# Generic instabilities in the relativistic Chapman-Enskog heat conduction law

## TL;DR

This paper demonstrates that the relativistic heat conduction law derived from Chapman-Enskog theory is generally unstable and non-hyperbolic, except in special cases like non-rotating fluids, raising concerns about its physical validity.

## Contribution

It shows the generic instabilities and non-hyperbolic nature of the relativistic Chapman-Enskog heat conduction law, highlighting limitations compared to Eckart's theory.

## Key findings

- The system is non-hyperbolic under general perturbations.
- Eigenvalues have unbounded real parts as wave-number increases.
- Instability is absent only when the fluid's velocity is non-rotating.

## Abstract

We address the well-posedness of the Cauchy problem corresponding to the relativistic fluid equations, when coupled with the heat-flux constitutive relation arising within the relativistic Chapman-Enskog procedure. The resulting system of equations is shown to be non hyperbolic, by considering general perturbations over the whole set of equations written with respect to a generic time direction. The obtained eigenvalues are not purely imaginary and their real part grows without bound as the wave-number increases. Unlike Eckart's theory, this instability is not present when the time direction is aligned with the fluid's direction. However, since in general the fluid velocity is not surface-forming, the instability can only be avoided in the particular case where no rotation is present.