Geometric Hydrodynamics in Open Problems — Problem 11

Definitions and mathematical statement

Let $M$ be a two-dimensional compact analytic Riemannian manifold with analytic boundary. The source denotes by $\mathscr{GM}(M)$ the set of generalized minimal flows and by $YU(M)$ the Yudovich space. The supplied record does not define either collection or specify the common ambient space in which their inclusion is to be understood. These symbols therefore retain their source-intended meanings, pending clarification.

The requested assertion is that, for every such $M$, \[ \varnothing\ne\mathscr{GM}(M)\subsetneq YU(M). \] Equivalently, with the source-intended objects and their identification understood, the assertion requires all three conditions \[ \begin{gathered} \exists g\in\mathscr{GM}(M),\\ \forall g\in\mathscr{GM}(M),\quad g\in YU(M),\\ \exists u\in YU(M)\quad\text{such that}\quad u\notin\mathscr{GM}(M). \end{gathered} \] A fully specified mathematical statement requires definitions of generalized minimal flows and the Yudovich space used in the cited problem, including their time domains, boundary conditions, and any identification between flows and velocity fields. No additional connectedness, orientability, or nonempty-boundary hypothesis is supplied; whether any of these is intended remains unresolved.

Short English statement

For every two-dimensional compact analytic Riemannian manifold with analytic boundary, the problem asks for at least one generalized minimal flow, for every such flow to belong to the Yudovich space, and for at least one Yudovich object outside that class. The supplied record leaves the definitions and the comparison between the two classes unspecified.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $800812$, exact source problem code catalogue entry; title: Geometric Hydrodynamics in Open Problems --- Problem 11. The registry attributes the problem to Khesin, Misiolek, and Modin, with proposed year 2022, and labels it open. This label is supplied registry metadata, not an independently verified mathematical status.
• Khesin, Misiolek, and Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 11. Supplied source URL: https://arxiv.org/abs/2205.01143. This is the reference supplied by the record; its contents have not been independently consulted here.
• Supplied provenance reports access on 2026-10-16 and extraction method arxiv_latex. The access date and extraction description are supplied provenance, not a claim of independent source verification.
• Supplied registry upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82; record creation and update timestamps: 2026-10-16T14:00:00Z.