| Contents |
| Smoluchowski's coagulation equation is a mean field model describing
%cluster growth. Let the function $c\left(x,t\right)$ represent the
%mean amount of $x$-mass polymers per unit volume at a given time
%$t$. Then, the variation of $c$ is expressed by a non linear, integrodifferential
%equation:
%\[
%\partial_{t}c\left(x,t\right)=\frac{1}{2}\intop_{0}^{x}K\left(x-y,y\right)c\left(x-y,t\right)c\left(y,t\right)dy-c\left(x,y\right)\intop_{0}^{\infty}K\left(x,y\right)c\left(y,t\right)dy
%\]
%where the first convolution integral accounts for clusters of smaller
%sizes aggregating to form a new cluster of size $x$ and the second
%integral represents the loss of $x$-mass clusters coagulating to
%form heavier ones. Smoluchowski's equation
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 \cite{key-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 \cite{key-1}-;
it can be fixed imposing the behaviour at the origin and infinity. This is joint work with Marco A. Fontelos.
%We present a numerical scheme to find the self-similar parameter corresponding
%to self-similar solutions with all its moments bounded. |
|