Nonlinear dynamics of three solvable aggregation models

Abstract

For three interesting kinetic models of clustering, we review results on dynamical phenomena related to the approach to self-similar form and their close connections to probability theory. For Smoluchowski's coagulation equation with additive rate kernel, we describe the scaling attractor and show how dynamics on it is trivialized in terms of Bertoin's Lévy–Khintchine-like representation. For a model motivated by domain coarsening dynamics in the Allen–Cahn equation, we describe the remarkable solution procedure found by Gallay and Mielke, and the ensuing classification of domains of attraction for self-similar solutions. And we describe the rigorous connection between Smoluchowski's equation and random shock coalescence in the inviscid Burgers equation. Recent work of Menon indicates that the latter problem is completely integrable for initial data comprising a spatial Markov process.