  We study the solutions of the Smoluchowski coagulation equation with a
regularisation term which removes clusters from the system when their mass
exceeds a specified cut-off size, M. We focus primarily on collision kernels
which would exhibit an instantaneous gelation transition in the absence of any
regularisation. Numerical simulations demonstrate that for such kernels with
monodisperse initial data, the regularised gelation time decreases as M
increases, consistent with the expectation that the gelation time is zero in
the unregularised system. This decrease appears to be a logarithmically slow
function of M, indicating that instantaneously gelling kernels may still be
justifiable as physical models despite the fact that they are highly singular
in the absence of a cut-off. We also study the case when a source of monomers
is introduced in the regularised system. In this case a stationary state is
reached. We present a complete analytic description of this regularised
stationary state for the model kernel, K(m_1,m_2)=max{m_1,m_2}^v, which gels
instantaneously when M tends to infinity if v>1. The stationary cluster size
distribution decays as a stretched exponential for small cluster sizes and
crosses over to a power law decay with exponent v for large cluster sizes. The
total particle density in the stationary state slowly vanishes as (Log
M^(v-1))^-1/2 when M gets large. The approach to the stationary state is
non-trivial : oscillations about the stationary state emerge from the interplay
between the monomer injection and the cut-off, M, which decay very slowly when
M is large. A quantitative analysis of these oscillations is provided for the
addition model which describes the situation in which clusters can only grow by
absorbing monomers.
