Hyperbolic extensions of constrained PDEs

TL;DR

This paper investigates hyperbolic extensions of constrained PDE systems, providing conditions for strong hyperbolicity and applying the theory to electromagnetism and magnetohydrodynamics models.

Contribution

It offers a systematic framework for analyzing hyperbolic extensions of PDEs with constraints, including sufficient conditions for strong hyperbolicity.

Findings

Derived sufficient conditions for strong hyperbolicity of extensions.

Applied the theory to electromagnetism and magnetohydrodynamics examples.

Enhanced understanding of well-posedness for constrained PDE systems.

Abstract

Systems of PDEs comprised of a combination of constraints and evolution equations are ubiquitous in physics. For both theoretical and practical reasons, such as numerical integration, it is desirable to have a systematic understanding of the well-posedness of the Cauchy problem for these systems. Presently we review the use of hyperbolic reductions, in which the evolution equations are singled out for consideration. We then examine in greater detail the extensions, in which constraints are evolved as auxiliary variables alongside the original variables. Assuming a particular structure of the original system, we give sufficient conditions for strong-hyperbolicity of an extension. This theory is then applied to the examples of electromagnetism and a toy for magnetohydrodynamics.