Does a bounded interface slope prevent two-phase Muskat singularities?
Let $f_0 \in \bigcap_{k \ge 0} H^k(\mathbb{R})$ be real and let $f$ be the classical solution, on its maximal interval $[0, T_*)$, of $$\partial_t f(t,x) = \operatorname{PV}\!\int_{\mathbb{R}} \frac{\alpha\,[\partial_x f(t,x) - \partial_x f(t,x-\alpha)]}{\alpha^2 + [f(t,x) - f(t,x-\alpha)]^2}\, d\alpha, \qquad f(0) = f_0.$$ This is the two-phase Muskat equation on the whole plane, with equal viscosities and no surface tension, for an interface $y = f(t,x)$ with the heavier fluid below the graph; the density jump is normalised to $2\pi$. Constantin, Gancedo, Shvydkoy and Vicol proved that for $f_0 \in H^k$, $k \ge 3$, the solution stays regular on $[0, T]$ if $\partial_x f$ is bounded on $[0, T]$ and obeys one modulus of continuity for all $x$ and all $t \in [0, T]$, and that a sufficiently small initial slope gives a global solution. Cameron proved global existence of smooth solutions when the initial slope is less than $1$ in absolute value. On the half-plane above an impermeable bottom, Zlatoš constructed solutions whose slope stays bounded and which become singular in finite time, from arbitrarily small smooth data.
Problem. If $T_* < \infty$ and $\sup_{t < T_*} \|\partial_x f(t)\|_{L^\infty} < \infty$, must $f$ extend as a classical solution beyond $T_*$? No common modulus of continuity of $\partial_x f(t)$ is assumed.
Source: Constantin, Gancedo, Shvydkoy and Vicol (2017), §1.3, who conjecture that the optimal well-posedness theory needs only a bound on the slope; see also Zlatoš (2024), §1. Stated in this form, as a continuation criterion, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, free boundaries, Muskat problem, porous media
References
@article{constantin2017,
author = {Constantin, Peter and Gancedo, Francisco and Shvydkoy, Roman and Vicol, Vlad},
title = {Global regularity for {2D} {M}uskat equations with finite slope},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {34},
number = {4},
year = {2017},
pages = {1041--1074},
doi = {10.1016/j.anihpc.2016.09.001},
eprint = {1507.01386},
archivePrefix = {arXiv},
}
@misc{zlatos2024,
author = {Zlato{\v{s}}, Andrej},
title = {The {2D} {M}uskat problem {II}: stable regime small data singularity on the half-plane},
year = {2024},
eprint = {2401.14660},
archivePrefix = {arXiv},
note = {Revised 2024 (v2)},
}
@article{cameron2019,
author = {Cameron, Stephen},
title = {Global well-posedness for the two-dimensional {M}uskat problem with slope less than 1},
journal = {Anal. PDE},
volume = {12},
number = {4},
year = {2019},
pages = {997--1022},
doi = {10.2140/apde.2019.12.997},
eprint = {1704.08401},
archivePrefix = {arXiv},
}
@misc{aim-muskat-bounded-slope-continuation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does a bounded interface slope prevent two-phase Muskat singularities?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/369-muskat-bounded-slope-continuation.md},
note = {Entry 369 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-00617, "Does a bounded interface slope prevent two-phase Muskat singularities?", clopen.solutions.