Liu’s proof proceeds from the observation that the field equations and the entropy equation are linear functions of the derivatives . By the Cauchy–Kowalewski theorem these derivatives are local representatives of an analytical thermodynamic process and therefore the entropy principle requires that the field equations and the entropy equation must hold for all . It is then a simple problem of linear algebra to prove that
Liu’s proof is not restricted to quasilinear systems of first order equations but here we need his result only
in that particularly simple case.
We may use the chain rule on and in (6) and obtain
The differential forms (8) represent a generalization of the Gibbs equation of equilibrium thermodynamics; the classical Gibbs equation for the entropy density is here generalized into four equations for the entropy four-flux. Relation (9) is the residual entropy inequality which represents the irreversible entropy production. Note that the entropy production is entirely due to the production terms in the balance equations.
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