Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with inverse power law kernels

TL;DR

This paper proves the uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with inverse power law kernels, enhancing understanding of the equation's solution structure.

Contribution

It establishes the uniqueness of solutions for a class of inverse power law kernels in Smoluchowski's coagulation equation, a problem previously unresolved.

Findings

Uniqueness of solutions for $K(x,x_*)=2(x x_*)^{-eta}$ with $eta>0$

Clarifies the solution structure for inverse power law kernels

Provides mathematical proof of solution uniqueness

Abstract

Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation is shown when the coagulation kernel $K$ is given by $K(x,x\_*)=2(x x\_*)^{-\alpha}$, $(x,x\_*)\in (0,\infty)^2$, for some $\alpha>0$.