LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The bounded-kernel continuous coagulation-fragmentation field is locally Lipschitz

Statement

Let $F (μ) = C (μ, μ) + F μ$ be the vector field from Continuous-size coagulation-fragmentation weak model. On the ball $∥ μ ∥_{w}, ∥ ν ∥_{w} \leq R$ it satisfies

$∥ F (μ) - F (ν) ∥_{w} \leq (3 K_{*} R + β_{*} (D + 1)) ∥ μ - ν ∥_{w} .$

Also, $F (0) = 0$ .

Facts & Assumptions

Given: The symmetric bilinear coagulation operator and linear fragmentation operator on $M_{w} (E)$ .

[F1]

Bounded coagulation is bilinear and bounded in weighted total variation: $∥ C (α, γ) ∥_{w} \leq \frac{3}{2} K_{*} ∥ α ∥_{w} ∥ γ ∥_{w}$ .

[F2]

Mass-normalized fragmentation is bounded in weighted total variation: $∥ F α ∥_{w} \leq β_{*} (D + 1) ∥ α ∥_{w}$ .

Proof

1.1givenF1algebra

Bilinearity and symmetry give $C (μ, μ) - C (ν, ν) = C (μ - ν, μ) + C (ν, μ - ν)$ . Applying [F1] to each term and the ball bounds yields at most $3 K_{*} R ∥ μ - ν ∥_{w}$ .

2.1givenF2step 1.1algebra∎

Linearity and [F2] bound the fragmentation difference by $β_{*} (D + 1) ∥ μ - ν ∥_{w}$ . Adding the coagulation and fragmentation estimates proves the stated Lipschitz bound. Both operators vanish at the zero measure, so $F (0) = 0$ .

Depends on

Used by

Dependency tree · two levels

5 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

Philippe Laurençot, Weak compactness techniques and coagulation equations, §§3.1–3.4 (standard reference, not scraped)
Philippe Laurençot and Stéphane Mischler, From the discrete to the continuous coagulation-fragmentation equations, §§1–2 (standard reference, not scraped)