Strict dissipativity of Cattaneo-Christov systems for compressible fluid flow

This work considers a compressible, heat-conducting fluid exhibiting thermal relaxation according to Christov's constitutive heat transfer law (Christov 2009 Mech. Res. Commun. 36 481-6), 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 in both the viscous and inviscid cases. 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 (1985 Hokkaido Math. J. 14 249-75). This property is used to obtain linear decay rates for solutions to the linearized equations around equilibrium states.