Abstract
In this chapter, we give a survey on the mathematical structure of the system of RET, which is strictly related to the mathematical problems of hyperbolic systems in balance form with a convex entropy density. We summarize the main results: The proof of the existence of the main field in terms of which a system becomes symmetric, and several properties derived from the qualitative analysis concerning symmetric hyperbolic systems. In particular, the Cauchy problem is well-posed locally in time, and if the so-called K-condition is satisfied, there exist global smooth solutions provided that the initial data are sufficiently small. Moreover the main field permits to identify natural subsystems and in this way we have a structure of nesting theories. The main property of these subsystems is that the characteristic velocities satisfy the so-called sub-characteristic conditions that imply, in particular, that the maximum characteristic velocity does not decrease when the number of equations increases. Another beautiful general property is the compatibility of the balance laws with the Galilean invariance that dictates the precise dependence of the field equations on the velocity.
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Notes
- 1.
The definition and the properties remain valid for prescribed values of \(\mathbf{w}_{{\ast}}^{{\prime}}\) that depend on x Ī± in an arbitrary manner. In this case the principal subsystem is not autonomous [13].
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Ruggeri, T., Sugiyama, M. (2015). Mathematical Structure. In: Rational Extended Thermodynamics beyond the Monatomic Gas. Springer, Cham. https://doi.org/10.1007/978-3-319-13341-6_2
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