Mass concentration in a spatially inhomogeneous coagulation model with fast sedimentation

Abstract

We study a spatially inhomogeneous coagulation model that contains a transport term in the spatial variable. The transport term models the vertical motion of particles due to gravity, thereby incorporating their fall into the dynamics. Local existence of mass-conserving solutions for a class of coagulation rates for which in the spatially homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs has been proved in [Cristian-Niethammer-Vel\'azquez, 2024]. In order to obtain some insight into how to prove global existence of solutions, we allow a fast sedimentation speed. For very fast sedimentation speed, we rigorously prove that solutions converge to a Dirac measure in the space variable. We also formally obtain in the limit a one-dimensional coagulation equation with diagonal kernel, i.e., only particles of the same size interact. This provides a physical intuition on how coagulation models with a diagonal kernel emerge.