A length-independent mean energy bound for Kuramoto–Sivashinsky
For $L \ge 1$ let $u$ solve the Kuramoto–Sivashinsky equation $$u_t + u u_x + u_{xx} + u_{xxxx} = 0$$ with spatial period $2\pi L$ and smooth mean-zero initial data; such solutions exist for all time and stay smooth (Nicolaenko, Scheurer and Temam). Its long-time mean energy is $$\overline{E}(u) = \limsup_{T \to \infty} \frac{1}{2\pi L T} \int_0^T \int_{-\pi L}^{\pi L} |u(x,t)|^2 \, dx \, dt .$$ The late-time maximum of the spatially averaged energy is bounded by $O(L^4)$ for odd solutions (Nicolaenko, Scheurer and Temam); Collet, Eckmann, Epstein and Stubbe, and Goodman, extended such bounds to all solutions, the former with $O(L^{11/5})$; Bronski and Gambill reached $O(L^2)$, and Giacomelli and Otto improved this to $o(L^2)$. Otto, and Goldman, Josien and Otto, proved bounds growing only logarithmically in $L$ for the long-time averages of $(|\partial_x|^\alpha u)^2$ with $1/3 < \alpha \le 2$, and as a corollary Otto obtained $\overline{E}(u) = O(L^{2/3 + \epsilon})$ for every $\epsilon > 0$. Bounds independent of $L$ are known for steady solutions (Michelson). For odd solutions, Goluskin and Fantuzzi computed bounds on $\overline{E}$ that are sharp for $L \le 10$, attained by the simplest nonzero steady states.
Problem. Is there an absolute constant $C$ such that $\overline{E}(u) \le C$ for every $L \ge 1$ and every such solution $u$? The coefficients of the equation are fixed and $L$ is unbounded.
Source: The conjecture that spatial averages in the Kuramoto–Sivashinsky equation are bounded independently of the length, going back to Pomeau, Pumir and Pelcé (1984) and Wittenberg and Holmes (1999); for the mean energy, see Goluskin and Fantuzzi (2019), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, energy bounds, Kuramoto–Sivashinsky equation, spatiotemporal chaos
References
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}
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@article{michelson1986,
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@misc{aim-kuramoto-sivashinsky-energy-density,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A length-independent mean energy bound for Kuramoto–Sivashinsky}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/225-kuramoto-sivashinsky-energy-density.md},
note = {Entry 225 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-00606, "A length-independent mean energy bound for Kuramoto–Sivashinsky", clopen.solutions.