The 3D Euler equations with inflow, outflow and vorticity boundary conditions

TL;DR

This paper establishes well-posedness and regularity results for the 3D incompressible Euler equations with inflow and outflow boundary conditions, extending classical results to more general boundary scenarios and multiply connected domains.

Contribution

It introduces new well-posedness results for Euler equations with inflow/outflow conditions, including cases with partial velocity data and vorticity, and extends regularity analysis to complex domains.

Findings

Proves well-posedness with inflow/outflow boundary conditions.

Derives compatibility conditions for regularity in Hölder spaces.

Extends results to multiply connected domains.

Abstract

The 3D incompressible Euler equations in a bounded domain are most often supplemented with impermeable boundary conditions, which constrain the fluid to neither enter nor leave the domain. We establish well-posedness with inflow, outflow of velocity when either the full value of the velocity is specified on inflow, or only the normal component is specified along with the vorticity (and an additional constraint). We derive compatibility conditions to obtain regularity in a H\"{o}lder space with prescribed arbitrary index, and allow multiply connected domains. Our results apply as well to impermeable boundaries, establishing higher regularity of solutions in H\"{o}lder spaces.