Multi-Sink Solutions to the Self-Similar Euler Equations

TL;DR

This paper constructs and classifies multi-stagnation point self-similar solutions to 2D incompressible Euler equations, revealing conditions for multiple stagnation points and their geometric features.

Contribution

It introduces new examples and a classification of self-similar Euler solutions with multiple stagnation points, contrasting with the single stagnation point case.

Findings

Constructed solutions with multiple stagnation points and velocity cusps.

Proved that bounded vorticity solutions have only one stagnation point.

Identified geometric structures of solutions with multiple stagnation points.

Abstract

We construct examples and provide a classification of self-similar solutions to the two-dimensional incompressible Euler equations whose pseudo-velocity fields possess more than one stagnation point. These solutions are also homogeneous steady states of the Euler equations. In contrast, we prove that any homogeneous self-similar solution with bounded vorticity away from the origin necessarily admits only a single stagnation point, located at the origin. The solutions we construct develop velocity cusps along rays from the origin, and this allows for additional stagnation points of the pseudo-velocity field.