Self-similar solutions for Smoluchowski's equations for coalescence

Smoluchowski's equations provide a mean-field description of several mass aggregation processes in nature such as the formation of clouds, stars and polymers. They also have mathematical applications to fields such as random graphs and population genetics. Nevertheless, little is known analytically about the existence of self-similar solutions.

In this talk, I will describe extremely simple proofs of self-similar blowup (gelation) and approach to equilibrium in one of the few solvable cases (additive and multiplicative kernels). The results have an appealing probabilistic interpretation. This is joint work with Bob Pego (Maryland).