Mass conservation and gelation for the Smoluchowski coagulation equation: a generalized moment approach

The Smoluchowski coagulation equation (SCE) is a population balance model that describes the time evolution of cluster size distributions resulting from particle aggregation. Although it is formally a mass-conserving system, solutions may exhibit a gelation phenomenon-a sudden loss of mass-when the coagulation kernel grows superlinearly. In this paper, we rigorously analyze mass conservation and gelation for weak solutions to the SCE with inhomogeneous coagulation kernels. By introducing a generalized moment framework, we derive sharp sufficient conditions for both mass conservation and gelation, expressed in terms of the initial data and the properties of the coagulation kernel.