We determine, for a generic dissipative hyperbolic system of balance laws, an upper bound such that for shock velocity greater than this limit no continuous shock-wave-structure solutions may exist. These general results are applied to the old and open problem of shock waves in classical and relativistic nonequilibrium thermodynamics. In this context, for the macroscopic theories of the extended thermodynamics related to the moment Grad procedure for the Boltzmann equation we can prove that, in contrast with a recent paper [D. Jou and D. Pavon, Phys. Rev. A 44, 6496 (1991)], this upper bound for critical Mach numbers is not influenced by adding other nonlinear terms. Moreover, taking into account the results of Weiss (Doctoral dissertation in Physics, Technical University Berlin, 1990), we can verify that our critical upper bound oscillates when the number of moments is increased. Therefore we conclude that the critical Mach number does not increase if we also consider more and more moments. As at the present the experiments do not put in evidence, also for high Mach numbers, a subshock formation in the shock structure, the natural conclusion of our result is that the shock thickness problem is not in the range of any hyperbolic continuum theory compatible with the Boltzmann equation.

• Received 30 October 1992

DOI:https://doi.org/10.1103/PhysRevE.47.4135