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.

Local characteristic rarefaction curve

Definition

On a neighborhood where an eigenvector field $r_{i}$ is $C^{2}$ , the local $i$ -rarefaction curve through $u_{0}$ is the unique local solution $\frac{d R_{i}}{d σ} (σ; u_{0}) = r_{i} (R_{i} (σ; u_{0})), R_{i} (0; u_{0}) = u_{0} .$ The parameter $σ$ is the signed integral-curve parameter for the chosen normalization of $r_{i}$ . Existence on a common small interval for nearby $u_{0}$ follows from Nearby initial values share one Picard–Lindelöf time interval and one state cylinder; the $C^{2}$ flow on a compact local tube follows from A $C^{2}$ vector field has a $C^{2}$ flow on a common compact tube. No global integral curve is asserted.

Depends on

Used by

Dependency tree · two levels

19 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

Alberto Bressan, Hyperbolic Conservation Laws: An Illustrated Tutorial (standard reference, not scraped)
Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)