The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on $\mathbb{T}^3$
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arXiv
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| Format: | Preprint |
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2025
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| author | Babb, Ryan L. Acosta Rout, Andrew |
| author_facet | Babb, Ryan L. Acosta Rout, Andrew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on $\mathbb{T}^3$ Babb, Ryan L. Acosta Rout, Andrew Analysis of PDEs Mathematical Physics 35Q55 (Primary) 35Q40, 37K06, 35R01 (Secondary) 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. |
| title | The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on $\mathbb{T}^3$ |
| topic | Analysis of PDEs Mathematical Physics 35Q55 (Primary) 35Q40, 37K06, 35R01 (Secondary) |
| url | https://arxiv.org/abs/2503.18520 |