Convergence of the discrete Redner-Ben-Avraham-Kahng coagulation equation

TL;DR

This work rigorously connects the discrete Redner–Ben-Avraham–Kahng (RBK) coagulation model to its continuous counterpart by constructing a sequence of discrete approximations with kernels $a_{i,j}^{\epsilon}$ that converge to the continuous kernel $\mathfrak{K}$. By establishing uniform a priori bounds and applying weak $L^1$-compactness via the Dunford–Pettis criterion and a de la Vallée-Poussin-type argument, the authors extract a subsequence $\mathfrak{f}_{\epsilon_n}$ that converges in $\mathcal{C}([0,T]; w-L^1(\mathbb{R}_+))$ to a limit $\mathfrak{f}$. They then pass to the limit in the weak formulation, leveraging the convergence of discretized objects $\psi_{\epsilon}$, $T_{\epsilon}(\psi_{\epsilon})$, and $\mathfrak{K}_{\epsilon}$ to show $\mathfrak{f}$ satisfies the weak CRBK equation, thus proving convergence of the discrete model to the continuous one. The results require a strictly subquadratic, symmetric kernel and initial data in $X_{0,1}^+$, and provide a rigorous justification for using discrete RBK models as convergent approximations to the CRBK system, with potential implications for numerical analysis and applications in coagulation processes.

Abstract

This article looks at the relationship between the discrete and the continuous Redner-Ben-Avraham-Kahng (RBK) coagulation models. On the basis of a priori estimation, a weak stability principle and the weak compactness in $L_1$ for the continuous RBK model is shown. By employing a sequence of discrete models to approximate the continuous one, we show that how discrete model eventually converges to the the modified continuous one using the stability principle.

Theorems & Definitions (19)

• Definition 3.1
• Definition 3.2
• Lemma 3.1
• proof
• Theorem 3.1
• Lemma 4.1
• proof
• Lemma 4.2
• Lemma 4.3
• proof