Transport Equation In Phase Space Research Articles

A commuting-vector-field approach to some dispersive estimates

We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain $L^1$---$L^\infty$ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space-time weighted energy estimates. Combined with a simple version of Sobolev's inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrodinger equations we can recover sharp pointwise decay; in the Schrodinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on $L^2$---$L^\infty$ decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.

Lie-transform theory of transport in plasma turbulence

From the Vlasov equation, a phase-space transport equation is derived by using the Lie-transform approach, and its connection with the quasilinear transport, nonlinear stochastic transport, and fractional transport equations are discussed. The phase-space transport equation indicates a particle redistribution in the real space induced by the inhomogeneity in the energy space distribution and by the correlation between the change of position and the change of energy.

Solar wind bulk velocity fluctuations acting as velocity space diffusion on comoving ions

From most in-situ plasma observations made in the outer heliosphere it became evident that above the injection border of pick-up ions (� 1 keV), an extended suprathermal ion tail is found which in most cases can be fitted by a power law with velocity power indices of (−6) ≤ γv ≤ (−4). As has been shown by theory such energetic ion tails cannot be explained by Fermi-2 type velocity diffusion, since in the outer heliosphere both Alfvenic and magnetoacoustic turbulences become too weak. Here we come to a new solution of this unsolved problem by studying the action of solar wind bulk velocity fluctuations on ions co-moving with the wind. As we show the passage of such fluctuations results in energization of each individual ion and systematic evolution of the ion distribution function towards suprathermal tails. From the basic knowledge that we can obtain on this process we can calculate the velocity divergence of the ion phasespace flow and thus can derive a velocity diffusion operator. As we can show here this operator leads to a velocity diffusion coefficient proportional to the square of the ion velocity and, when employed in the phasespace transport equation, together with terms for convective changes, cooling processes and pick-up ion injection, interestingly enough, permits to find solutions for suprathermal power law tails with power indices of γv �− 5 as very often observed.

A Discussion on the early days of ionospheric research and the theory of electric and magnetic waves in the ionosphere and magnetosphere - Waves in a hot plasma

In studying wave propagation in a hot plasma, we treat the dynamics of the medium by kinetic theory rather than by continuum mechanics. The theory thus combines Maxwell’s equations with a transport equation in phase space (the Vlasov equation). An outline of the required procedure will be given. Some of the results are in close agreement with those of the fluid treatment provided the specific heat ratio is appropriately chosen. This is generally the case if the phase speed of the waves well exceeds the thermal speed of the electrons and, for a magnetized plasma, the frequency is not close to a harmonic of the cyclotron frequency. New phenomena are found if there are particles whose unperturbed motion is in resonance with the wave field. In the unmagnetized case this results in Landau damping or in instabilities, the latter being analogous to the mechanism of the laser. In the magnetized case there are, in addition, completely new modes of propagation for waves travelling approximately normal to the applied field. Many of these phenomena find direct application in ionospheric phenomena and diagnostics.