Universality of the Anomalous Enstrophy Dissipation at the Collapse of Three Point Vortices on Euler--Poincar? Models
Universality of the Anomalous Enstrophy Dissipation at the Collapse of Three Point Vortices on Euler--Poincar? Models
复制标题
欧拉-庞加莱三点涡塌陷时反常熵耗散的普遍性？
DOI：
10.1137/17m1127855
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发表时间：
2018
影响因子：
1.9
通讯作者：
Sakajo Takashi
中科院分区：
文献类型：
--
作者：
Gotoda Takeshi;Sakajo Takashi
不可压缩流动在无粘极限处的反常熵耗散是表征二维湍流的一个重要性质。这表明对非光滑不可压缩和无粘流动的研究有助于从理论上理解湍流现象。在前面的研究中[J]。非线性科学。， 26 (2016), pp. 1525—1570]，考虑了欧拉方程的唯一全局弱解，即点涡初始数据的正则化欧拉方程，从而表明，在关键时刻，三个点涡的演化收敛于自相似的坍缩轨道，在分布意义上耗散了熵。在本文中，为了阐明奇异轨道是否可以在正则化方法上独立构造，我们考虑了欧拉方程的泛函推广，称为欧拉—庞卡罗模型，其中不可压缩速度场被平滑函数分散正则化。给出了奇异轨道存在的充分条件，该条件适用于许多平滑函数。作为算例，我们证实了在求解欧拉方程的数值格式中所使用的高斯正则化和涡团正则化是满足条件的。因此，三点涡塌缩引起的熵耗散是一种普遍现象，不是欧拉方程所特有的，而是欧拉-庞卡罗模型中的普遍现象。
Anomalous enstrophy dissipation of incompressible flows in the inviscid limit is a significant property characterizing two-dimensional turbulence. It indicates that the investigation of nonsmooth incompressible and inviscid flows contributes to the theoretical understanding of turbulent phenomena. In the preceding study [J. Nonlinear Sci., 26 (2016), pp. 1525--1570], a unique global weak solution to the Euler-equations, which is a regularized Euler equation, for point-vortex initial data is considered, and thereby it has been shown that, as, the evolution of three point vortices converges to a self-similar collapsing orbit dissipating the enstrophy in the sense of distributions at the critical time. In the present paper, to elucidate whether this singular orbit can be constructed independently on the regularization method, we consider a functional generalization of the Euler-equations, called theEuler--Poincaré models, in which the incompressible velocity field is dispersively regularized by a smoothing function. We provide a sufficient condition for the existence of the singular orbit, which is applicable to many smoothing functions. As examples, we confirm that the condition is satisfied with the Gaussian regularization and the vortex blob regularization that are both utilized in the numerical scheme solving the Euler equations. Consequently, the enstrophy dissipation via the collapse of three point vortices is a generic phenomenon that is not specific to the Euler-equations but universal within the Euler--Poincaré models.