Existence for the Discrete Nonlinear Fragmentation Equation with Degenerate Diffusion

TL;DR

This paper proves the existence of global weak solutions for a discrete nonlinear fragmentation equation with degenerate diffusion in any spatial dimension, extending previous results that required positive diffusion coefficients.

Contribution

It introduces a new approach to establish solutions without assuming strictly positive diffusion, broadening the mathematical understanding of fragmentation models.

Findings

Existence of global weak solutions in arbitrary dimensions.

Extension of previous results to degenerate diffusion cases.

Use of regularized systems and compactness methods in the proof.

Abstract

A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly positive lower bound on the diffusion coefficients, extending previous results that were restricted to one-dimensional domains and relied on uniformly positive diffusion. The analysis is carried out under boundedness assumptions on the collision and breakage kernels. The proof is based on the construction of a suitable regularized system, combined with weak $L^2$ a priori estimates and compactness arguments in $L^1$, which allow the passage to the limit in the nonlinear fragmentation operator.