Formation of singularities in relativistic theories of electromagnetism

本研究计划从经典和量子两个层次研究相对论电磁场非线性动力学理论中的奇异性形成问题，经典部分主要涉及两个问题：（1）分析著名的相对论Vlasov-Maxwell方程组，该方程组已被证明是一个整体适定的Cauchy问题。 这个项目将严格分析最近开发的一个反例的情况下，展示有限时间崩溃的一个家庭的解决方案。 (2)无点电荷时电磁场的非线性Maxwell-Born-Infeld方程的分析。本项目将研究最近发现的具有有限时间爆破的空间周期性平面波解，以确定相应的柯西数据集是否是一般坏集的一部分。研究项目的量子部分涉及具有代表粒子的点缺陷的Maxwell-Born-Infeld场方程的解。带电粒子根据量子速度场运动，该速度场是从耦合到一般电磁场的多体狄拉克形式获得的。本课题研究系统的狄拉克哈密顿量是否可以在以下变得无界。本课题研究现代科学和工程学的核心--电磁学理论中的基本问题。奇点形成的研究长期以来一直处于广义相对论和流体动力学研究的前沿。最近的发现表明，奇点也可能构成一个重大的概念挑战，在非线性电磁模型，已被提出作为候选人的一致制定的电磁理论没有人工正则化。首席研究员最近开发了一个电磁理论的一致性公式，结合粒子的内在自旋，在经典和量子水平上都是一致的。目前的项目研究可能的奇点形成在这个理论。 另一部分的工作审查有限时间崩溃的相对论性弗拉索夫-麦克斯韦模型。 这种坍缩将为非常小的天体的形成提供一种新的机制，这些天体的引力自吸引力太弱，无法帮助它们形成。该项目旨在建立这种可能性;其结果可能对行星系统形成理论产生重大影响。

This research project studies the question of singularity formation in nonlinear dynamical theories of relativistic electromagnetic fields, both at the classical and the quantum level.The classical part is concerned with two main problems: (1) Analysis of the well-known relativistic Vlasov-Maxwell equations, which had been conjectured to be globally well-posed as a Cauchy problem with suitable finite-energy classical data. This project will rigorously analyze a recently-developed scenario for a counterexample exhibiting finite-time collapse for a family of solutions. (2) Analysis of the nonlinear Maxwell-Born-Infeld equations for electromagnetic fields in the absence of point charges. This project will investigate recently-discovered spatially periodic plane wave solutions that exhibit finite-time blow-up to determine whether the corresponding set of Cauchy data is part of a generic bad set.The quantum part of the research project concerns solutions of the Maxwell-Born-Infeld field equations with point defects that represent particles. The charged particles move according to a quantum velocity field obtained from a many body Dirac formalism coupled to generic electromagnetic fields. This project will study whether the Dirac Hamiltonian for the system can become unbounded below.This project addresses fundamental issues in the theory of electromagnetism, which is central to modern science and engineering. The study of singularity formation has long been at the forefront of research in general relativity and in fluid dynamics. Recent discoveries suggest that singularities may also pose a major conceptual challenge in the nonlinear electromagnetic models that have been proposed as candidates for a consistent formulation of an electromagnetic theory without artificial regularizers. The principal investigator recently developed a consistent formulation of electromagnetic theory, incorporating intrinsic spin of particles, that is consistent at both classical and quantum levels. The current project investigates possible singularity formation in this theory. Another part of this work examines finite-time collapse in the relativistic Vlasov-Maxwell model. Such collapse would provide a novel mechanism for the formation of very small celestial bodies whose gravitational self-attraction is too weak to aid in their formation. This project aims to establish that possibility; the results could have a major impact on theories of planetary system formation.