Long-time behaviour of rouleau formation models

TL;DR

This paper analyzes a coagulation model for rouleaux formation in blood, characterizing initial conditions leading to gelation and describing the long-time behavior and self-similarity of solutions.

Contribution

It provides a detailed analysis of gelation conditions and the asymptotic localization and self-similar behavior of solutions in a two-component coagulation model.

Findings

Identification of initial data leading to gelation.

Solution localizes along a specific direction as gelation time approaches.

Convergence to a self-similar solution along the localization direction.

Abstract

In this paper we study a two-component coagulation equation that models the aggregation of rouleaux in blood. We consider product kernels that have homogeneity $2$ and we characterize the initial data that lead to gelation. We prove that, when gelation occurs, the solution to the two-component coagulation equation localizes along a direction of the space of cluster as $ t $ approaches the gelation time $0 < T_* < \infty $. The localization direction is determined by the initial datum. We also prove that the solution converges to a self-similar solution along the direction of localization.