Lagrangian Schemes for Irregular Porous-Medium Solutions

Definitions and mathematical statement

Let $\rho$ denote the density of a porous-medium solution considered in the supplied source, and write $S_t=\operatorname{supp}\rho(t,\cdot)$ for its spatial support at time $t$. In the notation of the record, the driving force is \[ v=-\nabla U'(\rho), \] where $U'$ denotes the derivative of the scalar function $U$, and $\nabla$ denotes the spatial gradient wherever this expression is defined. The function $U$, the spatial domain and dimension, the porous-medium exponent, the initial and boundary conditions, and the precise solution class are not specified in the supplied record.

Question (extension of a convergence result). Does the convergence result for the Lagrangian discretization scheme referred to in the source remain valid for less regular porous-medium solutions, particularly solutions for which the force $-\nabla U'(\rho)$ is nonzero at the boundary $\partial S_t$ of the support, as described in the source for the Barenblatt solution?

Here the reference to a nonzero boundary force must retain its source-specific meaning: the record does not specify whether it is a pointwise value, an interior limit, or a trace, nor at which boundary points or times it is required to be nonzero. It also does not specify the discretization, its parameters and refinement regime, the convergence topology, or the hypotheses and conclusion of the convergence theorem to be extended. Consequently, this is a qualified formulation of the supplied extension question; a fully quantified convergence assertion cannot be recovered from this record alone.

Short English statement

The question asks whether the cited Lagrangian scheme still converges for porous-medium solutions with weaker regularity, especially when the force at the edge of the support is nonzero, as in the source's Barenblatt example. The scheme, convergence criterion, admissible regularity, and meaning of the boundary force require clarification from the underlying source.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30004633$, title Lagrangian Schemes for Irregular Porous-Medium Solutions, exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection, records proposed year $2021$, and supplies no individual proposer attribution.
• Supplied report citation: Applications of Optimal Transportation in the Natural Sciences (2022), Oberwolfach Reports, pp. 507--587, DOI 10.4171/owr/2021/10. https://doi.org/10.4171/owr/2021/10. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Medina--Schmitzer--Bonafini, asymptotic analysis. https://arxiv.org/abs/2106.08084.
• Supplied literature source: Gallouet--Merigot--Natale, porous-media discretization. https://doi.org/10.1137/21M1422756.
• Supplied literature source: Oberwolfach report. https://doi.org/10.4171/owr/2021/10.
• Supplied literature assessment, checked on 2026-08-22: the record labels the problem open and reports convergence under smoothness assumptions for the discrete Wasserstein gradient-flow scheme, while describing convergence for less regular solutions with nonzero boundary force at the Barenblatt support as unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These supplied editorial labels do not establish the missing mathematical hypotheses or constitute independent verification here.