Existence and non-existence for the collision-induced breakage equation

TL;DR

This paper investigates the mathematical existence and non-existence of solutions to a collision-induced breakage equation, establishing conditions under which solutions conserve mass or fail to do so, depending on kernel parameters.

Contribution

It provides new existence results for weak solutions with specific collision kernels and identifies parameter regimes where solutions do not exist or are non-unique.

Findings

Existence of mass-conserving solutions when α + β ∈ [1,2].

Non-existence of global mass-conserving solutions when α + β ∈ [0,1) and α ≥ 0.

Non-existence of solutions for certain negative α and specific daughter distributions.

Abstract

A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functions, assuming the collision kernel $K$ to be given by $K(x,y)= x^{\alpha} y^{\beta} + x^{\beta} y^{\alpha}$ with $\alpha \le \beta \le 1$. When $\alpha + \beta \in [1,2]$, it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when $\alpha + \beta \in [0,1)$ and $\alpha \ge 0$, global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for $\alpha <0$ and a specific daughter distribution function, the non-existence of mass-conserving solutions is also established.