Self-similarity in Smoluchowski’s Coagulation Equation

Smoluchowski’s coagulation equation is one of the fundamental deterministic models that describe mass aggregation phenomena. A key question in the analysis of this equation is whether the large-time behaviour of solutions is universal and described by special self-similar solutions. This issue is however only well-understood for the small class of solvable kernels while the analysis for non-solvable kernels still poses many challenging problems. Our main focus in this article will be to describe recent progress in the analysis of self-similar solutions of Smoluchowski’s equation for non-solvable kernels. Existence results for self-similar solutions with finite mass have been available for some time for a large class of kernels. In contrast, the uniqueness of such solutions has been an open problem for some time. A first uniqueness result has recently been obtained for kernels that are in a certain sense close to constant. We present here a shorter proof under an additional assumption on the kernels that makes the analysis significantly simpler. We also give an overview of recent results on the existence of fat tail solutions for non-solvable kernels. © 2014, Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg.