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

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Local quasilinear characteristic graph construction

Statement

At $y_{0} \in V$ , suppose $[a (γ (y_{0}), ϕ (y_{0})), D γ (y_{0})]$ has rank $n$ . Then, after shrinking the characteristic strip, $Ψ (s, y) = X (s, y)$ is a $C^{1}$ diffeomorphism onto an open set $U$ , and

$u (x) : = Z (Ψ^{- 1} (x))$

is the unique $C^{1}$ function obtained by this inverse-projection construction. It attains $u (γ (y)) = ϕ (y)$ ; the next lemma verifies its PDE.

Facts & Assumptions

Given: The smooth coefficients, $C^{1}$ data, and the stated full-rank condition at $y_{0}$ .

Proof

technique · direct

1.1

The local ODE lemma supplies a $C^{1}$ strip, and the Jacobian lemma makes $D Ψ (0, y_{0})$ invertible.

givenalgebra

2.1

By The Euclidean inverse function theorem, shrink to a neighbourhood on which $Ψ$ has a $C^{1}$ inverse. Define $u = Z \circ Ψ^{- 1}$ there.

step 1.1construct

3.1

At $s = 0$ , $Ψ (0, y) = γ (y)$ and $Z (0, y) = ϕ (y)$ , hence $u (γ (y)) = ϕ (y)$ . Any inverse-projected function from this strip has the same formula and is therefore identical to $u$ .

step 2.1given∎

Depends on

Used by

The inverse-projected characteristic graph satisfies the quasilinear PDE Lemma

Dependency tree · two levels

20 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)