Two Dimensional Incompressible Ideal Flow Around a Small Curve

TL;DR

This paper investigates the asymptotic behavior of 2D incompressible Euler flows around shrinking curves, linking previous results and defining geometric conditions for obstacle convergence to ensure valid limit flows.

Contribution

It extends prior work by characterizing how obstacles shrinking to a curve influence the flow and establishing conditions for the limit flow to satisfy Euler equations.

Findings

Established geometric conditions for domain convergence.

Linked previous results on flow around small obstacles and curves.

Provided a framework for analyzing asymptotic flow behavior.

Abstract

We study the asymptotic behavior of solutions of the two dimensional incompressible Euler equations in the exterior of a curve when the curve shrinks to a point. This work links two previous results: [Iftimie, Lopes Filho and Nussenzveig Lopes, Two Dimensional Incompressible Ideal Flow Around a Small Obstacle, Comm. PDE, 28 (2003), 349-379] and [Lacave, Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve, Ann. IHP, Anl, 26 (2009), 1121-1148]. The second goal of this work is to complete the previous article, in defining the way the obstacles shrink to a curve. In particular, we give geometric properties for domain convergences in order that the limit flow be a solution of Euler equations.