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Interprétation probabiliste de l'équation de Landau.

Abstract : The aim of this PhD Thesis is to give a probabilistic interpretation of the Landau equation, also called the Fokker-Planck-Landau equation. This partial derivatives equation has been derived from the Boltzmann equation when the grazing collisions prevail in a gas. It describes the behaviour of the density of particles having the same velocity at the same time (we assume here that the density is spatially homogeneous). This equation has been studied from now with analytic tools, we give here a new approach. In the first part of this thesis, we prove the existence of a probability measure solution of the Landau equation for some gas, called 'moderately soft' potential gas, using some stochastic tools. Moreover, for some particular gas, we prove the uniqueness of the solution, and we deduce the existence of a density solution of the Landau equation thanks to the Malliavin calculus. The probabilistic approach allows generel initial data, which can be degenerated as Dirac measures. The second part of this thesis gives a probabilistic interpretation of the convergence of the Boltzmann equation to the Landau equation in the asymptotic of grazing collisions. We first extend the existence results, in a probabilistic sense, of the Boltzmann equation to the case of 'moderately soft' potential gas. Then we state the convergence of these solutions to a solution of the Landau equation when grazing collisions prevail. At last, using the Malliavin calculus, we obtain the pointwise convergnece of the densities for a Maxwell gas. The probabilistic approach gives a good understanding of the convergence of Boltzmann to Landau and allows to model it with a particle system. Some simulations are given in this thesis.
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Contributor : Hélène Guerin Connect in order to contact the contributor
Submitted on : Tuesday, December 3, 2002 - 4:15:22 PM
Last modification on : Thursday, October 21, 2021 - 3:16:03 PM
Long-term archiving on: : Friday, April 2, 2010 - 6:11:47 PM


  • HAL Id : tel-00002066, version 1


Hélène Guerin. Interprétation probabiliste de l'équation de Landau.. Mathématiques [math]. Université de Nanterre - Paris X, 2002. Français. ⟨tel-00002066⟩



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