Geometric Hydrodynamics in Open Problems — Problem 17

Definitions and mathematical statement

Let $\gamma(t)$ be a geodesic in the space denoted by $\mathscr{D}_\mu^s(M)$ in the supplied source, and write $\dot\gamma_0=\dot\gamma(0)$ for its initial velocity. The source asks to establish the bound \[ \mathcal{C}(\gamma(t))\lesssim \exp\!\left(t\|\dot\gamma_0\|_{H^s}\right) \] for every such geodesic, where $\mathcal{C}$ is called its absolute complexity. Here $\lesssim$ indicates an upper bound up to a positive multiplicative constant; the permitted dependence of that constant is not specified.

The supplied record does not define absolute complexity. The notation $\mathscr{D}_\mu^s(M)$ conventionally denotes a Sobolev group of volume-preserving diffeomorphisms, but the record does not specify the manifold $M$, volume structure $\mu$, admissible regularity $s$, boundary conditions, or precise group definition. It also does not specify the metric defining geodesics, the convention for the initial-velocity $H^s$ norm, or the time range of the estimate. Consequently, the displayed expression preserves the source's requested assertion, but these unresolved qualifications prevent a fully specified mathematical statement.

Short English statement

The source asks whether the absolute complexity along every geodesic in $\mathscr{D}_\mu^s(M)$ has an exponential upper bound with rate $\|\dot\gamma_0\|_{H^s}$. The complexity definition, geometric hypotheses, time range, and uniformity of the bound require clarification.

Sources

• Local UnsolvedMath registry: permanent ID $800818$, source problem code catalogue entry, title Geometric Hydrodynamics in Open Problems --- Problem 17. The supplied record attributes the problem to Khesin, Misiolek, and Modin, with proposed year 2022, and labels it open; that status is supplied metadata, not an independently verified mathematical assessment.
• Khesin, Misiolek, and Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 17. Supplied source URL: https://arxiv.org/abs/2205.01143.
• Supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. This access date is supplied provenance and does not indicate independent source verification for this draft.