Instabilities of the relativistic Vlasov--Maxwell system on unbounded domains

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 one-half 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 problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations. However, it is shown that under an additional compactness assumption the spectrum does vary continuously, and a family of symmetric finite-dimensional approximations is constructed. An important feature of these approximations is that they are valid for the entire family uniformly. An application of this result to the study of plasma instabilities is illustrated.Comment: 22 pages, final version to appear in Commun. Math. Phy

Continuous Family of Equilibria of the 3D Axisymmetric Relativistic Vlasov-Maxwell System

We consider the relativistic Vlasov-Maxwell system (RVM) on a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, assuming axisymmetry in the problem. We construct continuous global parametric solution sets for the time-independent RVM. The solutions in these sets have arbitrarily large electromagnetic field and the particle density functions have the form $f^\pm = \mu^\pm (e^\pm (x, v), p^\pm (x, v))$, where $e^\pm$ and $p^\pm$ are the particle energy and angular momentum, respectively. In particular, for a certain class of examples, we show that the spectral stability changes as the parameter varies from $0$ to $\infty$

Instability of Nonmonotone Magnetic Equilibria of the Relativistic Vlasov-Maxwell System

We consider the question of linear instability of an equilibrium of the Relativistic Vlasov-Maxwell (RVM) System that has a strong magnetic field. Standard instability results deal with systems where there are fewer particles with higher energies. In this paper we extend those results to the class of equilibria for which the number of particles does not depend monotonically on the energy. Without the standard sign assumptions, the analysis becomes significantly more involved.Comment: 46 page

Symmetry Results for Finite-Temperature, Relativistic Thomas-Fermi Equations

In the semi-classical limit, the quantum mechanics of a stationary beam of counter-streaming relativistic electrons and ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature / low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder.Comment: Final version. To appear in Commun. Math. Phys. (LaTeX, 26 pages