Convergence to equilibrium for a class of coagulation-fragmentation equations without detailed balance

TL;DR

This work studies convergence to equilibrium for coagulation-fragmentation equations without detailed balance by perturbing the constant kernels $K_\varepsilon=2+\varepsilon W$ and $F_\varepsilon=2+\varepsilon V$, with $0\le V\le 1/(x+y)$ and $0\le W\le 1$. Using a perturbative framework around the known constant-kernel case, the authors establish a spectral-gap-based exponential convergence to a unique equilibrium $Q_\varepsilon$ for small $\varepsilon$, together with stability and moment estimates that control time evolution in weighted spaces $L^1_\alpha$. They prove both local and global convergence to $Q_\varepsilon$ for solutions with appropriate initial data and entropy bounds, and they demonstrate that the equilibrium is unique for sufficiently small perturbations. Overall, the paper extends explicit convergence results beyond detailed balance, providing quantitative rates and a robust perturbative approach via the linearized operator and spectral theory for a broad class of coagulation-fragmentation kernels.

Abstract

We prove convergence to equilibrium for a class of coagulation-fragmentation equations that do not satisfy a detailed balance condition. More precisely, we consider perturbations of constant rate kernels. Our result provides in particular explicit convergence rates. Uniqueness of the stationary states is proven as well for the considered class of kernels.

Theorems & Definitions (31)

• Definition 1.1
• Remark 1.2
• Proposition 1.3
• Proposition 1.4
• proof
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9