Finite Larmor radius approximation for the Fokker-Planck-Landau equation

The subject matter of this paper concerns the derivation of the finite Larmor radius approximation, when collisions are taken into account. Several studies are performed, corresponding to different collision kernels. The main motivation consists in computing the gyroaverage of the Fokker-Planck-Landau operator, which plays a major role in plasma physics. We show that the new collision operator enjoys the usual physical properties; the averaged kernel balances the mass, momentum, kinetic energy and dissipates the entropy.Comment: 62 page

Impact of strong magnetic fields on collision mechanism for transport of charged particles

One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion is achieved by magnetic confinement i.e., the plasma is confined into a toroidal domain (tokamak) under the action of huge magnetic fields. Several models exist for describing the evolution of strongly magnetized plasmas, most of them by neglecting the collisions between particles. The subject matter of this paper is to investigate the effect of large magnetic fields with respect to a collision mechanism. We consider here linear collision Boltzmann operators and derive, by averaging with respect to the fast cyclotronic motion due to strong magnetic forces, their effective collision kernels

Mild solutions for the one dimensional Nordstrom-Vlasov system

International audienceThe Nordstrom-Vlasov system describes the evolution of a population of self-gravitating collisionless particles. We study the existence and uniqueness of mild solution for the Cauchy problem in one dimension. This approach does not require any derivative for the initial particle density. For any initial particle density uniformly bounded with respect to the space variable by some function with finite kinetic energy and any initial smooth data for the field equation we construct a global solution, preserving the total energy. Moreover the solution propagates with finite speed. The propagation speed coincides with the light speed

Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation

One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion requires the confinement of the plasma into a bounded domain and for this we appeal to the magnetic confinement. Several models exist for describing the evolution of strongly magnetized plasmas. The subject matter of this paper is to provide a rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large stationary inhomogeneous magnetic fields. The first order corrections are computed as well : electric cross field drift, magnetic gradient drift, magnetic curvature drift, etc. The mathematical analysis relies on average techniques and ergodicity

Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming

We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure

Homogenization of the 1D Vlasov-Maxwell equations

In this paper we investigate the homogenization of the one dimensional Vlasov-Maxwell system. We indicate the rate of convergence towards the limit solution. In the non relativistic case we compute explicitly the limit solution. The theoretical results are illustrated by some numerical simulations

The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime

We study here the finite Larmor radius regime for the Vlasov-Poisson equations with strong external magnetic field. One of the key points is to replace the particle distribution by the center distribution of the Larmor circles. The limit of these densities satisfies a transport equation, whose velocity is given by the gyro-average of the electric field

Transport equations with singular coefficients. Application to the gyro-kinetic models in plasma physics

The subject matter of this paper concerns the asymptotic regimes for transport equations with singular coefficients. Such models arise for example in plasma physics, when dealing with charged particles moving under the action of strong magnetic fields. These regimes are motivated by the magnetic confinement fusion. The stiffness of the coefficients comes from the multi-scale character of the problem. According to the different possible orderings between the typical physical scales (Larmor radius, Debye length, cyclotronic frequency, plasma frequency) we distinguish several regimes. From the mathematical point of view the analysis of such regimes reduces to stability properties for transport equations whose coefficients have different magnitude orders, depending on some small parameter. The main purpose is to derive limit models by letting the small parameter vanish. In the magnetic confinement context these asymptotics can be assimilated to homogenization procedures with respect to the fast cyclotronic movement of particles around the magnetic lines. We justify rigorously the convergence towards these limit models and we investigate the well-posedness of them

Periodic Solutions of the 1D Vlasov-Maxwell System with Boundary Conditions

We study the 1D Vlasov-Maxwell system with time periodic boundary conditions in its classical and relativistic form. For small data we prove existence of weak periodic solutions. It is necessary to impose non vanishing conditions for the incoming velocities in order to control the life-time of particles in the domain. In order to preserve the periodicity, another condition of vanishing the time average of the incoming current is imposed