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.

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.