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Long-time behaviour of rouleau formation models

Eugenia Franco, Bernhard Kepka

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.

Long-time behaviour of rouleau formation models

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 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 approaches the gelation time . 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.

Paper Structure

This paper contains 17 sections, 22 theorems, 251 equations, 1 figure.

Table of Contents

  1. Introduction
  2. State of the art
  3. Discrete and continuous coagulation equation for rouleaux
  4. Main results: localization and asymptotic self-similarity
  5. Plan of the paper.
  6. Notation
  7. Main results
  8. Heuristic arguments
  9. Well-posedness of coagulation models
  10. Localization and self-similar asymptotics of coagulation models
  11. Study of moments
  12. First order moments
  13. Second order moments
  14. Third order moments
  15. Fourth order moments
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\alpha \in \mathbb{R}_+^3$, $\alpha \neq 0$, $p\in \mathbb{N}$, $p\geq3$. Consider $f_0\in \mathscr{M}_{p,+}(\mathcal{S})$ and $T_*=T_{\alpha,*}(f_0)$ given in eq:sec1:BlowUpTime. Then, there exists a unique weak solution $f\in C^1([0,T_*); \mathscr{M}_{+}(\mathcal{S}))\cap L^\infty_{\operatorn

Figures (1)

  • Figure 4: Left: Initial configuration. the particles are localized in the directions $p_1 = p_{\beta_1}$ and $p_2 = p_{\beta_2 }$. All the clusters will remain between the lines $p_1$ and $p_2$ for all times. Right: possible directions after three coagulation events. In red (solid line) after one coagulation event. In green (dashed lines) after two coagulation events. In blue (dotted lines) after three coagulation events.

Theorems & Definitions (55)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 4.1: Weak solutions
  • Remark 4.2
  • Proposition 4.3
  • proof
  • Definition 4.4
  • ...and 45 more