Mathematical Sciences: Nonlinear Partial Differential Equations in Plasma Physics

点击翻译按钮获取中文摘要

This investigation will focus on the analysis of several specific nonlinear hyperbolic partial differential equations occurring in plasma physics. Questions of global solutions to the Cauchy problem, stability, possible development of shocks and collisional effects will be studied for the Vlasov-Maxwell system of equations. Collisionless plasmas are dilute ionized gases. In them, the interaction of particles takes place through the electromagnetic field which is generated by the particles themselves. Their trajectories are defined by certain differential equations involving relativistically high speeds. The principal investigator has shown recently that the Cauchy problem for the simpler Vlasov equations has a unique global smooth solution provided the momenta of the individual particles are uniformly bounded. Work will now be done on models in which the effects of collisions are taken into account. Heretofore, the electromagnetic effects were allowed to dominate the collisional. No sufficient condition for classical solvability is known at this time. A second line of investigation involves equations with rotational symmetry and the question of stability of solutions. Certain cases have affirmative answers (i.e. stability is confirmed); the applied literature goes further, stating - without proof - that stability is present under certain general conditions. What first must be done is a clarification of the distinctions, if any, between asymptotic stability and normal stability. Work will be done towards this end. Studies into the open question of the existence of shocks in collisionless plasma will be carried out. The work will begin with two dimensional problems where fields and densities remain bounded for all time. What is needed now are estimates on the derivatives of the fields and densities involved. There lacks a Huygens' Principle in two dimensions and it is unclear whether this is a real physical barrier (i.e. a shock really occur) or is simply a shortcoming of current methods. In addition to its contributions to the theory of partial differential equations, this work has important potential applications to mathematical physics and statistical mechanics.