Coagulation equations for non-spherical clusters

TL;DR

This paper investigates the long-term behavior of a coagulation model for particles with volume and surface area, revealing conditions under which particles tend to spherical shapes or ramified structures.

Contribution

It introduces a coagulation model incorporating collision and fusion phases, proving existence of self-similar profiles and analyzing shape evolution of particles.

Findings

Existence of self-similar profiles for certain fusion rates.

Particles tend to spherical shapes under specific conditions.

Alternative fusion mechanisms lead to ramified-like particle structures.

Abstract

In this work, we study the long time asymptotics of a coagulation model which describes the evolution of a system of particles characterized by their volume and surface area. The aggregation mechanism takes place in two stages: collision and fusion of particles. During the collision stage, the two particles merge at a contact point. The newly formed particle has volume and area equal to the sum of the respective quantities of the two colliding particles. After collision, the fusion phase begins and during it the geometry of the interacting particles is modified in such a way that the volume of the total system is preserved and the surface area is reduced. During their evolution, the particles must satisfy the isoperimetric inequality. Therefore, the distribution of particles in the volume and area space is supported in the region where $\{a\geq (36\pi)^{\frac{1}{3}}v^{\frac{2}{3}}\}$. We assume the coagulation kernel has a weak dependence on the area variable. We prove existence of self-similar profiles for some choices of the functions describing the fusion rate for which the particles have a shape that is close to spherical. On the other hand, for other fusion mechanisms and suitable choices of initial data, we show that the particle distribution describes a system of ramified-like particles.