Vanishing Viscosity and Propagating Terraces

Definitions and mathematical statement

Let $n$ denote the number of tubes in the incompressible porous-media model named in the supplied record, let $\nu$ denote its viscosity parameter, and let $l>0$ denote the tube separation in the two-tube model. The supplied record does not give the governing equations, spatial domain, initial or boundary conditions, admissible data, or notion of solution. Consequently, the following records the requested research targets; it does not specify a complete mathematical conjecture.

1. For the $n$-tube model with $\nu=0$, establish well-posedness. The class of data, solution space, time interval, and meaning of continuous dependence must be specified before this becomes a precise assertion.
2. Characterize the vanishing-viscosity limit as $\nu\to0$. This requires specifying the viscous equations, the family of solutions and data under consideration, the direction of approach of $\nu$, the topology of convergence, and the limiting solution concept and selection criterion. None of these is supplied.
3. For every tube separation $l>0$, establish existence of the propagating terrace referenced as equation (9) in the original source excerpt for the two-tube model, and study its stability. The terrace profile, propagation speeds, asymptotic states, admissible perturbations, and meaning of stability are not supplied. The original excerpt states that the result of its reference [1] is valid only for sufficiently small $l>0$; neither that reference nor its precise hypotheses and conclusion are included.

The original excerpt explicitly asks about the vanishing-viscosity limit and terrace existence for all $l>0$. Well-posedness at $\nu=0$ and terrace stability appear in the supplied reconstructed statement but are not explicit in that excerpt. The record labels the reconstruction as checked against the official report; that supplied verification claim has not been independently checked here.

Short English statement

The supplied problem asks for well-posedness of an inviscid tube model of incompressible porous media, a description of its vanishing-viscosity limit, and existence and stability of a propagating terrace in the two-tube model for every separation $l>0$. The equations and the referenced terrace definition are missing, so a complete precise formulation requires the original model and its hypotheses. The supplied excerpt reports an existing terrace result only for sufficiently small $l$.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005823$, title: Vanishing Viscosity and Propagating Terraces; exact source problem code: catalogue entry. The record assigns the proposed year 2024 and supplies no named proposer.
• Supplied authoritative report citation: Hyperbolic Balance Laws: Interplay between Scales and Randomness (2024), Oberwolfach Reports, pp. 567--656, DOI 10.4171/owr/2024/10, https://doi.org/10.4171/owr/2024/10. The supplied original excerpt identifies p. 623 and refers to equation (9) and reference [1], whose contents are not supplied.
• Supplied literature source: Lukacova-Medvidova and Rohde, Mathematical Challenges for Hyperbolic Balance Laws, https://link.springer.com/article/10.1365/s13291-024-00290-6.
• Supplied literature source: Oberwolfach Report 10/2024, https://doi.org/10.4171/owr/2024/10.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not an independently verified assessment. The record labels the problem open and states broadly that multidimensional hyperbolic balance-law limits and well-posedness remain substantially incomplete. This broad assessment does not itself establish the status of each tube-model question. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft.