Asymptotic localization in multicomponent mass conserving coagulation equations

TL;DR

This paper proves that solutions to multicomponent Smoluchowski coagulation equations tend to concentrate along a specific composition direction over time, with stability and uniqueness results for self-similar profiles.

Contribution

It establishes asymptotic localization and stability of solutions in multicomponent coagulation equations with non-constant kernels, extending understanding of long-term behavior.

Findings

Solutions concentrate along a direction determined by initial conditions

Uniqueness and global stability of self-similar profiles are proven

Results apply to kernels constant along composition directions

Abstract

In this paper we prove that the time dependent solutions of a large class of Smoluchowski coagulation equations for multicomponent systems concentrate along a particular direction of the space of cluster compositions for long times. The direction of concentration is determined by the initial distribution of clusters. These results allow to prove the uniqueness and global stability of the self-similar profile with finite mass in the case of coagulation kernels which are not identically constant, but are constant along any direction of the space of cluster compositions.