二维湍流的弱解、重整化解和拟能亏损
Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence
摘要：
涡度拟能是涡度平方积分的一半，它在二维湍流理论中的作用类似于动能在三维湍流的柯尔莫哥洛夫理论中的作用。因此，获得在高雷诺数下涡度拟能耗散方式的描述是有趣的。在这篇文章中，我们探讨的概念，粘性和运输拟能亏损，这模型的空间结构的耗散拟能。这些概念是由G. Eyink试图将Kraichnan-Batchelor的二维湍流理论与不可压缩和理想流体运动方程的弱解性质的现有知识相协调。Eyink理论引出了三个自然问题：（1）拟能亏损的存在性;（2）输运拟能亏损与粘性拟能亏损相等的条件;（3）拟能亏损消失的条件。在[Nonlinearity，v 14（2001）787-802]中，Eyink证明了一些与这些问题相关的结果，并提出了一个关于如何在物理上有意义的上下文中回答这些问题的猜想。本文改进和推广了Eyink的一些结果，并给出了他的猜想的一个反例。
英文摘要：
Enstrophy, half the integral of the square of vorticity, plays a role in 2D turbulence theory analogous to that played by kinetic energy in the Kolmogorov theory of 3D turbulence. It is therefore interesting to obtain a description of the way enstrophy is dissipated at high Reynolds number. In this article we explore the notions of viscous and transport enstrophy defect, which model the spatial structure of the dissipation of enstrophy. These notions were introduced by G. Eyink in an attempt to reconcile the Kraichnan-Batchelor theory of 2D turbulence with current knowledge of the properties of weak solutions of the equations of incompressible and ideal fluid motion. Three natural questions arise from Eyink's theory: (1) Existence of the enstrophy defects (2) Conditions for the equality of transport and viscous enstrophy defects (3) Conditions for the vanishing of the enstrophy defects. In [Nonlinearity, v 14 (2001) 787-802], Eyink proved a number of results related to these questions and formulated a conjecture on how to answer these problems in a physically meaningful context. In the present article we improve and extend some of Eyink's results and present a counterexample to his conjecture.
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