A necessary condition for strong hyperbolicity of general first order systems

TL;DR

This paper establishes a necessary condition for the strong hyperbolicity of first order PDE systems with constraints, using singular value analysis, and applies it to physical models showing some lack hyperbolization.

Contribution

It introduces a new tensor-based criterion for determining the non-existence of hyperbolic reductions in constrained PDE systems.

Findings

Certain physical systems do not admit hyperbolization.

The behavior of singular values under perturbations indicates hyperbolicity.

The condition can be checked invariantly and simply.

Abstract

We study strong hyperbolicity of first order partial differential equations for systems with differential constraints. In these cases, the number of equations is larger than the unknown fields, therefore, the standard Kreiss necessary and sufficient conditions of strong hyperbolicity do not directly apply. To deal with this problem one introduces a new tensor, called a reduction, which selects a subset of equations with the aim of using them as evolution equations for the unknown. If that tensor leads to a strongly hyperbolic system we call it a hyperbolizer. There might exist many of them or none. A question arises on whether a given system admits any hyperbolization at all. To sort-out this issue, we look for a condition on the system, such that, if it is satisfied, there is no hyperbolic reduction. To that purpose we look at the singular value decomposition of the whole system and study certain one parameter families ($\varepsilon $) of perturbations of the principal symbol. We look for the perturbed singular values around the vanishing ones and show that if they behave as $O\left( \varepsilon ^{l}\right) $, with $l\geq 2$, then there does not exist any hyperbolizer. In addition, we further notice that the validity or failure of this condition can be established in a simple and invariant way. Finally we apply the theory to examples in physics, such as Force-Free Electrodynamics in Euler potentials form and charged fluids with finite conductivity. We find that they do not admit any hyperbolization.