Solvability of the Cauchy Problem for the Vlasov- Poisson system with Collision Integrals of two types. Part I

In this work the Cauchy problems for the Vlasov-Poisson system of equation with a linear collision integral and for the same system with Uehling-Uhlenbeck collision integral in the case of Fermi-Dirac statistics are considered. In the first part of the paper global existence and uniqueness theorems are formulated and the proof of local solvability is given. The proof of global solvability and appendix are given in the second part of the paper.