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.

The study of boundary conditions of conservative type is now in progress.

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Last Updated: 20 October 1998 by Giovanni Frosali