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Mathematical Structure

In: Classical and Relativistic Rational Extended Thermodynamics of Gases

Author

Listed:
  • Tommaso Ruggeri

    (University of Bologna, Department of Mathematics and Research Center on Applied Mathematics)

  • Masaru Sugiyama

    (Nagoya Institute of Technology)

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.

Suggested Citation

  • Tommaso Ruggeri & Masaru Sugiyama, 2021. "Mathematical Structure," Springer Books, in: Classical and Relativistic Rational Extended Thermodynamics of Gases, chapter 0, pages 41-65, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-59144-1_2
    DOI: 10.1007/978-3-030-59144-1_2
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