Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories

TL;DR

This paper proves that under certain conditions, solutions to dissipative relativistic fluid theories on a globally hyperbolic manifold exist globally, decay exponentially, and stabilize near equilibrium states.

Contribution

It establishes conditions ensuring global existence and exponential decay for relativistic dissipative fluid theories, extending stability analysis to these complex systems.

Findings

Solutions exist globally for small initial perturbations.

Solutions decay exponentially to equilibrium.

All eigenvalues of the system's symbol have negative real parts.

Abstract

We consider dissipative relativistic fluid theories on a fixed flat, compact, globally hyperbolic, Lorentzian manifold. We prove that for all initial data in a small enough neighborhood of the equilibrium states (in an appropriate Sobolev norm), the solutions evolve smoothly in time forever and decay exponentially to some, in general undetermined, equilibrium state. To prove this, three conditions are imposed on these theories. The first condition requires the system of equations to be symmetric hyperbolic, a fundamental requisite to have a well posed and physically consistent initial value formulation. The second condition is a generic consequence of the entropy law, and is imposed on the non principal part of the equations. The third condition is imposed on the principal part of the equations and it implies that the dissipation affects all the fields of the theory. With these requirements we prove that all the eigenvalues of the symbol associated to the system of equations of the fluid theory have strictly negative real parts, which in fact, is an alternative characterization for the theory to be totally dissipative. Once this result has been obtained, a straight forward application of a general stability theorem due to Kreiss, Ortiz, and Reula, implies the results above mentioned.