VLASOV EQUATION 

The Vlasov equation models the transport of particles which move in the vacuum under the action of an electric or/and a magnetic field. Such an equation is studied in semiconductor physics, in plasma physics and in astrophysics. The study of the Vlasov equation has developed in many directions: asymptotic analysis, passage from microscopic models to macroscopic models, existence, uniqueness and approximation of solutions.

Interesting mathematical problems arise in the study of the Vlasov equation in a bounded domain. In fact, the various types of boundary conditions which describe the different physical behaviour of the particles at the boundaries need different mathematical approaches to be studied.

In a recent research (by S.Mancini and S.Totaro), the one-dimensional Vlasov equation, with constant electric field and null magnetic field, coupled with boundary conditions which describe an incoming flux of particles in the considered region, has been studied. Existence, uniqueness and positivity of a solution of this problem has been proved by means of theories of elliptic operators, of semigroups of operators and of affine operators. Also a possible approximation of the solutions is given. This problem has been studied also in a more general case. The boundary conditions are described through a linear bounded positive operator acting between incoming and outgoing densities of particles. Results on existence, uniqueness and positivity of the solutions are gained again and an approximation of the solution is given.

In the contest of kinetic problems defined on bounded region and of asymptotic analysis, P.Degond (Univ. of Toulouse) and S.Mancini study a problem arising from the modeling of a ionic thruster. This research is supported by the TMR team "Asymptotic Methods in Kinetic Theory". The goal is to present a mathematically rigorous derivation of a diffusion equation previously introduced by P.Degond to model the diffusion of charged-particles moving in the gap between two plane parallel plates. The particles are subject to crossed electric and magnetic fields and to collisions against the surface of the solid plates. The surface collisions are supposed to be elastic. In a first approach, the collisions of the charged-particles against the neutral molecules of the host medium (a ionized gas) are neglected. Thus, the equation to be studied is the Vlasov equation equipped with "accommodation" boundary conditions. Under appropriate scaling assumptions, the particle distribution function converges to a function of the energy and of the longitudinal position coordinates only, which evolves in time according to a diffusion equation. As it is well known the interest of deriving a diffusion model from a kinetic one stands in the fact that the variables diminish from seven to four, this fact leading to more efficient numerical simulations. This research yiealds to the submission of the paper by P.Degond and S.Mancini: ``Diffusion driven by collisions with the boundary''. Moreover, the study is now focused on the addition of a collision operator to the right hand side of the Vlasov equation, describing the collisions of the charged-particles with the neutral molecules of the host medium.

This research still is a work in progress.

S. Mancini, S. Totaro, Solution of the Vlasov equation in a slab with source terms on the boundaries, Riv. Mat. Universita' Parma, (to appear)

S.Mancini, S.Totaro, Vlasov equation with nonhomogeneous boundary conditions, Math. Meth. in Appl. Sci. (to appear)

P. Degond, S.Mancini, Diffusion driven by collisions with the boundary, Preprint

Last Updated: 8 November 1999 by Giovanni Frosali