THEORETICAL ANALYSIS OF A DISCRETE POPULATION BALANCE MODEL FOR SUM KERNEL

The paper establishes global existence of mild solutions for the Safronov–Dubovski (S–D) equation under Vi,j ≤ i + j and ψ0 ∈ X+ with finite µ1, using a mass-conserving truncation (summing the last term to n−1) and tail controls, and then proves density conservation for all solutions; see the model definition and Definition 1.1 (a)–(c) with its integral form (7) , the specific truncation (8)–(11) , the moment identity (14) and its corollaries (15)–(17) , the existence theorem (Theorem 4) with the compactness lemma (Lemma 3) , and the density conservation argument (Theorem 5) via (36)–(41) and tail estimates . The candidate solution also proves global existence via truncation, a priori µ1 bounds, Arzelà–Ascoli compactness, and passage to the limit in the integral formulation, which aligns conceptually with the paper. Differences: (i) the candidate truncates the last sum up to N (not N−1), thus losing mass conservation at the truncated level (the paper’s truncation ensures µ1 is conserved for the truncated system, dµn1/dt=0) ; (ii) the candidate’s passage to the limit for the infinite “loss” sum appeals to monotone convergence in a setting where pointwise monotonicity is not guaranteed. This step requires a standard tail-splitting/dominated-convergence argument as done in the paper through νn m controls, (25)–(33) . With this minor fix, the candidate’s proof yields the same existence conclusion. The paper additionally proves that all solutions conserve density under the same kernel bound, which the candidate did not claim .