Kinetic models for magnetized plasmas

Overview & Synopsis

This thesis is devoted to the study of kinetic models that describe collisionless plasmas subject to an external magnetic field. In a first part, we explore how in electrostatic plasmas subject to a constant external magnetic field certain waves are undamped, independently of the strength of the magnetic field. This phenomenon, known as the Bernstein–Landau paradox, highlights a discontinuity with the theory of unmagnetized plasmas, where the longitudinal space charge waves, which include the electric field, are damped. Usually, we study these physical properties by considering the linearized Vlasov–Poisson system. In this work, we manage to reinterpret the Bernstein–Landau paradox by reformulating this system as a Schrödinger type equation which leads us to consider the magnetized Vlasov–Ampère system. With this reformulation, we explain the Bernstein–Landau paradox with respect to the spectrum of a certain self-adjoint magnetized Vlasov–Ampère operator. Finally, we build semi-Lagrangian schemes to test the eigenfunctions of the magnetized Vlasov–Ampère operator. In a second part, we study certain mathematical properties of the magnetized Vlasov–Poisson system, focusing first on the propagation of velocity of moments for weak solutions. In the case of a constant magnetic field, we use a Eulerian approach in order to control the velocity moments directly. Because of the added magnetic field, we find singularities at multiples of the cyclotron period. In the case of a general magnetic field, we use a Lagrangian method, focusing on the characteristics of the system to obtain a priori estimates that imply propagation of moments. An original feature of this work is that, in both cases, we rely on an induction procedure involving the cyclotron period to prove propagation of velocity moments for all time. Finally, we find new uniqueness conditions for solutions to the magnetized Vlasov–Poisson system that include assumptions on the velocity moments.