Landau damping in relativistic plasmas

Abstract

We examine the phenomenon of Landau Damping in relativistic plasmas via a study of the relativistic Vlasov-Poisson system (rVP) on the torus for initial data sufficiently close to a spatially uniform steady state. We find that if the steady state is regular enough (essentially in a Gevrey class of degree in a specified range) and that the deviation of the initial data from this steady state is small enough in a certain norm, the evolution of the system is such that its spatial density approaches a uniform constant value sub-exponentially fast (i.e. like exp(−C|t| ν ) for ν ∈ (0, 1)). We take as a priori assumptions that solutions launched by such initial data exist for all times (by no means guaranteed with rVP, but reasonable since we are close to a spatially uniform state) and that the various norms in question are continuous in time (which should be a consequence of an abstract version of the Cauchy-Kovalevskaya Theorem). In addition, we must assume a kind of "reverse Poincaré inequality" on the Fourier transform of the solution. In spirit, this assumption amounts to the requirement that there exists 0 < κ < 1 so that the mass in the annulus κ ≤ |v| < 1 for the solution launched by the initial data is uniformly small for all t. Typical velocity bounds for solutions to rVP launched by small initial data (at least on R 6 ) imply this bound. We note that none of our results require spherical symmetry (which is a crucial assumption for most results known about rVP).

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

“…[49,16,51,40]). In [40], Landau's linearized analysis was confirmed, mathematically rigorously, to be accurate uniformly in time for the nonlinear Vlasov equations for initial data which was analytic or Gevrey class of sufficiently low index (see also earlier work of [17,31] and the more recent [12,58]). Landau damping is connected to the mixing in phase-space due to the transport operator, and can be considered a variant of velocity averaging (see e.g.…”

“…The gain of 1/2 derivative in the moment estimates without the time growth seen in (2.20a) is the same "free" gains in regularity seen in velocity averaging lemmas [27,26]. This and the hypoelliptic smoothing in x is crucial to making the proof work, as this is what allows to use σ + 1/2 in (2.21a) and σ in (2.22a) (note that a larger gap was used in [12,58,11,6], but this is not be possible due to the nonlinear collisions).…”

“…As has been done in several previous works on Landau damping and mixing in fluid mechanics [12,10,13,58,11,7,8,9], we apply a bootstrap argument. Bootstrap arguments typically require a local well-posedness and propagation of regularity result.…”

Landau Damping: Paraproducts and Gevrey Regularity

“…There is a pressing need for more flexible and powerful analytical tools as the 'basic' situation studied in [67] is only a tiny part of what remains to be understood and we believe that our work is a key step to addressing many interesting and qualitatively different questions ahead. Indeed, the tools developed in this paper have already been adapted to prove a nonlinear Landau damping result in relativistic plasmas [87]. Other examples where our methods may prove useful are damping in un-confined geometries by separating scales in frequencies, geometries and cyclotron effects imposed by external magnetic fields or more subtle self-created magnetic effects, damping in weakly collisional plasmas and finally the stability of self-created gravitational geometries [49,66] or nonlinear BGK waves [13].…”

Effects of ion motion on linear Landau damping

“…The linear Landau damping has been well demonstrated in Vlasov-Poisson simulation by many researchers in the past decades [2][3][4][5][6][7][8][9][10][11][12] . This collisionless damping has become very important in a wide context (see reviews by Ryutov [13] and Ivanov [14] ) and some recent works show that progress in understanding and modeling this problem is still being made [15][16][17][18][19][20][21][22] . In these studies, the ions are usually assumed to be immobile as in the original theory [1] .…”