Some open problems in non-linear extended thermodynamics and their possible solutions

In this article we aim to furnish arguments for further considerations on some procedures commonly used in Extended Thermodynamics, such as the Taylor's expansions around equilibrium or the transition to subsystems. The initial impulse for these considerations lies in the fact that we have found, for a 14 moments model, the exact closure to the conditions arising from the entropy principle and the material objectivity principle, without using Taylor's expansions. These generated some problems; one of these concerns the relationship between system and subsystems, another one concerns the convexity of entropy and hyperbolicity; a third problem deals with the correct moments to be used as independent variables, those suggested by the classical limit of the relativistic moment theory. We will give suggestions for the solution of these problems.