论文标题
具有$ C^α$初始涡度的轴对称Euler方程的潜在奇异性，大范围$α$
Potential Singularity of the Axisymmetric Euler Equations with $C^α$ Initial Vorticity for A Large Range of $α$
论文作者
论文摘要
我们提供了数值证据，证明3D轴对称Euler方程的潜在有限时间的自相似奇异性，没有漩涡，并且具有$ C^α$初始涡度，范围为$α$。我们采用一种高效的自适应网格方法来解决足够接近潜在爆炸时间的潜在奇异性。分辨率研究表明，我们的数值方法至少是二阶精度。提出了缩放分析和动态恢复方法，以定量研究潜在奇异性的缩放特性。我们证明，相对于初始数据的扰动，这种潜在的爆炸是稳定的。我们的数值研究表明，当Hölder指数$α$小于某些临界值$α^*$时，带有初始数据的3D轴对称Euler方程会形成有限的时间爆炸，这有可能为$ 1/3 $。我们还研究没有漩涡的$ n $维轴对称欧拉方程，并观察到关键的Hölder指数$α^*$接近$ 1- \ frac {2} {n} $。与Elgindi的爆炸结果相似，在类似的情况下\ cite {elgindi2021finite}，我们的潜在爆炸场景在初始数据中具有不同的Hölder连续性属性，并且两个初始数据的缩放属性也大不相同。我们还提出了一个相对简单的一维模型，并在数值上验证其近似与$ n $维轴对称的Euler方程。这个一维模型为我们理解了$ n $维欧拉方程的爆炸机制。
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.
