Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels

Abstract

We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying $C_1 \left(x^{-a}y^{b}+x^{b}y^{-a}\right)\leq K\left(x,y\right)\leq C_2\left(x^{-a}y^{b}+x^{b}y^{-a}\right)$ with $a>0$ and $b<1$. This covers especially the case of Smoluchowski's classical kernel $K(x,y)=(x^{1/3} + y^{1/3})(x^{-1/3} + y^{-1/3})$. For the proof of existence we first consider some regularized kernel $K_{\epsilon}$ for which we construct a sequence of solutions $h_{\epsilon}$. In a second step we pass to the limit $\epsilon\to 0$ to obtain a solution for the original kernel $K$. The main difficulty is to establish a uniform lower bound on $h_{\epsilon}$. The basic idea for this is to consider the time-dependent problem and choosing a special test function that solves the dual problem.