Potential Singularity of the Axisymmetric Euler Equations with $C^\alpha$ Initial Vorticity for A Large Range of $\alpha$. Part II: the $N$-Dimensional Case

TL;DR

This paper investigates finite-time singularity formation in n-dimensional axisymmetric Euler equations with $C^eta$ initial vorticity, identifying critical Hölder exponents and proposing a simplified model to understand blow-up mechanisms.

Contribution

It extends previous 3D analysis to n-dimensions, introduces a new stretching parameter, and develops a one-dimensional model to analyze blow-up behavior.

Findings

Finite-time blow-up occurs for $eta<eta^*$, with $eta^*$ approaching $1-rac{2}{n}$.

Numerical verification of the one-dimensional model approximating n-dimensional Euler equations.

The critical Hölder exponent $eta^*$ asymptotically approaches $1-rac{2}{n}$ as parameters vary.

Abstract

In Part II of this sequence to our previous paper for the 3-dimensional Euler equations \cite{zhang2022potential}, we investigate potential singularity of the $n$-diemnsional axisymmetric Euler equations with $C^\alpha$ initial vorticity for a large range of $\alpha$. We use the adaptive mesh method to solve the $n$-dimensional axisymmetric Euler equations and use the scaling analysis and dynamic rescaling method to examine the potential blow-up and capture its self-similar profile. Our study shows that the $n$-dimensional axisymmetric Euler equations with our initial data develop finite-time blow-up when the H\"{o}lder exponent $\alpha<\alpha^*$, and this upper bound $\alpha^*$ can asymptotically approach $1-\frac{2}{n}$. Moreover, we introduce a stretching parameter $\delta$ along the $z$-direction. Based on a few assumptions inspired by our numerical experiments, we obtain $\alpha^*=1-\frac{2}{n}$ by studying the limiting case of $\delta \rightarrow 0$. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the $n$-dimensional Euler equations. This one-dimensional model sheds useful light to our understanding of the blowup mechanism for the $n$-dimensional Euler equations. As shown in \cite{zhang2022potential}, the scaling behavior and regularity properties of our initial data are quite different from those of the initial data considered by Elgindi in \cite{elgindi2021finite}.