LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Local solvability and C1 parameter dependence for the augmented characteristic ODE

Statement

At every $y_{0} \in V$ , the system in The augmented characteristic system for a quasilinear first-order PDE has, after shrinking to $∣ s ∣ < ϵ$ and a neighbourhood $W$ of $y_{0}$ , a unique common solution $(X, Z)$ for $y \in W$ . It is $C^{1}$ in $(s, y)$ .

Facts & Assumptions

Given: The smooth coefficients and $C^{1}$ initial strip in the definition, and $y_{0} \in V$ .

Proof

technique · direct

1.1

Put $Y : = (X, Z)$ and $G (x, z) : = (a (x, z), b (x, z))$ . The coefficients in the defining Cauchy problem are smooth, so $G$ is smooth.

givenconstruct

2.1

Apply Smooth dependence of ODE solutions on parameters to $Y^{'} = G (Y)$ with initial value $Y (0, y) = (γ (y), ϕ (y))$ . It gives a common local interval, uniqueness, and $C^{1}$ dependence on $y$ ; including the ODE variable gives $C^{1}$ dependence on $(s, y)$ .

step 1.1given∎

Depends on

Used by

Quasilinear characteristics can cross before the lifted ODE blows up Counterexample
Characteristic crossing and caustic for a first-order PDE Definition
Jacobian of a characteristic strip at its initial surface Lemma
Local quasilinear characteristic graph construction Theorem

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

First order PDE: The Methods of Characteristics (standard reference, not scraped)