The Vlasov-Poisson system with infinite mass and energy.

摘要

The subject of this thesis is the study of the Vlasov-Poisson system of differential equations under the assumption of infinite charge and energy. We will show global existence of a unique, classical solution in both the radial three-dimensional and the general three-dimensional case, determine behavior of solutions for the three-dimensional problems and its one-dimensional analogue, and approximate solutions to the one-dimensional problem using a particle method.; We begin with an introductory chapter which sets notation and presents a deep historical background of problems in plasma dynamics, in particular, the Cauchy problem for the Vlasov-Poisson system. In addition, basic and well-known facts about the problem are introduced in this chapter.; In the second chapter, we provide a description of the one-dimensional analogue of the infinite charge problem and prove some new results regarding the behavior of global classical solutions. In particular, the charge density is shown to have compact support, and we discover that the electric field displays oscillatory behavior for large spatial values. We continue our analysis of the one dimensional problem in Chapter 3 by developing a particle method simulation to approximate solutions. Such a method is employed since particle methods have been shown to be more accurate and less costly when compared with traditional finite difference schemes or finite element approximations.; In Chapter 4, we turn our attention to the three-dimensional case in which the field is spherically symmetric. An adaptation of previously developed bounds on the velocity support and electric field are utilized in order to show suitable decay of both the field and charge density. Once this decay rate for the charge density is known, global existence of a unique, classical solution follows. Then, assuming some decay of the charge density, we are able to show stronger decay of this function without the previous assumptions of spherical symmetry.; Finally, in Chapter 5, global existence of a unique classical solution is shown for the full three-dimensional problem. This is achieved by adapting a previously known argument to bound the velocity support and then obtaining decay estimates of the field and its derivatives. Once these estimates are in place, we show that the charge density decays at the rate necessary to continue the local in time solution for all time.