Potential Singularity of the Axisymmetric Euler Equations with $C^\alpha$ Initial Vorticity for A Large Range of $\alpha$. Part II: the $N$Dimensional Case
Abstract
In Part II of this sequence to our previous paper for the 3dimensional 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 blowup and capture its selfsimilar profile. Our study shows that the $n$dimensional axisymmetric Euler equations with our initial data develop finitetime blowup when the Hölder 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 onedimensional model and numerically verify its approximation to the $n$dimensional Euler equations. This onedimensional 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}.
 Publication:

arXiv eprints
 Pub Date:
 December 2022
 DOI:
 10.48550/arXiv.2212.11924
 arXiv:
 arXiv:2212.11924
 Bibcode:
 2022arXiv221211924H
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis;
 35Q31;
 76B03;
 65M60;
 65M06;
 65M20