On the Ergodic theory of the Generalized incompressible flow

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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