Local and global strong solutions to continuous coagulation–fragmentation equations with diffusion

Abstract. We present models of the dynamics of phytoplankton aggregates. We start with an individual-based model in which aggregates can grow, divide, joint and move randomly. Passing to infinity with the number of individuals, we obtain a model which describes the space-size distribution of aggregates. The density distribution function satisfies a non-linear transport equation, which contains terms responsible for the growth of phytoplankton aggregates, their fragmentation, coagulation, and diffusion.

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“…[1,7,9,36,46]). Other results concerning this subject can be found in [4,20,30] and the papers quoted there.…”

Section: Introductionmentioning

confidence: 84%

Phytoplankton Dynamics: from the Behavior of Cells to a Transport Equation

Rudnicki

Wieczorek

2006

Math. Model. Nat. Phenom.

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“…The discrete Smoluchowski system is the system of a countable number of ordinary differential equations (1) with the right-hand side given by (4). In several cases it is preferable to consider the version of Smoluchowski equations for which the cluster masses can be any positive real number.…”

Section: Smoluchowski's Coagulation Equationsmentioning

confidence: 99%

“…Each particle have an integer mass m i 2 N C and is animated with a Brownian motion with diffusion constant 2d.m i /. When two particles of masses m i and m j are at a distance from one another equal to kx i x j k D " > 0 they can coagulate to made a particle of mass m i C m j , randomly located in any of the positions x i or x j , with probability dependent of the masses of the original particles 4 ; Due to this coagulation process the number of particles in the system diminishes with time and so the indexing set is time dependent , I q.t/ I. One assumes that the dynamics of this particle system…”

Section: Other Problems About Coagulation and Fragmentation Models: Rmentioning

confidence: 99%

“…We start by briefly presenting the most relevant spaces needed afterwards. 4 In other versions of this coagulation process of stochastic particles it is assumed the resulting particle is located at the centre of mass ximiCxjmj miCmj [176], in still others coagulation can happen within a whole interval of distances between the particles and not only at the distance " [104] …”

Section: Existence and Uniqueness Of Solutions To Discrete Coagulatiomentioning

confidence: 99%

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Mathematical Aspects of Coagulation-Fragmentation Equations

Costa

2015

CIM Series in Mathematical Sciences

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We give an overview of the mathematical literature on the coagulationlike equations, from an analytic deterministic perspective. In Sect. 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 Sect. 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 function 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 Sects. 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.

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“…Note yet that the methods developed in [2] and [8,9], should be sufficiently robust to deal with more general dynamics than (1.2) on bounded smooth domains as they essentially rely on some compactness property of the operator.…”

Section: Introductionmentioning

confidence: 99%