On global in time self-similar solutions of Smoluchowski equation with multiplicative kernel

TL;DR

This paper analyzes the self-similar solutions of the Smoluchowski coagulation equation with a multiplicative kernel for different parameter ranges, revealing distinct asymptotic behaviors and verifying results through numerical simulations.

Contribution

It provides a detailed characterization of global self-similar solutions for the Smoluchowski equation with multiplicative kernels, including asymptotic behaviors and singularities, supported by numerical verification.

Findings

Self-similar solutions exhibit three regions with distinct asymptotics.

Solutions have a Gamma distribution tail and a lognormal or Pareto-like intermediate region.

Numerical simulations confirm the theoretical asymptotic behaviors and convergence to self-similarity.

Abstract

We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<\frac{1}{2}$. When $s<0$% , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $s\in \left( 0,\frac{1}{2}\right) $, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution tail, an intermediate region of power-like (or Pareto distribution) decay and the region close to the origin where a singularity occurs. Finally, full numerical simulations of Smoluchowski equation serve to verify our theoretical results and show the convergence of solutions to the selfsimilar regime.