Gross–Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to initial value problems, in particular the Gross–Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations, and secondly, solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is U(1)-invariant and prove a general principle which can be stated formally as follows: In particular, several new well-posedness results, as well as a counterexample to uniqueness for the Gross–Pitaevskii hierarchy equation, are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces.

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关于 Gross-Pitaevskii 和 Hartree 型一般层次方程的适定性

Gross-Pitaevskii 和 Hartree 层次结构是从玻色气体的平均场理论中自然出现的耦合偏微分方程的无限系统。已知他们的解与初值问题有关，特别是 Gross-Pitaevskii 和 Hartree 方程。由于它们的物理和数学相关性，最近使用特定的非线性和组合技术彻底研究了这些层次结构的适定性和唯一性问题。在本文中，我们通过首先建立它们与某些Liouville方程之间的对偶性，然后解决后者的唯一性和存在性问题，引入了研究此类层次方程的新方法。结果，我们从任何 U(1) 不变的初值问题出发，制定了一个层次方程，并证明了一个可以正式表述如下的一般原则：特别是，几个新的适定性结果，以及唯一性的反例Gross-Pitaevskii 层次方程得到证明。我们工作的新颖之处在于上述对偶性和使用 Liouville 方程以及扩展到无限维功能空间的强大传输技术。