ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-05

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The stationary equation x dot Du = u is solved by radial characteristics

Example

On $R^{n} ∖ {0}$ , the stationary first-order equation

$x \cdot \nabla u (x) = u (x)$

has precisely the $C^{1}$ solutions of the form

$u (r ω) = r g (ω), r > 0, ω \in S^{n - 1},$

with arbitrary $C^{1}$ data $g$ on the unit sphere.

Facts & Assumptions

Given: The stationary equation $x \cdot \nabla u = u$ on $R^{n} ∖ {0}$ .

[L1]

Along a characteristic, a transport equation reduces to the scalar ODE from the transport lemma (A transport equation restricts to a linear ODE along each characteristic).

Verification

technique · direct

1.1

Regard the equation as transport with characteristic ODE $X^{'} (s) = X (s)$ ; its solutions are the rays $X (s) = e^{s} x_{0}$ , and by [L1] the restricted function $v (s) = u (X (s))$ satisfies $v^{'} (s) = v (s)$ , hence $v (s) = e^{s} v (0)$ .

2.1

Writing $x = r ω$ with $r = e^{s}$ and $ω \in S^{n - 1}$ gives $u (r ω) = r u (ω)$ , so every solution has the claimed form with $g (ω) = u (ω)$ ; conversely, for $u (r ω) = r g (ω)$ the radial derivative is $u / r$ , hence $x \cdot \nabla u = u$ .

step 1.1∎

Depends on

A transport equation restricts to a linear ODE along each characteristic

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

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