## Section: New Results

### Mathematical analysis of kinetic models

Participants : N. Besse, M. Bostan.

Contribution [13] concerns a one-dimensional version of the Vlasov equation dubbed the Vlasov−Dirac−Benney equation (in short V−D−B) where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full nonlinear problem and equations of fluids. In particular the connection with the so-called Benney equation leads to new stability results. Eventually the V−D−B appears to be at the cross road of several problems of mathematical physics which have as far as stability is concerned very similar properties.

The subject matter of paper (M. Bostan, Strongly anisotropic diffusion problems;
asymptotic analysis, in *J. Differential Equations*, vol. 256 pp. 1043-1092 (2013)) concerns
anisotropic diffusion equations: we consider heat equations whose diffusion matrices have
disparate eigenvalues. We determine first and second order approximations, we study their
well-posedness and then, we establish convergence results. The analysis relies on averaging
techniques, which have been used previously for studying transport equations whose advection
fields have disparate components.

In (M. Bostan, J.-A. Carrillo, Asymptotic fixed speed reduced dynamics for kinetic
equations in swarming, *Math. Models Methods Appl. Sci.*, Vol.23, No. 13 pp. 2353-2393 (2013))
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.

#### Gyrokinetic approximation

Participants : E. Frénod, M. Lutz.

Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build in [42] a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.