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.

翻译：我们为无旋三维轴对称欧拉方程在较大 $\alpha$ 范围内、具有 $C^\alpha$ 初值涡度时潜在的有限时间自相似奇异性提供了数值证据。我们采用高效的网格自适应方法来充分接近潜在的爆破时间以解析该潜在奇异性。分辨率研究表明我们的数值方法至少具有二阶精度。通过尺度分析和动态重标度方法，我们对潜在奇异性的尺度特性进行了定量研究。我们证明该潜在爆破对于初始数据的扰动是稳定的。我们的数值研究表明，当 Hölder 指数 $\alpha$ 小于某个临界值 $\alpha^*$（可能为 $1/3$）时，具有我们所用初始数据的三维轴对称欧拉方程会产生有限时间爆破。我们还研究了无旋的 $n$ 维轴对称欧拉方程，并观察到临界 Hölder 指数 $\alpha^*$ 接近 $1-\frac{2}{n}$。与 Elgindi 在类似设置下的爆破结果 \cite{elgindi2021finite} 相比，我们的潜在爆破情景在初始数据的 Hölder 连续性特性上有所不同，且两种初始数据的尺度特性也差异显著。我们还提出了一个相对简单的一维模型，并通过数值验证了其对 $n$ 维轴对称欧拉方程的近似性。该一维模型为我们理解 $n$ 维欧拉方程的爆破机制提供了有益的启示。