A macroscopic solution for a model suggested by the non relativistc limit of relativistic extended thermodynamics

Extended thermodynamics provides a good framework for studying the physics of fluids, 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 the Cauchy problem. For the case with many moments the model classically used leads to errors and problems: for example it includes models that do not have a relativistic counterpart or that cannot be represented in a kinetic way. To overcome these difficulties Pennisi and I proposed a model belonging from the relativistic one through its non relativistic limit. We proved that, even for this model, it is possible to use the classical procedures to close the system and the new methodology recently proposed by Pennisi and Ruggeri to exploit the Galilei relativity principle. We found the solution for our model by using a four-dimensional notation. It depends on a family of arbitrary scalar functions arising from integration. Here the solution will be found, by using the classical notation and it will proven that, by fixing a certain order $p$ up to equilibrium and only one scalar valued arbitrary function, everything is determined in terms of that 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 allowed 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 $p$ we can reach every desired degree of approximation even with the kinetic approach.