Cyclotron Damping along an Uniform Magnetic Field

We prove cyclotron damping for the collisionless Vlasov-Maxwell equations on ${\mathbb{T}}_x^3\times {\mathbb{R}}_v^3$ under the assumptions that the electric induction is zero and $(\mathbf{P}\mathbf{S}\mathbf{C})$ holds. It is a crucial step to solve the stability problem of the Vlasov-Maxwell equations. Our proof is based on a new dynamical system of the plasma particles, originating from Faraday Law of Electromagnetic induction and Lenz’s Law. On the basis of it, we use the improved Newton iteration scheme to show the damping mechanism.