# Selfsimilar solutions to Smoluchowski's coagulation equation in singular and non singular cases

Smoluchowski’s coagulation equation is a mean field model describing cluster growth that has been used in a very wide set of applications, ranging from physical chemistry to astrophysics and population dynamics. For a good introductory survey, see [2] and the references therein.

Many dynamical properties depend on the integration kernel $K\left(x,y\right)$, which determines the reactivity between couples of masses. It is known that, for certain kernels such as $K_{*}=xy$, a singularity in finite time occurs: the solution develops a heavy tail in finite time and the total mass is no longer conserved. This phenomenon is called gelation and represents the formation of a cluster with infinite density that drains mass from the coagulating system.

In this talk we will consider homogeneous kernels $K\left(x,y\right)=\left(xy\right)^{\lambda}$ with $\lambda \le1$ and present some results about selfsimilar solutions both in singular and non singular cases. Such self-similar solutions depend on a free exponent that cannot be determined from dimensional considerations -self-similar solution of the second kind, in the notation of Barenblatt [1]; it can be fixed imposing the behaviour at the origin and infinity. This is joint work with Marco A. Fontelos.

References

1. Barenblatt, G.I., Scaling, self-similarity, and intermediate asymptotics (Cambridge texts in applied Mathe- matics: Cambridge University Press) 1996.
2. Leyvraz, F., Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep. 383 (2003), 95 .
3. Smoluchowski, M., Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Phys Z, 17 (1916) 557—571 and 585—599.

This talk is part of the Analysis Seminar series.