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Fragmentation-coagulation processes with advection or diffusion in space

Jacek Banasiak, Nduduzo Majozi

TL;DR

This work develops a rigorous semigroup framework for spatially inhomogeneous fragmentation–coagulation with transport, proving generation of $C_0$-semigroups with parameter in weighted spaces $\mathcal{X}^i_r=L_1(\mathbb{R}_+, X^i_x, (1+m^r)dm)$. The authors introduce a dominating fragmentation kernel independent of space to obtain moment-regularising properties, and then apply a Desch–Voigt/Miyadera perturbation approach to handle unbounded coagulation, establishing well-posedness for the full transport–fragmentation–coagulation system. They treat advection and diffusion transport cases with explicit semigroups and extend the analysis to the $X^0_x$ setting via equi-integrability conditions on the fragmentation kernel; the results cover both $L_1$ and $C_0$ spatial frameworks and yield local solvability under broad, physically meaningful hypotheses. The findings provide a robust, operator-theoretic foundation for spatially dependent fragmentation–coagulation models, with potential implications for polymer science, aerosol physics, and population dynamics where transport, fragmentation, and coagulation interact with unbounded rates.

Abstract

In this paper, we consider a continuous fragmentation--coagulation model in which the reacting particles can be transported in physical space through either advection or diffusion. We prove new results on the generation of $C_0$-semigroups with parameter and use them to show that the Abstract Cauchy Problem associated with a more general version of the advection/diffusion--fragmentation problem generates a positive $C_0$-semigroup in spaces $L_1(\mathbb R_+, X_x, (1+m^r)dm),$ where $m$ is the particle mass, $X_x$ is either the space of integrable or continuous functions with respect to the spatial variable, and the weight exponent $r$ is sufficiently large. These results enable us to prove the classical solvability of a wide range of advection/diffusion--fragmentation--coagulation equations with unbounded coagulation kernels.

Fragmentation-coagulation processes with advection or diffusion in space

TL;DR

This work develops a rigorous semigroup framework for spatially inhomogeneous fragmentation–coagulation with transport, proving generation of -semigroups with parameter in weighted spaces . The authors introduce a dominating fragmentation kernel independent of space to obtain moment-regularising properties, and then apply a Desch–Voigt/Miyadera perturbation approach to handle unbounded coagulation, establishing well-posedness for the full transport–fragmentation–coagulation system. They treat advection and diffusion transport cases with explicit semigroups and extend the analysis to the setting via equi-integrability conditions on the fragmentation kernel; the results cover both and spatial frameworks and yield local solvability under broad, physically meaningful hypotheses. The findings provide a robust, operator-theoretic foundation for spatially dependent fragmentation–coagulation models, with potential implications for polymer science, aerosol physics, and population dynamics where transport, fragmentation, and coagulation interact with unbounded rates.

Abstract

In this paper, we consider a continuous fragmentation--coagulation model in which the reacting particles can be transported in physical space through either advection or diffusion. We prove new results on the generation of -semigroups with parameter and use them to show that the Abstract Cauchy Problem associated with a more general version of the advection/diffusion--fragmentation problem generates a positive -semigroup in spaces where is the particle mass, is either the space of integrable or continuous functions with respect to the spatial variable, and the weight exponent is sufficiently large. These results enable us to prove the classical solvability of a wide range of advection/diffusion--fragmentation--coagulation equations with unbounded coagulation kernels.
Paper Structure (22 sections, 24 theorems, 201 equations)

This paper contains 22 sections, 24 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Fragmentation and coagulation processes
  3. The model
  4. Mathematical preliminaries
  5. Spaces
  6. Semigroup associated with $\mathcal{T}_0$
  7. Solvability of the transport--fragmentation model
  8. The transport--absorption problem
  9. Full transport--fragmentation equation
  10. $L_1$ theory
  11. General $X_x$ theory
  12. Properties of $\beta$
  13. The generation result
  14. Analyticity of the fragmentation operator
  15. Applications
  16. ...and 7 more sections

Key Result

Proposition 2.1

Assume that there are $M$ and $\omega$ such that for all $m$ for which $M_m$ and $\omega_m$ are defined we have $M_m\leq M$ and $\omega_m\leq \omega$. With the above assumptions and notation, if $\Theta\ni m \to R(\lambda,T_m)\boldsymbol{f}(m)$ is measurable for any $\lambda>\omega$ and $\boldsymbol

Theorems & Definitions (46)

  • Remark 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.1
  • Proposition 3.1
  • proof
  • ...and 36 more