Continuity for degenerate fractional-pressure porous-medium flow
Fix $N \ge 2$, $m > 2$ and $0 < s < 1$. On $\mathbb{R}^N$ consider $$u_t = \nabla \cdot (u^{m-1} \nabla p), \qquad p = (-\Delta)^{-s} u,$$ where $(-\Delta)^{-s}$ has Fourier multiplier $|\xi|^{-2s}$, with a nonzero nonnegative initial density $u_0 \in L^1 \cap L^\infty(\mathbb{R}^N)$ of compact support. A bounded energy weak solution is a nonnegative distributional solution with initial trace $u_0$ and locally integrable flux, continuous in $t$ into $L^1$ with its weak topology, such that for every $t > 0$ it conserves mass, $\int u(t) = \int u_0$, satisfies $\|u(t)\|_\infty \le \|u_0\|_\infty$, the energy inequality $$\frac{1}{2} \|(-\Delta)^{-s/2} u(t)\|_2^2 + \int_0^t \!\! \int_{\mathbb{R}^N} u^{m-1} |\nabla (-\Delta)^{-s} u|^2 \le \frac{1}{2} \|(-\Delta)^{-s/2} u_0\|_2^2,$$ and, for every $q > 1$, the dissipation inequality $$\|u(t)\|_q^q + \frac{4q(q-1)}{(m+q-1)^2} \int_0^t \|(-\Delta)^{(1-s)/2} u^{(m+q-1)/2}\|_2^2 \, dr \le \|u_0\|_q^q.$$ Stan, del Teso and Vázquez constructed, for every $m > 1$, weak solutions with locally integrable flux that conserve mass and satisfy the $L^\infty$ bound, the energy inequality and the dissipation inequalities. For $m = 2$, the model of Caffarelli and Vázquez, Hölder continuity of such weak solutions was proved by Caffarelli, Soria and Vázquez for $s \ne 1/2$ and by Caffarelli and Vázquez for $s = 1/2$.
Problem. Does every bounded energy weak solution admit a representative continuous on $\mathbb{R}^N \times (0, \infty)$? Continuity must hold across the boundary of the set $\{u > 0\}$, not merely inside it.
Source: Stan, del Teso and Vázquez (2018), §7, who list continuity of weak solutions as a pending issue, Hölder continuity being known for m = 2. Stated by AIM — Open Applied Problems for m > 2 and bounded energy weak solutions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, continuity of weak solutions, fractional pressure, porous medium equations
References
@incollection{stan2018,
author = {Stan, Diana and del Teso, F{\'e}lix and V{\'a}zquez, Juan Luis},
title = {Porous medium equation with nonlocal pressure},
booktitle = {Current Research in Nonlinear Analysis},
editor = {Rassias, Themistocles M.},
series = {Springer Optim. Appl.},
volume = {135},
publisher = {Springer},
address = {Cham},
year = {2018},
pages = {277--308},
doi = {10.1007/978-3-319-89800-1_12},
eprint = {1801.04244},
archivePrefix = {arXiv},
}
@article{stan2019,
author = {Stan, Diana and del Teso, F{\'e}lix and V{\'a}zquez, Juan Luis},
title = {Existence of weak solutions for a general porous medium equation with nonlocal pressure},
journal = {Arch. Ration. Mech. Anal.},
volume = {233},
number = {1},
year = {2019},
pages = {451--496},
doi = {10.1007/s00205-019-01361-0},
eprint = {1609.05139},
archivePrefix = {arXiv},
}
@article{caffarelli2011,
author = {Caffarelli, Luis and V{\'a}zquez, Juan Luis},
title = {Nonlinear porous medium flow with fractional potential pressure},
journal = {Arch. Ration. Mech. Anal.},
volume = {202},
number = {2},
year = {2011},
pages = {537--565},
doi = {10.1007/s00205-011-0420-4},
eprint = {1001.0410},
archivePrefix = {arXiv},
}
@article{caffarelli2013,
author = {Caffarelli, Luis and Soria, Fernando and V{\'a}zquez, Juan Luis},
title = {Regularity of solutions of the fractional porous medium flow},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {15},
number = {5},
year = {2013},
pages = {1701--1746},
doi = {10.4171/JEMS/401},
eprint = {1201.6048},
archivePrefix = {arXiv},
}
@article{caffarelli2016,
author = {Caffarelli, Luis and V{\'a}zquez, Juan Luis},
title = {Regularity of solutions of the fractional porous medium flow with exponent 1/2},
journal = {St. Petersburg Math. J.},
volume = {27},
number = {3},
year = {2016},
pages = {437--460},
doi = {10.1090/spmj/1397},
eprint = {1409.8190},
archivePrefix = {arXiv},
}
@misc{aim-fractional-pressure-superquadratic-continuity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuity for degenerate fractional-pressure porous-medium flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/467-fractional-pressure-superquadratic-continuity.md},
note = {Entry 467 at commit 24edaad, 2026-10-15},
}
Resolution claims
We consider $u_t=\nabla\cdot(u^{m-1}\nabla(-\Delta)^{-s}u)$ on $\R^N$. For $N\ge2$, $m>2$ and $1/2<s<1$, we prove that a global bounded energy weak solution with nonzero compactly supported bounded integrable data cannot have a jointly continuous representative at all positive times. The pressure gradient is bounded, and the transport velocity vanishes with the density. A weak moving-halfspace estimate shows that joint continuity would confine the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00795, "Continuity for degenerate fractional-pressure porous-medium flow", clopen.solutions.