  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.
