On the hyperbolicity of Maxwell's equations with a local constitutive law

TL;DR

This paper analyzes the hyperbolicity properties of Maxwell's equations with local constitutive laws in a metric-free setting, providing criteria for symmetric hyperbolicity and exploring implications for well-posedness.

Contribution

It offers a new characterization of constitutive laws that ensure symmetric hyperbolicity of Maxwell's equations in a metric-free framework.

Findings

Characterization of constitutive laws for symmetric hyperbolicity

Analysis of hyperbolicity conditions for Maxwell's equations

Illustrations with biisotropic media and Born-Infeld theory

Abstract

Maxwell's equations are considered in metric-free form, with a local but otherwise arbitrary constitutive law. After splitting Maxwell's equations into evolution equations and constraints, we derive the characteristic equation and we discuss its properties in detail. We present several results that are relevant for the question of whether the evolution equations are hyperbolic, strongly hyperbolic or symmmetric hyperbolic. In particular, we give a convenient characterisation of all constitutive laws for which the evolution equations are symmetric hyperbolic. The latter property is sufficient, but not necessary, for well-posedness of the initial-value problem. By way of example, we illustrate our results with the constitutive laws of biisotropic media and of Born-Infeld theory.