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Mathematical Structure

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Classical and Relativistic Rational Extended Thermodynamics of Gases

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Abstract

In this chapter, we make a survey on the mathematical structure of the system of rational extended thermodynamics, 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 us 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 subcharacteristic 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. 1.

    The proof in [164] was done in a covariant formalism. Instead, here, we use a classical formalism for simplicity.

  2. 2.

    We recall that, due to the different sign of the entropy density in Definition 2.2, the word “positive definite” is changed into “negative definite”.

  3. 3.

    We recall that, in the paper [178], the sign of the quantities h α and h ′α is opposite to the one adopted here.

  4. 4.

    The definition and the properties remain valid for prescribed values of \( {\mathbf {w}}^\prime _*\) that depend on x α in an arbitrary manner. In this case the principal subsystem is not autonomous [165].

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Authors and Affiliations

  1. Department of Mathematics and Research Center on Applied Mathematics, University of Bologna, Bologna, Italy

    Tommaso Ruggeri

  2. Nagoya Institute of Technology, Nagoya, Japan

    Masaru Sugiyama

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  1. Tommaso Ruggeri
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  2. Masaru Sugiyama
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Ruggeri, T., Sugiyama, M. (2021). Mathematical Structure. In: Classical and Relativistic Rational Extended Thermodynamics of Gases. Springer, Cham. https://doi.org/10.1007/978-3-030-59144-1_2

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