Potential Singularity of the Axisymmetric Euler Equations with $C^α$ Initial Vorticity for A Large Range of $α$
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2022
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| author | Hou, Thomas Y. Zhang, Shumao |
| author_facet | Hou, Thomas Y. Zhang, Shumao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_11912 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Potential Singularity of the Axisymmetric Euler Equations with $C^α$ Initial Vorticity for A Large Range of $α$ Hou, Thomas Y. Zhang, Shumao Analysis of PDEs Numerical Analysis 35Q31, 76B03, 65M60, 65M06, 65M20 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. |
| title | Potential Singularity of the Axisymmetric Euler Equations with $C^α$ Initial Vorticity for A Large Range of $α$ |
| topic | Analysis of PDEs Numerical Analysis 35Q31, 76B03, 65M60, 65M06, 65M20 |
| url | https://arxiv.org/abs/2212.11912 |