Mihai Bostan

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.

Gyrokinetic Vlasov Equation in Three Dimensional Setting. Second Order Approximation

One of the main applications in plasma physics concerns the energy production through thermonuclear 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.

Existence and Uniqueness of the Mild Solution for the 1D Vlasov--Poisson Initial-Boundary Value Problem

We prove the existence and uniqueness of the mild solution for the 1D Vlasov--Poisson system with initial-boundary conditions by using iterated approximations. The same arguments yield existence and uniqueness for the free space or space periodic system. The major difficulty is the treatment of the boundary conditions. The main idea consists of splitting the velocities range by introducing critical velocities corresponding to each boundary. One of the crucial points is to estimate the critical velocity change in term of relative field. A result concerning the continuity of the mild solution upon the initial-boundary conditions is presented as well.

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 a Vicsek-like model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion reducing the Cucker–Smale model with diffusion to the non-normalized Vicsek model as in Ref. 29. 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.

PERMANENT REGIMES FOR THE 1D VLASOV-POISSON SYSTEM WITH BOUNDARY CONDITIONS ∗

We prove the existence of weak solutions for the Vlasov-Poisson problem problem with time periodic boundary conditions in one dimension. We consider boundary data with finite charge and current. This analysis is based upon the mild formulation for the regularized Vlasov-Poisson problem equations.

Periodic solutions of the 1D Vlasov–Maxwell system with boundary conditions

We study the ID Vlasov-Maxwell system with time-periodic boundary conditions in its classical and relativistic form. We are mainly concerned with existence of periodic weak solutions. We shall begin with the definitions of weak and mild solutions in the periodic case. The main mathematical difficulty in dealing with the Vlasov-Maxwell system consist of establishing L∞ estimates for the charge and current densities. In order to obtain this kind of estimates, we impose non-vanishing conditions for the incoming velocities, which assure a finite lifetime of all particles in the computational domain ]0, L[. The definition of the mild solution requires Lipschitz regularity for the electro-magnetic field. It would be enough to have a generalized flow but the result of DiPerna Lions (Invent. Math. 1989; 98: 511-547) does not hold for our problems because of boundary conditions. Thus, in the first time, the Vlasov equation has to be regularized. This procedure leads to the study of a sequence of approximate solutions. In the same time, an absorption term is introduced in the Vlasov equation, which guarantees the uniqueness of the mild solution of the regularized problem. In order to preserve the periodicity of the solution, a time-averaging vanishing condition of the incoming current is imposed: ∫ T 0 dt ∫ r ∫C x go(t, r x , v x )dv+∫ t 0 dt∫ rx ∫v x g z (t,c x ,v y )dv = 0 (1) where g O . g L are the incoming distributions f(t, 0, v x , v y ) = g O (t, v x , v y ), t ∈ R t , v x > 0, v y ∈ R v (2) f(t, L, v x , v y ) = g L (t, v x , v y ), t ∈ R t , v x < 0, v y ∈ R v (3) The existence proof uses the Schauder fixed point theorem and also the velocity averaging lemma of DiPerna and Lions (Comm. Pure Appl. Math. 1989; XVII: 729-757). In the last section we treat the relativistic case.

Mild solutions for the relativistic Vlasov-Maxwell system for laser-plasma interaction

We study a reduced 1D Vlasov-Maxwell system which describes the laser-plasma interaction. The unknowns of this system are the distribution function of charged particles, satisfying a Vlasov equation, the electrostatic field, verifying a Poisson equation and a vector potential term, solving a nonlinear wave equation. The nonlinearity in the wave equation is due to the coupling with the Vlasov equation through the charge density. We prove here the existence and uniqueness of the mild solution (i.e., solution by characteristics) in the relativistic case by using the iteration method.

Transport of Charged Particles Under Fast Oscillating Magnetic Fields

The energy production through thermonuclear fusion requires the confinement of the plasma within a bounded domain. In most cases, such configurations are obtained by using strong magnetic fields. Several models exist for describing the evolution of a strongly magnetized plasma, e.g., guiding-center approximation and the finite Larmor radius regime. The topic of this paper concerns a different approach leading to plasma confinement. More precisely, we are interested in mathematical models with fast oscillating magnetic fields. We provide rigorous derivations for this kind of model and analyze their properties.

Finite Larmor radius approximation for collisional magnetic confinement. Part II: the Fokker-Planck-Landau equation

This paper is devoted to the finite Larmor radius approximation of the Fokker-Planck-Landau equation, which plays a major role in plasma physics. We obtain a completely explicit form for the gyroaverage of the Fokker-Planck-Landau kernel, accounting for diffusion and convolution with respect to both velocity and (perpendicular) position coordinates. 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, globally in velocity and perpendicular position coordinates.

Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation

The subject matter of this paper concerns the derivation of the finite Lar-mor radius approximation, when collisions are taken into account. Several studies are performed, corresponding to different collision kernels : the relaxation and the Fokker-Planck operators. Gyroaveraging the relaxation operator leads to a position-velocity integral operator, whereas gyroaveraging the linear Fokker-Planck operator leads to diffusion in velocity but also with respect to the perpendicular position coordinates.