Stability and Alternative Costs in Spherical Optimal Transport

Definitions and mathematical statement

The supplied record concerns spherical semigeostrophic solutions on a sphere denoted by $S^2$. It uses $p_t$ for a scalar potential, $u_t$ for a velocity, and $\kappa$ for a curvature parameter. The governing equations, metric, solution class, initial data, and time interval are not supplied. In particular, no family of spherical geometries or identification of their domains as $\kappa\to0$ is specified.

1. Small-curvature stability. Establish stability properties of the spherical semigeostrophic solutions for $\kappa\ll1$. The curated statement describes this question as stability as $\kappa\to0$. Neither the quantities to be controlled nor the topology, comparison solutions, dependence on initial data, or required uniformity in $\kappa$ is specified. Thus the record does not determine a particular stability estimate or convergence assertion.
2. Alternative transport cost. Does there exist another transport cost $c(x,y)$, for $x,y\in S^2$, such that the map \[ T_t(x)=c\text{-}\exp_x\left(\nabla p_t(x)\right) \] yields a continuity equation for the associated transported density that does not involve $u_t$? Here $c\text{-}\exp_x$ is the cost exponential appearing in the source notation. Its defining sign convention, domain, and hypotheses ensuring that it is single-valued are not supplied; the metric defining $\nabla p_t$ is likewise unspecified. The record does not identify the original cost to be replaced, the measure transported by $T_t$, the reference measure for its density, or the permitted dependence of the resulting flux on other unknowns.

These are statement-only formulations of the two supplied questions. The missing specifications prevent a fully quantified stability claim or a precisely defined existence problem for the alternative cost; no particular missing hypothesis or closure condition is imposed here.

Short English statement

The supplied questions ask for stability of spherical semigeostrophic solutions at small curvature and for an alternative spherical transport cost whose induced density equation avoids the velocity $u_t$. The equations, stability criterion, geometric limit, and admissible transport costs need further specification before either question has a fully precise mathematical formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005793$, title: Stability and Alternative Costs in Spherical Optimal Transport; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2024; no proposer is identified.
• Supplied report reference: Applications of Optimal Transportation (2024), Oberwolfach Reports, pp. 309--388, DOI 10.4171/owr/2024/7. Source URL and supplied literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2024/7. No specific contribution, problem page, or equation reference is supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels both questions open and reports that no primary result resolving either question in the stated setting was found. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly and extraction artifacts removed. This verification claim is part of the supplied record; it has not been independently checked here.