A closed Lagrangian Euler trajectory on a surface of higher genus
Let $(M,g)$ be a closed connected oriented smooth Riemannian surface of genus at least two and let $s > 2$. For a divergence-free vector field $u_0 \in H^s(TM)$, let $u$ solve the incompressible Euler equations $$\partial_t u + \nabla_u u = -\operatorname{grad} p, \qquad \operatorname{div} u = 0, \qquad u(0) = u_0,$$ and let $\eta$ be its particle map, $\partial_t \eta = u \circ \eta$, $\eta(0) = \operatorname{id}_M$. By Arnold's observation, $t \mapsto \eta(t)$ is a geodesic of the right-invariant $L^2$ metric on the group of area-preserving diffeomorphisms of $M$, and a solution with $\eta(T) = \operatorname{id}_M$ and $u(T) = u_0$ is a closed geodesic of that group, not of $M$. Khesin, Misiołek and Shnirelman note that by the Kelvin–Helmholtz theorem every geodesic loop in this group is a closed geodesic.
Problem. For every such $(M,g)$ and every $s > 2$, is there a nonzero divergence-free $u_0 \in H^s(TM)$ and a $T > 0$ whose Euler solution satisfies $$\eta(T) = \operatorname{id}_M, \qquad u(T) = u_0,$$ so that every particle and the velocity return?
Source: Khesin, Misiołek and Shnirelman (2023), Problem 10; stated by AIM — Open Applied Problems for closed oriented surfaces of genus at least two.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, closed geodesics, diffeomorphism groups, Euler equations
References
@article{khesin2023,
author = {Khesin, Boris and Misio{\l}ek, Gerard and Shnirelman, Alexander},
title = {Geometric hydrodynamics in open problems},
journal = {Arch. Ration. Mech. Anal.},
volume = {247},
number = {2},
year = {2023},
pages = {15},
doi = {10.1007/s00205-023-01848-x},
}
@article{arnold1966,
author = {Arnold, Vladimir},
title = {Sur la g{\'e}om{\'e}trie diff{\'e}rentielle des groupes de {L}ie de dimension infinie et ses applications {\`a} l'hydrodynamique des fluides parfaits},
journal = {Ann. Inst. Fourier (Grenoble)},
volume = {16},
number = {1},
year = {1966},
pages = {319--361},
doi = {10.5802/aif.233},
}
@book{arnold1998,
author = {Arnold, Vladimir I. and Khesin, Boris A.},
title = {Topological Methods in Hydrodynamics},
series = {Applied Mathematical Sciences},
volume = {125},
publisher = {Springer},
address = {New York},
year = {1998},
doi = {10.1007/b97593},
}
@misc{aim-closed-euler-geodesic,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A closed Lagrangian Euler trajectory on a surface of higher genus}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/146-closed-euler-geodesic.md},
note = {Entry 146 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-00850, "A closed Lagrangian Euler trajectory on a surface of higher genus", clopen.solutions.