A note on instantaneous gelation for coagulation kernels vanishing on the diagonal

TL;DR

This paper proves that for certain coagulation equations with specific kernels, instantaneous gelation occurs, leading to immediate mass loss, except in the case of initial data being a single Dirac delta.

Contribution

It establishes the occurrence of instantaneous gelation for sum-type kernels with homogeneity greater than one that vanish on the diagonal, expanding understanding of coagulation dynamics.

Findings

Instantaneous gelation occurs for these kernels.

Solutions can be Radon measures, excluding initial Dirac delta.

Mass loss happens immediately for certain kernels.

Abstract

We prove that instantaneous gelation (i.e., instantaneous loss of mass) occurs for coagulation equations with sum-type kernels of homogeneity greater than one which vanish on the diagonal. Our proof includes solutions that are Radon measures if we exclude the case of initial data that are a single Dirac delta.