Local Constitutive Equations, Extended Thermodynamics and Maxwellian Iteration

The modern theory of extended thermodynamics, shows that the well known constitutive equations of continuum mechanics of non-local form in space are in reality approximations of balance laws when some relaxation times are neglected.We recall, for example, the Fourier’s equation, the Navier-Stokes’ equations, the Fick’s equation, the Darcy’s law and several others. This idea suggests that the “authentic” constitutive equations are local and, therefore, the differential systems of mathematical physics are hyperbolic rather than parabolic. Another consequence is that these equations do not need to satisfy the so called objectivity principle that on the contrary still continues to be valid only for the constitutive equations. Under suitable assumptions the conditions dictated by the entropy principle in the hyperbolic case guarantees the entropy principle validity also in the parabolic limit. Considerations are also made with regard to the formal limit between hyperbolic system and parabolic ones and from hyperbolic versus hyperbolic, between a system and a subsystem.