A necessary condition ensuring the strong hyperbolicity of first-order systems

Abstract

We study strong hyperbolicity of first order partial differential equationsfor systems with differential constraints. In these cases, the number ofequations is larger than the unknown fields, therefore, the standard Kreissnecessary and sufficient conditions of strong hyperbolicity do not directlyapply. To deal with this problem one introduces a new tensor, called areduction, which selects a subset of equations with the aim of using them asevolution equations for the unknown. If that tensor leads to a stronglyhyperbolic system we call it a hyperbolizer. There might exist many of themor none.A question arises on whether a given system admits any hyperbolization atall. To sort-out this issue, we look for a condition on the system, suchthat, if it is satisfied, there is no hyperbolic reduction. To that purposewe look at the singular value decomposition of the whole system and studycertain one parameter families ($arepsilon $) of perturbations of theprincipal symbol. We look for the perturbed singular values around thevanishing ones and show that if they behave as $Oleft( arepsilon^{l}ight) $, with $lgeq 2$, then there does not exist any hyperbolizer.In addition, we further notice that the validity or failure of thiscondition can be established in a simple and invariant way.Finally we apply the theory to examples in physics, such as Force-FreeElectrodynamics in Euler potentials form and charged fluids with finiteconductivity. We find that they do not admit any hyperbolization.