Instantaneous spreading for fractional-pressure flow in several dimensions
Fix $N \ge 2$, $1 < m < 2$ 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 weak solutions with the mass, $L^\infty$, energy and dissipation properties above in every dimension. In dimension one, where the weak solution is unique, they proved for $1 < m < 2$ that for every $t > 0$ and $R > 0$ the set $\{|x| \ge R : u(x,t) > 0\}$ has positive measure, even for compactly supported data, through the equation integrated in $x$; for $m \ge 2$ the solutions they construct keep compact support in every dimension. For $1 < m < 2$, self-similar solutions in every dimension have infinite speed of propagation, by their transformation to the fractional porous medium equation.
Problem. Does every bounded energy weak solution satisfy $$\int_{\{|x| > R\}} u(t,x) \, dx > 0 \qquad \text{for every } R > 0 \text{ and } t > 0?$$ This asks for unbounded essential support immediately after time zero, not for pointwise positivity, and $u_0$ is not assumed radially symmetric.
Source: Stan, del Teso and Vázquez (2016), Comments and open problems, and (2018), Remark 7(i), for infinite speed of propagation in dimension N ≥ 2 when 1 < m < 2. Stated by AIM — Open Applied Problems for bounded energy weak solutions, as unbounded essential support.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, fractional pressure, infinite speed of propagation, porous medium equations
References
@article{stan2016,
author = {Stan, Diana and del Teso, F{\'e}lix and V{\'a}zquez, Juan Luis},
title = {Finite and infinite speed of propagation for porous medium equations with nonlocal pressure},
journal = {J. Differential Equations},
volume = {260},
number = {2},
year = {2016},
pages = {1154--1199},
doi = {10.1016/j.jde.2015.09.023},
eprint = {1506.04071},
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},
}
@misc{aim-fractional-pressure-multidimensional-instantaneous-spread,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Instantaneous spreading for fractional-pressure flow in several dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/468-fractional-pressure-multidimensional-instantaneous-spread.md},
note = {Entry 468 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-00594, "Instantaneous spreading for fractional-pressure flow in several dimensions", clopen.solutions.