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

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Inviscid Burgers characteristic formula and first crossing time

Statement

For $u_{t} + u u_{x} = 0$ with $C^{1}$ datum $u (0, ξ) = u_{0} (ξ)$ , every classical characteristic has

$u (t, X (t, ξ)) = u_{0} (ξ), X (t, ξ) = ξ + t u_{0} (ξ) .$

Thus $X_{ξ} = 1 + t u_{0}^{'} (ξ)$ and its first forward crossing time is

$T_{*} : = \inf {t \geq 0 : \exists ξ, 1 + t u_{0}^{'} (ξ) = 0} \in [0, \infty],$

where the infimum of the empty set is $\infty$ .

Facts & Assumptions

Given: A classical Burgers solution with the stated $C^{1}$ initial datum.

Proof

technique · direct

1.1

Along $\dot{X} = u (t, X)$ , the chain rule and the PDE give $d (u (t, X)) / d t = u_{t} + u u_{x} = 0$ .

givenalgebra

2.1

The initial condition makes $u (t, X) = u_{0} (ξ)$ , hence $\dot{X} = u_{0} (ξ)$ and integration from $X (0, ξ) = ξ$ gives $X = ξ + t u_{0} (ξ)$ .

step 1.1givenalgebra

3.1

Differentiating in $ξ$ gives $X_{ξ} = 1 + t u_{0}^{'} (ξ)$ , so the definition of caustic gives exactly the displayed set and its stated infimum convention.

step 2.1given∎

Depends on

Used by

Monotone Burgers data have no forward characteristic crossing Corollary
Quasilinear characteristics can cross before the lifted ODE blows up Counterexample
Inviscid Burgers gradient catastrophe Example
Rarefying inviscid Burgers data Example

Dependency tree · two levels

6 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

Part III: Analysis of Partial Differential Equations (standard reference, not scraped)