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Discrete coagulation--fragmentation systems in weighted $\ell^1$ spaces

Lyndsay Kerr, Matthias Langer

TL;DR

The authors address the well-posedness of a discrete coagulation–fragmentation system in weighted $\ell^1$ spaces by formulating it as a semi-linear abstract Cauchy problem. They develop a novel abstract framework that accommodates a nonlinear coagulation term defined on a dense subspace $Y$ of a Banach space $X$, while the linear fragmentation part generates a positive (sub)stochastic semigroup; analytic interpolation spaces are leveraged to obtain regularity and compactness. The main contributions are local existence, uniqueness and positivity of mild solutions (with mass evolution described by a linear functional) and, under additional hypotheses, classical solutions and global existence. The results apply to time-dependent coagulation kernels and provide sufficient conditions ensuring mass non-increase, mass conservation, or global solvability, thereby extending the well-posedness theory for discrete C–F systems in a flexible functional-analytic setting.

Abstract

We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove existence, uniqueness and positivity of solutions of a corresponding semi-linear Cauchy problem in a weighted $\ell^1$ space. This requires the application of novel results, which we prove for abstract semi-linear Cauchy problems in Banach lattices where the non-linear term is defined only on a dense subspace.

Discrete coagulation--fragmentation systems in weighted $\ell^1$ spaces

TL;DR

The authors address the well-posedness of a discrete coagulation–fragmentation system in weighted spaces by formulating it as a semi-linear abstract Cauchy problem. They develop a novel abstract framework that accommodates a nonlinear coagulation term defined on a dense subspace of a Banach space , while the linear fragmentation part generates a positive (sub)stochastic semigroup; analytic interpolation spaces are leveraged to obtain regularity and compactness. The main contributions are local existence, uniqueness and positivity of mild solutions (with mass evolution described by a linear functional) and, under additional hypotheses, classical solutions and global existence. The results apply to time-dependent coagulation kernels and provide sufficient conditions ensuring mass non-increase, mass conservation, or global solvability, thereby extending the well-posedness theory for discrete C–F systems in a flexible functional-analytic setting.

Abstract

We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove existence, uniqueness and positivity of solutions of a corresponding semi-linear Cauchy problem in a weighted space. This requires the application of novel results, which we prove for abstract semi-linear Cauchy problems in Banach lattices where the non-linear term is defined only on a dense subspace.
Paper Structure (20 sections, 28 theorems, 158 equations)

This paper contains 20 sections, 28 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz continuity and Fréchet differentiability
  4. Positive semigroups
  5. Analytic semigroups and interpolation spaces
  6. Abstract Existence, Uniqueness and Positivity Results
  7. General existence, uniqueness and positivity results
  8. Proof of \ref{['C31']}
  9. Proof of \ref{['C32']}
  10. Special situations and additions
  11. The Coagulation--Fragmentation System
  12. Setting of the Problem
  13. The Fragmentation Operator
  14. The Coagulation Operator
  15. Sufficient Conditions
  16. ...and 5 more sections

Key Result

Lemma 2.2

Let $(Y,\lVert\,\cdot\,\rVert_{Y})$, $(X,\lVert\,\cdot\,\rVert_{X})$ be Banach spaces. Further, let $t_0$, $T \in \mathbb{R}$ be such that $t_0<T<\infty$ and let $\mathcal{I}$ be an interval of the form $[t_0,T)$, $[t_0,T]$ or $[t_0,\infty)$. Let $\widetilde{F}: \mathcal{I} \times Y \times Y \to X$ Define $F: \mathcal{I} \times Y \to X$ by $F(t,f) \coloneqq \widetilde{F}[t,f,f]$ for $t\in\mathcal

Theorems & Definitions (63)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 53 more