Instability of Multi-Peaked Orientational Steady States

Definitions and mathematical statement

Let $S^1=\{z\in{\mathbb{C}}:|z|=1\}$. A steady state means an admissible solution of the evolution equation that is independent of time. The supplied record refers to a noisy orientational aggregation model on $S^1$, but does not specify its equation, admissible states, interaction law, noise mechanism, or hypotheses ensuring the existence of states with $N$ peaks. It also does not define a peak or the intended notion of instability.

Conjecture as supplied. For an integer $N\geq 3$, under the hypotheses ensuring the existence of $N$-peaked steady states of the noisy orientational aggregation model on $S^1$ for sufficiently small noise, are all such steady states unstable?

For a schematic expression only, let $\eta$ denote the unspecified noise parameter and let $\mathcal S_N(\eta)$ denote the family of steady states described in the source as $N$-peaked, subject to its unspecified existence hypotheses. The proposed assertion is \[ \forall u_*\in\mathcal S_N(\eta),\qquad u_*\text{ is unstable}, \] for each integer $N\geq3$ and sufficiently small noise under those hypotheses. These symbols do not supply the missing model or stability definition. The record does not determine whether the small-noise bound may depend on $N$, the model parameters, or the particular steady-state family. Consequently, a fully specified mathematical conjecture cannot be recovered from this record alone.

Short English statement

The supplied conjecture asks whether every steady state with at least three peaks is unstable in the small-noise regime where the stated existence hypotheses apply. The model, existence hypotheses, meaning of a peak, and stability criterion are missing, so this remains a qualified statement requiring source consultation and human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001251$, title Instability of Multi-Peaked Orientational Steady States, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Mathematical Biology (2009), Oberwolfach Reports, pp. 1303--1374, DOI 10.4171/owr/2009/24. Supplied source URL and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2009/24. The background attributes the collection to mathematical workshop reports and their open-problem discussions; no individual proposer is supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and describes the assertion of instability for all such $N$-peaked states as a conjecture supported only by numerical evidence; it reports that no proof or counterexample was identified. This is supplied dated curation, not an independently verified literature assessment. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that this denotes meaning-preserving semantic, typographic, and LaTeX cleanup without unsupported source-dependent additions. This supplied label does not establish mathematical correctness or independent verification of the conjecture.