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Gross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEsemerging naturally from the mean field theory of Bose gases. Their solutionsare known to be related to an initial value problem, respectively theGross-Pitaevskii and Hartree equations. Due to their physical and mathematicalrelevance, the issues of well-posedness and uniqueness for these hierarchieshave recently been studied thoroughly using specific nonlinear andcombinatorial techniques. In this article, we introduce a new approach for thestudy of such hierarchy equations by firstly establishing a duality betweenthem and certain Liouville equations and secondly solving the uniqueness andexistence questions for the latter. As an outcome, we formulate a hierarchyequation starting from any initial value problem which is $U(1)$-invariant andprove a general principle which can be stated formally as follows:  (i) Uniqueness for weak solutions of an initial value problem implies theuniqueness of solutions for the related hierarchy equation.  (ii) Existence of solutions for the initial value problem implies existenceof solutions for the related hierarchy equation.  In particular, several new well-posedness results as well as a counterexampleto uniqueness for the Gross-Pitaevskii hierarchy equation are proved. Thenovelty in our work lies in the aforementioned duality and the use of Liouvilleequations with powerful transport techniques extended to infinite dimensionalfunctional spaces.

On well-posedness and uniqueness for general hierarchy equations of Gross-Pitaevskii and Hartree type