Geometric Hydrodynamics in Open Problems — Problem 14

Definitions and mathematical statement

The supplied record uses the term GM-flow without defining it. Write $\mathcal{G}_{\mathrm{GM}}$ solely as a placeholder for the intended class of such flows; its governing equations, spatial domain, time interval, boundary conditions, and admissibility conditions are not specified in the supplied text.

Regularity problem (source formulation). Determine the true regularity of GM-flows in the framework of anisotropic function spaces. The supplied description attributes to GM-flows an intrinsic structure consisting of analytic level lines of vorticity and a natural, finite regularity in the direction transverse to those lines. The intended task is to identify functional classes suitable for describing this directional regularity and to determine the regularity that the intended GM-flows possess in those classes.

Unresolved qualifications. The record does not specify which anisotropic spaces are intended, which flow quantity is to be measured, whether regularity is local or global, or what constitutes a sharp regularity conclusion. Analyticity of the level lines is a geometric assertion and does not by itself specify analytic regularity of the velocity or vorticity along those lines. The transverse direction also requires clarification at critical points of vorticity. Consequently, the supplied text does not determine a fully quantified regularity statement or numerical regularity thresholds.

The source explicitly distinguishes GM-flows from ancient solutions of the Navier--Stokes equations, despite their conceptual relationship. Its motivation concerns flows that have existed for a long time, illustrated by river flow, rather than flows posed at a definite initial time $t_0$ with an arbitrary initial velocity $u_0$. This motivation supplies no precise temporal hypothesis and must not be substituted for a definition of $\mathcal{G}_{\mathrm{GM}}$.

Short English statement

Identify anisotropic function spaces that capture the intended regularity of GM-flows, taking account of the source's description of analytic vorticity level lines and finite transverse regularity. The supplied excerpt leaves the definition of GM-flows and the precise regularity target unresolved, so a fully quantified statement requires further source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $800815$, source problem code catalogue entry; title: Geometric Hydrodynamics in Open Problems --- Problem 14. The supplied record attributes the problem to Khesin, Misiolek, and Modin, with proposed year 2022.
• Khesin, Misiolek, and Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 14. Supplied source link: https://arxiv.org/abs/2205.01143.
• Supplied provenance: extraction method arxiv_latex; source access date 2026-10-16. This access information is supplied provenance and does not represent an independent visit or verification for this draft.
• The local record labels the problem open and was updated on 2026-10-16. That label is supplied registry curation, not an independently verified assessment of the literature. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.