On the equations of compressible fluid dynamics with Cattaneo-type extensions for the heat flux: Symmetrizability and relaxation structure

TL;DR

This paper investigates the mathematical structure of compressible fluid dynamics equations with Cattaneo-type heat flux extensions, revealing non-symmetrizability and analyzing the relaxation effects on wave solutions.

Contribution

It demonstrates the existence of a non-symmetrizable hyperbolic system in fluid dynamics with Cattaneo extensions and analyzes the relaxation term's impact on wave persistence.

Findings

The system is not symmetrizable in Friedrichs' sense.

Linearizations around equilibrium are Friedrichs symmetrizable.

Persistent waves exist despite relaxation effects.

Abstract

The aim of this work is twofold. From a mathematical point of view, we show the existence of a hyperbolic system of equations that is not symmetrizable in the sense of Friedrichs. Such system appears in the theory of compressible fluid dynamics with Cattaneo-type extensions for the heat flux. In contrast, the linearizations of such system around constant equilibrium solutions have Friedrichs symmetrizers. Then, from a physical perspective, we aim to understand the relaxation term appearing in this system. By noticing the violation of the Kawashima-Shizuta condition, locally and smoothly, with respect to the Fourier frequencies, we construct persistent waves, i.e., solutions preserving the $L^{2}$ norm for all times that are not dissipated by the relaxation terms.