Existence and Uniqueness of Mass Conserving Solutions to Safronov-Dubovski Coagulation Equation for Product Kernel

TL;DR

This paper proves the existence, mass conservation, and uniqueness of solutions to the Safronov-Dubovski coagulation equation with product kernels, using advanced mathematical techniques for both conservative and non-conservative systems.

Contribution

It establishes the existence and uniqueness of solutions for the coagulation equation with specific kernels, extending previous results with rigorous proofs.

Findings

Solutions conserve density

Existence of solutions for kernels up to ext{min}\{i^\eta,j^\eta",

Uniqueness of solutions when \\phi_{i,j} \\leq ext{min}\{i^\eta,j^\eta"]],

Abstract

The article presents the existence and mass conservation of solution for the discrete Safronov-Dubovski coagulation equation for the product coalescence coefficients $\phi$ such that $\phi_{i,j} \leq ij$ $\forall$ $i,j \in \mathbb{N}$. Both conservative and non-conservative truncated systems are used to analyse the infinite system of ODEs. In the conservative case, Helly's selection theorem is used to prove the global existence while for the non-conservative part, we make use of the refined version of De la Vall\'ee-Poussin theorem to establish the existence. Further, it is shown that these solutions conserve density. Finally, the solutions are shown to be unique when the kernel $\phi_{i,j} \leq \text{min}\{i^\eta,j^{\eta}\}$ where $\eta \in [0,2]$.