navier-Stokes system in a cylinder, the diameter of which tends to zero. Our method is based on the relative energy inequality satisfied by any weak solution of the 3D Navier-Stokes system and a variant of Korn-Poincare's inequality on thin channels that may be of independent interest.