Transport equations with singular coefficients. Application to the gyro-kinetic models in plasma physics

The subject matter of this paper concerns the asymptotic regimes for transport equations with singular coefficients. Such models arise for example in plasma physics, when dealing with charged particles moving under the action of strong magnetic fields. These regimes are motivated by the magnetic confinement fusion. The stiffness of the coefficients comes from the multi-scale character of the problem. According to the different possible orderings between the typical physical scales (Larmor radius, Debye length, cyclotronic frequency, plasma frequency) we distinguish several regimes. From the mathematical point of view the analysis of such regimes reduces to stability properties for transport equations whose coefficients have different magnitude orders, depending on some small parameter. The main purpose is to derive limit models by letting the small parameter vanish. In the magnetic confinement context these asymptotics can be assimilated to homogenization procedures with respect to the fast cyclotronic movement of particles around the magnetic lines. We justify rigorously the convergence towards these limit models and we investigate the well-posedness of them. Transport equations with singular coefficients. Application to the gyro-kinetic models in plasma physics Résumé : Dans cet article nous étudions le comportement asymptotique des équations de transport dont les coefficients sont des différents ordres de grandeur. Parmi les applications visées on rappelle l'approximation centre-guide et le régime du rayon de Larmor fini dans la physique des plasmas magnétisés. L'objectif est d'identifier les modèles asymptotiques et de justifier rigoureusement ces limites.