Asymmetric Self-similar Spiral Solutions of 2-D Incomressible Euler Equations

TL;DR

This paper constructs nonradial, self-similar solutions to 2D incompressible Euler equations, extending previous symmetric spiral flow studies and relating to numerical simulations that explore solution non-uniqueness.

Contribution

It introduces asymmetric, self-similar spiral solutions without symmetry assumptions, advancing understanding of Euler flow structures and solution multiplicity.

Findings

Existence of nonradial, self-similar spiral solutions

Extension of symmetric spiral flow models

Connections to non-uniqueness in Euler solutions

Abstract

We construct nonradial, self-similar solutions to the two-dimensional incompressible Euler equations without assuming rotational symmetry. These solutions extend the study of self-similar algebraic spiral flows, initiated by Elling and further developed by Shao-Wei-Zhang [41], where m-fold symmetry with m>=2 was assumed. Moreover, they bear resemblance to the numerical simulations of Bressan-Shen [10], in connection with the ongoing investigation into non-uniqueness of solutions.