# A numerical and analytical study of kinetic models for particle-wave interaction in plasmas

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This dissertation presents a study of particle-wave interaction in plasmas. It focuses on a kinetic model called quasilinear theory, which is a reduction of VlasovMaxwell (or Vlasov-Poisson) system in the weak turbulence regime. The quantized waves in plasmas, known as plasmons, are absorbed or emitted by charged particles. Meanwhile, the particles change their states due to such emission/absorption process, therefore resulting in a nonlinear kinetic system for the pdf (probability density function) of particles and plasmons. The research presented here unfolds in two main topics: structure-preserving numerical solvers, and solvability of the kinetic model. On the first topic, we are interested in numerical simulation of non-uniform magnetized plasmas, which involves two processes: particle-wave interaction and wave propagation (plasmon advection). For particle-wave interaction in homogeneous magnetized plasmas, we propose a finite element scheme that preserves all the conservation laws. Firstly, an unconditionally conservative weak form is constructed. By “unconditional” we mean that conservation is independent of the transition probabilities. Then we design a discretization that preserves such unconditional conservation property, and discuss the conditions for positivity and stability. We present numerical examples with a “bump on tail” initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion of the particles. We generalize the strategy to obtain a conservative DG (discontinuous Galerkin) scheme. The evolution of plasmon pdf is governed by a Liouville equation with additional reaction term caused by particle-wave interaction, where the dominant Poisson bracket term necessitates trajectorial average. Hence, we propose a Galerkin approach for trajectorial average in dynamical systems. The weak form of averaged equation is derived, and the concept of trajectory bundle is introduced. To compute and store the trajectory bundles, we propose a novel algorithm, named connection-proportion algorithm, which transforms a continuous topological problem into a discrete graph theory problem. The conservative DG scheme, combined with our trajectorial average method, renders a structure-preserving solver for particle-wave interaction in non-uniform magnetized plasmas. We demonstrate that discrete weak form with/without average differs only in the choice of test/trial spaces. The complexity of each procedure is analyzed. Finally, a numerical example for a non-uniform magnetized plasma in an infinitely long symmetric cylinder is presented. It is verified that the connectionproportion algorithm allows to distinguish different trajectory bundles, and the proposed DG scheme rigorously preserves all the conservation laws. On the second topic, the existence of global weak solution to quasilinear theory for electrostatic plasmas is proved. In the one-dimensional case, both the particle pdf and the plasmon pdf can be expressed with the same auxiliary function. The auxiliary function itself, is the solution of a porous medium equation with nonlinear source terms, defined on an unbounded domain. The solvability is then proved in two steps: Firstly, the equation on finite cut-off domain with Dirichlet’s boundary condition is solved. Next, the solution, extended by zero outside the cut-off domain, turns out to be a solution to the same equation on the unbounded domain.