CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Support propagates along transport characteristics

Statement

Assume the homogeneous transport equation $u_{t} + a \cdot \nabla_{x} u = 0$ has a global $C^{1}$ flow map $Φ_{t}$ of spatial diffeomorphisms. If $u (\cdot, t) = u_{0} \circ Φ_{t}^{- 1}$ , then

$supp u (\cdot, t) = Φ_{t} (supp u_{0})$

for every time $t$ in the interval of existence.

Facts & Assumptions

Given: A global $C^{1}$ flow $Φ_{t}$ for the homogeneous transport equation and the representation $u (\cdot, t) = u_{0} \circ Φ_{t}^{- 1}$ .

[L1]

Homogeneous linear transport is represented by the inverse characteristic flow (Homogeneous linear transport is solved by the inverse characteristic flow).

Proof

technique · direct

1.1L1

By [L1], $u (x, t) = u_{0} (Φ_{t}^{- 1} (x))$ , so $u (x, t) \neq 0$ exactly when $u_{0} (Φ_{t}^{- 1} (x)) \neq 0$ , equivalently when $Φ_{t}^{- 1} (x) \in {u_{0} \neq 0}$ and therefore when $x \in Φ_{t} ({u_{0} \neq 0})$ .

2.1step 1.1∎

Because $Φ_{t}$ is a homeomorphism, it carries closures to closures, and taking closures in step 1.1 gives $supp u (\cdot, t) = Φ_{t} (supp u_{0})$ , so support can move only along the characteristic flow.

Depends on

Homogeneous linear transport is solved by the inverse characteristic flow

Used by

Nothing in the library uses this result yet.

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

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