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

✍ Scribed by Frederick Bloom

• Publisher: Chapman and Hall/CRC
• Year: 1993
• Tongue: English
• Leaves: 399
• Edition: 1
• Category: Library

✦ Synopsis

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

✦ Table of Contents

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