The Coagulation - Fragmentation Equation and Its Stochastic Counterpart
نویسنده
چکیده
We consider a coagulation multiple-fragmentation equation, which describes the concentration ct(x) of particles of mass x ∈ (0,∞) at the instant t ≥ 0 in a model where fragmentation and coalescence phenomena occur. We study the existence and uniqueness of measured-valued solutions to this equation for homogeneous-like kernels of homogeneity parameter λ ∈ (0, 1] and bounded fragmentation kernels, although a possibly infinite number of fragments is considered. We also study a stochastic counterpart of this equation where a similar result is shown. We ask to the initial state to have a finite λ-moment. This work relies on the use of a Wasserstein-type distance, which has shown to be particularly well-adapted to coalescence phenomena. It was introduced in previous works on coagulation and coalescence. Mathematics Subject Classification (2000): 45K05, 60K35.
منابع مشابه
Stochastic coagulation and fragmentation: incommensurability and first passage times
We develop a fully stochastic theory for coagulation and fragmentation in a finite system with a maximum cluster size constraint. The process is modeled using a high-dimensional Master equation for the probabilities of cluster configurations. For certain realizations of total mass and maximum cluster sizes, we are able to find exact analytical results for the expected equilibrium cluster distri...
متن کاملOn general kinetic equation for many particle systems with interaction, fragmentation and coagulation
We deduce the most general kinetic equation that describe the low density limit of general Feller processes for the systems of random number of particles with interaction, collisions, fragmentation and coagulation. This is done by studying the limiting as ε → 0 evolution of Feller processes on ∪∞n Xn with X = Rd or X = Zd described by the generators of the form ε−1 ∑K k=0 ε kB(k), K ∈ N , where...
متن کاملAbsence of Gelation and Self-Similar Behavior for a Coagulation-Fragmentation Equation
The dynamics of a coagulation-fragmentation equation with multiplicative coagulation kernel and critical singular fragmentation is studied. In contrast to the coagulation equation, it is proved that fragmentation prevents the occurrence of the gelation phenomenon and a mass-conserving solution is constructed. The large time behavior of this solution is shown to be described by a selfsimilar sol...
متن کاملStochastic particle methods for Smoluchowski coagulation equation: variance reduction and error estimations
Stochastic particle methods for the coagulation-fragmentation Smoluchowski equation are developed and a general variance reduction technique is suggested. This method generalizes the massow approach due to H. Babovsky, and has in focus the desired band of the size spectrum. Estimations of the variance and bias of the method are derived. A comparative cost and variance analysis is made for the k...
متن کاملExplosion Phenomena in Stochastic Coagulation – Fragmentation Models
First we establish explosion criteria for jump processes with an arbitrary locally compact separable metric state space. Then these results are applied to two stochastic coagulation–fragmentation models— the direct simulation model and the mass flow model. In the pure coagulation case, there is almost sure explosion in the mass flow model for arbitrary homogeneous coagulation kernels with expon...
متن کامل