Compressible Euler equations with second sound: Asymptotics of discontinuous solutions

We consider the compressible Euler equations in three space dimensions where heat conduction is modeled by Cattaneo’s law instead of Fourier’s law. For the arising purely hyperbolic system, the asymptotic behavior of discontinuous solutions to the linearized Cauchy problem is investigated. We give a description of the behavior as time tends to infinity and, in particular, as the relaxation parameter tends to zero. The latter corresponds to the singular limit and a formal convergence to the classical (i.e. Fourier law for the heat flux–temperature relation) Euler system. We recover a phenomenon observed for hyperbolic thermoelasticity, namely the dependence of the asymptotic behavior on the mean curvature of the initial surface of discontinuity; in addition, we observe a more complex behavior in general.