Uniqueness for Spatially Dependent Two-Phase Diffusion

Definitions and mathematical statement

The curated question asks whether Kružkov-type entropy and comparison techniques can prove uniqueness for multidimensional two-phase-flow equations with spatially dependent diffusion, without strong regularity assumptions. The original field instead asks whether finite-volume interface methods can be extended to spatially dependent diffusion functions under weak regularity. These related formulations have different immediate targets: uniqueness by entropy methods and extension of numerical/interface methods. Neither is silently substituted for the other.

Write the unspecified spatial diffusion law as $\Phi(x,s)$, where $s$ is the saturation or scalar state. Its occurrence in the differential operator is not given. In particular, a spatially dependent potential inside $\Delta\Phi(x,s)$ and a coefficient inside a divergence-form operator need not define the same equation. The record also gives no coupled pressure law, transport flux, degeneracy or monotonicity hypotheses, spatial coefficient class, interface transmission condition, spatial domain, initial/boundary data or entropy-solution definition. A particular porous-medium equation or interface law is therefore not invented.

Once those data are fixed on a domain $\Omega\subseteq{\mathbb{R}}^d$ with $d\geq2$ and time interval $(0,T)$, the uniqueness target is that any two admissible solutions $u,v$ with the same prescribed data satisfy \[ u=v\quad\text{almost everywhere on }(0,T)\times\Omega. \] An $L^1$ contraction would be a possible stronger route when the solution and boundary frameworks support it, not an additional theorem or requirement already stated by the source. "Without strong regularity" needs a specified regularity class for both coefficients and solutions before such a theorem is precise.

The supplied literature triage, checked 21 August 2026, reports finite-volume existence and uniqueness extensions for heterogeneous porous media with spatially discontinuous capillary or diffusion data, especially in one dimension. It explicitly says these results do not settle the broad multidimensional adaptation question. Their detailed hypotheses and interface conventions are not reproduced. The source labels the record partially solved and its edited wording corrected verified; this draft records that provenance without independently checking the report or literature.

Short English statement

The record asks for entropy-based uniqueness and also preserves an original finite-volume extension question. A precise theorem still needs the actual equations, weak-solution notion and interface/data hypotheses. The dated literature assessment reports important heterogeneous-media cases, especially one-dimensional ones, without claiming the full multidimensional result.

Sources

• Local UnsolvedMath registry, permanent ID 30000005, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026.
• Supplied citation: Hyperbolic Conservation Laws (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/18, pp. 915--962: https://doi.org/10.4171/owr/2004/18. The record separately gives proposed year 2004; this date discrepancy is not silently resolved.
• Supplied reference label "heterogeneous porous-media finite-volume scheme": https://arxiv.org/abs/0902.1873.
• Supplied reference label "capillary barriers in heterogeneous media": https://arxiv.org/abs/1005.5634.
• Literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. No authors or full theorem statements for the two preprints are supplied.