Regular and Singular Steady States of 2D incompressible Euler equations near the Bahouri-Chemin Patch

TL;DR

This paper constructs smooth and singular steady solutions to the 2D incompressible Euler equations on a torus, which converge to a known singular steady state called the Bahouri-Chemin patch, enhancing understanding of such flows.

Contribution

It introduces new families of steady solutions that approximate the Bahouri-Chemin patch, providing insights into the structure of singular and smooth steady states in 2D Euler flows.

Findings

Existence of smooth steady states near the Bahouri-Chemin patch.

Existence of singular steady states approximating the patch.

Convergence of these solutions to the Bahouri-Chemin patch.

Abstract

We consider steady states of the two-dimensional incompressible Euler equations in $\mathbb{T}^2$ and construct smooth and singular steady states around a particular singular steady state. More precisely, we construct families of smooth and singular steady solutions that converge to the Bahouri-Chemin patch.