Mass conservation for fractional porous-medium flow on noncompact manifolds
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 $\psi$.
For nonnegative $u_0 \in L^1(M)$, Berchio, Bonforte, Grillo and Muratori constructed a weak dual solution with $u(0) = u_0$ as the monotone limit of solutions with bounded integrable initial data, and its $L^1(M)$ norm does not exceed that of $u_0$. On manifolds with nonnegative Ricci curvature on which the fractional Green function is finite, von Criegern, Grillo and Monticelli proved that weak dual solutions obtained in this way from nonnegative data in $L^1(M)$ conserve their total mass.
Problem. For every $u_0 \ge 0$ in $L^1(M)$ and every nonnegative weak dual solution $u$ with $u(0) = u_0$ and $u(t) \in L^1(M)$, is $$\int_M u(t,x)\, d\mathrm{Vol}(x) = \int_M u_0(x)\, d\mathrm{Vol}(x) \qquad (t > 0)?$$ Conservation at every finite time is asked, not merely an upper bound on the total mass, and the solution need not be one constructed by approximation.
Source: Berchio, Bonforte, Grillo and Muratori (2024), §7, the open problem on mass conservation, 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, for weak dual solutions that are integrable at each time.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, fractional porous medium equations, mass conservation, Riemannian manifolds
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{voncriegern2026,
author = {von Criegern, Dorothea-Enrica and Grillo, Gabriele and Monticelli, Dario Daniele},
title = {Fractional porous medium equation on manifolds with nonnegative {R}icci curvature: existence of solutions and smoothing effects via potential methods},
year = {2026},
eprint = {2609.12954},
archivePrefix = {arXiv},
}
@misc{aim-manifold-fractional-pme-mass-conservation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Mass conservation for fractional porous-medium flow on noncompact manifolds}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/473-manifold-fractional-pme-mass-conservation.md},
note = {Entry 473 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove mass conservation for the spectral fractional porous-medium equation on complete noncompact Riemannian manifolds with Ricci curvature bounded below and a Euclidean-dimensional Faber--Krahn inequality. The result holds for every nonnegative weak dual solution that is spatially integrable at each finite time, for arbitrary nonnegative integrable initial data, all exponents \(m>1\), and all fractional orders \(0<s<1\). Thus the prescribed weak …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/mass_conservation.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00809, "Mass conservation for fractional porous-medium flow on noncompact manifolds", clopen.solutions.