Stationary Kolmogorov Solutions of the Smoluchowski Aggregation Equation with a Source Term

TL;DR

This paper uses Zakharov transformations to analyze stationary solutions of the Smoluchowski aggregation equation, deriving conditions for Kolmogorov spectra and applying them to a family of kernels to understand gelation transitions.

Contribution

It introduces a locality criterion for Kolmogorov solutions and computes exact stationary states for a family of kernels, including gelling and non-gelling cases.

Findings

Derived a locality criterion for admissible Kolmogorov solutions.

Computed exact stationary states for a family of kernels.

Found the Kolmogorov constant is the same across the kernel family.

Abstract

In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. We derive a ``locality criterion'', expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels,$K_\zeta(m_1,m_2)=(m_1m_2)^{\zeta/2}$ which includes both gelling and non-gelling cases, reproducing the known solution in the case $\zeta=0$. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.