Calculus of Variations and Geometric Measure Theory

P. R. A. S. A. N. T. A. K. U. M. A. R. BARIK - A. K. Giri - P. Laurençot

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

created by barik on 23 Aug 2018
modified by barik1 on 19 Jun 2019

[BibTeX]

preprint

Inserted: 23 aug 2018
Last Updated: 19 jun 2019

Journal: Proceedings of Royal Society Edinburgh Section A: Mathematics
Year: 2018
Doi: 10.1017/prm.2018.15

ArXiv: 1804.00853 PDF

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.