Approach to Equilibrium for the Coagulation-Fragmentation Equation via a Lyapunov Functional

Existence, Uniqueness and Mass Conservation for the Coagulation-Fragmentation Equation

“…For both non-zero kernels K and F the global existence and uniqueness of solutions to (1.1) has been investigated in [1,2,3,5,12,16,. For the case K = a, F = b with a, b constants the convergence to equilibrium of solutions has been studied via a Lyapunov function by Aizenman and Bak [ 13 and by the authors [23]. It is our aim to prove existence, uniqueness and mass conservation theorems for the initial value problem (l.l), (1.2) with K d k ( l + x + y) and a very large class of fragmentation kernels including bounded ones; in this paper we use and generalize the approach of [9,11].…”

The coagulation-fragmentation hierarchy with homogeneous rates and underlying stochastic dynamics

“…This equation is very similar to (1.1) with K and F being constants, the solution of which has been studied extensively in [1] and [31]. However, it is not clear whether or not there is any direct connection between solutions of these two equations.…”

Diffusion, fragmentation, and coagulation processes: Analytical and numerical results

“…The existence of steady states in the model with no coagulation has recently been identified for a wider class of fragmentation kernels [23], where connections to the pure fragmentation equation [24] and the so-called "shattering" transition [25] related to the formation of dust particles, is further discussed. In the absence of diffusion, the model has been applied to the kinetics of reacting polymers in [26,27], where the uniqueness of solutions and convergence to equilibrium in the limit t → ∞ have been proved. As will be demonstrated, the inclusion of a diffusion term in the equation may lead to solutions radically different from the ones found in [26,27].…”

“…In the absence of diffusion, the model has been applied to the kinetics of reacting polymers in [26,27], where the uniqueness of solutions and convergence to equilibrium in the limit t → ∞ have been proved. As will be demonstrated, the inclusion of a diffusion term in the equation may lead to solutions radically different from the ones found in [26,27].…”