Two-scale expansion of a singularly perturbed convection equation

In magnetic fusion, a plasma is constrained by a very large magnetic field, which introduces a new time scale, namely the period of rotation of the particles around the magnetic field lines. This new time scale is very restrictive for numerical simulation, which makes it important to find approximate models of the Vlasov-Poisson equation where it is removed. The gyrokinetic models aim at exactly this. Such models have been derived in the physics literature for several decades now, but only in the last few years there have been rigorous mathematical derivations. Those have only addressed the limit when the magnetic field becomes infinite. We consider here the Vlasov equation in different physical regimes for which small parameters are identified, and cast the obtained dimensionless equations into the abstract framework of a singularly perturbed convection equation. In this framework we derive an asymptotic expansion with respect to the small parameter of its solution, and characterize the terms of the expansion. The proofs make use of Allaire's two-scale convergence.