Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition

The coagulation–fragmentation equation describes the concentration fi(t) of particles of size i ∈ N/{0} at time t ⩾ 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break–up. We show that when the rate of fragmentation is sufficiently stronger than that of coalescence, (fi(t))i ∈ N/{0} tends to a unique equilibrium as t tends to infinity. Although we suppose that the initial datum is sufficiently small, we do not assume a detailed balance (or reversibility) condition. The rate of convergence we obtain is, furthermore, exponential.

[1]  D. Aldous.  Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists , 1999 .

[2]  Miguel Escobedo Martínez,et al.  Gelation in coagulation and fragmentation models , 2002 .

[3]  Benoît Perthame,et al.  Gelation and mass conservation in coagulation-fragmentation models , 2003 .

[4]  F. P. da Costa.  Existence and Uniqueness of Density Conserving Solutions to the Coagulation-Fragmentation Equations With Strong Fragmentation , 1995 .

[5]  John L. Spouge,et al.  An existence theorem for the discrete coagulation–fragmentation equations , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  I. W. Stewart,et al.  Trend to Equilibrium for the Coagulation–Fragmentation Equation , 1996 .

[7]  Intae Jeon,et al.  Existence of Gelling Solutions for Coagulation-Fragmentation Equations , 1998 .

[8]  J. M. BalP.  The Discrete Coagulation-Fragmentation Equations: Existence, Uniqueness, and Density Conservation , 2004 .

[9]  F. P. Dacosta,et al.  Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation , 1995 .

[10]  P. Jabin,et al.  On the rate of convergence to equilibrium in the Becker–Döring equations , 2003 .

[11]  Jack Carr,et al.  The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions , 1986 .

[12]  Jack Carr,et al.  Asymptotic behaviour of solutions to the coagulation–fragmentation equations. I. The strong fragmentation case , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  Michael Aizenman,et al.  Convergence to equilibrium in a system of reacting polymers , 1979 .

[14]  Jack Carr,et al.  Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation , 1994 .