TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Eigenstructure of the admissible ideal-gas Euler system

Statement

For $U = (ρ, m, E) \in G_{γ}$ , put $u = m / ρ$ , $H = (E + p) / ρ$ , and $c = \sqrt{γ p / ρ}$ . The flux Jacobian has the three simple real eigenvalues and right eigenvectors $λ_{-} = u - c, r_{-} = (1, u - c, H - u c)^{T},$ $λ_{0} = u, r_{0} = (1, u, u^{2} / 2)^{T},$ $λ_{+} = u + c, r_{+} = (1, u + c, H + u c)^{T} .$ The middle field is linearly degenerate; the acoustic fields are genuinely nonlinear, with $D λ_{-} r_{-} = - (γ + 1) c / (2 ρ)$ and $D λ_{+} r_{+} = (γ + 1) c / (2 ρ)$ .

Facts & Assumptions

Given: $γ > 1$ and an admissible ideal-gas Euler state.

[F1]

Admissible states and pressure for one-dimensional ideal-gas Euler: $p = (γ - 1) (E - m^{2} / (2 ρ))$ , $F = (m, m^{2} / ρ + p, u (E + p))$ , and $c^{2} = γ p / ρ$ .

[F2]

Strictly hyperbolic conservation systems and characteristic families: eigenpairs are right eigenvectors of $D F$ ; genuine nonlinearity and linear degeneracy are tested by $D λ_{i} r_{i}$ .

Proof

1.1givenF1algebra

Write $U = (ρ, ρ u, E)$ and use [F1] to obtain $H = u^{2} / 2 + c^{2} / (γ - 1)$ . Differentiating the flux components with respect to $(ρ, m, E)$ gives $D F (U) = (\begin{array}{ccc} 0 & 1 & 0 \\ (γ - 3) u^{2} / 2 & (3 - γ) u & γ - 1 \\ u ((γ - 1) u^{2} / 2 - H) & H - (γ - 1) u^{2} & γ u \end{array})$ .

2.1step 1.1algebra

Multiplying $D F$ by $r_{0}$ gives $(u, u^{2}, u^{3} / 2)^{T} = u r_{0}$ . For $r_{\pm}$ , the first row gives $u \pm c$ ; the second gives $(3 - γ) u^{2} / 2 + (γ - 1) H \pm 2 u c = u^{2} + c^{2} \pm 2 u c = (u \pm c)^{2}$ ; and the third gives $γ u H - (γ - 1) u^{3} / 2 \pm c (H + u^{2}) = u (H + c^{2}) \pm c (H + u^{2}) = (u \pm c) (H \pm u c)$ . Each row therefore equals the matching component of $(u \pm c) r_{\pm}$ , using $H = u^{2} / 2 + c^{2} / (γ - 1)$ . Since $c > 0$ , the eigenvalues are distinct and strictly ordered.

2.2step 1.1F1F2algebra

For a state increment $(d ρ, d m, d E)$ , differentiating the definitions gives $d u = (d m - u d ρ) / ρ$ and $d p = (γ - 1) (d E - u d m + u^{2} d ρ / 2)$ . Along $r_{\pm}$ , these yield $d u = \pm c / ρ$ and $d p = c^{2}$ ; hence $d c = \frac{1}{2} c (d p / p - d ρ / ρ) = (γ - 1) c / (2 ρ)$ . Along $r_{0}$ they give $d u = d p = 0$ and $d c = - c / (2 ρ)$ ; in particular $D λ_{0} r_{0} = d u = 0$ . Therefore the stated directional derivatives of $u - c, u, u + c$ follow.

3.1step 2.2F2algebra∎

Because $c, ρ, γ$ are positive, the acoustic directional derivatives are nonzero with the displayed signs, while the middle one is zero. This proves genuine nonlinearity of the two acoustic fields and linear degeneracy of the contact field.

Depends on

Used by

Dependency tree · two levels

4 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

Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)