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Censored Glauber dynamics for the mean field Ising model. (English) Zbl 1267.82110
The paper analyzes the evolution of dust particles in a rarefied atmosphere. This topic can be found in the context of safety studies for the ITER project of nuclear fusion. When the typical size of a dust particle becomes too large compared to that of a molecule of the gas, the corresponding numerical simulation becomes too expensive. Consequently, one needs to introduce an asymptotic model, in which the ratio of the gas-molecule mass to the dust-particle mass tends to zero. In order to obtain such model, the paper considers a system of two Boltzmann’s equations for the distribution function of the gas molecules and distribution function of the dust particles. The latter function depends not only on the position and velocity but also on the particle radius, which can vary in a certain interval. The main result of the work is that, when the proposed model is spatially homogeneous, the model is, in the aforementioned limit case, reduced to a system of two equations of the Vlasov-Boltzmann type. The system is simpler and admits more effective numerical treatments. The passage to the limit is proven. The proof includes a new version of Povzner’s inequality in which the vanishing mass ratio is taken into account.

82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D05 Statistical mechanical studies of gases
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35Q20 Boltzmann equations
35Q83 Vlasov equations
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