A Mathematical Analysis of Coarsening Processes Driven by Vanishing

Abstract

In this thesis, we investigate several models of coarsening processes with the property that an agent, commonly referred to as particle, vanishes from the system once it reaches size zero and hence the average particle size tends to increase over time. We show how different types of interaction (local vs. mean-field interaction, deterministic vs. stochastic modeling) affect the general coarsening behavior and require different mathematical tools and strategies to analyze. <br /> In Chapter 1 we investigate a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by a discrete backward parabolic equation. By an analysis of the time-reversed evolution we prove existence of solutions with optimal coarsening rate. In particular we establish positivity estimates and long-time equilibrium properties for discrete parabolic equations with bounded initial data. <br /> In Chapter 2 we consider a class of nonlocal coarsening models after Lifshitz, Slyozov and Wagner with singular particle interaction. For these equations we establish existence of general measure valued solutions by approximation with empirical measures. Furthermore, we show that there exists a one-parameter family of self-similar solutions, all of which have compact support but only one of them being smooth, a phenomenon that is typical for LSW models. <br /> In Chapter 3 we study the exchange-driven growth model that arises as mean-field limit of a stochastic particle system and describes a process in which pairs of clusters exchange atomic particles. For the product kernel we rigorously establish coarsening rates and convergence to the self-similar profile by linking the evolution to a discrete weighted heat equation on the positive integers. We establish a new weighted Nash inequality that yields scaling-invariant decay and continuity estimates from which we derive a discrete-to-continuum scaling limit for the weighted heat equation and deduce coarsening rates and self-similar convergence.