Semigeostrophic Equations on the Whole Rotating Sphere

Definitions and mathematical statement

Let $S_R^2=\{x\in{\mathbb{R}}^3:|x|=R\}$, where $R>0$, denote the spherical surface, and let $e\in{\mathbb{R}}^3$, $|e|=1$, denote its rotation axis. Its equator is \[ E=\{x\in S_R^2:x\cdot e=0\}. \] Write $f$ for the Coriolis parameter; the supplied formulation identifies its degeneracy on $E$ as the obstruction of interest. The record does not supply the semigeostrophic system or its normalization.

Problem. Extend the local well-posedness theory for the semigeostrophic equation from a closed spherical subdomain contained strictly in one hemisphere to the entire rotating sphere $S_R^2$, including its equator $E$, where the Coriolis parameter degenerates. The intended hemispheres appear to be those determined by the rotation axis, \[ H_\pm=\{x\in S_R^2:\pm x\cdot e>0\}, \] but the exact domain hypotheses of the antecedent result are not supplied.

Here local well-posedness is understood to require existence on a positive time interval for every admissible initial datum, uniqueness in a specified solution class, and continuous dependence on the initial data in specified topologies. These requirements must be formulated for the intended spherical semigeostrophic system and must include evolution across the equator. The whole spherical surface has no spatial boundary.

Unresolved formulation requirements. The record does not identify the precise equations, unknowns, admissible initial data, regularity and stability assumptions, solution class, or topologies for continuous dependence. It also does not state the boundary conditions or exact hypotheses of the hemisphere result to be extended. Consequently, the supplied target cannot yet be written as a fully specified quantified initial-value problem. In particular, no existence, uniqueness, or continuous-dependence theorem is asserted here, and no particular interpretation of weak or classical solutions is selected.

Short English statement

The question asks whether local well-posedness for semigeostrophic dynamics on a spherical region strictly within one hemisphere extends to the entire rotating sphere, including the equator where the Coriolis parameter degenerates. The governing equations and analytical hypotheses are missing, so this remains a qualified statement of the target requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30005792$, title Semigeostrophic Equations on the Whole Rotating Sphere, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2024 and does not identify a proposer.
• Applications of Optimal Transportation (2024), Oberwolfach Reports, pp. 309--388, DOI 10.4171/owr/2024/7. Supplied source report and literature reference labelled "original OWR report": https://doi.org/10.4171/owr/2024/7. No problem-specific page or speaker is supplied. The retained original passage describes well-posedness in a more accurate physical context as open to the speaker's knowledge, including among meteorologists; it does not itself state the detailed spherical formulation.
• Supplied literature reference labelled "Silini, local existence on curved domains": https://doi.org/10.1137/22M1532846. The record does not provide its full bibliographic details or the hypotheses of the result being invoked.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports weak and local well-posedness results in restricted geometries, while describing extension to the full rotating sphere and across the equator as unresolved. This is supplied dated curation, not an independently verified assessment; the record expressly 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 that the clean statement restores definitions and removes extraction artifacts. These are supplied provenance claims, not verification performed for this draft.