Can constitutive relations be represented by non-local equations?

Ruggeri, Tommaso

doi:10.1090/S0033-569X-2012-01314-3

Author: Tommaso Ruggeri
Journal: Quart. Appl. Math. 70 (2012), 597-611
MSC (2010): Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/S0033-569X-2012-01314-3
Published electronically: May 9, 2012
MathSciNet review: 2986136
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Abstract: 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 objectivity principle that on the contrary still continues to be valid only for the constitutive equations. We prove that under suitable assumptions the conditions dictated by the entropy principle in the hyperbolic case guarantee the validity of the entropy principle also in the parabolic limit. Considerations are also made with regard to the formal limit between hyperbolic systems and parabolic ones and from hyperbolic versus hyperbolic, between a system and a subsystem. We end the paper with a discussion of the main analytical properties concerning the global existence of smooth solutions for dissipative hyperbolic systems.

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