##
**A macroscopic solution for a model suggested by the non-relativistic limit of relativistic extended thermodynamics.**
*(English)*
Zbl 1225.74007

Extended thermodynamics provides a good framework for studying the physics, because it leads to symmetric hyperbolic systems of field laws having important properties such as finite propagation speeds of shock waves and well-posedness of Cauchy problem. For the case with many moments the model classically used leads to errors: for example, it includes models that do not have a relativistic counterpart or that cannot be represented in a kinetic way.

In this paper the author proposes a relativistic model belonging in the limit to relativistic models. The author proves that it is possible to use classical procedures to close the system. The author also uses a new methodology recently proposed by S. Pennisi and T. Ruggeri to exploit the Galilei relativity principle [Ric. Mat. 55, No. 2, 319–339 (2006; Zbl 1378.74006)].

In order to find the solution for the presented model, the author uses a four-dimensional notation. It depends on a family of arbitrary scalar functions arising from integration. From these point of view, the solution can be described, by using the classical notation, and the author proves that, by fixing a certain order \(p\) up to equilibrium and only one scalar valued arbitrary function, all relevant physical quantities can be determined in terms of a single function. The same result has been found also with a generalized kinetic approach. Up to a fixed order \(p\), the two methods lead to the same solution, and then we are able to use the generalized kinetic method whose results are expressed in an easier and handier way. This is not the case for orders greater than \(p\), but because of the arbitrariness of \(p\), the author can reach every desired degree of approximation even with the kinetic approach.

In this paper the author proposes a relativistic model belonging in the limit to relativistic models. The author proves that it is possible to use classical procedures to close the system. The author also uses a new methodology recently proposed by S. Pennisi and T. Ruggeri to exploit the Galilei relativity principle [Ric. Mat. 55, No. 2, 319–339 (2006; Zbl 1378.74006)].

In order to find the solution for the presented model, the author uses a four-dimensional notation. It depends on a family of arbitrary scalar functions arising from integration. From these point of view, the solution can be described, by using the classical notation, and the author proves that, by fixing a certain order \(p\) up to equilibrium and only one scalar valued arbitrary function, all relevant physical quantities can be determined in terms of a single function. The same result has been found also with a generalized kinetic approach. Up to a fixed order \(p\), the two methods lead to the same solution, and then we are able to use the generalized kinetic method whose results are expressed in an easier and handier way. This is not the case for orders greater than \(p\), but because of the arbitrariness of \(p\), the author can reach every desired degree of approximation even with the kinetic approach.

Reviewer: Jerzy Gawinecki (Warszawa)