Mass-conserving weak solutions to the continuous nonlinear fragmentation equation in the presence of mass transfer

TL;DR

This paper proves the existence and uniqueness of mass-conserving weak solutions to a nonlinear fragmentation equation with mass transfer, extending previous results to more general collision kernels and growth conditions.

Contribution

It establishes the existence of global and local weak solutions for nonlinear fragmentation equations with specific collision kernels, including cases with linear and sublinear growth.

Findings

Existence of at least one global weak solution for linearly growing kernels.

Existence of a local weak solution for sublinear kernels.

Finite superlinear moment bounds are obtained without initial moment finiteness.

Abstract

A mathematical model for the continuous nonlinear fragmentation equation is considered in the presence of mass transfer. In this paper, we demonstrate the existence of mass-conserving weak solutions to the nonlinear fragmentation equation with mass transfer for collision kernels of the form $\Phi(x,y) = \kappa(x^{{\sigma_1}} y^{{\sigma_2}} + y^{{\sigma_1}} x^{{\sigma_2}})$, $\kappa>0$, $0 \leq {\sigma_1} \leq {\sigma_2} \leq 1$, and ${\sigma_1} \neq 1$ for $(x, y) \in \mathbb{R}_+^2$, with integrable daughter distribution functions, thereby extending previous results obtained by Giri \& Lauren\c cot (2021). In particular, the existence of at least one global weak solution is shown when the collision kernel exhibits at least linear growth, and one local weak solution when the collision kernel exhibits sublinear growth. In both cases, finite superlinear moment bounds are obtained for positive times without requiring the finiteness of initial superlinear moments. Additionally, the uniqueness of solutions is confirmed in both cases.