A 2 $\times$ 2 simple model in which the sub-shock exists when the shock velocity is slower than the maximum characteristic velocity

Abstract

For a generic hyperbolic system of balance laws, the shock-structure solution is not continuous and a discontinuous part (sub-shock) arises when the velocity of the front s is greater than a critical value. In particular, for systems compatible with the entropy principle, continuous shock-structure solutions cannot exist when s is larger than the maximum characteristic velocity evaluated in the unperturbed state $s >\lambda ^{\max }_0$. This is the typical situation of systems of Rational Extended Thermodynamics (ET). Nevertheless, in principle, sub-shocks may exist also for s smaller than $\lambda ^{\max }_0$. This was proved with a simple example in a recent paper by Taniguchi and Ruggeri (Int J Non-Linear Mech 99:69, 2018). In the present paper, we offer another simple case that satisfies all requirements of ET, that is, the entropy inequality, convexity of the entropy, sub-characteristic condition and Shizuta-Kawashima condition, however, there exists a sub-shock with s slower than $\lambda ^{\max }_0$. Therefore there still remains an open question which other property makes the systems coming from ET have this beautiful property that the sub-shock exists only for s greater than the unperturbed maximum characteristic velocity.