Fractional porous medium equation on manifolds with nonnegative Ricci curvature: existence of solutions and smoothing effects via potential methods
Abstract
We study the fractional porous medium equation on complete noncompact Riemannian manifolds with nonnegative Ricci curvature for s∈(0,1] and n>2s. Assuming that the manifold is s-nonparabolic, so that the fractional Laplacian admits a suitable positive minimal Green function, we use this Green function to introduce a natural weighted space of initial data, strictly larger than L1. This leads to a weak dual, or potential, formulation of the equation, for which we prove existence for nonnegative initial data in the weighted space. We then establish quantitative local smoothing estimates for initial data in either L1 or the Green-weighted space. Under additional noncollapsing and uniform volume-growth assumptions, we obtain global smoothing estimates. We also show that, even in ℝn, data in the Green-weighted space need not generate bounded solutions unless a suitable uniform weighted integrability condition is imposed. The short- and long-time behaviors predicted by our estimates are shown to be optimal in appropriate senses. When s∈(0,1/2), the results require only Ric≥0, together with a noncollapsing assumption where needed. Our estimates also cover the case s=1, several of them being new in that setting. Finally, we extend the approach to more general filtration equations.
Related
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BibTeX
@misc{criegern2026fractional,
title = {{Fractional porous medium equation on manifolds with nonnegative Ricci curvature: existence of solutions and smoothing effects via potential methods}},
author = {Dorothea-Enrica von Criegern and Gabriele Grillo and Dario Daniele Monticelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12954},
eprint = {2609.12954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12954},
}
RIS
TY - UNPB
TI - Fractional porous medium equation on manifolds with nonnegative Ricci curvature: existence of solutions and smoothing effects via potential methods
AU - Dorothea-Enrica von Criegern
AU - Gabriele Grillo
AU - Dario Daniele Monticelli
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12954
UR - https://arxiv.org/abs/2609.12954
AB - We study the fractional porous medium equation on complete noncompact Riemannian manifolds with nonnegative Ricci curvature for s∈(0,1] and n>2s. Assuming that the manifold is s-nonparabolic, so that the fractional Laplacian admits a suitable positive minimal Green function, we use this Green function to introduce a natural weighted space of initial data, strictly larger than L^1. This leads to a weak dual, or potential, formulation of the equation, for which we prove existence for nonnegative initial data in the weighted space. We then establish quantitative local smoothing estimates for initial data in either L^1 or the Green-weighted space. Under additional noncollapsing and uniform volume-growth assumptions, we obtain global smoothing estimates. We also show that, even in ℝ^n, data in the Green-weighted space need not generate bounded solutions unless a suitable uniform weighted integrability condition is imposed. The short- and long-time behaviors predicted by our estimates are shown to be optimal in appropriate senses. When s∈(0,1/2), the results require only Ric≥0, together with a noncollapsing assumption where needed. Our estimates also cover the case s=1, several of them being new in that setting. Finally, we extend the approach to more general filtration equations.
ER -