The Smoluchowski coagulation equation is a classical equation describing the distribution of particle sizes undergoing binary coagulation. Despite extensive study for over a century, it continues to pose significant analytical challenges, particularly concerning long-time behavior for general coagulation kernels.
Recently, an existence theory for nontrivial steady states has been developed for a large class of kernels. We construct a time-dependent solution that is expected to converge to one of such nontrivial steady states, and prove this convergence for the constant kernel. The solution satisfies a nonzero flux boundary condition describing a constant input of zero-sized particles, called dust. We show that the dust is instantaneously converted into particles of positive size, so that no dust remains in the system, and the total mass grows linearly in time. The construction applies to a broad class of non-gelling kernels for which stationary solutions exist; in the complementary regime, no such type of solutions with flux exist. (Based on a joint work with Aleksis Vuoksenmaa - Sapienza University)