Global Solutions and Sharp-Interface Limits in Three Dimensions

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^3$ and consider the source's coupled quasistatic-elasticity/phase-field system \[ -\operatorname{div}_x T=b,\qquad T=D\left(\varepsilon(\nabla_xu)-\bar\varepsilon S\right), \] \[ S_t=-c\left(\psi_S(\varepsilon(\nabla_xu),S)-\nu\Delta_xS\right) |\nabla_xS|,\qquad \nu>0. \] In the usual interpretation, $u(t,x)\in{\mathbb{R}}^3$ is displacement, $S(t,x)\in{\mathbb{R}}$ is the phase variable, $T$ is stress, $b$ a body force, and \[ \varepsilon(\nabla_xu)=\tfrac12(\nabla_xu+(\nabla_xu)^{\mathsf T}). \] The elasticity operator $D$ acts on symmetric matrices, $\bar\varepsilon$ is a transformation strain, and $\psi_S$ is the derivative of a free-energy density with respect to its phase argument, with strain held fixed. The kinetic coefficient is $c$; its positivity, the coercivity of $D$, the form of $\psi$, and their regularity are customary modeling data but are not explicitly supplied as hypotheses here. The factor $|\nabla_xS|$ makes the phase equation degenerate even for $\nu>0$.

The record asks both:

1. For positive $\nu$, does the associated initial-boundary-value problem have global-in-time solutions in three spatial dimensions?
2. Do the resulting solutions converge as $\nu\to0^+$ to solutions of the corresponding sharp-interface model?

The second asks for an actual limiting evolution, not just a formal substitution $\nu=0$. A sharp-interface description requires identifying phase values, interface geometry, elastic transmission and interface-velocity laws, together with a notion of convergence; these are absent. If a family $(u_\nu,S_\nu)$ is constructed, full-sequence versus subsequential convergence and its time interval are likewise unspecified.

The source calls this an initial-boundary-value problem but does not reproduce the initial phase, displacement/traction or phase boundary conditions, domain regularity, force assumptions, or strong/weak/viscosity solution definition. Consequently no assertion for arbitrary data, nor a substituted Dirichlet, Neumann or periodic problem, is made here.

The supplied literature triage checked 21 August 2026 says that no authoritative result resolving both three-dimensional existence and the sharp-interface limit was located. It cites only the original report; that finding does not determine separately the current status of every existence or limit variant.

Short English statement

For the stated three-dimensional elastic phase-field system, do global solutions exist for each positive regularization parameter, and do they converge to the intended moving-interface model as that parameter vanishes? Initial/boundary data, solution spaces and the interface law are missing, so the exact theorem requires further source context.

Sources

• Local UnsolvedMath registry, permanent ID 30000159, exact source code catalogue entry; Global Solutions and Sharp-Interface Limits in Three Dimensions. Proposed year 2004, status open. The supplied corrected_verified statement restores the full displayed system and both questions.
• Supplied report citation: Thermodynamische Materialtheorien (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/55, pp. 2971--3036, https://doi.org/10.4171/owr/2004/55. Title year differs from DOI/proposed year. The verification's claimed primary-report inspection is source metadata, not an inspection performed here.
• Supplied literature assessment checked 2026-08-21, citing only "original OWR report", https://doi.org/10.4171/owr/2004/55. This dated OWR-derived triage requests independent verification of time-sensitive status and cited claims; no separate later result is supplied.