Abstract

The dynamics of a continuous medium in a pipe is not exhausted by spontaneous unsteady turbulence vortices (first seen in flashes of light and generated at high Reynolds numbers), which permanently level the parabolic velocity profile in the pipe. The ambient space also includes steady swirls and curls, which are usually approximated by analytical dependences decomposable in power series. It turns out that, in the Kochin–Yudovich boundary value flow problem, the existence of an ideal incompressible steady flow that is arbitrarily smooth, but not analytic (and, hence, a phantom, i.e., it cannot be classically approximated by polynomials with any prescribed degree of accuracy or, in other words, cannot be computed exactly, but is established over time) is also reduced to vortices of this type. Specifically, the existence analysis is reduced to finding an infinitely smooth uncomputable mass rate of such vortices in the form of a stream function solving the two-dimensional Dirichlet problem for the negative Laplacian with a right-hand side specified as an infinitely smooth Sobolev cutoff function, which was introduced as early as the 1930s and later became known as a Friedrichs mollifier. This problem is briefly discussed below.

中文翻译：

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静止 Kochin-Yudovich 流问题中的涡流模型

摘要

管道中连续介质的动力学不会被自发的不稳定湍流漩涡（首先出现在闪光中并在高雷诺数下产生）耗尽，这些涡流永久地使管道中的抛物线速度分布保持水平。环境空间还包括稳定的漩涡和卷曲，它们通常由可分解为幂级数的分析相关性来近似。事实证明，在 Kochin-Yudovich 边界值流问题中，存在一个理想的不可压缩稳态流，该流是任意光滑的，但不是解析的（因此，一个幻象，即它不能被具有任何规定的准确度，或者换句话说，不能精确计算，但随着时间的推移而建立）也减少到这种类型的漩涡。具体来说，存在性分析被简化为以流函数的形式找到此类涡流的无限平滑不可计算质量速率，解决负拉普拉斯算子的二维狄利克雷问题，右侧指定为无限平滑 Sobolev 截止函数，其中早在 1930 年代就被引入，后来被称为弗里德里希缓和剂。下面简要讨论这个问题。