Fragmentation-coagulation processes with advection or diffusion in space

TL;DR

This paper develops a mathematical framework for fragmentation-coagulation models with particle transport via advection or diffusion, proving semigroup generation and classical solvability for complex equations.

Contribution

It introduces new semigroup generation results for advection/diffusion fragmentation models and establishes classical solutions for equations with unbounded kernels.

Findings

Generated positive $C_0$-semigroups in weighted $L_1$ spaces.

Proved classical solvability for a broad class of equations.

Extended the theory to models with unbounded coagulation kernels.

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.