The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on <inline-formula><tex-math id="M1">$ \mathbb{T}^3 $</tex-math></inline-formula>

In this article, we consider Hartree equations generalised to $ 2p+1 $ order nonlinearities. These equations arise in the study of the mean-field limits of Bose gases with $ p $–body interactions. We study their well-posedness properties in $ H^{s_c}( \mathbb{T}^3) $, where $ \mathbb{T}^3 $ is the three dimensional torus and $ s_c = 3/2 - 1/p $ is the scaling-critical regularity. The convergence of solutions of the Hartree equation to solutions of the nonlinear Schrödinger equation is proved. We also consider the case of mixed nonlinearities, proving local well-posedness in $ s_c $ by considering the problem as a perturbation of the higher-order Hartree equation. In the particular case of the (defocusing) quintic-cubic Hartree equation, we also prove global well-posedness for all initial conditions in $ H^1( \mathbb{T}^3) $. This is done by viewing it as a perturbation of the local quintic NLS.