Global smoothness of the one-dimensional surface-growth equation
Let $\mathbb{T} = \mathbb{R}/(2\pi\mathbb{Z})$ and consider the surface growth equation $$\partial_t h + \partial_x^4 h + \partial_x^2\bigl((\partial_x h)^2\bigr) = 0, \qquad h(0,x) = h_0(x),$$ a model of epitaxial growth, for real-valued $h_0 \in C^\infty(\mathbb{T})$ of zero mean. Blömker and Romito proved that for data in $\dot H^{1/2}(\mathbb{T})$ there is a unique local smooth solution, which is global when the $\dot H^{1/2}$ norm of $h_0$ is small and otherwise, if its maximal time is finite, blows up there in $\dot H^\beta$ for every $\beta > 1/2$; and that the set of singular times of a weak solution has zero $\tfrac14$-dimensional Hausdorff measure. Ożański and Robinson proved that the space-time singular set of a suitable weak solution has box-counting dimension at most $7/6$ and zero one-dimensional parabolic Hausdorff measure. Blömker, Nolde and Robinson verified global existence and uniqueness of the smooth solution for certain initial data by a rigorous computer-assisted method.
Problem. For every real-valued $h_0 \in C^\infty(\mathbb{T})$ of zero mean, with no smallness assumption on $h_0$, does the maximal classical solution exist for all $t \ge 0$? Equivalently, is $\sup_{0 \le t \le T} \|h(t)\|_{H^m(\mathbb{T})} < \infty$ for every finite $T$ and every integer $m \ge 0$?
Source: Blömker and Romito (2009), §1.1, who obtain global smooth solutions only for small data; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, fourth-order parabolic equations, global regularity, surface growth models
References
@article{blomker2009,
author = {Bl{\"o}mker, Dirk and Romito, Marco},
title = {Regularity and blow up in a surface growth model},
journal = {Dyn. Partial Differ. Equ.},
volume = {6},
number = {3},
year = {2009},
pages = {227--252},
doi = {10.4310/DPDE.2009.v6.n3.a2},
eprint = {0902.1409},
archivePrefix = {arXiv},
}
@article{ozanski2019,
author = {O{\.z}a{\'n}ski, Wojciech S. and Robinson, James C.},
title = {Partial regularity for a surface growth model},
journal = {SIAM J. Math. Anal.},
volume = {51},
number = {1},
year = {2019},
pages = {228--255},
doi = {10.1137/18M1166821},
eprint = {1710.07625},
archivePrefix = {arXiv},
}
@article{blomker2015,
author = {Bl{\"o}mker, Dirk and Nolde, Christian and Robinson, James C.},
title = {Rigorous numerical verification of uniqueness and smoothness in a surface growth model},
journal = {J. Math. Anal. Appl.},
volume = {429},
number = {1},
year = {2015},
pages = {311--325},
doi = {10.1016/j.jmaa.2015.04.025},
eprint = {1311.2205},
archivePrefix = {arXiv},
}
@misc{aim-surface-growth-global-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global smoothness of the one-dimensional surface-growth equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/374-surface-growth-global-regularity.md},
note = {Entry 374 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-00647, "Global smoothness of the one-dimensional surface-growth equation", clopen.solutions.