Uniqueness of all weak dual solutions of fractional porous-medium flow
Let $(M,g)$ be a complete, connected, noncompact Riemannian manifold of dimension $N \ge 2$ with $\operatorname{Ric} \ge -(N-1)k$ for some $k \ge 0$, satisfying the uniform Faber–Krahn inequality $\lambda_1(D) \ge c \operatorname{Vol}(D)^{-2/N}$, with $c > 0$, for every relatively compact smooth domain $D \subset M$, where $\lambda_1(D)$ is the first Dirichlet eigenvalue of $-\Delta_M$ on $D$. Fix $m > 1$ and $0 < s < 1$ and consider $$\partial_t u + (-\Delta_M)^s (u^m) = 0,$$ with the spectral fractional Laplace–Beltrami operator. Let $G_s(x,y) = \Gamma(s)^{-1} \int_0^\infty p_t(x,y)\, t^{s-1}\, dt$, where $p_t$ is the heat kernel, and $$\|f\|_{1,x_0,G_s} = \int_{B_1(x_0)} |f| + \int_{M \setminus B_1(x_0)} |f(x)|\, G_s(x,x_0)\, dx,$$ all integrals with respect to Riemannian volume; $L^1_{x_0,G_s}$ is the space where this norm is finite. A nonnegative weak dual solution is a nonnegative $u$ with $u \in C([0,T]; L^1_{x_0,G_s})$ for every $x_0 \in M$ and $T > 0$, $u^m \in L^1((0,T); L^1_{\mathrm{loc}}(M))$, and $$\int_0^T\!\!\int_M \bigl[(-\Delta_M)^{-s}u\, \partial_t \psi - u^m \psi\bigr] = 0$$ for every compactly supported smooth space-time test function $\psi$; its trace is $u(0)$.
Berchio, Bonforte, Grillo and Muratori proved that for every $u_0 \ge 0$ with $\sup_{x_0 \in M} \|u_0\|_{1,x_0,G_s} < \infty$ a weak dual solution with trace $u_0$ exists, obtained as the monotone limit of solutions with bounded integrable initial data, and that this limit does not depend on the approximating sequence.
Problem. Given any $u_0 \ge 0$ with $\sup_{x_0 \in M} \|u_0\|_{1,x_0,G_s} < \infty$, are any two nonnegative weak dual solutions with trace $u_0$ identical? Solutions are not restricted to those constructed by a monotone sequence of bounded integrable approximations.
Source: Berchio, Bonforte, Grillo and Muratori (2024), §7, among their open problems, for the weak dual solutions of their Definition 2.1 on the manifolds of their Assumption 2.1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, fractional porous medium equations, Riemannian manifolds, weak dual solutions
References
@article{berchio2024,
author = {Berchio, Elvise and Bonforte, Matteo and Grillo, Gabriele and Muratori, Matteo},
title = {The fractional porous medium equation on noncompact {R}iemannian manifolds},
journal = {Math. Ann.},
volume = {389},
number = {4},
year = {2024},
pages = {3603--3651},
doi = {10.1007/s00208-023-02731-6},
eprint = {2109.10732},
archivePrefix = {arXiv},
}
@misc{aim-manifold-fractional-pme-weak-dual-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of all weak dual solutions of fractional porous-medium flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/472-manifold-fractional-pme-weak-dual-uniqueness.md},
note = {Entry 472 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-00613, "Uniqueness of all weak dual solutions of fractional porous-medium flow", clopen.solutions.