Section: New Results

Mathematical analysis of kinetic models

Gyrokinetic and Finite Larmor radius approximations

Participants : Mihai Bostan, Céline Caldini, Emmanuel Frénod, Mathieu Lutz.

In a work in progress by E. Frénod and M. Lutz, the deduction of the Geometrical Gyro-Kinetic Approximation, which was originally obtained by Littlejohn in [75] , [76] , [77] using a physical approach which was mathematically formal, is done. The rigorous mathematical theory is built and explained in a form for providing it, especially, for analysts, applied mathematicians and computer scientists.

In the Note [16] , we present the derivation of the finite Larmor radius approximation, when collisions are taken into account. We concentrate on the Boltzmann relaxation operator whose study reduces to the gyroaverage computation of velocity convolutions, which are detailed here. We emphasize that the resulting gyroaverage collision kernel is not local in space anymore and that the standard physical properties (i.e., mass balance, entropy inequality) hold true only globally in space and velocity. This work is a first step in this direction and it will allow us to handle more realistic collisional mechanisms, like the Fokker-Planck or Fokker-Planck-Landau kernels.

The subject matter of the paper [34] 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.

Singularities of the stationary Vlasov–Poisson system in a polygon

Participant : Simon Labrunie.

In collaboration with Fahd Karami (Université Cadi Ayyad, Morocco) and Bruno Pinçon (Université de Lorraine and project-team CORIDA), we conducted in [43] a theoretical and numerical study of the so-called “point effect” in plasma physics. The model (stationary Vlasov–Poisson system with external potential) corresponds a fully ionised plasma considered on a time scale much smaller than that of ions, but much larger than that of electrons. It appears as the relevant non-linear generalisation of the electrostatic Poisson equation. This may be a first step toward a quasi-equilibrium model valid on a larger time scale, where the equilibrium description of the electrons would be coupled to a kinetic or fluid model for the ions. This approximation is classical in plasma physics. We proved a general existence result for our model in a bounded domain $\Omega \subset {ℝ}^N$, which is not assumed to be smooth. When $\Omega$ is a polygonal domain of ${ℝ}^2$, we described the singular behavior of the solution near a reentrant corner. In the important case of the Maxwell–Boltzmann distribution, we established a link between various asymptotics of the problem and the (suitably extended) theory of large solutions to nonlinear elliptic problems (for a review of this theory, see e.g. [50] ). This allowed us to determine the the dependence of the singularity coefficients on the parameters of the problem, such as the total mass of the distribution, or the boundary conditions of the potential. Numerical tests confirmed the theory.