AUTHOR=Abalos Fernando , Reula Oscar , Hilditch David 

TITLE=Hyperbolic extensions of constrained PDEs

JOURNAL=Frontiers in Physics

VOLUME=Volume 12 - 2024

YEAR=2025

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2024.1517192

DOI=10.3389/fphy.2024.1517192

ISSN=2296-424X

ABSTRACT=Systems of partial differential equations (PDEs) comprising 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. In this article, we first review the use of hyperbolic reductions, where the evolution equations are singled out for consideration. We then examine in greater detail the extensions, namely, systems in which constraints are evolved as auxiliary variables alongside the original variables, resulting in evolution systems with no constraints. Assuming a particular structure of the original system, we provide sufficient conditions for the strong hyperbolicity of an extension. Finally, this theory is applied to the examples of electromagnetism and a toy model of magnetohydrodynamics.