Potential Singularity of the Axisymmetric Euler Equations with $C^\alpha$ Initial Vorticity for A Large Range of $\alpha$

TL;DR

This paper provides numerical evidence for a potential finite-time singularity in 3D axisymmetric Euler equations with $C^eta$ initial vorticity, exploring the critical H"older exponent and comparing with existing blow-up results.

Contribution

It introduces a numerical study of potential singularities for a range of initial vorticity regularities, including a new one-dimensional model to understand the blow-up mechanism.

Findings

Potential blow-up occurs when $eta < ext{critical value}$, approximately 1/3.

The critical H"older exponent $eta^*$ varies with dimension, near $1 - 2/n$.

The proposed 1D model approximates the $n$-dimensional Euler equations and aids in understanding blow-up.

Abstract

We provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with $C^\alpha$ initial vorticity for a large range of $\alpha$. We employ a highly effective adaptive mesh method to resolve the potential singularity sufficiently close to the potential blow-up time. Resolution study shows that our numerical method is at least second-order accurate. Scaling analysis and the dynamic rescaling method are presented to quantitatively study the scaling properties of the potential singularity. We demonstrate that this potential blow-up is stable with respect to the perturbation of initial data. Our numerical study shows that the 3D axisymmetric Euler equations with our initial data develop finite-time blow-up when the H\"older exponent $\alpha$ is smaller than some critical value $\alpha^*$, which has the potential to be $1/3$. We also study the $n$-dimensional axisymmetric Euler equations with no swirl, and observe that the critical H\"older exponent $\alpha^*$ is close to $1-\frac{2}{n}$. Compared with Elgindi's blow-up result in a similar setting \cite{elgindi2021finite}, our potential blow-up scenario has a different H\"older continuity property in the initial data and the scaling properties of the two initial data are also quite different. We also propose a relatively simple one-dimensional model and numerically verify its approximation to the $n$-dimensional axisymmetric Euler equations. This one-dimensional model sheds useful light to our understanding of the blow-up mechanism for the $n$-dimensional Euler equations.