Work

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

Year: 2025

Type: article

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.

Source: Discrete and Continuous Dynamical Systems

DOI: 10.3934/dcds.2025145

Authors: Ryan L. Acosta Babb, Andrew Rout

Institutions: University of Warwick, Centre National de la Recherche Scientifique, Institut de recherche mathématique de Rennes, Université de Rennes

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Deutsche Forschungsgemeinschaft, Agence Nationale de la Recherche

Award: Effective Approximation and Dynamics of Many Body Quantum Systems