The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on ${\mathbb{T}}^3$

Abstract

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.