Geometric Hydrodynamics in Open Problems — Problem 20

Definitions and mathematical statement

The supplied source record names four entropy functions, \[ H_\epsilon,\qquad H_{\epsilon,\delta},\qquad \tilde{H}_\epsilon,\qquad \tilde{H}_{\epsilon,\delta}, \] but does not supply their definitions. It also omits the section referred to in the source statement, so the underlying objects, domains, parameter ranges, and hypotheses are unspecified.

The stated research task is to investigate the properties of these four entropy functions as defined in that omitted section and to explain the relations between them. No particular property or relation is specified. Without the referenced definitions and context, this task cannot be formulated as a precise quantified mathematical statement. The notation alone does not determine the intended entropy constructions or justify assumptions about their parameters.

Short English statement

Investigate the four entropy functions named in Problem 20 and explain their relationships. Their definitions and the referenced section are absent from the supplied record, so a precise mathematical formulation remains unresolved.

Sources

• Local UnsolvedMath registry: supplied source record with permanent ID $800821$, exact source code catalogue entry, titled Geometric Hydrodynamics in Open Problems --- Problem 20.
• Khesin, Misiolek, Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 20. Supplied source URL: https://arxiv.org/abs/2205.01143. The record attributes the problem to these authors with proposed year 2022.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; neither the source nor its surrounding section was independently inspected for this draft.
• The local record labels the problem open. This is a supplied status label, not an independently verified mathematical or literature assessment.