We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the energy is also constructed, as well as the corresponding flow. Poincare recurrence theorem is used to show that the flow returns infinitely many times in a neighbourhood of the initial state.

中文翻译：

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二维平均欧拉方程的不变测度

我们为二维平均欧拉方程定义了一个高斯不变测度，并在该测度的支持下显示了其解的存在性和初始条件。还构建了能量水平集上的不变表面测度，以及相应的流。Poincare 递推定理用于证明流在初始状态的邻域内无限多次返回。