Finite-Time Singularities in Free-Boundary Fluid Models

Definitions and mathematical statement

The supplied record concerns the following five settings: vortex patches, water waves, interface flows in porous media, Hele--Shaw cells, and atmospheric front dynamics. These names identify families of evolution problems; the record does not specify particular governing equations or admissible configurations.

A finite-time singularity concerns a failure of the required regularity or admissibility of an evolution at some time $T$ with $0<T<\infty$. Its precise meaning depends on the chosen equations, solution concept, regularity class, and geometric conditions. For example, once these choices are fixed, one possible formulation is that a solution defined for $0\leq t<T$ cannot be extended past $T$ within the specified solution class. This is only a possible formalization: the supplied record does not select this criterion or identify it with any particular norm blow-up, interface collision, or geometric degeneration.

Source-level problem. Investigate whether finite-time singularities form in each of the five settings listed above. The source describes singularity formation as a fundamental open question in these settings, but supplies a research theme rather than a single quantified mathematical proposition.

Unresolved formulation. No particular system of equations, spatial dimension, spatial domain, initial-data class, boundary conditions, physical parameters, or singularity criterion is supplied. The record also does not distinguish existence of singularity-forming data from singularity formation for all data, generic data, or a prescribed family of data. Consequently, no unique precise theorem or conjecture can be extracted without additional hypotheses and a specified target. In particular, the five settings must not be treated as one fully specified evolution problem.

Short English statement

The record asks about finite-time singularity formation in vortex patches, water waves, porous-media interfaces, Hele--Shaw cells, and atmospheric fronts. It states a broad research direction. A precise mathematical question requires a chosen model, admissible initial and boundary data, a solution class, a definition of singularity, and explicit quantifiers; these are absent from the supplied record.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30005447$, title Finite-Time Singularities in Free-Boundary Fluid Models, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied source citation: Mini-Workshop: Free Boundary Problems Arising in Fluid Mechanics (2023), Oberwolfach Reports, pp. 607--638, DOI 10.4171/owr/2023/11. Source URL: https://doi.org/10.4171/owr/2023/11. The same URL is supplied in the literature sources under the label "original Oberwolfach report." No individual proposer is identified in the record.
• The supplied original passage identifies the five settings and describes finite-time singularity formation as a fundamental open question. Its Mathematics Subject Classification (2020) codes are 35Q35, 35R35, and 76B03.
• Supplied literature assessment, checked on 2026-08-22: the record is a broad statement concerning several models rather than a single formal problem, and the assessment reports no general resolution across the listed models. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the record open and the statement status corrected_verified; its verification note says that the statement was reconstructed and checked against the official Oberwolfach report. These are supplied registry assertions, not verification performed for this draft.