A Review of Some Results of Kac’s Program in Kinetic Theory

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Saberbaghi, Hamid Reza | 2019

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51871 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Safdari, Mohammad
Abstract:
This thesis is devoted to the study of suitable functional frameworks for proving propagation of chaos and mean-field limit of nonlinear Vlasov-type equations for indistinguishable particles. The Kac model is a classic example and the motivation of this functional structure. On the one hand, Kac considered a spatially homogeneous gas, and on the other hand, he reduced the problem to one-dimensional collisions and dropped the conservation of momentum assumption. Therefore, he obtained a simplified version of the Boltzmann equation, which is named after him, the Kac equation. The main results which we study are quantitative estimates
on the decay of fluctuations around the deterministic limit and the decay
of correlations between particles, as the number of particles goes to infinite. To this end, we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules)
Keywords:
Mean-Field Limit ; Boltzman Equation ; Kinetic Theory ; Vlasov Equation