CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.
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

1.1

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 $x \in Φ_{t} ({u_{0} \neq 0})$ .

2.1

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.

step 1.1∎

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

4 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)