The aim of this paper is to show that procedure of maximum entropy principle for closure of moments equations for rarefied monatomic gases can be extended also to polyatomic gases. The main difference with respect to the usual procedure is existence of two hierarchies of macroscopic equations for moments of suitable distribution function, in which the internal energy of a molecule is taken into account. The field equations for 14 moments of distribution function, which include dynamic pressure, are derived. The entropy and the entropy flux are shown to be a generalization of the ones for classical Grad's distribution. The results are in perfect agreement with the recent macroscopic approach of extended thermodynamics for real gases.

A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal transformation group (basic Nöther's identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of Nöther's theorem leading to conservation laws for fractional Euler-Lagrangian equation is obtained as well. Results are illustrated by several concrete examples. Finally, an approximation of a fractional Euler-Lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws. (2000): Primary: 49K05; secondary: 26A33 PACS numbers: 02.30 Xx, 45.10 Hj Mathematics Subject Classification

SUMMARYThe first rational model of homogeneous mixtures of fluids was proposed by Truesdell in the context of rational thermodynamics. Afterwards, two different theories were developed: one with a single-temperature (ST) field of the mixture and the other one with several temperatures. The two systems are from the mathematical point of view completely different and the relationship between their solutions was not clarified.In this paper, the hyperbolic multi-temperature (MT) system of a mixture of Eulerian fluids will be explained and it will be shown that the corresponding single-temperature differential system is a principal subsystem of the MT one. As a consequence, the subcharacteristic conditions for characteristic speeds hold and this gives an upper-bound esteem for pulse speeds in an ST model. Global behaviour of smooth solutions for large time for both systems will also be discussed through the application of the ShizutaKawashima condition. Finally, as an application, the particular case of a binary mixture is considered.

This paper treats the nonequilibrium processes in mixtures of fluids under the assumption that each constituent is characterized by its own velocity and temperature field. First we discuss the concept of the average temperature of mixture based upon considerations that the internal energy of the mixture is the same as in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. An illustrative example of homogeneous mixtures is given to support the theoretical considerations. Through the procedure of Maxwellian iteration a new constitutive equation for nonequilibrium temperatures of constituents is obtained in a classical limit, together with the Fick's law for the diffusion flux. These results obtained for n-species are in perfect agreement with a recent classical approach of thermodynamics of irreversible processes in multitemperature case due to Gouin and Ruggeri and generalize our previous papers concerning the case of a binary mixture.