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

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.

💡 Research Summary

The paper addresses the long‑time behavior of the continuous coagulation‑fragmentation equation without assuming the detailed‑balance condition, which traditionally guarantees the existence of an explicit equilibrium and a Lyapunov (entropy) functional. The authors consider kernels that are small perturbations of the constant kernels K₀ = 2 and F₀ = 2, namely
K_ε(x,y) = 2 + ε W(x,y), F_ε(x,y) = 2 + ε V(x,y)
with ε > 0 small, 0 ≤ W ≤ 1 and 0 ≤ V ≤ 1/(x + y). This class does not satisfy detailed balance, yet the kernels remain bounded and physically admissible.

The analysis proceeds in several steps. First, the authors establish boundedness of the bilinear coagulation operator C_K and the linear fragmentation operator F_F on weighted L¹ spaces L¹_α (α ≥ 0). Using these bounds together with uniform moment estimates, they prove global existence, uniqueness, and mass conservation for mild solutions with initial data in L¹_α (α ≥ 2). For stationary solutions Q_ε they show that all moments M_m(Q_ε) are uniformly bounded by constants C_m(ρ,ε_) that depend only on the total mass ρ and the perturbation size ε_.

Next, they recall the well‑known results for the unperturbed constant kernels: the equilibrium is explicit, Q(x)=e^{-x/√ρ}, and the linearized operator L₀ = 2 C₂(Q,·)+F₂(·) possesses a spectral gap 2√ρ in L² and, via Laplace transform techniques, also in L¹_α for any α ≥ 1. This yields exponential decay e^{-2√ρ t} for the linearized dynamics.

The core contribution is a perturbative argument that transfers the spectral gap from L₀ to the full nonlinear operator with kernels K_ε, F_ε. By showing that the difference between the full operator and L₀ is O(ε) in the appropriate operator norm, the authors apply a fixed‑point and stability analysis to obtain:

1. Uniqueness of equilibrium: For each mass ρ and sufficiently small ε (ε < ε_*), there exists at most one stationary solution Q_ε ∈ L¹₁ with total mass ρ.

2. Local exponential convergence: If the initial datum f₀ satisfies ‖f₀ − Q_ε‖{L¹_α} ≤ δ (δ small), then the corresponding solution satisfies
   ‖f(t) − Q_ε‖{L¹_α} ≤ C_* ‖f₀ − Q_ε‖{L¹_α} e^{-(2√ρ − c ε) t}
   for all t ≥ 0, where C*, c are explicit constants independent of the solution.

3. Global exponential convergence: Using an entropy functional H(f|Q_ε)=∫ f log(f/Q_ε) − f + Q_ε and assuming only that the initial entropy and the zeroth moment are bounded (i.e., ‖f₀ log f₀‖_{L¹}+M₀(f₀) ≤ R), the same exponential rate holds for any solution, without any smallness condition on the distance to equilibrium.

The convergence rate is explicitly given by 2√ρ − c ε, which reduces to the optimal 2√ρ when ε→0, matching the known results for constant kernels. The paper also provides detailed moment bounds for both transient and stationary solutions, ensuring that all required integrals are finite and the perturbative framework is well‑posed.

Overall, the work extends the entropy‑based and spectral‑gap techniques, previously limited to detailed‑balance or discrete models (e.g., Beck‑Döring), to a class of continuous coagulation‑fragmentation equations lacking detailed balance. The results are significant for applications where symmetry between coagulation and fragmentation rates cannot be assumed, such as cloud formation, polymerisation, and biological aggregation processes. The explicit rates and uniform moment estimates make the theory amenable to numerical verification and further extensions to more general kernels.