Sharp Waiting-Time Estimates for Variational PDE Discretizations

Definitions and mathematical statement

The supplied problem concerns the waiting-time phenomenon for the porous-medium and thin-film equations and their variational/Lagrangian spatial discretizations. Here waiting time refers informally to an initial period during which a relevant free boundary remains stationary.

An abstract definition, conditional on specifying the relevant interface, is the following. If $\Gamma(t)$ denotes that interface at time $t\geq 0$, its waiting time can be defined by \[ T_{\mathrm{wait}}=\sup\left\{T\geq 0: \Gamma(t)=\Gamma(0)\text{ for every }t\in[0,T]\right\}, \] allowing the value $+\infty$. This is an explanatory convention, not an attributed definition from the report: the record does not specify whether the intended interface is an entire support boundary, one endpoint, or a local boundary portion, nor does it define its discrete counterpart.

The supplied research question is: can the variational/Lagrangian approach to the porous-medium and thin-film equations be sharpened to obtain mesh-independent, quantitatively sharper waiting-time estimates and to account quantitatively for the observed behavior of adapted discretization meshes? Mesh independence would require the relevant estimate constants to be uniform over a specified admissible family of meshes.

A fully quantified mathematical statement cannot be recovered from this record alone. The equations and their parameters, spatial domain and dimension, initial and boundary conditions, solution concept, discretization, admissible meshes, comparison estimates, and meaning of agreement with observed mesh behavior are unspecified. In particular, neither a particular waiting-time inequality nor a convergence claim is supplied. These choices must be resolved before the research question can be stated as a definite theorem or conjecture.

Short English statement

The problem asks for sharper waiting-time estimates for variational/Lagrangian discretizations of porous-medium and thin-film equations, with control uniform over admissible meshes and quantitative agreement with adapted-mesh observations. The supplied record does not specify the hypotheses or numerical comparison needed for a fully precise conjecture. Its open status is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30004511$, source problem code catalogue entry, title: Sharp Waiting-Time Estimates for Variational PDE Discretizations. The record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is supplied. The proposed year is recorded as 2020.
• Supplied report citation: Variational Methods for Evolution (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/29, pp. 1391--1467. https://doi.org/10.4171/owr/2020/29. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Supplied literature source: Fischer--Matthes, SIAM Journal on Numerical Analysis. https://doi.org/10.1137/19M1300017.
• Supplied literature source: Fischer--Matthes, preprint. https://arxiv.org/abs/1911.04185.
• Literature assessment dated exactly 2026-08-22: the record labels the problem open and states that waiting-time behavior is rigorously analyzed for spatially discretized porous-medium and thin-film equations, while the broader numerical-fidelity claim is not a general theorem. This is supplied dated curation derived from the supplied OWR material, not independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry supplies the statement-status label corrected_verified and reports manual semantic, typographic, and LaTeX review of its clean statement. Those are supplied review metadata, not independent verification of the mathematical formulation or cited literature.