关于广义不可压流的遍历论
On the Ergodic theory of the Generalized incompressible flow
摘要：
为了研究与不可压缩流体力学有关的变分问题，Brenier提出了广义流的概念，并指出广义不可压缩流（GIF）与不可压缩欧拉方程的经典解有着深刻的联系。本文将研究GIF的遍历论，这将有助于我们理解不可压Euler方程的经典解的动力学性质。首先，我们证明了GIF具有弱递归性而不是经典的递归性。然后，我们定义了GIF的遍历性，并讨论了它与经典遍历流的关系。接下来，我们证明了GIF的一些遍历定理。最后，我们给出了一个关于所有GIF的集合的结构的定理。
英文摘要：
To study the variation problem related to the incompressible fluid mechanics, Brenier brings the concept of generalized flow and shows that the generalized incompressible flow (GIF) is deeply related to the classical solution of the incompressible Euler equations. In this paper, we will study the ergodic theory of the GIF which may help us understand the dynamic property of the classical solution of the incompressible Euler equations. First, we show that the GIF has the weak recurrent property rather than the classical one. Then, we define the ergodicity of the GIF and discuss its relation with the classical ergodic flow. Next, we prove some ergodic theorems of the GIF. Finally, we give a theorem about the structure of the set of all GIFs.
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