The discrete collision-induced breakage equation with mass transfer: well-posedness and stationary solutions

TL;DR

This paper studies the well-posedness and stationary solutions of a discrete collision-induced breakage equation modeling cluster dynamics with mass transfer, establishing existence, uniqueness, and boundedness of solutions.

Contribution

It introduces new results on the existence, uniqueness, and properties of solutions for a class of collision kernels with mass transfer, including stationary solutions construction.

Findings

Global mass-conserving solutions exist for specified kernels.

All algebraic superlinear moments of solutions are bounded over time.

Non-trivial stationary solutions are constructed via a dynamical approach.

Abstract

The discrete collisional breakage equation, which captures the dynamics of cluster growth when clusters encounter binary collisions with possible matter transfer, is discussed in this article. The existence of global mass-conserving solutions is investigated for the collision kernels $a_{i,j}=A(i^{\alpha} j^{\beta} + i^{\beta}j^{\alpha})$, $i, j \ge 1$, with $\alpha \in (-\infty,1)$, $\beta\in [\alpha,1]\cap (0,1]$, and $A>0$ and for a large class of possibly unbounded daughter distribution functions. All algebraic superlinear moments of these solutions are bounded on time intervals $[T,\infty)$ for any $T>0$. The uniqueness issue is further handled under additional restrictions on the initial data. Finally, non-trivial stationary solutions are constructed by a dynamical approach.