On necessary and sufficient conditions for strong hyperbolicity in systems with constraints

Abstract

In this work, we study constant-coefficient first order systems of partial differential equations and give necessary and sufficient conditions for those systems to have a well-posed Cauchy problem. In many physical applications, due to the presence of constraints, the number of equations in the PDE system is larger than the number of unknowns, thus the standard Kreiss conditions can not be directly applied to check whether the system admits a well-posed initial value formulation. In this work, we find necessary and sufficient conditions such that there exists a reduced set of equations, of the same dimensionality as the set of unknowns, which satisfy Kreiss conditions and so are well defined and properly behaved evolution equations. We do that by studying the systems using the Kronecker decomposition of matrix pencils and, once the conditions are met, finding specific families of reductions which render the system strongly hyperbolic. We show the power of the theory in some examples: Klein Gordon, the ADM, and the BSSN equations by writing them as first order systems, and studying their Kronecker decomposition and general reductions.