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 C1(x−ayb+xby−a)≤K(x,y)≤C2(x−ayb+xby−a) with a>0 and b<1. This covers especially the case of Smoluchowski's classical kernel K(x,y)=(x1/3+y1/3)(x−1/3+y−1/3).For the proof of existence we take a self-similar solution hε for a regularized kernel Kε and pass to the limit ε→0 to obtain a solution for the original kernel K. The main difficulty is to establish a uniform lower bound on hε. The basic idea for this is to consider the time-dependent problem and to choose a special test function that solves the dual problem.