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Finite-grid mass conservation does not bound the product-kernel continuum second moment

Statement

For every cutoff $N$ , the finite state $n_{N} = 1 / N$ at the sole occupied size $N$ has mass $1$ and second moment $N$ . Hence finite-grid mass conservation alone yields no second-moment bound uniform in the cutoff. Separately, the infinite product-kernel solution from The discrete product-kernel solution has finite-time second-moment blow-up has conserved first moment $1$ but $M_{2} (t) = 1 / (1 - t) \to \infty$ as $t ↑ 1$ . Thus finite-grid augmented-mass conservation does not prove a global continuum second-moment estimate or rule out gelation for a superlinear kernel.

Facts & Assumptions

Given: The finite sectional mass identity and the product-kernel pre-gelation solution.

[F1]

Finite sectional event dynamics preserve positivity and retained-plus-reservoir mass: finite sectional event dynamics conserve retained-plus-reservoir mass exactly.

[F2]

The discrete product-kernel solution has finite-time second-moment blow-up: for the product kernel, $M_{1} (t) = 1$ and $M_{2} (t) = 1 / (1 - t)$ on $0 \leq t < 1$ .

Proof

1.1givenF1algebra

On a grid with top pivot $N$ , the state with $n_{N} = 1 / N$ and all other retained coordinates zero has first moment $N (1 / N) = 1$ and second moment $N^{2} (1 / N) = N$ . Its reservoir may be zero, so the conserved augmented mass is $1$ . As $N$ increases these admissible mass-one states have unbounded second moment, proving the finite mass identity alone gives no cutoff-independent moment bound.

2.1givenF2algebra∎

The continuum product-kernel solution in [F2] has $M_{2} (t) = 1 / (1 - t)$ , which diverges at the finite time $1$ while its first moment remains $1$ . The kernel $K (x, y) = x y$ is superlinear and unbounded, outside the bounded-kernel global theorem's hypotheses; therefore this example warns against inferring continuum mass-conserving global moment control from finite-grid conservation.

Remark

Laurençot's full-text lecture notes separately state a post-gel mass-loss theorem for nontrivial weak solutions with $K (x, y) \geq κ (x y)^{λ / 2}$ , $λ \in (1, 2]$ , under the solution hypotheses there (Theorem 3.13, §3.3); that source result is contextual and is not used to prove the pre-gelation formula here. Kumar's dissertation reports fixed-pivot moment deviations from analytical product-kernel comparisons near gelation in its numerical examples (Chapter 3, §3.2.2); this is a numerical warning, not a substitute for the local proof above. (Laurençot, §§3.3; Kumar, Chapter 3)

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Sources

Philippe Laurençot, Weak compactness techniques and coagulation equations, §§1 and 3.3 (standard reference, not scraped)
Jacek Kumar, Numerical approximations of population balance equations in particulate systems, Chapter 2, §2.2 (standard reference, not scraped)