Self-similar Singularity of a 1D Model for the 3D Axisymmetric Euler Equations

TL;DR

This paper proves the existence of self-similar singularities in a 1D model inspired by 3D axisymmetric Euler equations, providing insights into singularity formation with numerical and stability analysis.

Contribution

It introduces a new 1D model for 3D Euler equations and demonstrates the existence and properties of self-similar singularity profiles.

Findings

Existence of a discrete family of self-similar profiles.

Profiles match direct numerical simulations.

Profiles exhibit some stability.

Abstract

We investigate the self-similar singularity of a 1D model for the 3D axisymmetric Euler equations, which is motivated by a particular singularity formation scenario observed in numerical computation. We prove the existence of a discrete family of self-similar profiles for this model and analyze their far-field properties. The self-similar profiles we find agree with direct simulation of the model and seem to have some stability.