Chemotactic Cell Sorting in Keller–Segel Models

Definitions and mathematical statement

The supplied record concerns a two-species Keller--Segel chemotaxis model in two spatial dimensions. It denotes one model parameter by $\chi_2$ and asks about the qualitative regime \[ \chi_2\ll 1. \] The equations, the definition and normalization of $\chi_2$, its admissible values, and the parameters held fixed in this regime are not supplied. Thus the displayed relation records the source's qualitative smallness condition, not a specified inequality or limiting procedure.

The source question is: can chemotaxis be the principal mechanism of cell sorting in this model, and what behavior occurs in the regime $\chi_2\ll 1$? Here neither "cell sorting" nor "principal mechanism" is given a mathematical definition. In particular, the record does not specify a criterion for spatial separation, the relevant time scale, or the comparison needed to identify a principal mechanism.

A precise quantified problem remains unresolved at the level of the supplied formulation: the model equations, spatial domain, initial and boundary conditions, solution class, parameter hypotheses, and sorting observable must be specified before a theorem or conjecture can be stated. No particular Keller--Segel system or such hypotheses are imposed in this draft.

The supplied background reports that radial estimates provide regimes of global existence and blow-up, and that blow-up of one species forces blow-up of the other in a cited theorem. The theorem's hypotheses and conclusion are not reproduced, so this implication is retained only as a qualified report of background results, not as an assertion for every two-species Keller--Segel model. The supplied assessment says the small-$\chi_2$ sorting question is not fully resolved.

Short English statement

The question asks whether chemotaxis can drive cell sorting in a two-species, two-dimensional Keller--Segel model, especially when $\chi_2\ll 1$. The supplied record does not define the system, the parameter, or a mathematical criterion for sorting, so it does not determine a precise quantified conjecture. Its partially solved label and descriptions of radial results are supplied curation requiring independent review.

Sources

• Local UnsolvedMath registry, permanent ID $30003940$, source problem code catalogue entry, title "Chemotactic Cell Sorting in Keller--Segel Models," in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2018 and names no individual proposer.
• Supplied source citation: Reactive Flows in Deformable, Complex Media (2019), Oberwolfach Reports, pp. 2385--2473, DOI 10.4171/owr/2018/39. The literature-source label is "Oberwolfach Report 39/2018." https://doi.org/10.4171/owr/2018/39. Both year designations are retained as supplied.
• Supplied literature reference: Sorting Phenomena in a Mathematical Model for Two Species. https://doi.org/10.48550/arXiv.1704.04179. The record supplies no authors or theorem identifier.
• Literature assessment dated exactly 2026-08-21: the supplied status is "partially solved." The assessment reports radial global-existence and blow-up regimes, a theorem in which blow-up of one species forces the other, and an incompletely resolved small-$\chi_2$ sorting question. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. Time-sensitive status and cited mathematical claims require independent verification.