Inventiones mathematicae

, Volume 173, Issue 3, pp 497–546 | Cite as

A sharp stability criterion for the Vlasov–Maxwell system

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Abstract

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 \(1\frac{1}{2}\)D case. The new formulation introduced here is applicable as well to the non-relativistic case, to other symmetries, and to general equilibria.

Keywords

Stability Criterion Matrix Operator Negative Eigenvalue Essential Spectrum Linear Instability 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of MissouriColumbiaUSA
  2. 2.Department of Mathematics and Lefschetz Center for Dynamical SystemsBrown UniversityProvidenceUSA

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