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Contributions to the Derivation and Well-posedness Theory of Kinetic Equations
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.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
Zero-electron-mass and quasi-neutral limits for bipolar Euler-Poisson systems
We consider a set of bipolar Euler-Poisson equations and study two asymptotic
limiting processes. The first is the zero-electron-mass limit, which formally
results in a non-linear adiabatic electron system. In a second step, we analyse
the combined zero-electron-mass and quasi-neutral limits, which together lead
to the compressible Euler equations. Using the relative energy method, we
rigorously justify these limiting processes for weak solutions of the
two-species Euler-Poisson equations that dissipate energy, as well as for
strong solutions of the limit systems that are bounded away from vacuum. This
justification is valid in the regime of initial data for which strong solutions
exist. To deal with the electric potential, in the first case we use elliptic
theory, whereas in the second case we employ the theory of Riesz potentials and
properties of the Neumann function
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB)
model of plasma physics. This model consists of the pressureless gas dynamics
equations coupled with the Poisson equation and where the Boltzmann relation
relates the potential to the electron density. If the quasi-neutral assumption
is made, the Poisson equation is replaced by the constraint of zero local
charge and the model reduces to the Isothermal Compressible Euler (ICE) model.
We compare a numerical strategy based on the EPB model to a strategy using a
reformulation (called REPB formulation). The REPB scheme captures the
quasi-neutral limit more accurately
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