Can constitutive relations be represented by nonlocal equations?
Author:
Tommaso Ruggeri
Journal:
Quart. Appl. Math. 70 (2012), 597611
MSC (2010):
Primary 54C40, 14E20; Secondary 46E25, 20C20
Published electronically:
May 9, 2012
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Abstract: Using the modern theory of extended thermodynamics, it is possible to show that the wellknown constitutive equations of continuum mechanics in nonlocal form with respect to space variables such as Fourier's, NavierStokes'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 nonlocal equations are not useful not only because there are situations where nonlocal equations may be an effective approximation, but also because nonlocality permits us to obtain the evaluation of nonobservable 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 socalled 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.
 1.
C.
Truesdell and W.
Noll, The nonlinear field theories of mechanics, Handbuch der
Physik, Band III/3, SpringerVerlag, Berlin, 1965, pp. 1–602.
MR
0193816 (33 #2030)
 2.
Bernard
D. Coleman and Walter
Noll, The thermodynamics of elastic materials with heat conduction
and viscosity, Arch. Rational Mech. Anal. 13 (1963),
167–178. MR 0153153
(27 #3122)
 3.
Ingo
Müller, On the entropy inequality, Arch. Rational Mech.
Anal. 26 (1967), 118–141. MR 0214336
(35 #5187)
 4.
Ingo
Müller, On the frame dependence of stress and heat flux,
Arch. Rational Mech. Anal. 45 (1972), no. 4,
241–250. MR
1553565, http://dx.doi.org/10.1007/BF00251375
 5.
A. Bressan, On relativistic heat conduction in the stationary and nonstationary cases, the objectivity principle and piezoelasticity, Lett. Nuovo Cimento, 33 (4), 108 (1982).
 6.
Tommaso
Ruggeri, Generators of hyperbolic heat equation in nonlinear
thermoelasticity, Rend. Sem. Mat. Univ. Padova 68
(1982), 79–91 (1983). MR 702148
(85e:73066)
 7.
Ingo
Müller and Tommaso
Ruggeri, Rational extended thermodynamics, 2nd ed., Springer
Tracts in Natural Philosophy, vol. 37, SpringerVerlag, New York,
1998. With supplementary chapters by H. Struchtrup and Wolf Weiss. MR 1632151
(99h:80001)
 8.
Harold
Grad, On the kinetic theory of rarefied gases, Comm. Pure
Appl. Math. 2 (1949), 331–407. MR 0033674
(11,473a)
 9.
IShih
Liu and Ingo
Müller, Extended thermodynamics of classical and degenerate
ideal gases, Arch. Rational Mech. Anal. 83 (1983),
no. 4, 285–332. MR 714978
(85j:80001), http://dx.doi.org/10.1007/BF00963838
 10.
E.
Ikenberry and C.
Truesdell, On the pressures and the flux of energy in a gas
according to Maxwell’s kinetic theory. I, J. Rational Mech.
Anal. 5 (1956), 1–54. MR 0075725
(17,796c)
 11.
T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama, Extended thermodynamics of dense gases, Continuum Mech. Thermodyn. DOI 10.1007/s001610110213x (2011).
 12.
C.
Truesdell, Rational thermodynamics, McGrawHill Book Co., New
York, 1969. A course of lectures on selected topics; With an appendix on
the symmetry of the heatconduction tensor by C. C. Wang. MR 0366236
(51 #2484)
 13.
T.
Ruggeri, Galilean invariance and entropy principle for systems of
balance laws. The structure of extended thermodynamics, Contin. Mech.
Thermodyn. 1 (1989), no. 1, 3–20. MR 1001434
(90c:80003), http://dx.doi.org/10.1007/BF01125883
 14.
Tommaso
Ruggeri and Srboljub
Simić, On the hyperbolic system of a mixture of Eulerian
fluids: a comparison between single and multitemperature models,
Math. Methods Appl. Sci. 30 (2007), no. 7,
827–849. MR 2310555
(2008e:35158), http://dx.doi.org/10.1002/mma.813
 15.
T. Ruggeri and S. Simić, Average temperature and Maxwellian iteration in multitemperature mixtures of fluids. Phys. Rev. E 80, 026317 (2009).
 16.
H. Gouin and T. Ruggeri, Identification of an average temperature and a dynamical pressure in a multitemperature mixture of fluids. Phys. Rev. E 78, 01630, (2008).
 17.
T. Ruggeri and S. Simić, Nonequilibrium temperatures in the mixture of gases via Maxwellian iteration. Submitted in Phys. Rev. E. (2012).
 18.
Krzysztof
Wilmanski, Continuum thermodynamics. Part I, Series on
Advances in Mathematics for Applied Sciences, vol. 77, World
Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. Foundations. MR 2482665
(2010f:74002)
 19.
K.
R. Rajagopal, On a hierarchy of approximate models for flows of
incompressible fluids through porous solids, Math. Models Methods
Appl. Sci. 17 (2007), no. 2, 215–252. MR 2292356
(2007k:76156), http://dx.doi.org/10.1142/S0218202507001899
 20.
C. Dafermos, Private Communication (Bologna 2011).
 21.
Guy
Boillat, Sur l’existence et la recherche
d’équations de conservation supplémentaires pour les
systèmes hyperboliques, C. R. Acad. Sci. Paris Sér. A
278 (1974), 909–912 (French). MR 0342870
(49 #7614)
 22.
Tommaso
Ruggeri and Alberto
Strumia, Main field and convex covariant density for quasilinear
hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H.
Poincaré Sect. A (N.S.) 34 (1981), no. 1,
65–84. MR
605357 (82b:76077)
 23.
Guy
Boillat and Tommaso
Ruggeri, Hyperbolic principal subsystems: entropy convexity and
subcharacteristic conditions, Arch. Rational Mech. Anal.
137 (1997), no. 4, 305–320. MR 1463797
(98h:82055), http://dx.doi.org/10.1007/s002050050030
 24.
Henning
Struchtrup and Manuel
Torrilhon, Regularization of Grad’s 13 moment equations:
derivation and linear analysis, Phys. Fluids 15
(2003), no. 9, 2668–2680. MR 2060065
(2005a:76142), http://dx.doi.org/10.1063/1.1597472
 25.
Yasushi
Shizuta and Shuichi
Kawashima, Systems of equations of hyperbolicparabolic type with
applications to the discrete Boltzmann equation, Hokkaido Math. J.
14 (1985), no. 2, 249–275. MR 798756
(86k:35107)
 26.
B.
Hanouzet and R.
Natalini, Global existence of smooth solutions for partially
dissipative hyperbolic systems with a convex entropy, Arch. Ration.
Mech. Anal. 169 (2003), no. 2, 89–117. MR 2005637
(2004h:35135), http://dx.doi.org/10.1007/s0020500302576
 27.
WenAn
Yong, Entropy and global existence for hyperbolic balance
laws, Arch. Ration. Mech. Anal. 172 (2004),
no. 2, 247–266. MR 2058165
(2005c:35195), http://dx.doi.org/10.1007/s0020500303043
 28.
Stefano
Bianchini, Bernard
Hanouzet, and Roberto
Natalini, Asymptotic behavior of smooth solutions for partially
dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl.
Math. 60 (2007), no. 11, 1559–1622. MR 2349349
(2010i:35227), http://dx.doi.org/10.1002/cpa.20195
 29.
Constantine
M. Dafermos, Hyperbolic conservation laws in continuum
physics, Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences], vol. 325, SpringerVerlag,
Berlin, 2000. MR
1763936 (2001m:35212)
 30.
Tommaso
Ruggeri and Denis
Serre, Stability of constant equilibrium state for dissipative
balance laws system with a convex entropy, Quart. Appl. Math.
62 (2004), no. 1, 163–179. MR 2032577
(2004k:35257)
 31.
Jie
Lou and Tommaso
Ruggeri, Acceleration waves and weak ShizutaKawashima
condition, Rend. Circ. Mat. Palermo (2) Suppl. 78
(2006), 187–200. MR 2210603
(2006k:35254)
 32.
T. Ruggeri, Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics, in Trends and Applications of Mathematics to Mechanics, STAMM 2002. SpringerVerlag 215, (2005).
 33.
T.
Ruggeri, Entropy principle and relativistic extended
thermodynamics: global existence of smooth solutions and stability of
equilibrium state, Nuovo Cimento Soc. Ital. Fis. B
119 (2004), no. 79, 809–821. MR 2136908
(2006g:80002)
 34.
T. Ruggeri, Extended Relativistic Thermodynamics. Section inserted in the book of Yvonne Choquet Bruhat, General Relativity and the Einstein equations, pp. 334340. Oxford Univ. Press, ISBN 9780199230723, (2009).
 1.
 C. Truesdell, W. Noll and S. S. Antman, The nonlinear field theories of mechanics, Volume 3. Springer, 1602 (2004). MR 0193816 (33:2030)
 2.
 B. Coleman and W. Noll, The thermomechanics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13, 167178 (1963). MR 0153153 (27:3122)
 3.
 I. Müller, On the entropy inequality, Arch. Rational Mech. Anal. 26, 118141 (1967). MR 0214336 (35:5187)
 4.
 I. Müller, On the frame dependence of stress and heat flux, Arch. Rational Mech. Anal. 45, 241 (1972). MR 1553565
 5.
 A. Bressan, On relativistic heat conduction in the stationary and nonstationary cases, the objectivity principle and piezoelasticity, Lett. Nuovo Cimento, 33 (4), 108 (1982).
 6.
 T. Ruggeri, Generators of hyperbolic heat equation in nonlinear thermoelasticity, Rend. Sem. Mat. Padova, 68, 79 (1982). MR 702148 (85e:73066)
 7.
 I. Müller and T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy 37, 2nd Ed., Springer Verlag (1998). MR 1632151 (99h:80001)
 8.
 H. Grad, On the kinetic theory of rarefied gases. Comm. Appl. Math. 2, 331 (1949). MR 0033674 (11:473a)
 9.
 IS. Liu and I. Müller, Extended thermodynamics of classical and degenerate ideal gases, Arch. Rational Mech. Anal. 83, (4), 285332 (1983). MR 714978 (85j:80001)
 10.
 E. Ikenberry and C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, J. Rational Mech. Anal. 5, 1 (1956). MR 0075725 (17:796c)
 11.
 T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama, Extended thermodynamics of dense gases, Continuum Mech. Thermodyn. DOI 10.1007/s001610110213x (2011).
 12.
 C. Truesdell, Rational Thermodynamics, McGrawHill, New York, 1969. MR 0366236 (51:2484)
 13.
 T. Ruggeri, Galilean Invariance and Entropy Principle for Systems of Balance Laws. The Structure of the Extended Thermodynamics, Contin. Mech. Thermodyn. 1, 3 (1989). MR 1001434 (90c:80003)
 14.
 T. Ruggeri and S. Simić, On the Hyperbolic System of a Mixture of Eulerian Fluids: A Comparison Between Single and MultiTemperature Models. Math. Meth. Appl. Sci., 30, 827 (2007). MR 2310555 (2008e:35158)
 15.
 T. Ruggeri and S. Simić, Average temperature and Maxwellian iteration in multitemperature mixtures of fluids. Phys. Rev. E 80, 026317 (2009).
 16.
 H. Gouin and T. Ruggeri, Identification of an average temperature and a dynamical pressure in a multitemperature mixture of fluids. Phys. Rev. E 78, 01630, (2008).
 17.
 T. Ruggeri and S. Simić, Nonequilibrium temperatures in the mixture of gases via Maxwellian iteration. Submitted in Phys. Rev. E. (2012).
 18.
 K. Wilmanski, Continuum Thermodynamics  Part 1: Foundations. World Scientific, Singapore, 2008. MR 2482665 (2010f:74002)
 19.
 R. Rajakopal, On a Hierarchy of Approximate Models for Flows of Incompressible Fluids through Porous Solids. Mathematical Models and Methods in Applied Sciences Vol. 17, No.2, 215252 (2007). MR 2292356 (2007k:76156)
 20.
 C. Dafermos, Private Communication (Bologna 2011).
 21.
 G. Boillat, Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques. C.R. Acad. Sc. Paris, 278A, 909912 (1974). Non Linear Fields and Waves. In CIME Course, Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics 1640, T. Ruggeri Ed. SpringerVerlag, 103152 (1995). MR 0342870 (49:7614)
 22.
 T. Ruggeri and A. Strumia, Main field and convex covariant density for quasilinear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincaré 34 A, 65 (1981). MR 605357 (82b:76077)
 23.
 G. Boillat and T. Ruggeri, Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic conditions. Arch. Rat. Mech. Anal. 137, 305320 (1997). MR 1463797 (98h:82055)
 24.
 H. Struchtrup and M. Torrilhon, Regularization of Grad's moment equations: Derivation and linear analysis, Phys. Fluids, 15 pp. 26682680, (2003). MR 2060065 (2005a:76142)
 25.
 Y. Shizuta and S. Kawashima, Systems of equations of hyperbolicparabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14, 249 (1985). MR 798756 (86k:35107)
 26.
 B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rat. Mech. Anal. 169 89 (2003). MR 2005637 (2004h:35135)
 27.
 WenAn Yong, Entropy and global existence for hyperbolic balance laws. Arch. Rat. Mech. Anal. 172 , no. 2, 247266 (2004). MR 2058165 (2005c:35195)
 28.
 S. Bianchini, B. Hanouzet and R.Natalini, Asymptotic Behavior of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy. Comm. Pure Appl. Math., Vol. 60, 1559 (2007). MR 2349349 (2010i:35227)
 29.
 C. Dafermos, Hyperbolic conservation laws in continuum physics, Springer, Berlin, 2000. MR 1763936 (2001m:35212)
 30.
 T. Ruggeri and D. Serre, Stability of constant equilibrium state for dissipative balance laws system with a convex entropy, Quart. Appl. Math, 62 (1), 163 (2004). MR 2032577 (2004k:35257)
 31.
 J. Lou and T. Ruggeri, Acceleration Waves and Weak ShizutaKawashima Condition, Suppl. Rend. Circ. Mat. Palermo Nonlinear Hyperbolic Fields and Waves. A tribute to Guy Boillat, Serie II, Numero 78, pp. 187200 (2006). MR 2210603 (2006k:35254)
 32.
 T. Ruggeri, Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics, in Trends and Applications of Mathematics to Mechanics, STAMM 2002. SpringerVerlag 215, (2005).
 33.
 T. Ruggeri, Entropy principle and Relativistic Extended Thermodynamics: Global existence of smooth solutions and stability of equilibrium state. Il Nuovo Cimento B, 119 (79), 809821 (2004). Lecture notes of the International Conference in honour of Y. ChoquetBruhat: Analysis, Manifolds and Geometric Structures in Physics, G. Ferrarese and T. Ruggeri Eds. (2004). MR 2136908 (2006g:80002)
 34.
 T. Ruggeri, Extended Relativistic Thermodynamics. Section inserted in the book of Yvonne Choquet Bruhat, General Relativity and the Einstein equations, pp. 334340. Oxford Univ. Press, ISBN 9780199230723, (2009).
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Additional Information
Tommaso Ruggeri
Affiliation:
Department of Mathematics & Research Center of Applied Mathematics, University of Bologna, Via Saragozza 8, 40123 Bologna, Italy
Email:
tommaso.ruggeri@unibo.it
DOI:
http://dx.doi.org/10.1090/S0033569X2012013143
PII:
S 0033569X(2012)013143
Keywords:
Constitutive equations,
objective principle,
hyperbolic systems of balance laws
Received by editor(s):
January 19, 2012
Published electronically:
May 9, 2012
Additional Notes:
The author was supported in part by GNFM of INdAM
Dedicated:
Dedicated to Constantine Dafermos for his 70th birthday
Article copyright:
© Copyright 2012 Brown University
