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

Ryan L. Acosta Babb; Andrew Rout

doi:10.3934/dcds.2025145

Article Contents

2026, Volume 48: 376-403. Doi: 10.3934/dcds.2025145

This volume Previous Article On lattice points, short-time estimates, and global well-posedness of the quintic NLS on $ \mathbb{T} $ Next Article Ruelle's inequality and Pesin's formula for Anosov geodesic flows in non-compact manifolds

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

Ryan L. Acosta Babb^1, , and
Andrew Rout^2,

1.
University of Warwick, UK
2.
Univ Rennes, [UR1], CNRS, IRMAR, France

^*Corresponding author: Ryan L. Acosta Babb
^*Corresponding author: Ryan L. Acosta Babb

Received: April 2025

Early access: September 2025

Published: March 2026

The second author is supported by the ANR-DFG project (ANR-22-CE92-0013, DFG PE 3245/3-1 and BA 1477/15-1).

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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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q40, 37K06, 35R01.

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