论文标题

具有$ C^α$初始涡度的轴对称Euler方程的潜在奇异性,范围为$α$。第二部分:$ n $二维案件

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

论文作者

Hou, Thomas Y., Zhang, Shumao

论文摘要

在此序列的第二部分中,我们先前的3维欧拉方程\ cite {zhang2022Potential},我们研究了$ n $ diemnssimational轴对称的Euler方程的潜在奇异性,其$ C^α$ $ c^α$初始涡流的大型范围为$ $ $ lage $ lagence。我们使用自适应网格方法来求解$ n $维轴对称的Euler方程,并使用缩放分析和动态恢复方法来检查潜在的爆炸并捕获其自相似的轮廓。我们的研究表明,当Hölder指数$α<α^*$和此上限$α^*$渐近地接近$ 1- \ frac {2} {n} $时,带有初始数据的$ n $维轴对称欧拉方程会形成有限的时间爆炸。此外,我们在$ z $方向上引入了拉伸参数$δ$。基于我们数值实验启发的一些假设,我们通过研究$δ\ rightarrow 0 $的限制情况来获得$α^*= 1- \ frac {2} {n} $。对于一般情况,我们提出了一个相对简单的一维模型,并在数值上验证了其与$ n $维的Euler方程的近似值。这个一维模型为我们对$ n $维欧拉方程的爆炸机制的理解提供了有用的光线。如\ cite {zhang2022电位}中所示,我们初始数据的缩放行为和规律性属性与elgindi在\ cite {elgindi2021finite}中考虑的初始数据的最初数据完全不同。

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 $C^α$ initial vorticity for a large range of $α$. 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 Hölder exponent $α<α^*$, and this upper bound $α^*$ can asymptotically approach $1-\frac{2}{n}$. Moreover, we introduce a stretching parameter $δ$ along the $z$-direction. Based on a few assumptions inspired by our numerical experiments, we obtain $α^*=1-\frac{2}{n}$ by studying the limiting case of $δ\rightarrow 0$. 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}.

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