Weak uniqueness for fourth-order aggregation–diffusion
Fix $d \ge 2$ and $\chi > 0$, and consider on $\mathbb{R}^d$ $$\partial_t \rho = -\nabla \cdot (\rho \nabla \Delta \rho) - \chi \Delta (\rho^2), \qquad \rho(0) = \rho_0,$$ where $\rho_0 \ge 0$, $\int \rho_0 = 1$, $\rho_0 \in H^1(\mathbb{R}^d)$ and $\int |x|^2 \rho_0(x)\,dx < \infty$. A weak solution on $[0, T]$ is a curve $t \mapsto \rho(t)$ of probability measures with finite second moment, continuous for narrow convergence (against bounded continuous functions), with $$\rho \in L^\infty(0, T; H^1(\mathbb{R}^d)) \cap L^2(0, T; H^2(\mathbb{R}^d)),$$ such that for every $\phi \in C_c^2(\mathbb{R}^d)$ and $0 \le a < b \le T$ $$\int \phi\, [\rho(b) - \rho(a)]\,dx = -\int_a^b \!\! \int \bigl( \rho \Delta \rho \Delta \phi + \Delta \rho \nabla \rho \cdot \nabla \phi + \chi \rho^2 \Delta \phi \bigr)\,dx\,dt.$$ The equation is the Wasserstein gradient flow of $\frac12 \int |\nabla \rho|^2 - \chi \int \rho^2$, and arises as the short-range limit of nonlocal models of cell–cell adhesion (Falcó, Baker and Carrillo). Carrillo, Esposito, Falcó and Fernández-Jiménez proved that a weak solution exists on every $[0, T]$ for every such $\rho_0$, in every dimension, and more generally with $\rho^2$ replaced by $\rho^m$ for $1 \le m < 2 + 2/d$ (2024).
Problem. Is the nonnegative weak solution unique in this class on every finite interval $[0, T]$? Vacuum is allowed: $\rho$ is not assumed to be bounded away from zero.
Source: Carrillo, Esposito, Falcó and Fernández-Jiménez (2024), Introduction, for the family with backward diffusion −χΔρ^m; see Falcó, Baker and Carrillo (2025), §4. Stated by AIM — Open Applied Problems for m = 2 and d ≥ 2, in their class of weak solutions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- aggregation-diffusion equations, AIM Open Applied Problems, thin-film equations, Wasserstein gradient flows
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-weak-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Weak uniqueness for fourth-order aggregation–diffusion}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/480-fourth-order-aggregation-weak-uniqueness.md},
note = {Entry 480 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00591, "Weak uniqueness for fourth-order aggregation–diffusion", clopen.solutions.