Can constitutive relations be represented by non-local equations?

Transcription

1 Can constitutive relations be represented by non-local equations? Tommaso Ruggeri Dipartimento di Matematica & Centro di Ricerca per le Applicazioni della Matematica (CIRAM) Universitá di Bologna Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments A workshop in honor of Francesco Mainardi Bilbao - Basque Country - Spain 6-8 November 2013

2 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

3 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

4 The Physical laws in continuum theories are balance laws: Let F 0 (x, t); x Ω, t R +, a generic density. The time derivative in the domain Ω is expressed by d dt Ω F 0 dω = Σ G i n i dσ + fdω, Ω where the first integral on the r.h.s. represents the flux of some quantities G i trough the surface Σ of unit normal n and velocity v, while the last integral represents the productions. Under regularity assumptions the system can be put in the local form: F 0 t + Fi x i = f, F i = F 0 v i + G i

5 For example in the case of fluids: ρ t + ρv i x i = 0 (mass balance) (ρv j ) + t x i (ρv iv j t ij ) = 0 (balance of momentum) E t + x i (Ev i + q i t ij v j ) = 0 (energy conservation). where E = ρ v ρε and ρ, v (v i), t (t ij ), q (q i ), ε are the density, the velocity, the stress tensor, the heat flux and the internal energy. Of course the system is not closed and we need the so called constitutive equations.

6 Constitutive Equations A very roughly definition of constitutive equations can be considered as the equations that need to close the system i.e., choosing a field u R N we have to give the relations between the 5N vectors F 0, F i and f and the field of the unknowns u. This definition have physical meaning? For long time the constitutive equation was done in empiric way and they belong substantially in three big class:

7 Local constitutive equations: Examples are: 1 stress-deformation relations in non-linear elasticity t t(e) (the Hook law in the linear case); 2 The caloric and thermal equations of state in Euler fluids that connect internal energy and pressure as function of density and temperature e e(ϱ, T ), p p(ϱ, T ).

8 Non local type (in space) Examples are i) the Fourier law q = L grad(1/t ), ii) the Navier-Stokes equations σ = 2 ν D D + λ divv I in the case of a single dissipative fluids D = 1/2( v + ( v) T ), iii) the Fourier, Navier-Stokes and Fick law in the case of a mixture of dissipative fluids: σ = 2 ν D D + λ divv I, ( ) 1 n 1 ( µb µ n q = L grad + L b grad T T b=1 ( ) 1 n 1 ( µb µ n J a = L a grad L ab grad T T b=1 ), (1) ) ;

9 Non local type (in time): Constitutive equations with memory Examples are: the visco-elasticity or in general all materials in which the stress depends not only of deformations but also of the history of the deformation. In the modern constitutive theory all the constitutive equations must obey the two principles:

10 The objectivity principle: the balance laws are invariant with respect Galilean transformations and the proper constitutive equations are independent of the Observer; The second principle of thermodynamics that in the Rational Thermodynamics requires that any solutions of the full system satisfies the inequality (Coleman-Noll Müller 1967): ρs t + x i ( ρsv i + Φ i) 0 for all processes

11 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

12 The Frame-Dependence of the Heat Flux Müller (1972):

13 Fourier law is a constitutive equation? Question (Aldo Bressan Lett. Nuovo Cimento (1982) & T. Ruggeri - Rend. Sem. Mat. Padova (1982)): It is wrong the Objective Principle or the heat equation is not a constitutive equation? The answer was done 10 years later by Extended Thermodynamics!

14 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

15 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T The Extended Thermodynamics The kinetic theory describes the state of a rarefied gas through the phase density f (x, t, c), where f (x, t, c)dc is the number density of atoms at point x and time t that have velocities between c and c + dc. The phase density obeys the Boltzmann equation f t + f ci x i = Q (2) where Q represents the collisional terms. Most macroscopic thermodynamic quantities are identified as moments of the phase density F k1 k 2 k j = fc k1 c k2 c kj dc, and due to the Boltzmann equation (2), the moments satisfy an infinity hierarchy of balance laws in which the flux in one equation becomes the density in the next one:

16 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T t F + i F i = 0 t F k1 + i F ik1 = 0 t F k1 k 2 + i F ik1 k 2 = P <k1 k 2 > t F k1 k 2 k 3 + i F ik1 k 2 k 3 = P k1 k 2 k 3. t F k1 k 2...k n + i F ik1 k 2...k n = P k1 k 2...k n.

17 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Taking into account that P kk = 0, the first five equations are conservation laws and coincides with the mass, momentum and energy conservation respectively, while the remaining ones are balance laws. When we cut the hierarchy at the density with tensor of rank n, we have the problem of closure because the last flux end the production terms are not in the list of the densities.

18 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T The idea of Rational Extended Thermodynamics Müller and Ruggeri (Springer - Verlag 1993, 1998) was to view the truncated system as a phenomenological system of continuum mechanics and then we consider the new quantities as constitutive functions: F k1 k 2...k nk n+1 F k1 k 2...k nk n+1 (F, F k1, F k1 k 2,... F k1 k 2...k n ) P k1 k 2...k j P k1 k 2...k j (F, F k1, F k1 k 2,... F k1 k 2...k n ) 2 j n. According with the continuum theory, the restrictions on the constitutive equations come only from universal principles, i.e.: Entropy principle, Objectivity Principle and Causality and Stability (convexity of the entropy).

19 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T The restrictions are so strong (in particular the entropy principle) that, at least, for processes not to far from the equilibrium the system is completely closed. In the case of 13 moments the results are in perfect agreement with the kinetic closure procedure proposed by Grad (1949) (Liu- Müller (1983)).

20 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T In the one-dimensional case and usual symbols the equations are: ρ + ρv x = 0 ρ v + (p σ) x = 0 ρ ε + q x + (p σ)v x = 0 (3) [ τ σ σ 8 15 q x σv ] x 4 3 µv x = σ [ ] τ q q qv x 7 2 ( p ρ ) xσ 1 ρ (p + σ)σ x + σ ρ p x + χt x = q

21 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Maxwellian Iteration To reveal the relation between extended and classical model, a formal iterative scheme known as Maxwellian iteration will be applied. The first iterates q (1) and σ (1) are calculated from the right hand sides of balance laws by putting "zero th " iterates equilibrium values q (0) = 0 and σ (0) = 0 on the left hand sides. In the next step second iterates q (2) and σ (2) are obtained from the right hand sides of the same Eqs. by putting first q (1) and σ (1) on their left hand sides, an so on. Therefore the Fourier and Navier Stokes equations are a limit case of the previous balance law and not true constitutive equations!

22 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

23 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T The Mixture theory α = 1,..., n; c α = ρ n α ρ, ρ = α=1 ρ α Tα P ρ α vα

24 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T - Classical theory with a single Temperature - CST Fields: ρ α, v, T (4 + n) v = 1 ρ n ρ α v α α=1 Balance Laws: Balance law of mass of each constituents; Global balance law of total momentum; Global energy conservation. The diffusion flux vector J α = ρ α (v α v) is given by constitutive equation: the FICK law.

25 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T - Classical theory with a single Temperature - CST Fields: ρ α, v, T (4 + n) v = 1 ρ n ρ α v α α=1 Balance Laws: Balance law of mass of each constituents; Global balance law of total momentum; Global energy conservation. The diffusion flux vector J α = ρ α (v α v) is given by constitutive equation: the FICK law.

26 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T - Classical theory with a single Temperature - CST Fields: ρ α, v, T (4 + n) v = 1 ρ n ρ α v α α=1 Balance Laws: Balance law of mass of each constituents; Global balance law of total momentum; Global energy conservation. The diffusion flux vector J α = ρ α (v α v) is given by constitutive equation: the FICK law.

27 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T - Classical theory with a single Temperature - CST Fields: ρ α, v, T (4 + n) v = 1 ρ n ρ α v α α=1 Balance Laws: Balance law of mass of each constituents; Global balance law of total momentum; Global energy conservation. The diffusion flux vector J α = ρ α (v α v) is given by constitutive equation: the FICK law.

28 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T - Classical theory with a single Temperature - CST Fields: ρ α, v, T (4 + n) v = 1 ρ n ρ α v α α=1 Balance Laws: Balance law of mass of each constituents; Global balance law of total momentum; Global energy conservation. The diffusion flux vector J α = ρ α (v α v) is given by constitutive equation: the FICK law.

29 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T - Classical theory with a single Temperature - CST Fields: ρ α, v, T (4 + n) v = 1 ρ n ρ α v α α=1 Balance Laws: Balance law of mass of each constituents; Global balance law of total momentum; Global energy conservation. The diffusion flux vector J α = ρ α (v α v) is given by constitutive equation: the FICK law.

30 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T - Classical theory with a single Temperature - CST Fields: ρ α, v, T (4 + n) v = 1 ρ n ρ α v α α=1 Balance Laws: Balance law of mass of each constituents; Global balance law of total momentum; Global energy conservation. The diffusion flux vector J α = ρ α (v α v) is given by constitutive equation: the FICK law.

31 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T In correspondence of the field U (ρ α, v, T ) T we have the usual field equations of the classical approach: dρ + ρ div v = 0, dt ρ dv dt div t = 0, ρ dε dt t grad v + div q = 0, ρ dc b dt + div J b = 0, (b = 1, n 1),

32 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T and "constitutive equations" given by Fourier, Navier-Stokes and Fick: σ = 2 ν D D + λ divv I, ( ) 1 n 1 ( µb µ n q = L grad + L b grad T T b=1 ( ) 1 n 1 ( µb µ n J a = L a grad L ab grad T T b=1 ), (4) ) ;

33 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T The Extended Theory of Mixture Description of simple homogeneous mixtures in the context of rational thermodynamics relies on the postulate that each constituent obeys the same balance laws as a single fluid. They express rates of change of mass, momentum and energy with appropriate production terms due to mutual interaction of constituents: ρ α + div(ρ α v α ) = τ α t (ρ α v α ) + div(ρ α v α v α t α ) = m α, (α = 1, 2,..., n) (5) t ( 1 2 ρ αvα 2 ) {( ) } + ρ α ε α 1 + div t 2 ρ αvα 2 + ρ α ε α v α t α v α + q α = e α, with ρ α the density, v α the velocity, ε α the internal energy, t α the stress tensor and q α the heat flux of the αconstituent of the mixture.

34 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T The production terms must satisfy the following relations: n τ α = 0; α=1 n m α = 0; α=1 n e α = 0, α=1 due to conservation of mass, momentum and energy of the mixture. By the proper choice of field variables balance laws (5) have to be adjoined with the constitutive equations in order to close the system and make it compatible with the second principle of thermodynamics. Let us give a brief review of the models which could be proposed in the mixture theory.

35 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Firstly, one may obtain from (5) conservation laws of mass, momentum and energy of the mixture (as a whole) through summation of appropriate balance laws and introduction of following quantities: ρ = n α=1 ρ α total mass density, v = 1 ρ n α=1 ρ αv α mixture velocity, u α = v α v ( n α=1 ρ αu α = 0 ) diffusion velocity, t = n α=1 (t α ρ α u α u α ) stress tensor ε = 1 n ρ α=1 ρ ( ) α εα u2 α internal energy q = n α=1 { qα + ρ α ( εα u2 α) uα t α u α } flux of internal energy.

36 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T In such a way conservation laws read: ρ + div(ρv) = 0; t (ρv) + div(ρv v t) = 0; (6) t ( 1 2 ρv 2 + ρε ) {( ) } 1 + div t 2 ρv 2 + ρε v tv + q = 0.

37 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Multi temperatures Mixture: ρ + div(ρv) = 0; t (ρv) + div(ρv v t) = 0 t ( 1 2 ρv 2 + ρε ) {( 1 + div t 2 ρv 2 + ρε ) } v tv + q = 0; ρ b t + div(ρ bv b ) = τ b ; (7) (ρ b v b ) + div(ρ b v b v b t b ) = m b ; (b = 1,..., n 1); {( ) } 1 2 ρ bvb 2 + ρ b ε b v b t b v b + q b = e b t ( 1 2 ρ bvb 2 + ρ ) bε b + div t The unknown field is u = (ρ α, v α, T α ) T

38 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T In the classical approach we have instead: ρ + div(ρv) = 0; t (ρv) + div(ρv v t) = 0 t ( 1 2 ρv 2 + ρε ) {( 1 + div t 2 ρv 2 + ρε ) } v tv + q = 0; ρ b t + div(ρ bv b ) = τ b ; (8) (ρ b v b ) + div(ρ b v b v b t b ) = m b ; t ( 1 2 ρ bvb 2 + ρ ) {( ) } bε b 1 + div t 2 ρ bvb 2 + ρ b ε b v b t b v b + q b = e b The unknown field is u = (ρ α, v, T ) T

39 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T The first constraints for the production terms come from the requirement of Galilean invariance for the balance equations and we can prove: Galilean Invariance: τ b = ˆτ b ; m j b = ˆτ bv j + ˆm j b ; (9) v 2 e b = ˆτ b 2 + ˆmk b v k + ê b.

40 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Another important restriction comes from entropy inequality, that in the case without chemical reaction implies: Entropy Principle: n 1 ˆm b = c=1 ( uc ψ bc u ) n T c T n n 1 ; ê b = c=1 θ bc ( 1 T c + 1 T n ), where ψ bc and θ bc are semi definite positive matrix functions of ρ α and T α.

41 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T Maxwellian Iteration By applying the Maxwellian iteration scheme to the momentum equation of single constituents (ρ b v b ) t + div(ρ b v b v b t b ) = m b after some simple manipulations with the one obtains the first iterate of diffusion flux that coincide with the Fick law: n 1 ( ) ( ) µn µ b 1 J a = L ab grad + L a grad. T T b=1

42 Balance Laws and Constitutive equations The Frame-Dependence Maxwellian of theiteration Heat Flux. The Extended Thermodynamics T While we have also as limit of energy balances ( 1 2 ρ bvb 2 + ρ ) {( ) } bε b 1 + div t 2 ρ bvb 2 + ρ bε b v b t b v b + q b = e b the new constitutive equations for the temperature differences: Θ a n 1 = M ab r b divv, (10) b=1 where for perfect gases r b = ρ b c (b) V (γ b γ n ) Eq. (10) is not obtained in classical theory.

43 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

44 Using Maxwellian iteration we can obtain by Extended Thermodynamics- at least formally - the usual non-local constitutive equations of Classical theory. Therefore the parabolic systems of classical theories appears, from physical point of view, approximations of hyperbolic systems when some relaxation times are negligible. Examples are: Fourier and Navier Stokes as limit case of Grad 13 moments equations of Extended Thermodynamics; Fick law as limit case of momentum equations of each species in mixture with single-temperature; The new equation for difference of temperatures in mixture with multi-temperature. The Darcy law for porous material is a limit case of momentum equation.

45 The advantage of the non local equations Nevertheless that previous non-local equations also if are not constitutive equations but approximations of balance law are very useful in practice. In fact first in many normal physical situation the relaxation time are very negligible and they are important only in limit case: rarefied gas, low temperature for example. Another big advantage that the non local equations permit to measure non-observable quantities like heat flux, viscosity stress and in particular the velocities and the temperature of each species in a mixture of fluids.

46 Balance Laws and Constitutive equations The Frame-Dependence of the Heat Flux. The Extended Thermodynamics T Still remain open the rigorous prove of convergence of the solutions via Maxwellian iterations. Moreover a very subtle point arise: Entropy Principle and Maxwellian Iteration The questions is: The entropy principle is preserved in the Maxwellian iteration? In other words the conditions such that the hyperbolic theory satisfy the entropy inequality implies that the corresponding parabolic limit system satisfy also an entropy inequality? It is possible to prove a simple counter-example (Dafermos Ruggeri) that in general this is not true: T. Ruggeri Can constitutive relations be represented by non-local equa

47 Let u R n, v R m t v i + λa iα t u α + µ x v i + B iα x u α = 0 t u α + λa iα t v i + B iα x v i = u α Putting u = 0 in the left side t v i + µ x v i = 0 λa iα t v i + B iα x v i = u α eliminating t v i u α (λµa iα B iα ) x v i. Substituting into the first equation: t v j B jα x u α = (B iα B jα λµa iα B jα ) 2 x v i Fix A and B, take λ << 1 so entropy u 2 + v 2 convex, then choose µ >> 1 with sign that destroys positive definitess.

48 We need to add others physical requirements: the conservative quantities are equilibrium quantities and process are not far from equilibrium (Ruggeri 2011). A sketch is done in the linear case: Let u R n, v R m t v + A x v + B x u = 0 t u + B T x v + C t v = L u where A(m m), C(n n) Sym and L(n n) Sym + for the entropy principle. Then after Maxwellian iteration we obtain: t v + A x v D xx v = 0 and the diffusion matrix D = BL 1 B T Sym +.

49 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

50 The Mathematical Situation From mathematical point of view seem on the contrary that hyperbolic systems are particular case of parabolic ones. The prototype is Burgers equation with artificial viscosity to restore uniqueness of weak solutions: u t + uu x = µu xx. Moreover parabolic equations seem from mathematical point more realistic because usual exists regular solutions at the contrary of hyperbolic one for which soon regular solution become shock or they blow up. I want now to convince that this is not completely true also from mathematical point of view. The first question is in reality in my opinion a misunderstanding because at least are the hyperbolic sub-systems a particular case of the parabolic system and not the system. I want to explain this with the simple example of Grad system.

51 Grad System: ρ + ρv x = 0 ρ v + (p σ) x = 0 ρ ε + q x + (p σ)v x = 0 [ τ σ σ 8 15 q x σv ] x 4 3 µv (11) x = σ [ ] τ q q qv x 7 2 ( p ρ ) xσ 1 ρ (p + σ)σ x + σ ρ p x + χt x = q τ σ εµ; τ q εχ

52 EULER χ > 0 χ > 0 μ > 0 μ > 0 GRAD ε > 0 FNS

53 EULER χ > 0 χ > 0 µ > 0 µ > 0 GRAD ε > 0 FNS τ > 0 τ > 0 More Moments e.g. 21 Moments Maxwellian Iteration Regularized Moments e.g. Regularized Grad

54 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

55 Systems of mixed type In physical dissipative case the hyperbolic systems are of mixed type, some equations are conservation laws and other ones are real balance laws, i.e., we are in the case in which with F(u) u t + i F i (u) = F(u) ( 0 g(u) ) ; g R N M. In this case the coupling condition discovered for the first time by Kawashima and Shizuta (K-condition) such that the dissipation present in the second block have effect also to the first block of equation plays a very important role in this case for global existence of smooth solutions.

56 Global Existence of smooth solutions In fact, if the system of balance law is endowed with a convex entropy law, and it is dissipative, then the K-condition becomes a sufficient condition for the existence of global smooth solutions provided that the initial data are sufficiently smooth (Hanouzet and Natalini 2003, Wen-An Yong 2004, Bianchini, Hanouzet and Natalini 2007) Theorem (Global Existence) Assume that the system of balance laws is strictly dissipative and the K-condition is satisfied. Then there exists δ > 0, such that, if u(x, 0) 2 δ, there is a unique global smooth solution, which verifies u C 0 ( [0, ); H 2 (R) C 1 ( [0, ); H 1 (R). )

57 Moreover Ruggeri and Serre have proved in the one-dimensional case that the constant states are stable: Theorem (Stability of Constant State) Under natural hypotheses of strongly convex entropy, strict dissipativeness, genuine coupling and zero mass initial for the perturbation of the equilibrium variables, the constant solution stabilizes ( u(t) 2 = O t 1/2). Lou and Ruggeri have observed that the weaker K-condition in which we require the K-condition only for the right eigenvectors corresponding to genuine non linear is a necessary (but not sufficient) condition for the global existence of smooth solutions.

58 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

59 In conclusion Extended Thermodynamics seem to indicate in clear manner that non local relations are not constitutive equations but approximations of balance laws. The true constitutive equations are in local from and they obey the material frame difference. The physical systems are hyperbolic in agreement with relativity principle that any disturbance propagate with finite speed. Nevertheless the usual Fourier, Navier-Stokes, Fick, Darcy and others non local equations are useful to measure non-observable quantities and they are good approximation in many practical problems. Hyperbolic systems with dissipation (balance laws with production terms) can have global smooth solutions provide the initial data are small.

60 In conclusion Extended Thermodynamics seem to indicate in clear manner that non local relations are not constitutive equations but approximations of balance laws. The true constitutive equations are in local from and they obey the material frame difference. The physical systems are hyperbolic in agreement with relativity principle that any disturbance propagate with finite speed. Nevertheless the usual Fourier, Navier-Stokes, Fick, Darcy and others non local equations are useful to measure non-observable quantities and they are good approximation in many practical problems. Hyperbolic systems with dissipation (balance laws with production terms) can have global smooth solutions provide the initial data are small.

61 In conclusion Extended Thermodynamics seem to indicate in clear manner that non local relations are not constitutive equations but approximations of balance laws. The true constitutive equations are in local from and they obey the material frame difference. The physical systems are hyperbolic in agreement with relativity principle that any disturbance propagate with finite speed. Nevertheless the usual Fourier, Navier-Stokes, Fick, Darcy and others non local equations are useful to measure non-observable quantities and they are good approximation in many practical problems. Hyperbolic systems with dissipation (balance laws with production terms) can have global smooth solutions provide the initial data are small.

62 In conclusion Extended Thermodynamics seem to indicate in clear manner that non local relations are not constitutive equations but approximations of balance laws. The true constitutive equations are in local from and they obey the material frame difference. The physical systems are hyperbolic in agreement with relativity principle that any disturbance propagate with finite speed. Nevertheless the usual Fourier, Navier-Stokes, Fick, Darcy and others non local equations are useful to measure non-observable quantities and they are good approximation in many practical problems. Hyperbolic systems with dissipation (balance laws with production terms) can have global smooth solutions provide the initial data are small.

63 In conclusion Extended Thermodynamics seem to indicate in clear manner that non local relations are not constitutive equations but approximations of balance laws. The true constitutive equations are in local from and they obey the material frame difference. The physical systems are hyperbolic in agreement with relativity principle that any disturbance propagate with finite speed. Nevertheless the usual Fourier, Navier-Stokes, Fick, Darcy and others non local equations are useful to measure non-observable quantities and they are good approximation in many practical problems. Hyperbolic systems with dissipation (balance laws with production terms) can have global smooth solutions provide the initial data are small.

64 Outline 1 Balance Laws and Constitutive equations 2 The Frame-Dependence of the Heat Flux. 3 The Extended Thermodynamics 4 The Mixture theory 5 Maxwellian iteration and constitutive equations 6 The Mathematical Situation 7 Qualitative Analysis 8 Conclusions 9 Bibliography

65 Bibliography B. Coleman, W. Noll, The thermomechanics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13, 167 (1963) I, Müller, On the entropy inequality. Arch. Rational Mech. Anal. 26, 118 (1967) I, Müller, On the frame dependence of stress and heat flux. Arch. Rational Mech. Anal.45, 241 (1972)

66 A. Bressan, On relativistic heat conduction in the stationary and nonstationary cases, the objectivity principle and piezoelasticity Lett. Nuovo Cimento, 33 (4), 108 (1982) T. Ruggeri, Generators of hyperbolic heat equation in nonlinear thermoelasticity Rend. Sem. Mat. Padova, 68, 79 (1982) I. Müller, T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy 37, 2nd Ed., Springer Verlag (1998). H. Grad, On the kinetic theory of rarefied gases. Comm. Appl. Math. 2, 331 (1949)

67 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). T. Ruggeri, S. Simić, On the Hyperbolic System of a Mixture of Eulerian Fluids: A Comparison Between Single- and Multi-Temperature Models. Math. Meth. Appl. Sci., 30, 827 (2007). H. Gouin and T. Ruggeri, Identification of an average temperature and a dynamical pressure in a multi-temperature mixture of fluids. Phys. Rev. E 78, 01630, (2008). T. Ruggeri and S. Simić, Average temperature and Maxwellian iteration in multitemperature mixtures of fluids. Phys. Rev. E 80, (2009).

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