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Hyperbolic extensions of constrained PDEs

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arxiv 2410.18286 v1 pith:TF5BMFKY submitted 2024-10-23 math.AP gr-qc

Fernando Abalos , Oscar Reula , David Hilditch This is my paper · ORCID
classification math.APgr-qc
keywords constraintsequationsevolutionextensionshyperbolicoriginalpdessystems
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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.

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Cited by 2 Pith papers

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  1. Higher-derivative gravitational effective field theories are generically weakly hyperbolic

    gr-qc 2026-07 conditional novelty 6.5 of 10

    Any pure-metric higher-derivative gravity EFT with derivative-independent characteristics has a weakly hyperbolic physical spin-2 block that gauge fixing and constraint addition cannot remove.

  2. Relativistic dissipative fluids in the trace-fixed particle frame: Strongly hyperbolic quasi-linear first-order evolution equations

    gr-qc 2024-12 conditional novelty 6.0 of 10

    A new first-order relativistic dissipative fluid theory in the trace-fixed particle frame is proved to be strongly hyperbolic, causal, and locally well-posed as a constrained quasilinear system.

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