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Fast fusion in a two-dimensional coagulation model

Iulia Cristian, Juan J. L. Velázquez

Abstract

In this work, we study a particular system of coagulation equations characterized by two values, namely volume $v$ and surface area $a$. Compared to the standard one-dimensional models, this model incorporates additional information about the geometry of the particles. We describe the coagulation process as a combination between collision and fusion of particles. We prove that we are able to recover the standard one-dimensional coagulation model when fusion happens quickly and that we are able to recover an equation in which particles interact and form a ramified-like system in time when fusion happens slowly.

Fast fusion in a two-dimensional coagulation model

Abstract

In this work, we study a particular system of coagulation equations characterized by two values, namely volume and surface area . Compared to the standard one-dimensional models, this model incorporates additional information about the geometry of the particles. We describe the coagulation process as a combination between collision and fusion of particles. We prove that we are able to recover the standard one-dimensional coagulation model when fusion happens quickly and that we are able to recover an equation in which particles interact and form a ramified-like system in time when fusion happens slowly.
Paper Structure (7 sections, 14 theorems, 103 equations, 2 figures)

This paper contains 7 sections, 14 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and plan of the paper
  3. Setting and main results
  4. The case of fast fusion
  5. Existence of a limit of solutions of coagulation equations with fast fusion
  6. Reduction to the one-dimensional coagulation model in the case of fast fusion
  7. The case of negligible fusion

Key Result

Theorem 1.4

Let $K:(0,\infty)^{4}\rightarrow [0,\infty)$ be a continuous kernel satisfying (kersym1), (lower_bound_kernel) and (alpha non neg). Assume the fusion kernel $r\in\textup{C}^{1}(\mathbb{R}_{>0}^{2})$ satisfies (fusion_form) and (ode_fusion) with $\mu>0$. Assume in addition that $\int_{(0,\infty)^{2}} Moreover, there exists a subsequence (which we do not relabel) and $\overline{f}\in\textup{C}((0,T]

Figures (2)

  • Figure 1: Coagulation process: collision of particles followed by fusion
  • Figure 2: System of particles under different fusion times

Theorems & Definitions (41)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 31 more