On constraint preservation and strong hyperbolicity

TL;DR

This paper establishes conditions under which PDE systems preserve constraints during evolution, focusing on hyperbolic equations, with applications to Maxwell electrodynamics and wave equations in arbitrary space-times.

Contribution

It provides a framework for deriving strongly hyperbolic constraint evolution equations and analyzes their parameter dependence for physical PDE systems.

Findings

Constraint evolution equations are unique for Maxwell electrodynamics.

Constraint equations form a family parametrized by free parameters in wave equations.

The framework ensures constraint preservation under certain hyperbolicity conditions.

Abstract

We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the Free-evolution approach, which consists in obtaining the solutions of the entire system by solving only the evolution equations. Certainly, this is valid only when the chosen initial data satisfies the constraints and the constraints are preserved in the evolution. In this paper, we establish the sufficient conditions required for the PDEs of the system to guarantee the constraint preservation. This is achieved by considering quasi-linear first-order PDEs, assuming the sufficient condition and deriving strongly hyperbolic first-order partial differential evolution equations for the constraints. We show that, in general, these constraint evolution equations correspond to a family of equations parametrized by a set of free parameters. We also explain how these parameters fix the propagation velocities of the constraints. As application examples of this framework, we study the constraint conservation of the Maxwell electrodynamics and the wave equations in arbitrary space-times. We conclude that the constraint evolution equations are unique in the Maxwell case and a family in the wave equation case.