A simple model of 1D coarsening dynamics in the Allen-Cahn equation is a `one-dimensional bubble bath' which coarsens by two simple rules: (i) The two nearest domain walls `pop,' annihilating each other. (ii) Repeat indefinitely.
We study mean-field models for a class of `min-driven' clustering processes of this kind. By extending a remarkable solution procedure found by Gallay and Mielke, and using the sharp exponential Tauberian theorem of de Haan, we
establish: a well-posedness theorem for measure-valued size distributions; necessary and sufficient conditions for approach to self-similar form; and a Levy-Khintchine representation formula for eternal solutions. This is joint work with Govind Menon and Barbara Niethammer.