Mathematical Sciences: Nonlinear Partial Differential Equations in Plasma Physics

本研究将集中分析等离子体物理中出现的几个特定的非线性双曲型偏微分方程。对于Vlasov-Maxwell方程组，将研究柯西问题的全局解、稳定性、冲击的可能发展和碰撞效应。无碰撞等离子体是稀释的电离气体。在它们中，粒子之间的相互作用是通过粒子自身产生的电磁场进行的。它们的轨迹是由某些涉及相对高速的微分方程定义的。首席研究员最近证明，如果单个粒子的动量是均匀有界的，则较简单的Vlasov方程的Cauchy问题具有唯一的全局光滑解。现在的工作将是建立考虑碰撞效应的模型。迄今为止，电磁效应被认为是主导碰撞的因素。目前还不知道经典可解性的充分条件。第二项研究涉及到具有旋转对称的方程和解的稳定性问题。某些情况有肯定的答案（即稳定性得到确认）；应用文献更进一步，在没有证据的情况下指出，在某些一般条件下存在稳定性。首先必须做的是澄清渐近稳定性和正常稳定性之间的区别（如果有的话）。我们将为此而努力。将对无碰撞等离子体中是否存在激波这一悬而未决的问题进行研究。这项工作将从二维问题开始，在二维问题中，场和密度永远是有界的。现在需要的是对所涉及的场和密度的导数的估计。在二维空间中缺乏惠更斯原理，目前尚不清楚这是一个真正的物理障碍（即冲击真的发生了）还是仅仅是当前方法的一个缺点。除了对偏微分方程理论的贡献外，这项工作在数学物理和统计力学方面也有重要的潜在应用。

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.