Strict dissipativity of Cattaneo-Christov systems for compressible fluid flow

TL;DR

This paper proves that Cattaneo-Christov systems for one-dimensional compressible fluid flow are strictly dissipative, enabling the derivation of linear decay rates for solutions near equilibrium states.

Contribution

It demonstrates strict dissipativity of Cattaneo-Christov systems for compressible fluids, using coupling conditions and compensating functions, which was not previously established.

Findings

Cattaneo-Christov systems are strictly dissipative.

Existence of compensating functions for the system.

Linear decay rates for solutions near equilibrium.

Abstract

This work considers a compressible, viscous, heat-conducting fluid exhibiting thermal relaxation according to Christov's constitutive heat transfer law (C. I. Christov, Mech. Research Comm. 36 (2009)), which is of Cattaneo type. The resulting evolution equations are known as Cattaneo-Christov systems. In this contribution, it is shown that Cattaneo-Christov systems for one-dimensional compressible fluid flow are strictly dissipative. The proof is based on the verification of a genuine coupling condition for hyperbolic-parabolic systems with viscous and relaxation effects combined as well as on showing the existence of compensating functions of the state variables in the sense of Shizuta and Kawashima (Y. Shizuta and S. Kawashima, Hokkaido Math. J. 14 (1985)). This property is used to obtain linear decay rates for solutions to the linearized equations around equilibrium states.