Multidimensional weak uniqueness for porous-medium flow with fractional pressure
Fix $N \ge 2$, $0 < s < 1$ and $m = 2$. 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 $$\|u(t)\|_q^q + \frac{4q(q-1)}{(m+q-1)^2} \int_0^t \|(-\Delta)^{(1-s)/2} u^{(m+q-1)/2}\|_2^2 \, dr \le \|u_0\|_q^q.$$ With $m = 2$ this is the model of Caffarelli and Vázquez, who constructed weak solutions with finite speed of propagation. Stan, del Teso and Vázquez constructed, for every $m > 1$ and every $N$, weak solutions with the mass, $L^\infty$, energy and dissipation properties above, and proved uniqueness of weak solutions in one dimension; for $m = 2$ uniqueness in one dimension is due to Biler, Karch and Monneau, through the equation integrated in $x$ and its viscosity solutions. Without uniqueness, the weak solutions of Caffarelli and Vázquez and the gradient-flow solutions of Lisini, Mainini and Segatti might differ, as Stan, del Teso and Vázquez note.
Problem. Is there at most one bounded energy weak solution for each such initial density $u_0$, without assuming strict positivity or further smoothness?
Source: Caffarelli and Vázquez (2011), §8, who list uniqueness and comparison for their model among the pending questions; for every m > 1, Stan, del Teso and Vázquez (2018), §1 and §7. 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, fractional pressure, porous medium equations, uniqueness of weak solutions
References
@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},
}
@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{biler2010,
author = {Biler, Piotr and Karch, Grzegorz and Monneau, R{\'e}gis},
title = {Nonlinear diffusion of dislocation density and self-similar solutions},
journal = {Comm. Math. Phys.},
volume = {294},
number = {1},
year = {2010},
pages = {145--168},
doi = {10.1007/s00220-009-0855-8},
eprint = {0812.4979},
archivePrefix = {arXiv},
}
@article{lisini2018,
author = {Lisini, Stefano and Mainini, Edoardo and Segatti, Antonio},
title = {A gradient flow approach to the porous medium equation with fractional pressure},
journal = {Arch. Ration. Mech. Anal.},
volume = {227},
number = {2},
year = {2018},
pages = {567--606},
doi = {10.1007/s00205-017-1168-2},
eprint = {1606.06787},
archivePrefix = {arXiv},
}
@misc{aim-fractional-pressure-pme-multidimensional-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Multidimensional weak uniqueness for porous-medium flow with fractional pressure}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/466-fractional-pressure-pme-multidimensional-uniqueness.md},
note = {Entry 466 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-00595, "Multidimensional weak uniqueness for porous-medium flow with fractional pressure", clopen.solutions.