On the self-similar behaviour of coagulation systems with injection

TL;DR

This paper establishes the existence and properties of self-similar solutions for coagulation equations with constant particle injection, providing insights into their long-term behavior and profile estimates.

Contribution

It proves the existence of smooth self-similar solutions for coagulation equations with injection and characterizes their asymptotic behavior.

Findings

Self-similar solutions exist for the coagulation equations with constant flux.

Profiles are smooth if the kernel is smooth.

Profiles behave like x^{-(bamma+3)/2} near zero and decay exponentially at infinity.

Abstract

In this paper we prove the existence of a family of self-similar solutions for a class of coagulation equations with a constant flux of particles from the origin. These solutions are expected to describe the longtime asymptotics of Smoluchowski's coagulation equations with a time independent source of clusters concentrated in small sizes. The self-similar profiles are shown to be smooth, provided the coagulation kernel is also smooth. Moreover, the self-similar profiles are estimated from above and from below by $x^{-(\gamma+3)/2}$ as $x \to 0$, where $\gamma <1 $ is the homogeneity of the kernel, and are proven to decay at least exponentially as $x \to \infty$.