Full-space minimizers for subcritical fourth-order aggregation energy
For $d \ge 2$, $\chi > 0$ and $1 < m < 2 + 2/d$, let $$\mathcal{E}_m(\rho) = \frac12 \int_{\mathbb{R}^d} |\nabla \rho|^2\,dx - \frac{\chi}{m - 1} \int_{\mathbb{R}^d} \rho^m\,dx$$ on $$\mathcal{A} = \Bigl\{\rho \in H^1(\mathbb{R}^d) : \rho \ge 0,\ \int_{\mathbb{R}^d} \rho\,dx = 1,\ \int_{\mathbb{R}^d} |x|^2 \rho(x)\,dx < \infty\Bigr\}.$$ The fourth-order aggregation–diffusion equation $\partial_t \rho = -\operatorname{div}(\rho \nabla \Delta \rho) - \chi \Delta \rho^m$ is a Wasserstein gradient flow of this energy. Carrillo, Esposito, Falcó and Fernández-Jiménez proved that the energy is bounded below on probability densities with square-integrable gradient when $1 < m < 2 + 2/d$ (and, with $\rho\log\rho$ in place of $\rho^m/(m-1)$, when $m = 1$). At the critical exponent $m = 2 + 2/d$ there is a critical value $\chi_c$: the energy is bounded below for $\chi \le \chi_c$, its infimum is attained at $\chi = \chi_c$, and it is unbounded below for $\chi > \chi_c$; for $m > 2 + 2/d$ it is unbounded below.
Problem. Is $\inf_{\rho \in \mathcal{A}} \mathcal{E}_m(\rho)$ attained in $\mathcal{A}$ for every $d \ge 2$, $\chi > 0$ and $1 < m < 2 + 2/d$? No confining potential, prescribed support or symmetry constraint is imposed, and only existence of a minimizer is asked.
Source: Carrillo, Esposito, Falcó and Fernández-Jiménez (2024), Introduction, who note that existence of minimisers in the subcritical case in the whole space is not clear. Stated by AIM — Open Applied Problems for d ≥ 2 and m > 1, with a finite second moment.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- aggregation-diffusion equations, AIM Open Applied Problems, energy minimizers
References
@article{carrillo2024,
author = {Carrillo, Jos{\'e} Antonio and Esposito, Antonio and Falc{\'o}, Carles and Fern{\'a}ndez-Jim{\'e}nez, Alejandro},
title = {Competing effects in fourth-order aggregation--diffusion equations},
journal = {Proc. Lond. Math. Soc. (3)},
volume = {129},
number = {2},
year = {2024},
pages = {e12623},
doi = {10.1112/plms.12623},
eprint = {2307.14706},
archivePrefix = {arXiv},
}
@article{falco2025,
author = {Falc{\'o}, Carles and Baker, Ruth E. and Carrillo, Jos{\'e} A.},
title = {A nonlocal-to-local approach to aggregation-diffusion equations},
journal = {SIAM Rev.},
volume = {67},
number = {2},
year = {2025},
pages = {353--372},
doi = {10.1137/25M1726248},
eprint = {2505.08443},
archivePrefix = {arXiv},
}
@misc{aim-fourth-order-aggregation-full-space-minimizer,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Full-space minimizers for subcritical fourth-order aggregation energy}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/659-fourth-order-aggregation-full-space-minimizer.md},
note = {Entry 659 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
Yes: for every d ≥ 2, χ > 0 and 1 < m < 2 + 2/d the infimum over A is attained, by a radially nonincreasing, compactly supported minimizer..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 488-aggregation-minimizer/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00684, "Full-space minimizers for subcritical fourth-order aggregation energy", clopen.solutions.