DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Convex entropy pairs, entropy variables, and entropy potentials for systems

Definition

Use the conservation-system convention of Strictly hyperbolic conservation systems and characteristic families: on an open convex state domain $A \subseteq R^{m}$ , the flux is $F : A \to R^{m}$ . A convex entropy pair is a pair $(η, q)$ with $η \in C^{2} (A)$ , $q \in C^{1} (A)$ , positive-semidefinite Hessian $D^{2} η$ , and $D q (U) = D η (U) D F (U) (U \in A) .$ The entropy variable is the column vector $v (U) = \nabla η (U)$ ; gradients are represented as columns, so the compatibility equation above means $\nabla q (U) = D F (U)^{T} v (U)$ . The entropy potential is $ψ (U) = v (U)^{T} F (U) - q (U)$ . This definition does not assert that every convex $η$ admits a flux $q$ ; the compatibility equation is part of the hypothesis.

Depends on

Strictly hyperbolic conservation systems and characteristic families

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Tadmor, Entropy Stable Schemes for Hyperbolic Conservation Laws, Handbook of Numerical Methods for Hyperbolic Problems, Vol. 17 (standard reference, not scraped)
R. Bürger, P. E. Méndez, and C. Parés, On Entropy Stable Schemes for Degenerate Parabolic Multispecies Kinematic Flow Models (standard reference, not scraped)