ANALYSIS OF THE VLASOV-POISSON EQUATION BY USING A VISCOSITY TERM

The well-known Vlasov-Poisson equation describes plasma physics as nonlinear ¯rst-order partial di？？erential equations. Be-cause of the nonlinear condition from the self consistency of the Vlasov-Poisson equation, many problems occur: the existence, the numerical solution, the convergence of the numerical solution, and so on. To solve the problems, a viscosity term (a second-order par-tial di？？erential equation) is added. In a viscosity term, the Vlasov-Poisson equation changes into a parabolic equation like the Fokker-Planck equation. Therefore, the Schauder ¯xed point theorem and the classical results on parabolic equations can be used for analyz- ing the Vlasov-Poisson equation. The sequence and the convergence results are obtained from linearizing the Vlasove-Poisson equation by using a ¯xed point theorem and Gronwall s inequality. In nu- merical experiments, an implicit ¯rst-order scheme is used. The numerical results are tested using the changed viscosity terms.