Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems

TL;DR

This paper proves the global existence and decay of small solutions for a class of dissipative quasi-linear hyperbolic systems, using para-differential calculus, with applications to relativistic fluid dynamics.

Contribution

It introduces a novel application of para-differential operators to establish global solutions for non-symmetric hyperbolic systems with dissipation.

Findings

Global-in-time existence of solutions near reference states

Asymptotic stability with decay rates

Application to relativistic viscous fluid models

Abstract

This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Its main theorem establishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states. Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic symmetrizers is assumed. The dissipation is characterized by conditions equivalent to the uniform decay of all Fourier modes at the reference state. On a technical level, the theory developed herein uses para-differential operators as its main tool. Apparently being the first to apply such operators in the context of global-in-time existence for quasi-linear hyperbolic systems, the present work contains new results in the field of para-differential calculus. In the context of theoretical physics, the theorem applies to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids notably such as that of Bemfica, Disconzi and Noronha (Phys. Rev. D, 98:104064, 2018.)