Matemática e Estatística | Capítulos/artigos em livros internacionais / Book chapters/papers in international books
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Browsing Matemática e Estatística | Capítulos/artigos em livros internacionais / Book chapters/papers in international books by Author "Costa, Fernando Pestana da"
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- Convergence to equilibria of solutions to the coagulation-fragmentation equationsPublication . Costa, Fernando Pestana daThe Coagulation-Fragmentation equations are a model for the dynamics of cluster growth and consist of a countable number of non-locally coupled ordinary differential equations, modelling the concentration of the various clusters. The framework for the study of these equations is to see them as a nonlinear ODE in an appropriate infinite dimensional Banach space. A number of results about existence, uniqueness, density conservation, and asymptotic behaviour of solutions have been obtained in recent years. In the present paper we review some of the more recent results on the asymptotic behaviour of solutions as time tends to infinity. We pay special attention to the different behaviour of solutions in the weak-star and in the strong topologies of the phase space. This distinction can be interpreted in terms of a dynamic phase transition in the physical system modelled by the equations. It is shown that a balance between the relative strength of coagulation and fragmentation is crucial for that distinct behaviour to take place.
- Mathematical aspects of coagulation-fragmentation equationsPublication . Costa, Fernando Pestana daWe 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.