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We present a survey on some recent results concerning the different models
of a mixture of compressible fluids. In particular we
discuss the most realistic case of a mixture when each constituent has
its own temperature (MT) and we first compare the solutions of this
model with the one with a unique common temperature (ST). In the case of
Eulerian fluids it will be shown that the corresponding (ST)
differential system is a principal subsystem of the (MT) one. Global
behavior of smooth solutions for large time for both systems will also
be discussed through the application of the Shizuta-Kawashima condition.
Then we introduce the concept of the average temperature of mixture
based upon the consideration 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. Through the procedure of Maxwellian iteration a
new constitutive equation for non-equilibrium temperatures of
constituents is obtained in a classical limit, together with the Fick's
law for the diffusion flux. Finally, to justify the Maxwellian
iteration, we present for dissipative fluids a possible approach of a
classical theory of mixture with multi-temperature and we prove that the
differences of temperatures between the constituents imply the existence
of a new dynamical pressure even if the fluids have a zero bulk
viscosity.
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