Hyperbolicity of second-order in space systems of evolution equations

TL;DR

This paper explores the concept of hyperbolicity in second-order evolution equations, providing criteria, alternative definitions, and boundary conditions, and establishing equivalences among different hyperbolicity notions in three dimensions.

Contribution

It introduces necessary and sufficient criteria for hyperbolicity, compares multiple definitions, and extends results to constraint systems without explicit reduction.

Findings

Criteria for strong/symmetric hyperbolicity in second-order systems

Equivalence of different hyperbolicity definitions in 3D

Method for imposing boundary conditions on symmetric hyperbolic systems

Abstract

A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a reduction to first order and give necessary and sufficient criteria for strong/symmetric hyperbolicity of the reduction in terms of the principal part of the original second-order system. An alternative definition of strong hyperbolicity is based on the existence of a complete set of characteristic variables, and an alternative definition of symmetric hyperbolicity is based on the existence of a conserved (up to lower order terms) energy. Both these definitions are made without any explicit reduction. Finally, strong hyperbolicity can be defined through a pseudo-differential reduction to first order. We prove that both definitions of symmetric hyperbolicity are equivalent and that all three definitions of strong hyperbolicity are equivalent (in three space dimensions). We show how to impose maximally dissipative boundary conditions on any symmetric hyperbolic second order system. We prove that if the second-order system is strongly hyperbolic, any closed constraint evolution system associated with it is also strongly hyperbolic, and that the characteristic variables of the constraint system are derivatives of a subset of the characteristic variables of the main system, with the same speeds.