Strongly hyperbolic quasilinear systems revisited, with applications to relativistic fluid dynamics
Marcelo M. Disconzi, Yuanzhen Shao
TL;DR
The paper develops a self-contained local well-posedness theory for first-order quasilinear hyperbolic systems with diagonalizable principal parts and real eigenvalues, under minimal regularity. It builds a robust energy-estimate framework for the associated linear system, then constructs a convergent approximation sequence to obtain existence, uniqueness, and continuous dependence in Sobolev spaces, plus a continuation criterion. The method is then applied to relativistic fluid models—relativistic Euler, out-of-equilibrium bulk dynamics, and first-order causal relativistic viscous fluids—showing these systems satisfy strong hyperbolicity and thereby admit locally well-posed Cauchy problems under physically reasonable conditions. The results provide a rigorous mathematical foundation for the analysis and numerical treatment of relativistic fluids in curved spacetimes and in coupling with Einstein’s equations, with implications for astrophysical and gravitational-wave contexts.
Abstract
We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest.