Global existence to the discrete Safronov-Dubovski\v{i} coagulation equations and failure of mass-conservation

TL;DR

This paper proves the global existence of solutions to a class of discrete coagulation equations and demonstrates the failure of mass conservation, indicating gelation phenomena in these models.

Contribution

It establishes the existence of solutions for a broad class of coagulation kernels and analyzes the conditions leading to mass loss, advancing understanding of gelation in coagulation processes.

Findings

Global solutions exist for kernels with linearly or superlinearly growing sequences.

Mass conservation fails under certain conditions, confirming gelation occurrence.

The class of kernels includes those with specific growth and boundedness properties.

Abstract

This paper presents the existence of global solutions to the discrete Safronov-Dubvoski\v{i} coagulation equations for a large class of coagulation kernels satisfying $\Lambda_{i,j} = \theta_i \theta_j + \kappa_{i,j}$ with $\kappa_{i,j} \leq A\theta_i \theta_j, \ \ \forall \ \ i,j\ge 1$ where the sequence $(\theta_i)_{i\geq 1}$ grows linearly or superlinearly with respect to $i$. Moreover, the failure of mass-conservation of the solution is also addressed which confirms the occurrence of the gelation phenomenon.