On the sub-shock formation in extended thermodynamics

In hyperbolic dissipative systems, the solution of the shock structure is not always continuous and a discontinuous part (sub-shock) appears when the velocity of the shock wave is greater than a critical value. In principle, the sub-shock may occur when the shock velocity $s$ reaches one of the characteristic eigenvalues of the hyperbolic system. Nevertheless, Rational Extended Thermodynamics (ET) for a rarefied monatomic gas predicts the sub-shock formation only when $s$ exceeds the maximum characteristic velocity of the system evaluated in the unperturbed state $\lambda^{\max}_0$. This fact agrees with a general theorem asserting that continuous shock structure cannot exist for $s >\lambda^{\max}_0$. In the present paper, first, the shock structure is numerically analyzed on the basis of ET for a rarefied polyatomic gas with $14$ independent fields. It is shown that, also in this case, the shock structure is still continuous when $s$ meets characteristic velocities except for the maximum one and therefore the sub-shock appears only when $s >\lambda^{\max}_0$. This example reinforces the conjecture that, the differential systems of ET theories have the special characteristics such that the sub-shock appears only for $s$ greater than the unperturbed maximum characteristic velocity. However, in the second part of the paper, we construct a counterexample of this conjecture by using a simple $2 \times 2$ hyperbolic dissipative system which satisfies all requirements of ET. In contrast to previous results, we show the clear sub-shock formation with a slower shock velocity than the maximum unperturbed characteristic velocity.