We present a novel method of analysis and prove finite time asymptotically self‐similar blowup of the De Gregorio model [13, 14] for some smooth initial data on the real line with compact support. We also prove self‐similar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on ℝ or for Hölder‐continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self‐similar singularity into the problem of establishing the nonlinear stability of an approximate self‐similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self‐similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well‐posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations. © 2021 Wiley Periodicals LLC.

中文翻译：

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关于3D欧拉方程的De Gregorio模型的有限时间爆破

我们提出了一种新颖的分析方法，并证明了在紧凑支持下真实线上一些平滑初始数据的De Gregorio模型的有限时间渐近自相似爆炸[13，14]。我们也证明了广义德格雷戈里奥模型[41]的参数对整个范围内的自相似爆破结果ℝ或具有紧密支持的Hölder连续初始数据。我们的策略是将使用有限时间渐近自相似奇异性证明的问题，改成使用动态重标度方程建立具有很小残留误差的近似自相似轮廓的非线性稳定性的问题。我们使用具有适当奇异权重函数的能量方法，从线性算子的近似自相似轮廓中提取阻尼效果，并考虑各种非局部项之间的抵消，以建立稳定性分析。我们注意到，我们的分析并不排除原始De Gregorio模型在圆上平滑初始数据时的正确性。本文提出的分析方法为分析非线性非局部偏微分方程组的有限时间奇异性提供了一个有希望的新框架。版权©2021 Wiley Periodicals LLC。