On the discrete Safronov-Dubovskii coagulation equation: well-posedness, mass-conservation and asymptotic behaviour

The global existence of mass-conserving weak solutions to the Safronov-Dubovskii coagulation equation is shown for the coagulation kernels satisfying the at most linear growth for large sizes. In contrast to previous works, the proof mainly relies on the de la Vallee-Poussin theorem [8, Theorem 7.1.6], which only requires the finiteness of the first moment of the initial condition. By showing the necessary regularity of solutions, it is shown that the weak solutions con-structed herein are indeed classical solutions. Under additional restrictions on the initial data, the uniqueness of solutions is also shown. Finally, the continuous dependence on the initial data and the large-time behaviour of solutions are also addressed

Weak solutions to the discrete Redner-Ben-Avraham-Kahng coagulation model

This study examines the global existence of solutions to the discrete Redner-Ben-Avraham-Kahng coagulation equations for a wide range of coagulation kernels $\theta_{i,j}$ defined as $\theta_{i,j} =\omega_i \omega_j +\kappa_{i,j}$ and $\kappa_{i,j} \le A \omega_i \omega_j, \ \ i,j \ge 1$ when $(\omega_i)_{i\ge 1}$ grows linearly or super-linearly with respect to $i$

Mathematical aspects of coagulation-fragmentation equations

We give an overview of the mathematical literature on the coagulation-like equations, from an analytic deterministic perspective. In Section 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In Section 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the functional spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sections 3 and 4 we are concerned with several aspects of the solutions behaviour.We pay special attention to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof

An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach

We summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity (an integer multiple of some basic unit size). In particular, we discuss Smoluchowski's equation for aggregation, the Becker-Döring model of simultaneous aggregation and fragmentation, and more general models involving coagulation and fragmentation

The continuous Redner–Ben-Avraham–Kahng coagulation system: well-posedness and asymptotic behaviour

This paper examines the existence of solutions to the continuous Redner-Ben-Avraham-Kahng coagulation system under specific growth conditions on unbounded coagulation kernels at infinity. Moreover, questions related to uniqueness and continuous dependence on the data are also addressed under additional restrictions. Finally, the large-time behaviour of solutions is also investigated.Council of Scientific and Industrial Research (CSIR), India, grant No. 09/143(0901)/2017-EMR-I. Fundação para a Ciência e Tecnologia (FCT), Portugal, projects CAMGSD UIDB/04459/2020 and UIDP/04459/2020.info:eu-repo/semantics/acceptedVersio

An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach

We summarise the properties and the fundamental mathematical resultsassociated with basic models which describecoagulation and fragmentation processes in a deterministic mannerand in which cluster size is a discrete quantity (an integermultiple of some basic unit size).In particular, we discuss Smoluchowski's equation for aggregation,the Becker-Döring model of simultaneous aggregation and fragmentation,and more general models involving coagulation and fragmentation

Existence of self-similar profile for a kinetic annihilation model

We show the existence of a self-similar solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard-spheres such that, whenever two particles meet, they either annihilate with probability $\alpha \in (0,1)$ or they undergo an elastic collision with probability $1 - \alpha$. For such a model, the number of particles, the linear momentum and the kinetic energy are not conserved. We show that, for $\alpha$ smaller than some explicit threshold value $\alpha_*$, a self-similar solution exists.Comment: This new version supersedes and replaces the previous one. We found a mistake in the previous (and published) version of the manuscript and explained how to fix it in "Erratum to "Existence of self-similar profile for a kinetic annihilation model" [J. Differential Equations 254 (7) (2013) 3023-3080]. J. Differential Equations 257 (2014), no. 8, 3071-3074." This version provides a complete and corrected version of the previous manuscrip

A kinetic equation for repulsive coalescing random jumps in continuum

A continuum individual-based model of hopping and coalescing particles is introduced and studied. Its microscopic dynamics are described by a hierarchy of evolution equations obtained in the paper. Then the passage from the micro- to mesoscopic dynamics is performed by means of a Vlasov-type scaling. The existence and uniqueness of solutions of the corresponding kinetic equation are proved