On gelation for the Smoluchowski equation

TL;DR

This paper revisits and simplifies the proof that certain coagulation kernels cause finite-time mass loss (gelation) in solutions to the Smoluchowski equation, extending previous results to more general kernels.

Contribution

It provides a concise proof of gelation for a broad class of homogeneous kernels, including those with logarithmic factors, building on and simplifying earlier work.

Findings

Weak solutions lose mass in finite time for specified kernels.

Gelation depends on the kernel's degree and shape, especially for degree one.

Logarithmic factors influence the gelation threshold.

Abstract

Motivated by the recent results of Andreis-Iyer-Magnanini (2023), we provide a short proof, revisiting the one of Escobedo-Mischler-Perthame (2002), that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation looses mass in finite time. The class of kernels we consider is essentially the same as the one of Andreis-Iyer-Magnanini (2023): homogeneous kernels of degree $\gamma>1$ not vanishing on the diagonal, or homogeneous kernels of degree $\gamma=1$ not vanishing on the diagonal with some additional logarithmic factor. We also show that when $\gamma=1$, the power of the logarithmic factor ensuring gelation may depend on the shape of the kernel.