Uniqueness of Yudovich's solutions to the 2D incompressible Euler equation despite the presence of sources and sinks

TL;DR

This paper extends the uniqueness results of Yudovich's solutions for the 2D incompressible Euler equations to domains with multiple internal sources and sinks, addressing an open problem in fluid dynamics.

Contribution

It adapts previous results to smooth domains with internal sources and sinks, establishing uniqueness of solutions with bounded vorticity in this setting.

Findings

Proves uniqueness of Yudovich's solutions with bounded vorticity in complex domains.

Extends prior results from rectangular to smooth domains with multiple sources and sinks.

Addresses an open problem in the mathematical theory of fluid flows.

Abstract

In $1962$, Yudovich proved the existence and uniqueness of classical solutions to the 2D incompressible Euler equations in the case where the fluid occupies a bounded domain with entering and exiting flows on some parts of the boundary. The normal velocity is prescribed on the whole boundary, as well as the entering vorticity. The uniqueness part of Yudovich's result holds for H\"older vorticity, by contrast with his 1961 result on the case of an impermeable boundary, for which the normal velocity is prescribed as zero on the boundary, and for which the assumption that the initial vorticity is bounded was shown to be sufficient to guarantee uniqueness. Whether or not uniqueness holds as well for bounded vorticities in the case of entering and exiting flows has been left open until $2014$, when Weigant and Papin succeeded to tackle the case where the domain is a rectangle. In this paper we adapt Weigant and Papin's result to the case of a smooth domain with several internal sources and sinks.