Energy conservation at the exact Onsager Hölder exponent
Let $T > 0$ and let $u \in C([0,T]; C^{1/3}(\mathbb{T}^3; \mathbb{R}^3))$ solve the incompressible Euler equations $$\partial_t u + \nabla \cdot (u \otimes u) + \nabla p = 0, \qquad \nabla \cdot u = 0$$ in the sense of distributions. Here $C^{1/3}$ means that $\sup_{x \ne y} |u(x) - u(y)|/d(x,y)^{1/3}$ is finite, $d$ being the distance on $\mathbb{T}^3$, and $E(t) = \tfrac12 \int_{\mathbb{T}^3} |u(t,x)|^2\,dx$ is the kinetic energy. Onsager asserted that Euler flows with Hölder exponent above $1/3$ conserve energy and predicted that energy dissipation is possible below it. Constantin, E and Titi proved conservation for every exponent above $1/3$, and Cheskidov, Constantin, Friedlander and Shvydkoy for weak solutions in $L^3([0,T]; B^{1/3}_{3,c(\mathbb{N})})$, a Besov-type class of exponent exactly $1/3$ with a summability condition at high frequencies. Isett constructed, for every $\alpha < 1/3$, weak solutions in $C_t C^\alpha_x$ with nonempty compact support in time, which therefore do not conserve energy; he later constructed one, again with nonempty compact support in time and so not conserving energy, that lies in $C^\alpha_{t,x}$ for every $\alpha < 1/3$ and satisfies $$|u(t, x + h) - u(t, x)| \le C |h|^{\frac13 - B \sqrt{\log\log |h|^{-1} / \log |h|^{-1}}} \qquad (|h| \le 10^{-2})$$ for some constants $B$ and $C$.
Problem. Must $E(t)$ be constant on $[0,T]$ for every such solution? Equivalently, does a weak solution of the Euler equations in this class exist whose energy changes?
Source: Onsager (1949), who asserted energy conservation for Hölder exponents above 1/3 and predicted dissipation below; the endpoint case as formulated by Isett (2024), Introduction. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, energy conservation, Euler equations, Onsager's conjecture
References
@article{onsager1949,
author = {Onsager, L.},
title = {Statistical hydrodynamics},
journal = {Il Nuovo Cimento},
volume = {6},
number = {Suppl. 2},
year = {1949},
pages = {279--287},
doi = {10.1007/BF02780991},
}
@article{isett2024,
author = {Isett, Philip},
title = {On the endpoint regularity in {O}nsager's conjecture},
journal = {Anal. PDE},
volume = {17},
number = {6},
year = {2024},
pages = {2123--2159},
doi = {10.2140/apde.2024.17.2123},
eprint = {1706.01549},
archivePrefix = {arXiv},
}
@article{constantin1994,
author = {Constantin, Peter and E, Weinan and Titi, Edriss S.},
title = {Onsager's conjecture on the energy conservation for solutions of {E}uler's equation},
journal = {Comm. Math. Phys.},
volume = {165},
number = {1},
year = {1994},
pages = {207--209},
doi = {10.1007/BF02099744},
}
@article{cheskidov2008,
author = {Cheskidov, A. and Constantin, P. and Friedlander, S. and Shvydkoy, R.},
title = {Energy conservation and {O}nsager's conjecture for the {E}uler equations},
journal = {Nonlinearity},
volume = {21},
number = {6},
year = {2008},
pages = {1233--1252},
doi = {10.1088/0951-7715/21/6/005},
eprint = {0704.0759},
archivePrefix = {arXiv},
}
@article{isett2018,
author = {Isett, Philip},
title = {A proof of {O}nsager's conjecture},
journal = {Ann. of Math. (2)},
volume = {188},
number = {3},
year = {2018},
pages = {871--963},
doi = {10.4007/annals.2018.188.3.4},
eprint = {1608.08301},
archivePrefix = {arXiv},
}
@misc{aim-onsager-endpoint,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Energy conservation at the exact Onsager Hölder exponent}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/144-onsager-endpoint.md},
note = {Entry 144 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-00884, "Energy conservation at the exact Onsager Hölder exponent", clopen.solutions.