Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion

TL;DR

This paper introduces a new a-priori estimate for discrete coagulation-fragmentation equations with diffusion, enabling simplified existence proofs and demonstrating mass conservation in complex models, thus extending understanding of gelation phenomena.

Contribution

It provides a novel a-priori estimate based on duality, which simplifies existence proofs and proves mass conservation for a broad class of coagulation-fragmentation models with diffusion.

Findings

Established a global $L^2$ bound on mass density.

Proved mass conservation and absence of gelation in complex models.

Simplified the existence theory for generalized coagulation-fragmentation systems.

Abstract

We present a new a-priori estimate for discrete coagulation-fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global $L^2$ bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.