PreprintarXiv licencearXiv
Free boundary regularity for the porous medium equation
Abstract
In this paper, we resolve the local cone conjecture proposed by Kienzler, Koch, and Vázquez concerning the free-boundary regularity of the porous medium equation. We prove that the local cone condition guarantees the smoothness of the free boundary lying outside the initial support. The proof relies on new gradient estimates obtained through maximum principle arguments and comparison principles. As an application, we establish the global-in-time smoothness of the free boundary for solutions with convex initial support under standard assumptions on the initial data.
Related
- Boundary regularity for fractional porous medium-type equations in bounded domains
- Integral Harnack inequality and expansion of positivity for fractional porous medium equation
- The Porous Medium Equation and Convexity
- Sharp Convergence Rate for stationary states of the Porous Medium Equation
- Interface condition for the Poisson equation in a partially porous medium
BibTeX
@misc{liu2026free,
title = {{Free boundary regularity for the porous medium equation}},
author = {Jiakun Liu and Junqing Wang and Maolin Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25740},
eprint = {2610.25740},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25740},
}
RIS
TY - UNPB
TI - Free boundary regularity for the porous medium equation
AU - Jiakun Liu
AU - Junqing Wang
AU - Maolin Zhou
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.25740
UR - https://arxiv.org/abs/2610.25740
AB - In this paper, we resolve the local cone conjecture proposed by Kienzler, Koch, and Vázquez concerning the free-boundary regularity of the porous medium equation. We prove that the local cone condition guarantees the smoothness of the free boundary lying outside the initial support. The proof relies on new gradient estimates obtained through maximum principle arguments and comparison principles. As an application, we establish the global-in-time smoothness of the free boundary for solutions with convex initial support under standard assumptions on the initial data.
ER -