Long-Time Behavior of the Electric Potential and Stability in the Linearized Vlasov Theory

Abstract

In this paper we study in a mathematically rigorous manner how the electric potential, produced by small electronic charge density oscillations of definite wavenumber vector k in a plasma, behaves in the long-time limit and the connection between this behavior and the stability of a given steady, spatially uniform, distribution of the plasma electrons. Our work is based on the linearized Vlasov equation and on the associated Poisson equation. We formulate a very general initial-value problem concerning this system of equations, writing the above electric potential at a given position vector r and time t as φ(t)eik·r multiplied by a suitable constant, where φ(t) is independent of r. We establish the existence and uniqueness of solution of this problem by exploiting the fact that, in the linear theory, φ(t) obeys an inhomogeneous Volterra integral equation of convolution type, which is rigorously derived here. A detailed study of the asymptotic properties of the solutions of this equation for t → ∞ is made, including the establishment of necessary and sufficient conditions on the initial perturbations (perturbations of the steady electron distribution function at t = 0) for φ(t) to be of negative exponential order as t → ∞. As a byproduct of this asymptotic investigation, we give a precise discussion of the Landau damping of long wavelength plasma oscillations in an initially Maxwellian plasma, concluding that in this case φ(t) exhibits such damping for a broad range of initial perturbations and that the damping decrement is essentially that first computed by Landau. We introduce criteria of stability and instability based on the boundedness and unboundedness, respectively, in the limit t → ∞ of certain nonnegative quantities Wp(t), which are defined as suitable norms of the perturbed electron distribution function. New sufficient conditions for stability and instability are proved for extensive classes of initial distributions and initial perturbations. These results are compared with conclusions on stability and instability reached by Backus.

On Landau damping

“…The Landau damping has been since long understood at the linearized level [3,8,10], but the study of the full (nonlinear) equation poses important conceptual and technical problems. As a consequence, up to now the only existing results were proving existence of some damped solutions with prescribed behavior as t → ±∞ [2,5].…”

Integral equation for electrostatic waves generated by a point source in a spatially homogeneous magnetized plasma

“…Integral equation formulations of the Vlasov-Poisson system in unmagnetized plasmas are well known [44][45][46][47][48][49][50][51][52]. However, to the authors' knowledge, the integral equation formulation for a magnetized plasma given by (8) and (9) is new.…”

The work of Cédric Villani

“…The Landau damping for the linearized Vlasov-Poisson was already understood in the sixties by the work of A. Saenz [42]; for the quasi-linear case only nonrigorous results were available. On the other hand, it was pointed out by G. Backus [4] that the linear approximation is not expected to be valid for the full nonlinear equation in the large time regime.…”