Weak Solutions of Zero-Mobility Diffuse and Sharp Interfaces

Definitions and mathematical statement

The supplied record concerns a diffuse-interface system and its limiting sharp-interface system. Here $\varepsilon$ denotes the interface thickness, $m$ the diffuse-interface mobility, and $m_0$ the limiting mobility.

The two requested existence statements are:

1. For positive interface thickness $\varepsilon>0$ and zero mobility $m=0$, establish existence of weak solutions of the diffuse-interface system.
2. For zero limiting mobility $m_0=0$, establish existence of global-in-time weak solutions of the limiting sharp-interface system.

The supplied record does not specify either system of equations, its unknowns, spatial domain or dimension, initial or boundary conditions, admissible data, constitutive assumptions, or definition of weak solution. It also does not specify the limiting procedure relating the two systems. Consequently, these are source-qualified existence requests rather than complete mathematical propositions: the missing specifications must be recovered before precise solution spaces, hypotheses, and quantifiers can be stated. Global-in-time existence is explicitly requested only for the sharp-interface system; the time interval intended for the diffuse-interface request is unspecified.

Short English statement

The problem asks for weak solutions with zero mobility in two settings: a diffuse interface of positive thickness, and its sharp-interface limit. Global-in-time existence is requested for the sharp-interface setting. The governing systems and admissibility conditions are missing, so a precise, self-contained existence statement cannot yet be given.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30001412$, source problem code catalogue entry, title "Weak Solutions of Zero-Mobility Diffuse and Sharp Interfaces." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2010; no individual proposer is identified.
• Supplied report citation: New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries (2010), Oberwolfach Reports, pp. 253--324, DOI 10.4171/owr/2010/07. Source URL and supplied literature reference "Oberwolfach Report 7/2010": https://doi.org/10.4171/owr/2010/07. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked 2026-08-21: the original report posed both weak-solution regimes as open, and the supplied triage found no later primary source settling either regime for the extracted model. This is supplied dated curation, not an independently verified literature assessment. The record labels the problem open; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified, with a claim that manual semantic, typographic, and LaTeX review confirmed a self-contained, meaning-preserving cleanup. That supplied review label does not resolve the model specifications absent from this record.