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Bounded-kernel measure solutions conserve mass and control number and second moments

Statement

Let $μ_{t}$ be a positive local solution of Continuous-size coagulation-fragmentation weak model with initial number $N_{0}$ , mass $M_{1}$ , and second moment $M_{2, 0} < \infty$ . Then for every time in its interval of existence,

$M_{1} (μ_{t}) = M_{1}, M_{0} (μ_{t}) \leq N_{0} e^{β_{*} D t}, M_{2} (μ_{t}) \leq M_{2, 0} + K_{*} M_{1}^{2} t .$

Facts & Assumptions

Given: The weak equation with symmetric bounded $K$ , daughter mass normalization, and a positive local solution in $M_{w} (E)$ .

[F1]

Continuous-size coagulation-fragmentation weak model: the coagulation increment for a test $φ$ is $φ (x + y) - φ (x) - φ (y)$ ; fragmentation increment is $\int φ (z) B_{x} (d z) - φ (x)$ , with $\int z B_{x} (d z) = x$ and $z \leq x$ on its support.

[F2]

Bounded kernels give a local positive measure-valued solution: a positive local solution exists in $M_{w} (E)$ , so the tests $1, x, x^{2}$ are integrable.

Proof

1.1givenF1F2algebra

Take $φ (x) = x$ in [F1]. The coagulation increment is $(x + y) - x - y = 0$ , and the fragmentation increment is $\int z B_{x} (d z) - x = 0$ . Thus $M_{1}^{'} (μ_{t}) = 0$ , and integration gives $M_{1} (μ_{t}) = M_{1}$ .

1.2F1F2algebra

With $φ = 1$ , each coagulation event changes number by $- 1$ , while fragmentation changes it by $B_{x} (E) - 1 \leq D$ . Hence $M_{0}^{'} (μ_{t}) \leq β_{*} D M_{0} (μ_{t})$ . The derivative of $e^{- β_{*} D t} M_{0} (μ_{t})$ is nonpositive, giving $M_{0} (μ_{t}) \leq N_{0} e^{β_{*} D t}$ .

2.1F1F2step 1.1algebra∎

With $φ = x^{2}$ , coagulation contributes $\frac{1}{2} \iint K (x, y) [(x + y)^{2} - x^{2} - y^{2}] d μ_{t} d μ_{t} = \iint K (x, y) x y d μ_{t} d μ_{t} \leq K_{*} M_{1}^{2}$ . Fragmentation contributes $\int β (x) [\int z^{2} B_{x} (d z) - x^{2}] d μ_{t} \leq 0$ , since $z^{2} \leq x z$ and $\int z B_{x} = x$ . Therefore $M_{2}^{'} (μ_{t}) \leq K_{*} M_{1}^{2}$ , and integration proves the last bound.

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Sources

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