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.

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Acknowledgements

The results contained in the present paper have been partially presented in Wascom 2017. This work was partially supported by JSPS KAKENHI Grant Number JP16K17555 (S. T.) and by National Group of Mathematical Physics GNFM-INdAM (T. R.).

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Correspondence to Shigeru Taniguchi.

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Taniguchi, S., Ruggeri, T. A 2 \(\times \) 2 simple model in which the sub-shock exists when the shock velocity is slower than the maximum characteristic velocity. Ricerche mat 68, 119–129 (2019). https://doi.org/10.1007/s11587-018-0380-1

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Keywords

  • Shock waves
  • Extended thermodynamics
  • Hyperbolic systems with relaxation
  • Sub-shock formation