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# Contribution to the probabilistic and numerical study of homogeneous Coagulation - Fragmentation equations

Abstract : This thesis is devoted to the study of systems of particles undergoing successive coagulations and fragmentations. In the deterministic case, we deal with measure-valued solutions of the coagulation - multifragmentation equation. We also study, on the other hand, its stochastic counterpart: coalescence - multifragmentation Markov processes. In Chapter 1 we only take into account coagulation phenomena. We consider the Smoluchowski equation (which is deterministic) and the Marcus-Lushnikov process (the stochastic version) which can be seen as an approximation of the Smoluchowski equation. We derive a satisfying rate of convergence of the Marcus-Lushnikov process toward the solution to Smoluchowski's coagulation equation. \noindent The result applies to a class of homogeneous-like coagulation kernels with homogeneity degree ranging in $(-\infty,1]$. It relies on the use of the Wasserstein-type distance $d_{\lambda}$, which has shown to be particularly well-adapted to coalescence phenomena. It was introduced and used in preceding works. In Chapter 2 we perform some simulations in order to confirm numerically the rate of convergence deduced in Chapter 1 for the kernels studied in this chapter. Finally, in Chapter 3 we add a fragmentation phenomena and consider a coagulation multiple-fragmentation equation, which describes the concentration $c_t(x)$ of particles of mass $x \in (0,+\infty)$ at the instant $t \geq 0$. We study the existence and uniqueness of measured-valued solutions to this equation for homogeneous-like kernels of homogeneity parameter $\lambda \in (0,1]$ and bounded fragmentation kernels, although a non-finite measure giving the mass distribution of fragments and a possibly infinite number of fragments are considered. We also study a stochastic counterpart of this equation where a similar result is shown. We prove existence of such a process for a larger set of fragmentation kernels, namely we relax the boundedness hypothesis. In both cases, the initial state has a finite $\lambda$-moment.
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Cited literature [68 references]

https://tel.archives-ouvertes.fr/tel-00833475
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Submitted on : Wednesday, June 12, 2013 - 6:21:01 PM
Last modification on : Saturday, January 15, 2022 - 4:01:35 AM
Long-term archiving on: : Tuesday, April 4, 2017 - 9:39:02 PM

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• HAL Id : tel-00833475, version 1

### Citation

Eduardo Cepeda. Contribution to the probabilistic and numerical study of homogeneous Coagulation - Fragmentation equations. Probability [math.PR]. Université Paris-Est, 2013. English. ⟨NNT : ⟩. ⟨tel-00833475⟩

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