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

TL;DR

The paper investigates potential finite-time self-similar blow-up in the 3D axisymmetric Euler equations with no swirl and $C^\alpha$ initial vorticity across a broad range of $\alpha$, using adaptive mesh refinement and a dynamic rescaling framework to resolve near-singularity structures. It demonstrates evidence of self-similar blow-up for $\alpha$ below a critical value $\alpha^*$ (near $1/3$ in 3D) and extends the analysis to $n$-dimensional axisymmetric cases, where $\alpha^*$ is near $1-2/n$, all while providing a simple 1D model that captures the leading blow-up behavior. The study leverages scaling relations, operator splitting, and high-resolution computations to show robust blow-up signals consistent with Beale-Kato-Majda-type criteria and to extract scaling exponents $c_l$ and $c_\omega$ that satisfy the expected self-similar relations. Overall, the results support conjectures on axisymmetric blow-up, reveal how regularity near the axis governs the mechanism, and offer a tractable reduced model to illuminate the blow-up dynamics in higher dimensions.

Abstract

We provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with $C^α$ initial vorticity for a large range of $α$. 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 Hölder exponent $α$ is smaller than some critical value $α^*$, 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 Hölder exponent $α^*$ 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 Hölder 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.

Figures (40)

• Figure 1: 3D profiles of the initial value $-\omega^\circ_1$ and $-\psi^\circ_1$.
• Figure 1: Curves of $\|\omega_1\|_{L^\infty}$, $\|\omega\|_{L^\infty}$, $\log\log\|\omega\|_{L^\infty}$, $\int_0^t\|\omega(s)\|_{L^\infty}\mathrm{d}s$, $\|\psi_{1,z}\|_{L^\infty}$ as functions of time $t$.
• Figure 1: Decay of the derivatives of $\psi_1$.
• Figure 1: Steady states of $-\tilde{\omega}_1$ with different $\alpha$ in $\mathbb{R}^3$.
• Figure 1: Profiles of the initial data in all three cases.