Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited

We study the solutions of a regularised Smoluchowski coagulation equation with instantaneously gelling kernels. Regularisation is done by introducing a cut-off, $M_{\rm max}$, which physically corresponds to the removal from the system of clusters having mass greater than $M_{\rm max}$. Careful numerical simulations demonstrate that, for monodisperse initial data, the gelation time for $\nu>1$ {\em decreases}, albeit logarithmically slowly, as $M_{\max}$ increases. We thereby clearly demonstrate the instantaneous gelation transition numerically for the first time. The slow dependence on $M_{\rm max}$ explains previous difficulties in characterising the instantaneous gelation transition in simulations and justifies the use of instantaneously gelling kernels as physical models. We also consider solutions with a source of monomers which ultimately reach a stationary state. Approach to the stationary state is non-trivial. Oscillations results from the interplay between the monomer injection and the cut-off which decay very slowly when the cut-off is large.