Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation

TL;DR

This paper constructs new types of vortex sheet solutions for the 2D incompressible Euler equations, supported on small closed curves, expanding the known classes of solutions using advanced mathematical techniques.

Contribution

It introduces co-rotating and traveling vortex sheets supported on small closed curves, a novel class beyond previously known solutions, using Birkhoff-Rott operator and implicit function theorem.

Findings

New vortex sheet solutions supported on small closed curves

Application of Birkhoff-Rott operator in construction

Use of implicit function theorem at point vortex solutions

Abstract

We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.