Mathematical Problems of Classical Nonlinear Electromagnetic Theory: 63 (Monographs and Surveys in Pure and Applied Mathematics)

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A survey of some problems of current interest in the realm of classical nonlinear electromagnetic theory. Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface 1 Elements of Classical Electromagnetic Theory 1.0 Introduction 1.1 Electrostatics and Dielectric Media 1.2 Electric Currents and Electromagnetic Induction 1.3 Maxwell's Equations and the Propagation of Electromagnetic Waves 1.4 Transmission Lines: Basic Concepts 2 Wave-Dielectric Interactions I: Formation of Singularities 2.0 Introduction 2.1 Hyperbolic Systems and Conservation Laws 2.2 Shock Formation: Single Equations in One Space Dimension 2.3 Weak Solutions of Hyperbolic Conservation Laws 2.4 Riemann Invariants and Genuine Nonlinearity 2.5 Shock Formation in 2 x 2 Systems 2.6 Basic Equations for Wave-Dielectric Interactions 2.7 Shock Formation in Dielectrics 2.8 Multidimensional Problems: Shock Formation A. Unidirectional Propagation B. Bidirectional Propagation 3 Wave-Dielectric Interactions II: Growth Estimates and Existence of Smooth Solutions 3.0 Introduction 3.1 Logarithmic Convexity and Concavity 3.2 Growth Estimates for Wave-Dielectric Interactions: Differential Inequalities 3.3 Growth Estimates for Wave-Dielectric Interactions: Riemann Invariants 3.4 Global Existence for Small Data: An Example 3.5 Global Existence of Solutions in the Wave-Dielectric Interaction Problem 3.6 A Model of Superconductivity in a Nonlinear Dielectric 3.7 Multidimensional and Nonstrictly Hyperbolic Problems Governing Wave-Dielectric Interactions 4 Distributed Parameter Nonlinear Transmission Lines I: Shock Formation and Existence of Smooth Solutions 4.0 Introduction 4.1 Shock Formation in Nonlinear Transmission Lines 4.2 Global Existence with Small Data: Riemann Invariants Arguments 4.3 Global Existence with Small Data: Energy Estimates 5 Distributed Parameter Nonlinear Transmission Lines II: Existence of Weak Solutions 5.0 Introduction 5.1 Weak Solutions of the Burgers' Equation I 5.2 The Concept of the Young Measure 5.3 Weak Solutions of the Burgers' Equation II 5.4 Systems of Conservation Laws and DiPerna's Theorem 5.5 The Nonlinear Transmission Line: Some Preliminaries 5.6 The Regularized Transmission Line: Existence, Uniqueness, and Uniform Boundedness of Solutions 5.7 Existence of Global Weak Solutions for the Transmission Line 6 Some Nonlocal Electromagnetic Problems 6.0 Introduction 6.1 Equilibrium States of an Elastic Conductor in a Magnetic Field: Non-Self Interaction Case 6.2 The Self-Interacting Current Bearing Wire 6.3 Self-Interacting Wires: Existence of Straight Equilibrium States 6.4 Self-Interacting Wires: Existence of Plane Circular States 6.5 Helical States of Self-Interacting Wires 6.6 Perturbations of Equilibrium States of Self-Interacting Current Bearing Wires 6.7 Nonlocal Problems for Conducting Bodies Bibliography