Qualitative Analysis Of Several Kinetic Equations With Force Fields

This Ph.D. thesis studies several kinetic models with force fields:Cometary flow equa-tion, Vlasov-poisson system and Vlasov-Helmholtz system. They have very important ap-plications in the research of astrophysical, plasma, neutral gas, neutron transport and semi-conductors.This Ph.D. thesis is divided into five chapters. In Chapter1, we introduce the physical background, the mathematical models and the current situation of the objects.In Chapter2, the Cauchy problem of a cometary flow equation with a force field is discussed. Two kinds of existence results for weak solutions are established for initial data having finite mass and finite kinetic energy. The first one concerns a given force field which is assumed to be divergence free with respect to the velocity variable, it is shown that there exists a nonnegative weak solution to the Cauchy problem when the initial datum and the force field have reasonable integrability. As a special case, a Lorentz field is also considered and a similar existence result is obtained. The second one deals with self-consistent elec-trostatic field, we show that when the initial datum has an L2integrability the system has a global nonnegative solution.In Chapter3, the Cauchy problem of a three dimensional cometary flow equation with a self-consistent electromagnetic field is studied. This kinetic model comes from the theory of astrophysical plasmas and can be viewed as a perturbation by a wave-particle collision operator, of the classical Vlasov-Maxwell system. The main result is a global existence theorem of a nonnegative weak solution for initial data meeting the physical significance, namely for initial data having finite mass and total energy and verifying some compatibility conditions.In Chapter4, the Vlasov-Poisson equation in three space dimensions in the repulsive case is considered. For smooth solution with compactly supported initial datum, the growth estimate of its velocity support is improved to t11/2+ε for any ε>0. As a consequence, we obtain a better decay estimate of the electrical field namely|‖E‖∞=O(t-11/1+ε) as tâ†'∞. In Chapter5, a collisionless plasma modeled by the Vlasov-Helmholtz equation in two dimensions is concerned. A fixed background of positive charge, which is independent of time and space, is assumed. Smooth solutions with infinite mass are shown to exist locally in time, and criteria for the continuation of these solutions are established.