Weak-Solution Uniqueness for Coupled Thermomechanical Systems

Definitions and mathematical statement

Uniqueness question. Are weak solutions unique for the particular coupled thermomechanical initial-boundary-value system (1)--(4) in the report, modeling steel heat treatment through quasistatic force balance, phase-fraction evolution, transformation strain, and an energy balance containing a quadratic stress source? All four coupled components and the quadratic term belong to the stated target; removing that term or decoupling the balances changes the problem.

Denote by $\mathcal U$ the full tuple of unknown fields in that source model. Once its material laws, spatial domain $\Omega$, time interval $(0,T)$, prescribed loads and control inputs, initial and boundary data, and weak-solution class have been fixed, the uniqueness assertion is\[\mathcal U^{(1)},\mathcal U^{(2)}\text{ are weak solutions of (1)--(4) for the same data}\quad\Longrightarrow\quad\mathcal U^{(1)}=\mathcal U^{(2)}\text{ in the stated solution spaces}.\] For ordinary function components this means equality almost everywhere on their respective domains. Any quasistatic displacement determined only up to rigid motions must have the source's normalization or quotient convention specified. No boundary condition or normalization is supplied here.

The report equations (1)--(4), constitutive coefficients, exact quadratic source, number of phases, admissible data, regularity, and weak formulation are absent from the record. Consequently the question cannot be converted into a fully explicit PDE theorem without inventing the model. In particular, a strong-solution or weak--strong uniqueness result would not automatically be the requested uniqueness among all weak solutions.

The supplied literature triage checked 2026-08-21 labels the problem open. It reports a weak-existence theorem for the cited model but says uniqueness remains unresolved for the coupled system with the quadratic stress term. Its assumptions and solution spaces are not reproduced, so this is a dated report requiring independent verification, not an independently established current status.

Short English statement

Determine whether two weak solutions of the specified steel heat-treatment system with identical data must coincide. The full coupling and quadratic stress source must be retained. The actual equations, boundary conditions, and weak-solution spaces are missing from the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID 30000217, code catalogue entry; Weak-Solution Uniqueness for Coupled Thermomechanical Systems, Oberwolfach Reports Open Problems; original target is the report's system (1)--(4).
• Optimal Control of Coupled Systems of PDE (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/18, pp. 995--1072. https://doi.org/10.4171/owr/2005/18. Proposed year and year tag: 2005; no proposer named.
• Supplied literature label Original OWR report: https://ems.press/content/serial-article-files/45992?nt=1.
• Supplied literature label Underlying coupled-model paper: https://doi.org/10.1007/BF02647190.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports checking the official report, not independently checked here.