Contributions to the Derivation and Well-posedness Theory of Kinetic Equations
University of Cambridge
Pure Mathematics and Mathematical Statistics
Doctor of Philosophy (PhD)
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Griffin-Pickering, M. (2020). Contributions to the Derivation and Well-posedness Theory of Kinetic Equations (Doctoral thesis). https://doi.org/10.17863/CAM.49109
This thesis is concerned with certain partial differential equations, of kinetic type, that are involved in the modelling of many-particle systems. The Vlasov-Poisson system is a model for a dilute plasma in an electrostatic regime. The classical version describes the electrons in the plasma. The first part of this thesis focuses on a variant known as the Vlasov-Poisson system with massless electrons (VPME), which instead describes the ions. Compared to the classical system, VPME includes an additional exponential nonlinearity, with the consequence that several results known for the classical system were not previously available for VPME. In particular, global well-posedness had not been proved. In this thesis, we prove that VPME has unique global-in-time solutions in two and three dimensions, for a general class of initial data matching results currently available for the classical system. The quasi-neutral limit is an important approximation of Vlasov equations in plasma physics, in which the Debye screening length of the plasma tends to zero; the formal limiting system is a kinetic Euler equation. For a rigorous passage to the limit, a restriction on the initial data is required. In this thesis, we prove the quasi-neutral limit from the VPME system to the kinetic isothermal Euler system, for a certain class of rough data. We then investigate the rigorous connection between these Vlasov equations and the associated particle systems. We derive VPME and the two kinetic Euler models associated respectively to the classical Vlasov-Poisson and VPME systems rigorously from systems of extended charges.
Partial differential equations, Kinetic equations, Plasma, Vlasov-Poisson system, Massless electrons regime, Kinetic Euler system, Quasi-neutral limit, Mean field derivation
This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.
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