Can constitutive relations be represented by non-local equations?

Using the modern theory of extended thermodynamics, it is possible to show that the well-known constitutive equations of continuum mechanics in non-local form with respect to space variables such as Fourier’s, Navier-Stokes’s, Fick’s and Darcy’s laws are in reality an approximation of the general balance laws when some suitable relaxation times are neglected. In the present paper we conjecture that this fact is completely general and indeed all the “real” constitutive equations of mathematical physics are local in nature and, therefore, the corresponding differential systems of balance equations are hyperbolic rather than parabolic. This does not means that non-local equations are not useful not only because there are situations where non-local equations may be an effective approximation, but also because non-locality permits us to obtain the evaluation of non-observable quantities such as the velocity and the temperature of each constituent of a mixture of fluids. An important consequence is that these equations do not need to satisfy the so-called