Mathematical Problems of Classical Nonlinear Electromagnetic Theory

A survey of some problems of current interest in the realm of classical nonlinear electromagnetic theory.

Frederick Bloom, Northern Illinois University

Preface, ELEMENTS OF CLASSICAL ELECTROMAGNETIC THEORY, Introduction, Electrostatics and Dielectric Media, Electric Currents and Electromagnetic Induction, Maxwell's Equations and the Propagation of Electromagnetic Waves, Transmission Lines: Basic Concepts, WAVE-DIELECTRIC INTERACTIONS I: FORMATION OF SINGULARITIES, Introduction, Hyperbolic Systems and Conservation Laws, Shock Formation: Single Equations in One Space Dimension, Weak Solutions of Hyperbolic Conservation Laws, Riemann Invariants and Genuine Nonlinearity, Shock Formation in 2 x 2 Systems, Basic Equations for Wave-Dielectric Interactions, Shock Formation in Dielectrics, Multidimensional Problems: Shock Formation, Unidirectional Propagation, Bidirectional Propagation, WAVE-DIELECTRIC INTERACTIONS 11: GROWTH ESTIMATES AND EXISTENCE OF SMOOTH SOLUTIONS, Introduction, Logarithmic Convexity and Concavity, Growth Estimates for Wave-Dielectric Interactions: Differential Inequalities, Growth Estimates for Wave-Dielectric Interactions: Riemann Invariants, Global Existence for Small Data: An Example, Global Existence of Solutions in the Wave-Dielectric Interaction Problem, A Model of Superconductivity in a Nonlinear Dielectric, Multidimensional and Nonstrictly Hyperbolic Problems Governing Wave-Dielectric Interactions, DISTRIBUTED PARAMETER NONLINEAR TRANSMISSION LINES I: Shock Formation and Existence of Smooth Solutions, Introduction, Shock Formation in Nonlinear Transmission Lines, Global Existence with Small Data: Riemann Invariants Arguments, Global Existence with Small Data: Energy Estimates, DISTRIBUTED PARAMETER NONLINEAR TRANSMISSION LINES II: Existence of Weak Solutions, Introduction, Weak Solutions of the Burgers' Equation I, The Concept of the Young Measure, Weak Solutions of the Burgers' Equation 11, Systems of Conservation Laws and DiPerna's Theorem, The Nonlinear Transmission Line: Some Preliminaries, The Regularized Transmission Line: Existence, Uniqueness, and Uniform Boundedness of Solutions, Existence of Global Weak Solutions for the Transmission Line, SOME NONLOCAL ELECTROMAGNETIC PROBLEMS, Introduction, Equilibrium States of an Elastic Conductor in a Magnetic Field: Non-Self Interaction Case, The Self-Interacting Current Bearing Wire, Self-Interacting Wires: Existence of Straight Equilibrium States, Self-Interacting Wires: Existence of Plane Circular States, Helical States of Self-Interacting Wires, Perturbations of Equilibrium States of Self-Interacting Current Bearing Wires, Nonlocal Problems for Conducting Bodies, Bibliography