A sharp stability criterion for the Vlasov–Maxwell system

We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov–Maxwell system that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically with the particle energy, we obtained a linear stability criterion in our previous paper [24]. Here we prove that this criterion is sharp; that is, there would otherwise be an exponentially growing solution to the linearized system. We also treat the considerably simpler periodic D case. The new formulation introduced here is applicable as well to the non-relativistic case, to other symmetries, and to general equilibria.