On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations
On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations
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DOI：
10.1002/cpa.21991
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发表时间：
2019-05
影响因子：
3
通讯作者：
Jiajie Chen;T. Hou;De Huang
中科院分区：
文献类型：
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作者：
Jiajie Chen;T. Hou;De Huang
我们提出了一种新的分析方法，并证明了De Gregorio模型[13，14]在具有紧支撑的真实的直线上的某些光滑初始数据的有限时间渐近自相似爆破。我们还证明了广义De Gregorio模型[41]的自相似爆破结果，对于具有紧支撑的Hölder连续初始数据，在λ或S1上的整个参数范围内。我们的策略是将证明有限时间渐近自相似奇异性的问题转化为使用动态重标度方程建立具有小残差的近似自相似轮廓的非线性稳定性的问题。我们使用能量方法和适当的奇异权函数从近似自相似轮廓周围的线性化算子中提取阻尼效应，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的德格雷戈里奥模型对于圆上的光滑初始数据是适定的可能性。本文提出的分析方法为非线性非局部偏微分方程组的有限时间奇异性分析提供了一个新的框架。© 2021 Wiley Periodicals LLC.
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 S1 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.