On Solutions to the Vlasov Equations

Abstract

In this study we consider the mixed boundary value problem for the Vlasov-Poisson equations in an infinite cylinder, a problem describing the evolution of the density distribution of ions and electrons in a high temperature plasma under an external magnetic field. We examine the way to find a solution presented by Skubachevskii in his articles. We then present the alternative way consisting in the following. The system is reduced to an inhomogeneous form by replacing the unknown distribution function. After this, the fixed-point iteration method is used twice. First, we find function $f^{\beta }( x,p,t )$ as the limit for the sequence $f_{n}^{\beta }(x,p,t )$, then we use it to construct the solution $\varphi ( x,t )$ as the limit for $\varphi_{n}(x,t)$. The found classical solution for which the supports of the charged-particle density distributions are at a distance from the cylindrical boundary is shown to exist and to be unique in some neighbourhood of the stationary solution.