CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Characteristic Cauchy data may be nonunique or incompatible

Statement refuted

Cauchy data for a first-order linear transport equation always determine a unique local classical solution, even when the data surface is characteristic.

Facts & Assumptions

Given: The transport equation $u_{t} + u_{x} = 0$ and the line $Σ = {(x, t) : x = t}$ .

[L1]

The local uniqueness theorem for linear transport assumes the data surface is noncharacteristic (Local linear transport has a unique solution from noncharacteristic Cauchy data).

[L2]

A noncharacteristic first-order Cauchy surface is one transverse to the space-time transport vector (Noncharacteristic Cauchy surfaces for first-order transport).

Counterexample

technique · direct

1.1

For $u_{t} + u_{x} = 0$ , the space-time transport vector is $(1, 1)$ , which is tangent to $Σ$ , so $Σ$ is characteristic rather than noncharacteristic by [L2], and [L1] does not apply.

L1L2

2.1

Every $C^{1}$ function of the form $u (x, t) = F (x - t)$ solves $u_{t} + u_{x} = 0$ , and on $Σ$ one has $x - t = 0$ , so the restriction is the constant $F (0)$ . Taking $F_{0} (s) = 0$ and $F_{1} (s) = s$ gives two different local classical solutions with the same constant data on $Σ$ , proving nonuniqueness. On the other hand, every classical solution restricts to a constant on $Σ$ , so nonconstant prescribed data such as $g (t) = t$ are incompatible.

step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Victor Ivrii, Partial Differential Equations (standard reference, not scraped)