We analyse the 2-dimensional Euler point vortices dynamics in the Koopman-Von Neumann approach. Classical results provide well-posedness of this dynamics involving singular interactions for a finite number of vortices, on a full-measure set with respect to the volume measure $dx^N$ on the phase space, which is preserved by the measurable flow thanks to the Hamiltonian nature of the system. We identify a core for the generator of the one-parameter group of Koopman-Von Neumann unitaries on $L^2(dx^N)$ associated to said flow, the core being made of observables smooth outside a suitable set on which singularities can occur.

中文翻译：

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二维欧拉点涡的 Liouville 算子的基本自伴随性

我们在 Koopman-Von Neumann 方法中分析了二维欧拉点涡流动力学。经典结果提供了这种动力学的适定性，该动力学涉及有限数量涡旋的奇异相互作用，在相对于相空间上的体积测量 $dx^N$ 的完整测量集上，由于可测量的流动而保持系统的哈密顿性质。我们在与所述流相关的 $L^2(dx^N)$ 上为 Koopman-Von Neumann 幺正的单参数组的生成器确定了一个核心，核心由在合适的集合之外平滑的可观察物组成，在该集合上奇点可以发生。