## Section: New Results

### Mathematical analysis of kinetic models

Participants : Aurore Back, Nicolas Besse, Emmanuel Frénod, Mathieu Lutz.

#### Asymptotic analysis of gyrokinetic models

Proceeding [52] presents the method that allows us to get the gyrokinetic Approximation of the Dynamical System satisfied by the trajectory of a particle submitted to a Strong Magnetic Field. The goal of the method is to build a change of coordinates in order to make the fast dynamics of two components of the trajectory to disappear. This change of coordinates is based on a Darboux mathematical Algorithm and on a Lie Transform. It is the first work of a forthcoming series of papers which goal is to make the Geometrical gyrokinetic Approximation a mathematically affordable theory. Review paper [51] presents the results of Two-Scale Convergence Theory and an application to Homogenization of linear Singularly Perturbed Hyperbolic Partial Differential Equations. It consists in the theoretical basis of the Two-Scale Numerical Methods.

#### Two-scale convergence with differential form

In the framework of the thesis of Aurore Back [11] , we developed at two-scale convergence theory using the tools of exterior calculus and differential forms. A geometric formulation of the Vlasov-Maxwell equations was introduced and some geometric conservation properties were proved.

#### Analysis of multi-water-bag models

In the case of toroidal geometry, thanks to the strong anisotropy between parallel (to the magnetic field) and transverse direction, we could perform an asymptotic analysis of the eigenvalue problem for the integro-differential gyrowaterbag operator with two independent dimensions, the third being represented by Fourier toroidal modes. This analysis enabled us to reduce the two-dimensional integro-differential operator to a series of one-dimensional integro-differential operators the solution of which enables to obtain as well the poloidal as the radial envelope of the global eigenmodes as well as their local frequency. Note that the terms of the series can be computed numerically independently from each other as the differential variables decouple, which leads to an embarassingly parallel algorithm. The global dispersion relation for the global eigenfrequency appears as an integral quantification relation involving the local frequency due to the property of conservation of the action. On the other hand the mathematical analysis, in particular the spectral properties, of the obtained operators has been performed. Several mathematical results on the well-posedness of gyrowaterbag models have been obtained as an exact geometric reduction of the Vlasov equation, the notion of solution differing depending on the nature of the problem being considered [14] , [15] , [16] .