Surjectivity of the two-dimensional Euler endpoint map
Let $(M,g)$ be a closed connected oriented Riemannian surface 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, $\partial_t \eta(t,x) = u(t, \eta(t,x))$, $\eta(0,x) = x$. 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 $M$, so $u_0 \mapsto \eta(1)$ is its Riemannian exponential map at the identity. Ebin and Marsden showed that on the group of area-preserving $H^s$ diffeomorphisms this map is smooth near zero; by the inverse function theorem it is a local diffeomorphism there, so every such diffeomorphism close enough to the identity is reached. Ebin, Misiołek and Preston proved that on a closed surface the exponential map is a nonlinear Fredholm map of index zero.
Problem. For every such $M$, every $s > 2$ and every area-preserving $H^s$ diffeomorphism $\phi$ of $M$ in the identity component, is there a divergence-free $u_0 \in H^s(TM)$ whose Euler solution has particle map $\eta(1) = \phi$?
Source: Khesin, Misiołek and Shnirelman (2023), Problem 8(i), the surjectivity problem; stated by AIM — Open Applied Problems for closed oriented surfaces and diffeomorphisms in the identity component.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, diffeomorphism groups, Euler equations, exponential maps
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},
}
@article{ebin1970,
author = {Ebin, David G. and Marsden, Jerrold},
title = {Groups of diffeomorphisms and the motion of an incompressible fluid},
journal = {Ann. of Math. (2)},
volume = {92},
number = {1},
year = {1970},
pages = {102--163},
doi = {10.2307/1970699},
}
@article{ebin2006,
author = {Ebin, D. G. and Misio{\l}ek, G. and Preston, S. C.},
title = {Singularities of the exponential map on the volume-preserving diffeomorphism group},
journal = {Geom. Funct. Anal.},
volume = {16},
number = {4},
year = {2006},
pages = {850--868},
doi = {10.1007/s00039-006-0573-8},
}
@misc{aim-euler-endpoint-map,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Surjectivity of the two-dimensional Euler endpoint map}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/145-euler-endpoint-map.md},
note = {Entry 145 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-00860, "Surjectivity of the two-dimensional Euler endpoint map", clopen.solutions.