Steady three-dimensional ideal flows with nonvanishing vorticity in domains with edges

TL;DR

This paper proves the existence of steady three-dimensional ideal fluid flows with vorticity in domains with edges, extending previous results from smooth to nonsmooth boundary domains with specific geometric configurations.

Contribution

It extends the existence theory of stationary Euler equations to nonsmooth domains with edges, analyzing compatibility conditions for the Poisson equation in such settings.

Findings

Existence of solutions in nonsmooth domains with edges

Analysis of compatibility conditions for the Poisson equation

Extension of previous smooth domain results

Abstract

We prove an existence result for solutions to the stationary Euler equations in a domain with nonsmooth boundary. This is an extension of a previous existence result in smooth domains by Alber (1992). The domains we consider have a boundary consisting of three parts, one where fluid flows into the domain, one where the fluid flows out, and one which no fluid passes through. These three parts meet at right angles. An example of this would be a right cylinder with fluid flowing in at one end and out at the other, with no fluid going through the mantle. A large part of the proof is dedicated to studying the Poisson equation and the related compatibility conditions required for solvability in this kind of domain.