Instabilities of the Relativistic Vlasov--Maxwell System on Unbounded Domains

Abstract

The relativistic Vlasov-Maxwell system describes the evolution of a collisionless plasma. The problem of linear instability of this system is considered in two physical settings: the so-called "one and onehalf" dimensional case, and the three dimensional case with cylindrical symmetry. Sufficient conditions for instability are obtained in terms of the spectral properties of certain Schrödinger operators that act on the spatial variable alone (and not in full phase space). An important aspect of these conditions is that they do not require any boundedness assumptions on the domains, nor do they require monotonicity of the equilibrium.

Approximations of Strongly Continuous Families of Unbounded Self-Adjoint Operators

“…The two families A λ and K λ satisfy: (i) Sectoriality: The families {A λ ± } λ∈D are holomorphic of type (B). 1 That is, they are families of sectorial operators and the associated sesquilinear forms a λ ± are holomorphic of type (a): all {a λ ± } λ∈D are sectorial and closed, with domains that are independent of λ and dense in H ± , 2 and…”

“…Finally, in Sect. 6 we give a brief description of an application of these results related to plasma instabilities, which is the subject of [2], where one can find the full details.…”

“…We are interested in the following problem: Problem 1. Consider the family of self-adjoint unbounded operators This problem is motivated by Maxwell's equations, which in the Lorenz gauge may be written as the following elliptic system for the electromagnetic potentials φ and A (after taking a Laplace transform in time): 2) where ρ and j are the charge and current densities, respectively. The specific problem we have in mind, treated separately in [2], is that of instabilities of the relativistic VlasovMaxwell system describing the evolution of collisionless plasmas and it is outlined in Sect.…”

“…A significant obstacle has been the existence of an essential spectrum extending to both ±∞. Let us briefly outline the problem, which is treated in detail in [2]. Plasmas are typically modelled by the relativistic Vlasov-Maxwell system: Letting f = f (t, x, v) be a probability density function measuring the density of electrons that at time t ≥ 0 are located at the point x ∈ R d , have momentum v ∈ R d and velocitŷ v = v/ 1 + |v| 2 , the (relativistic) Vlasov equation…”

“…Theorem 3 is then invoked to show that one can also take the two limits n → ∞ and ε → 0 to conclude that M λ ∞ has a nontrivial kernel. We refer to [2] for full details.…”