Rates and Self-Similarity in Energy-Driven Coarsening

Definitions and mathematical statement

The supplied record poses a research question about energy-driven coarsening systems: evolving patterns whose characteristic spatial scale grows over time in connection with an energy-driven evolution. This description is qualitative; the record does not specify a class of equations, an energy functional, or a mathematical definition of coarsening.

For a specified system, write $u(t)$ for its evolving state and, if a characteristic length has been defined, write $L(t)>0$ for that length. These symbols are placeholders for quantities that must be supplied by the particular model; they do not introduce additional hypotheses or a universal scaling law.

Source question. For energy-driven coarsening systems:

1. Determine the rate of coarsening and establish it rigorously. Where a characteristic length $L(t)$ is available, this requires determining its dependence on time and proving estimates that justify the proposed rate.
2. Characterize the features of the evolving spatial patterns.
3. Determine whether the statistics of those patterns are self-similar in time, with the statistics and the relevant rescaling specified for the system under consideration.

Unresolved qualifications. The source does not specify the evolution law, spatial domain or dimension, boundary conditions, admissible initial states, solution concept, characteristic length, pattern observables, or statistical framework. It also does not specify the time regime, the order of any large-domain and long-time limits, or whether self-similarity means exact invariance or asymptotic convergence after rescaling. Consequently, this is a model-dependent research program rather than a single fully quantified conjecture. No universal rate, power law, probability distribution, or rescaling is asserted here. The supplied literature assessment refers to matching upper and lower rates, but gives no model-specific estimates or quantifiers from which such a theorem could be formulated.

Short English statement

Determine how rapidly energy-driven patterns coarsen, prove the rate rigorously, describe their evolving structure, and decide whether their statistics become or remain self-similar after an appropriate rescaling. The supplied record leaves the model, characteristic scale, statistics, and meaning of self-similarity unspecified, so a fully quantified statement requires further source context.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001482$, title Rates and Self-Similarity in Energy-Driven Coarsening, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2010; no individual proposer is identified.
• Source report: Phase Transitions (2010), Oberwolfach Reports, pp. 1431--1484, DOI 10.4171/owr/2010/24, https://doi.org/10.4171/owr/2010/24. The supplied background describes Oberwolfach Reports as reports of mathematical workshops and their open-problem discussions.
• Supplied literature reference labeled Brenier--Otto--Seis: https://doi.org/10.1137/090775142.
• Supplied literature reference labeled Kohn--Otto: https://doi.org/10.1007/s00220-002-0693-4.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and reports rigorous one-sided coarsening estimates while describing matching upper and lower rates and the full pattern-statistics picture as unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement status as exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. These are supplied verification claims; no independent source consultation is asserted here.