LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Mixed shock–rarefaction wave curves are continuously differentiable

Statement

Under the hypotheses of Mixed local wave curve for a genuinely nonlinear or linearly degenerate family, the map $(σ, u) \mapsto Ψ_{i} (σ; u)$ is $C^{1}$ near $(0, \overset{ˉ}{u})$ , with $Ψ_{i} (0; u) = u$ and $\partial_{σ} Ψ_{i} (0; u) = r_{i} (u)$ .

Facts & Assumptions

Given: The local rarefaction and shock curves, their parameterizations, and the mixed-curve convention.

[F1]

Local characteristic rarefaction curve: $R_{i} (0; u) = u$ and $\partial_{σ} R_{i} (0; u) = r_{i} (u)$ ; its dependence on $(u, σ)$ is $C^{1}$ locally.

[F2]

Local shock curve by an augmented Rankine–Hugoniot equation: $S_{i} (0; u) = u$ , $\partial_{σ} S_{i} (0; u) = r_{i} (u)$ , and the shock curve is $C^{1}$ jointly in base state and strength locally.

[F3]

Mixed local wave curve for a genuinely nonlinear or linearly degenerate family: the two branches are joined at $σ = 0$ ; a linearly-degenerate family uses only $R_{i}$ .

Proof

1.1givenF1F2F3

For a linearly-degenerate family, [F1] directly gives the claim. For a genuinely nonlinear family, each branch in [F3] is $C^{1}$ on its closed half-neighborhood by [F1] and [F2]. At $σ = 0$ both values equal $u$ , both state derivatives equal the identity, and both strength derivatives equal $r_{i} (u)$ .

2.1step 1.1algebra

At a boundary point $(u, 0)$ , the first-order expansions of either branch are $u + h + r_{i} (u) σ + o (∥ h ∥ + ∣ σ ∣)$ for state increment $h$ ; the remainder is uniform over the two branches because there are only two and each is $C^{1}$ . Thus the joined function is differentiable there with derivative $(h, σ) \mapsto h + r_{i} (u) σ$ . The branch derivatives tend to this common derivative as $σ \to 0$ , so the joined derivative is continuous.

3.1step 1.1step 2.1∎

The derivatives and values just obtained prove $C^{1}$ regularity and the asserted value and strength derivative at zero.

Depends on

Used by

Local small-data Riemann solver near a strictly hyperbolic state Theorem

Dependency tree · two levels

11 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)