Global regularity for two-dimensional Kuramoto–Sivashinsky on arbitrary square tori
For $L > 0$ let $\mathbb{T}_L^2 = (\mathbb{R}/(2L\mathbb{Z}))^2$ and consider the Kuramoto–Sivashinsky equation $$\partial_t h + \Delta^2 h + \Delta h + |\nabla h|^2 - \frac{1}{4L^2}\int_{\mathbb{T}_L^2} |\nabla h(t,y)|^2\,dy = 0, \qquad h(0) = h_0,$$ for real-valued $h_0 \in C^\infty(\mathbb{T}_L^2)$ of zero mean; subtracting the mean of $|\nabla h|^2$ keeps the mean of $h$ zero. When $L > \pi$ the linearised operator $-\Delta^2 - \Delta$ has positive eigenvalues, the growing modes. In one space dimension Tadmor proved global existence. In two dimensions global existence is known on thin or strongly anisotropic domains; on two-dimensional tori $[0, L_1] \times [0, L_2]$ Ambrose and Mazzucato proved that small solutions exist for all time when there are no growing modes (2019), and that small, evenly symmetric data give solutions on every finite time interval when both periods are slightly larger than $2\pi$, so that there is a linearly growing mode in each direction (2021). Ambrose, Mazzucato and Montalto proved that for every $L > \pi$ there are data of arbitrarily small size $\varepsilon$ in $H^s$ whose solutions reach size of order one by a time of order $\log(1/\varepsilon)$ (2026).
Problem. For every $L > \pi$ and every real-valued $h_0 \in C^\infty(\mathbb{T}_L^2)$ of zero mean, with no restriction on the size of $h_0$, does the equation have a global smooth solution, that is, one with $h \in C([0,T]; H^m(\mathbb{T}_L^2))$ for every finite $T$ and every integer $m \ge 0$?
Source: A long-standing question on the two-dimensional Kuramoto–Sivashinsky equation; see Ambrose and Mazzucato (2019), §1, and Ambrose, Mazzucato and Montalto (2026), §1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, global regularity, growing modes, Kuramoto–Sivashinsky equation
References
@article{ambrose2019,
author = {Ambrose, David M. and Mazzucato, Anna L.},
title = {Global existence and analyticity for the {2D} {K}uramoto--{S}ivashinsky equation},
journal = {J. Dynam. Differential Equations},
volume = {31},
number = {3},
year = {2019},
pages = {1525--1547},
doi = {10.1007/s10884-018-9656-0},
eprint = {1708.08752},
archivePrefix = {arXiv},
}
@misc{ambrose2026,
author = {Ambrose, David M. and Mazzucato, Anna L. and Montalto, Riccardo},
title = {Nonlinear instability in the {2D} {K}uramoto--{S}ivashinsky equation},
year = {2026},
eprint = {2607.19887},
archivePrefix = {arXiv},
}
@article{tadmor1986,
author = {Tadmor, Eitan},
title = {The well-posedness of the {K}uramoto--{S}ivashinsky equation},
journal = {SIAM J. Math. Anal.},
volume = {17},
number = {4},
year = {1986},
pages = {884--893},
doi = {10.1137/0517063},
}
@article{ambrose2021,
author = {Ambrose, David M. and Mazzucato, Anna L.},
title = {Global solutions of the two-dimensional {K}uramoto--{S}ivashinsky equation with a linearly growing mode in each direction},
journal = {J. Nonlinear Sci.},
volume = {31},
number = {6},
year = {2021},
pages = {96},
doi = {10.1007/s00332-021-09748-8},
eprint = {2102.05093},
archivePrefix = {arXiv},
}
@misc{aim-two-dimensional-kuramoto-sivashinsky-global,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global regularity for two-dimensional Kuramoto–Sivashinsky on arbitrary square tori}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/375-two-dimensional-kuramoto-sivashinsky-global.md},
note = {Entry 375 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-00652, "Global regularity for two-dimensional Kuramoto–Sivashinsky on arbitrary square tori", clopen.solutions.