Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel

TL;DR

This paper constructs global weak solutions to the Smoluchowski coagulation equation with singular kernels, allowing for infinite initial moments and proving mass conservation under broader conditions than previous studies.

Contribution

It extends the existence of mass-conserving solutions to the SCE with singular kernels, including linear growth at infinity and infinite initial second moments.

Findings

Constructed global weak solutions with singular kernels.

Proved all weak solutions are mass-conserving.

Included kernels with linear growth at infinity.

Abstract

Global weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo \& Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in $L^1$ and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.