Potential Singularity of the 3D Euler Equations in the Interior Domain
Potential Singularity of the 3D Euler Equations in the Interior Domain
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DOI：
10.1007/s10208-022-09585-5
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发表时间：
2021-07
影响因子：
3
通讯作者：
T. Hou
中科院分区：
文献类型：
--
作者：
T. Hou
三维不可压Euler方程能否从光滑初值发展出有限时间奇异性是非线性偏微分方程中最具挑战性的问题之一。本文给出了一些新的数值证据，证明了具有有限能量光滑初值的三维轴对称不可压Euler方程在原点处存在一个潜在的有限时间奇异性。这种潜在的奇点与Luo和Hou（111：12968-12973，2014）和（12：1722-1776，2014）揭示的发生在边界上的爆破情景不同。我们的初始条件有一个简单的形式，并分享了侯和黄在（arXiv：2102.06663，2021）和（435：133257，2022）中构造的更复杂的初始条件的几个有吸引力的特征。这两种爆破场景之间的一个重要区别是，我们初始数据的解决方案具有单尺度结构，而不是Hou和Huang（arXiv：2102.06663，2021）和（435：133257，2022）中报告的双尺度结构。更重要的是，该解决方案似乎开发了几乎自相似的缩放特性，与3D Navier-Stokes方程兼容。我们将提出数值证据表明，三维欧拉方程似乎开发一个潜在的有限时间奇异性。此外，几乎自相似的轮廓似乎是非常稳定的初始数据的小扰动。
Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo and Hou (111:12968–12973, 2014) and (12:1722–1776, 2014), which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou and Huang in (arXiv:2102.06663, 2021) and (435:133257, 2022). One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in Hou and Huang (arXiv:2102.06663, 2021) and (435:133257, 2022). More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.