Unbounded Coagulation Research Articles

Instantaneous gelation and non-existence of weak solutions for the Oort–Hulst–Safronov coagulation model

The possible occurrence of instantaneous gelation to the Oort–Hulst–Safronov (OHS) coagulation equation is investigated for a certain class of unbounded coagulation kernels. The existence of instantaneous gelation is confirmed by showing the non-existence of mass-conserving weak solutions. Finally, it is shown that for such kernels, there is no weak solution to the OHS coagulation equation at any time interval.

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.

Growth-fragmentation-coagulation equations with unbounded coagulation kernels.

In this paper, we prove the global in time solvability of the continuous growth-fragmentation-coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth-fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic. This article is part of the theme issue 'Semigroup applications everywhere'.

A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation

In general, the non-conservative approximation of coagulation-fragmentation equations (CFEs) may lead to the occurrence of gelation phenomenon. In this article, it is shown that the non-conservative approximation of CFEs can also provide the existence of mass conserving solutions to CFEs for large classes of unbounded coagulation and fragmentation kernels.

The singular coagulation equation with multiple fragmentation

In this paper, we prove the global existence and uniqueness of the solutions to the initial-value problem for the coagulation–fragmentation equation with singular coagulation kernel and multiple fragmentation kernel. The solution obtained in this case also satisfies the mass conservation law. The proof is based on strong convergence methods applied to suitably chosen unbounded coagulation kernels having singularities in both the coordinate axes and satisfying certain growth conditions, which can possibly reach up to a quadratic growth at infinity, and the fragmentation kernel covers a very large class of unbounded functions.

Existence and uniqueness theorem for the Safronov–Dubovski coagulation equation

The solutions of the discrete Safronov–Dubovski coagulation equation are investigated. We prove global existence for a class of unbounded coagulation kernels. We also show that for sub-linear unbounded kernels, the mass conservation law holds. Finally, we show that for bounded kernels, this equation has a unique global solution that is continuously dependent on the initial data.

Strong fragmentation and coagulation with power-law rates

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when both fragmentation and coagulation are described by power-law rates which are commonly used in the engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L1 spaces.

On the uniqueness for coagulation and multiple fragmentation equation

In this article, the uniqueness of weak solutions to the continuous coagulation and multiple fragmentation equation is proved for alarge range of unbounded coagulation and multiple fragmentation kernels. The multiple fragmentation kernels may have a singularity at origin.This work generalizes the preceding ones, by including some physically relevant coagulation and fragmentation kernels which were not considered before.

Analytic fragmentation semigroups and continuous coagulation–fragmentation equations with unbounded rates

In this paper we show that continuous fragmentation operators are sectorial for a large range of physically relevant fragmentation rates and use this fact to prove classical solvability of the combined coagulation–fragmentation equation with unbounded coagulation kernels.

Weak solutions to the continuous coagulation equation with multiple fragmentation

The existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels, the fragmentation kernel having possibly a singularity at the origin. This result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel was assumed.

Global classical solutions of coagulation–fragmentation equations with unbounded coagulation rates

We study discrete fragmentation coagulation equations in spaces X p , p > 1 , consisting of distributions having the p th moments finite. We show that for sufficiently regular fragmentation laws the fragmentation semigroup is analytic in X p , and fully characterize the domain of its generator. This allows for explicit characterization of the domains of the fractional powers of the generator through real interpolation. Finally, we use the linear results to show the existence of global classical solutions to fragmentation coagulation equations for a class of unbounded coagulation kernels.

The central limit theorem for the Smoluchovski coagulation model

The general model of coagulation is considered. For basic classes of unbounded coagulation kernels the central limit theorem (CLT) is obtained for the fluctuations around the dynamic law of large numbers (LLN). A rather precise rate of convergence is given both for LLN and CLT.

Weak solutions to the Cauchy problem for the diffusive discrete coagulation–fragmentation system

The initial value problem for the discrete coagulation–fragmentation system with diffusion is studied. This is an infinite countable system of reaction–diffusion equations describing coagulation and fragmentation of discrete clusters moving by spatial diffusion in all space R d . The model considered in this work is a generalization of Smoluchowski's discrete coagulation equations. Existence of global-in-time weak solutions to the Cauchy problem is proved under natural assumptions on initial data for unbounded coagulation and fragmentation coefficients. This work extends existence theory for this system from the case of clusters distribution on bounded domain subject to no-flux boundary condition to the case of all R d.

Approximative solution of the coagulation–fragmentation equation by stochastic particle systems

This paper studies stochastic particle systems related to the coagulation fragmentation equation. For a certain class of unbounded coagulation kernels and fragmentation rates, relative compactness of the stochastic systems is established and weak accumulation points are characterized as solutions. These results imply a new existence theorem. Finally a simulation algorithm based on the particle systems is proposed

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

We prove global existence and uniqueness to the initial value problem for the coagulation fragmentation equation for an unbounded coagulation kernel with possible linear growth at infinity and a fragmentation kernel from a very large class of unbounded functions. We show that the solutions satisfy the mass conservation law.

Similarity solutions for homogeneous and nonhomogeneous aerosol balance equations

This paper is concerned with similarity solutions for homogeneous and nonhomogeneous aerosol balance equations. The systems describe the time evolution of an aerosol population in fluid media under the influences of volumetric sources and sinks, coagulation of particles, condensational growth, a vertical wind field, gravitational settling, and vertical diffusion. Applying transformation equations, the physical space is mapped into a one-dimensional similarity space. Direct and indirect methods are used to solve the governing equations for the number densities in the similarity space. The forms of the solutions vary with the changes in the system parameters. Hence, several relationships between the parameters and the similarity spectra and the integral moments of the spectra are obtained. For example, the heavier-tailed spectra correspond to the unbounded coagulation parameters and the lighter-tailed spectra correspond to the bounded kernels.