B05 – Self-similarity in Smoluchowski’s coagulation equation
In this project we investigate the fundamental question of emergence of self-similarity in Smoluchowski’s mean-field model for coagulation. This issue is by now well understood for so-called solvable kernels that can be treated by explicit solution methods. The problem is however still completely open for all non-solvable kernels.
We will first continue our recently started study on the existence of self-similar solutions with fat tails with the goal to extend it to singular kernels such as the one that was derived in Smoluchowski’s classical paper. Apart from existence we will also rigorously derive regularity results as well as detailed asymptotics of the solutions for small and large clusters.
The key issues we plan to investigate in this project are the uniqueness of self-similar solutions within the class of functions with a prescribed decay behaviour and their domains of attraction under the evolution. No standard methods yet exist to attack these problems and thus we will embark upon a range of different ideas, such as deriving detailed estimates of the corresponding dual problem and exploring the connection to stochastic particle models by considering the continuum limit in the associated entropy-dissipation relation.
In the long term we will also investigate the analogous questions for the case of gelation, that is when infinitely large clusters are created in finite time. We also plan to further deepen the connection to stochastics by investigating self-similarity in discrete coarsening models and by considering the derivation of coagulation models as large scale limits of spin systems.
|Velázquez, Juan José||IAM||En60email@example.com|