# Open Mathematical Aspects of Continuum Thermodynamics: Hyperbolicity, Boundaries and Nonlinearities

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Classical Irreversible Thermodynamics

- (1)
- to ensure a unified thermodynamical background for the classical transport equations, e.g., for Fourier heat conduction, Fick diffusion, and Newton and Stokes viscosity laws; and,
- (2)
- to discuss different cross-effects systematically, such as Soret-, Dufour-, Peltier-, and Seebeck-effects.

- Conservation of mass implies the so-called continuity equation$$\begin{array}{c}\hfill \dot{\rho}+\rho {\partial}_{i}{v}_{i}=0.\end{array}$$
- Cauchy’s first equation of motion$$\begin{array}{c}\hfill \rho {\dot{v}}_{i}={\partial}_{j}{\sigma}_{ij}+\rho {g}_{i}\end{array}$$
- The conservation of angular momentum implies the symmetry of the Cauchy stress, i.e.,$$\begin{array}{c}\hfill {\sigma}_{ij}={\sigma}_{ji},\end{array}$$
- Starting with the conservation of total energy, one can derive the balance of internal energy or the first law of thermodynamics, which reads as$$\begin{array}{c}\hfill \rho \dot{e}=-{\partial}_{i}{q}_{i}+{\sigma}_{ij}{\partial}_{j}{v}_{i}+{q}_{V},\end{array}$$

- There exists independent thermodynamical bodies, which can be described by extensive state quantities and intensive state functions. One body is characterized by N extensive state quantities (${X}_{a},\phantom{\rule{4pt}{0ex}}a\in 1,2,\cdots ,N$). The thermodynamical state space is spanned by the Descartes product of all ${X}_{a}$. We will denote the intensive state functions with ${Y}_{a},\phantom{\rule{4pt}{0ex}}a\in 1,2,\cdots ,N$, which are the functions of the extensive ones. For fluids and gases with one component, the extensive state quantities are the mass M, the volume V, and the internal energy E.
- The entropy is the potential function of the vector space spanned by the intensive thermodynamical state quantities, characterized by the Gibbs relation as$$\begin{array}{c}\hfill \mathrm{d}S\left({X}_{1},{X}_{2},\cdots ,{X}_{N}\right)=\sum _{a=1}^{N}{Y}_{a}\mathrm{d}{X}_{a}.\end{array}$$In other words, the intensive state functions are the partial derivatives of entropy, i.e.,$$\begin{array}{cccc}\hfill {Y}_{a}\left({X}_{1},{X}_{2},\cdots ,{X}_{N}\right)& ={\left.\frac{\partial S}{\partial {X}_{a}}\right|}_{{X}_{b},\phantom{\rule{4pt}{0ex}}a\ne b},\hfill & \hfill a,b& =1,2,\cdots ,N.\hfill \end{array}$$For fluids and gases, the Gibbs relation is$$\begin{array}{c}\hfill \mathrm{d}S(E,V,m)=\frac{1}{T}\mathrm{d}E+\frac{p}{T}\mathrm{d}V-\frac{\mu}{T}\mathrm{d}M,\end{array}$$$$\begin{array}{cccccc}\hfill \frac{1}{T}(E,V,m)& ={\left.\frac{\partial S}{\partial E}\right|}_{V,M},\hfill & \hfill \frac{p}{T}(E,V,m)& ={\left.\frac{\partial S}{\partial V}\right|}_{E,M},\hfill & \hfill -\frac{\mu}{T}(E,V,m)& ={\left.\frac{\partial S}{\partial M}\right|}_{E,V},\hfill \end{array}$$
- The entropy is an extensive thermodynamical function, called locality condition. More precisely,
- (a)
- the entropy is an Euler homogeneous function of any of its variables, e.g.,$$\begin{array}{ccc}\hfill S\left(\lambda {X}_{1},{X}_{2},\cdots ,{X}_{N}\right)& =\lambda S\left({X}_{1},{X}_{2},\cdots ,{X}_{N}\right),\hfill & \hfill \lambda \in {\mathbb{R}}^{+}.\end{array}$$
- (b)
- for all scalar ${X}_{A}$, one can be introduce the ${X}_{A}$-specific entropy, which is the function of the corresponding specific quantities, e.g., the ${X}^{1}$-specific entropy s is defined as$$\begin{array}{c}\hfill S\left({X}_{1},{X}_{2},\cdots ,{X}_{N}\right)={X}_{1}s\left(\frac{{X}_{2}}{{X}_{1}},\cdots ,\frac{{X}_{N}}{{X}_{1}}\right).\end{array}$$Usually, the mass specific entropy $s(e,v)$ and the entropy density ${\rho}_{S}\left({\rho}_{E},\rho \right)$ is used, where e is the (mass) specific internal energy, v is the specific volume, and ${\rho}_{E}=\frac{E}{V}$ is the internal energy density.
- (c)
- The Euler relation for entropy reads as$$\begin{array}{c}\hfill S\left({X}_{1},{X}_{2},\cdots ,{X}_{N}\right)=\sum _{a=1}^{N}{Y}_{a}\left({X}_{1},{X}_{2},\cdots ,{X}_{N}\right){X}_{a},\end{array}$$$$\begin{array}{c}\hfill S(E,V,M)=\frac{1}{T}E+\frac{p}{T}V-\frac{\mu}{T}M.\end{array}$$In the light of (13), the entropy can be given as$$\begin{array}{c}\hfill S(E,V,M)=Ms(e,v)=M\left(\frac{1}{T}e+\frac{p}{T}v-\frac{\mu}{T}\right),\end{array}$$

As a consequence of statement 3, the Gibbs relation follows for $s(e,v)$:$$\begin{array}{c}\hfill \mathrm{d}s=\frac{1}{T}\mathrm{d}e+\frac{p}{T}\mathrm{d}v.\end{array}$$ - The ${X}_{1}$-specific entropy is a concave function of its variables, i.e.,$$\begin{array}{cccc}\hfill det{\partial}_{{x}_{a},{x}_{b}}s({x}_{2},\cdots ,{x}_{N})& \le 0,\hfill & \hfill a,b& =2,\cdots ,N.\hfill \end{array}$$This statement leads to the internal or material stability criteria. Particularly for fluids and gases, these criteria are$$\begin{array}{cccc}\hfill {c}_{v}& :={\left.\frac{\partial e(T,v)}{\partial T}\right|}_{v}>0,\hfill & \hfill {\chi}_{T}& :=-\frac{1}{v}{\left.\frac{\partial p(T,v)}{\partial v}\right|}_{T}>0,\hfill \end{array}$$

#### Exploiting the Second Law

**Example**

**1.**

**Heat conduction in rigid bodies—Fourier heat conduction.**Here, we consider a rigid body at rest w.r.t. a given reference frame, thus both the velocity field and the volume change are identically zero. Furthermore, we also neglect the volumetric heat source density. Consequently, the balance of internal energy and the Gibbs relation are

- First kind or Dirichlet boundary condition: the temperature is prescribed on the boundary:$$\begin{array}{c}\hfill T(t,{r}_{i}){|}_{\partial \mathcal{B}}={T}_{\partial \mathcal{B}}\left(t\right),\end{array}$$here, t and ${r}_{i}$ are the time and space coordinates, respectively.
- Second kind or Neumann boundary condition: the normal component of the heat flux is prescribed on the boundary, which is,$$\begin{array}{c}\hfill \left({q}_{i}(t,{r}_{i}){n}_{i}\right){|}_{\partial \mathcal{B}}={q}_{\partial \mathcal{B}}\left(t\right).\end{array}$$Applying the Fourier Equation (26), (29) can be rewritten to$$\begin{array}{c}\hfill \left({\partial}_{i}T(t,{r}_{i}){n}_{i}\right){|}_{\partial \mathcal{B}}=-\frac{1}{\lambda}{q}_{\partial \mathcal{B}}\left(t\right).\end{array}$$
- Third kind or Robin boundary condition (convection type): the linear combination of the temperature and the normal component of heat flux is prescribed on the boundary. For instance, the Newtonian cooling boundary is$$\begin{array}{c}\hfill \left({q}_{i}(t,{r}_{i}){n}_{i}\right){|}_{\partial \mathcal{B}}=h\left[T(t,{r}_{i}){|}_{\partial \mathcal{B}}-{T}_{\infty}\right],\end{array}$$where h is the heat transfer coefficient and ${T}_{\infty}$ is the ambient temperature.

## 3. Non-Equilibrium Thermodynamics with Internal Variables

#### 3.1. Maxwell-Cattaneo-Vernotte Equation

#### 3.2. Guyer-Krumhansl Equation

#### 3.3. Further Remarks about Boundary Conditions

#### 3.4. Coupled Heat and Mass Transport

## 4. Rational Extended Thermodynamics

#### 4.1. Formalism of RET

- governing equations are of balance type; they are also invariant with respect to Galilean transformations;
- constitutive functions depend on local (in point) values of field variables; and,
- governing equations are adjoined with the entropy balance law with convex entropy density and non-negative entropy production.

#### 4.2. Molecular Extended Thermodynamics—Application of the Maximum Entropy Principle

#### 4.3. Note on Boundary Conditions

## 5. Study Closure

#### Outlook: Experiments as Benchmarks

_{2}[109]. Experimental data for thin layer are still lacking, although the analysis based upon solutions of the Boltzmann equation give similar predictions as RET [110]. The second result is related to the shock structure in multi-temperature model of mixtures [95,96]. Although the model is rather simple—it neglects viscous and heat conducting effects, but takes into account mutual momentum and energy exchange between the constituents through source terms—it yields good agreement with available experimental data [111] and direct simulations [112].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RET | Rational Extended Thermodynamics |

EIT | Extended Irreversible Thermodynamics |

CIT | Classical Irreversible Thermodynamics |

NET-IV | Non-Equilibrium Thermodynamics with Internal Variables |

MEP | Maximum Entropy Principle |

## References

- Ackerman, C.C.; Bertman, B.; Fairbank, H.A.; Guyer, R.A. Second sound in solid Helium. Phys. Rev. Lett.
**1966**, 16, 789–791. [Google Scholar] [CrossRef] - McNelly, T.F. Second Sound and Anharmonic Processes in Isotopically Pure Alkali-Halides. Ph.D. Thesis, Cornell University, Ithaca, NY, USA, 1974. [Google Scholar]
- Both, S.; Czél, B.; Fülöp, T.; Gróf, G.; Gyenis, Á.; Kovács, R.; Ván, P.; Verhás, J. Deviation from the Fourier law in room-temperature heat pulse experiments. J. Non Equilib. Thermodyn.
**2016**, 41, 41–48. [Google Scholar] [CrossRef] [Green Version] - Józsa, V.; Kovács, R. Solving Problems in Thermal Engineering: A Toolbox for Engineers; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- Prigogine, I. Etude Thermodinamique des Phénomènes Irréversibles; Desoer: Liège, Belgium, 1947. [Google Scholar]
- De Groot, S.R.; Mazur, P. Non-Equilibrium Thermodynamics; North-Holland Publishing Company: Amsterdam, The Netherlands, 1962. [Google Scholar]
- Gyarmati, I. Non-Equilibrium Thermodynamics—Field Theory and Variational Pronciples; Springer-Verlag: Berlin, Germany, 1970. [Google Scholar]
- Verhás, J. Thermodynamics and Rheology; Akadémiai Kiadó-Kluwer Academic Publisher: Budapest, Hungary, 1997. [Google Scholar]
- Berezovski, A.; Ván, P. Internal Variables in Thermoelasticity; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar] [CrossRef]
- Müller, I.; Ruggeri, T. Rational Extended Thermodynamics; Springer: New York, NY, USA, 1998. [Google Scholar]
- Ruggeri, T.; Sugiyama, M. Rational Extended Thermodynamics beyond the Monatomic Gas; Springer: New York, NY, USA, 2015. [Google Scholar]
- Jou, D.; Casas-Vázquez, J.; Lebon, G. Extended Irreversible Thermodynamics. Rep. Prog. Phys.
**1988**, 51, 1105. [Google Scholar] [CrossRef] [Green Version] - Kovács, R.; Madjarević, D.; Simić, S.; Ván, P. Non-equilibrium theories of rarefied gases: Internal variables and extended thermodynamics. Contin. Mech. Thermodyn.
**2020**, arXiv:1812.10355. [Google Scholar] [CrossRef] - Cimmelli, V.A.; Jou, D.; Ruggeri, T.; Ván, P. Entropy Principle and Recent Results in Non-Equilibrium Theories. Entropy
**2014**, 16, 1756–1807. [Google Scholar] [CrossRef] [Green Version] - Morrison, P.J. Bracket formulation for irreversible classical fields. Phys. Lett. A
**1984**, 100, 423–427. [Google Scholar] [CrossRef] - Grmela, M. Particle and bracket formulations of kinetic equations. Contemp. Math.
**1984**, 28, 125–132. [Google Scholar] - Grmela, M.; Öttinger, H.C. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E
**1997**, 56, 6620–6632. [Google Scholar] [CrossRef] - Öttinger, H.C.; Grmela, M. Dynamics and thermodynamics of complex fluids. II. Illustration of a general formalism. Phys. Rev. E
**1997**, 56, 6633–6655. [Google Scholar] [CrossRef] - Öttinger, H.C. Beyond Equilibrium Thermodynamics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2005. [Google Scholar]
- Pavelka, M.; Klika, V.; Grmela, M. Multiscalce Thermo-Dynamics—Introduction to GENERIC; De Gruyter: Berlin, Germany, 2018. [Google Scholar]
- Dzyaloshinskii, I.E.; Volovick, G.E. Poisson brackets in condensed matter physics. Ann. Phys.
**1980**, 125, 67–97. [Google Scholar] [CrossRef] - Shang, X.; Öttinger, H.C. Structure-preserving integrators for dissipative systems based on reversible— Irreversible splitting. Proc. R. Soc. A
**2020**, 476, 20190446. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pavelka, M.; Klika, V.; Grmela, M. Ehrenfest regularization of Hamiltonian systems. Phys. D
**2019**, 399, 193–210. [Google Scholar] [CrossRef] [Green Version] - Portillo, D.; García Orden, J.C.; Romero, I. Energy-Entropy-Momentum integration schemes for general discrete non-smooth dissipative problems in thermomechanics. Int. J. Numer. Methods Eng.
**2017**, 112, 776–802. [Google Scholar] [CrossRef] - Betsch, P.; Schiebl, M. Energy-momentum-entropy consistent numerical methods for large-strain thermoelasticity relying on the GENERIC formalism. Int. J. Numer. Methods Eng.
**2019**, 119, 1216–1244. [Google Scholar] [CrossRef] - Szücs, M.; Fülöp, T. Kluitenberg–Verhás Rheology of Solids in the GENERIC Framework. J. Non Equilib. Thermodyn.
**2019**, 44, 247–259. [Google Scholar] [CrossRef] [Green Version] - Öttinger, H.C.; Struchtrup, H.; Torrilhon, M. Formulation of moment equations for rarefied gases within two frameworks of non-equilibrium thermodynamics: RET and GENERIC. Proc. R. Soc. A
**2020**, 378, 20190174. [Google Scholar] [CrossRef] [Green Version] - Onsager, L. Reciprocal relations of irreversible processes I. Phys. Rev.
**1931**, 37, 405–426. [Google Scholar] [CrossRef] - Onsager, L. Reciprocal relations of irreversible processes II. Phys. Rev.
**1931**, 38, 2265–2279. [Google Scholar] [CrossRef] [Green Version] - Eckart, C. The thermodynamics of irreverible processes I. The simple fluid. Phys. Rev.
**1940**, 58, 267–269. [Google Scholar] [CrossRef] - Eckart, C. The thermodynamics of irreverible processes II. Fluid mixtures. Phys. Rev.
**1940**, 58, 269–275. [Google Scholar] [CrossRef] - Eckart, C. The thermodynamics of irreverible processes III. Relativistic theory of the simple fluid. Phys. Rev.
**1940**, 58, 919–924. [Google Scholar] [CrossRef] - Eckart, C. The thermodynamics of irreverible processes IV. The theory of elasticity and anelasticity. Phys. Rev.
**1948**, 73, 373–382. [Google Scholar] [CrossRef] - Ván, P.; Pavelka, M.; Grmela, M. Extra Mass Flux in Fluid Mechanics. J. Non Equilib. Thermodyn.
**2017**, 42, 133–151. [Google Scholar] [CrossRef] [Green Version] - Ván, P. Non-Equilibrium Thermomechanics [in Hungarian: Nemegyensúlyi termomechanika]. Ph.D. Thesis, Hungarian Academy of Sciences, Budapest, Hungary, 2018. [Google Scholar]
- Ván, P.; Abe, S. Emergence of modified Newtonian gravity from thermodynamics. arXiv
**2019**, arXiv:1912.00252. [Google Scholar] - Ván, P. Thermodynamically consistent gradient elasticity with an internal variable. Theor. Appl. Mech.
**2020**, 47, 1–17. [Google Scholar] [CrossRef] - Coleman, B.D.; Noll, W. The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. Vol.
**1963**, 13, 167–178. [Google Scholar] [CrossRef] - Liu, I.-S. Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal.
**1972**, 46, 131–148. [Google Scholar] [CrossRef] - Ván, P. Exploiting the Second Law in weakly nonlocal continuum physics. Period. Polytech. Ser. Mech. Eng.
**2005**, 49, 79–94. [Google Scholar] - Ván, P.; Fülöp, T. Weakly nonlocal fluid mechanics—The Schrödinger equation. Proc. R. Soc. Lond. A
**2006**, 462, 541–557. [Google Scholar] [CrossRef] [Green Version] - Cimmelli, V.A. An extension of Liu procedure in weakly nonlocal thermodynamics. J. Math. Phys.
**2007**, 48, 113510. [Google Scholar] [CrossRef] - Ván, P.; Papenfuss, C.; Berezovski, A. Thermodynamic approach to generalized continua. Contin. Mech. Thermodyn.
**2014**, 25, 403–420. [Google Scholar] [CrossRef] [Green Version] - Rogolino, P.; Cimmelli, V.A. Differential consequences of balance laws in extended irreversible thermodynamics of rigid heat conductors. Proc. R. Soc. A
**2019**, 475, 20180482. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gorgone, M.; Oliveri, F.; Rogolino, P. Continua with non-local constitutive laws: Exploitation of entropy inequality. Int. J. Non Linear Mech.
**2020**, 103573. [Google Scholar] [CrossRef] - Ván, P.; Kovács, R. Variational principles and nonequilibrium thermodynamics. Philos. Trans. R. Soc. A
**2020**, 378, 20190178. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ván, P.; Papenfuss, C.; Muschik, W. Mesoscopic dynamics of microcracks. Phys. Rev. E
**2000**, 62, 6206. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Maxwell, J.C. On the dynamical theory of gases. Philos. Trans. R. Soc. Lond.
**1867**, 157, 49–88. [Google Scholar] - Cattaneo, C. Sur une forme de lequation de la chaleur eliminant le paradoxe dune propagation instantanee. Comptes Rendus Hebd. Seances De L’Academie Sci.
**1958**, 247, 431–433. [Google Scholar] - Vernotte, P. Les paradoxes de la théorie continue de léquation de la chaleur. Comptes Rendus Hebd. Seances De L’Academie Sci.
**1958**, 246, 3154–3155. [Google Scholar] - Guyer, R.A.; Krumhansl, J.A. Solution of the Linearized Phonon Boltzmann Equation. Phys. Rev.
**1966**, 148, 766–778. [Google Scholar] [CrossRef] - Guyer, R.A.; Krumhansl, J.A. Thermal Conductivity, Second Sound and Phonon Hydrodynamic Phenomena in Nonmetallic Crystals. Phys. Rev.
**1966**, 148, 778–788. [Google Scholar] [CrossRef] - Gyarmati, I. On the Wave Approach of Thermodynamics and some Problems of Non-Linear Theories. J. Non Equilib. Thermodyn.
**1977**, 2, 233–260. [Google Scholar] [CrossRef] - Kovács, R.; Rogolino, P. Numerical treatment of nonlinear Fourier and Maxwell-Cattaneo-Vernotte heat transport equations. Int. J. Heat Mass Transf.
**2020**, 150, 119281. [Google Scholar] [CrossRef] [Green Version] - Di Domenico, M.; Jou, D.; Sellitto, A. Nonlinear heat waves and some analogies with nonlinear optics. Int. J. Heat Mass Transf.
**2020**, 156, 119888. [Google Scholar] [CrossRef] - Nyíri, B. On the entropy current. J. Non Equilib. Thermodyn.
**1991**, 16, 179–186. [Google Scholar] [CrossRef] - Kovács, R.; Ván, P. Generalized heat conduction in heat pulse experiments. Int. J. Heat Mass Transf.
**2015**, 83, 613–620. [Google Scholar] [CrossRef] [Green Version] - Coleman, B.D.; Newman, D.C. Implications of a nonlinearity in the theory of second sound in solids. Phys. Rev. B
**1988**, 37, 1492. [Google Scholar] [CrossRef] - Zhukovsky, K. Operational approach and solutions of hyperbolic heat conduction equations. Axioms
**2016**, 5, 28. [Google Scholar] [CrossRef] [Green Version] - Zhukovsky, K. Violation of the maximum principle and negative solutions for pulse propagation in Guyer–Krumhansl model. Int. J. Heat Mass Transf.
**2016**, 98, 523–529. [Google Scholar] [CrossRef] - Kovács, R. Analytic solution of Guyer-Krumhansl equation for laser flash experiments. Int. J. Heat Mass Transf.
**2018**, 127, 631–636. [Google Scholar] [CrossRef] [Green Version] - Rieth, Á.; Kovács, R.; Fülöp, T. Implicit numerical schemes for generalized heat conduction equations. Int. J. Heat Mass Transf.
**2018**, 126, 1177–1182. [Google Scholar] [CrossRef] [Green Version] - Fülöp, T.; Kovács, R.; Szücs, M.; Fawaier, M. Thermodynamical extension of a symplectic numerical scheme with half space and time shifts demonstrated on rheological waves in solids. Entropy
**2020**, 22, 155. [Google Scholar] [CrossRef] [Green Version] - Alvarez, F.X.; Jou, D. Boundary conditions and evolution of ballistic heat transport. J. Heat Transf.
**2010**, 132, 012404. [Google Scholar] [CrossRef] - Cimmelli, V.A. Boundary conditions in the presence of internal variables. J. Non Equilib. Thermodyn.
**2002**, 27, 327–334. [Google Scholar] [CrossRef] - Klika, V.; Pavelka, M.; Benziger, J. Functional constraints on phenomenological coefficients. Phys. Rev. E
**2017**, 95, 022125. [Google Scholar] [CrossRef] [PubMed] - Rana, A.S.; Gupta, V.K.; Struchtrup, H. Coupled constitutive relations: A second law based higher-order closure for hydrodynamics. Proc. R. Soc. A
**2018**, 474, 20180323. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kovács, R. On the rarefied gas experiments. Entropy
**2019**, 21, 718. [Google Scholar] [CrossRef] [Green Version] - Müller, I. Zum paradox der Wärmeleitungstheorie. Zeitschrift für Physik
**1967**, 198, 329–344. [Google Scholar] [CrossRef] - Grad, H. On the kinetic theory of rarefied gases. Commun. Pure Appl. Math.
**1949**, 2, 331–407. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics II. Phys. Rev.
**1957**, 108, 171. [Google Scholar] [CrossRef] - Ruggeri, T. Galilean Invariance and Entropy Principle for Systems of Balance Laws. The Structure of the Extended Thermodynamics. Contin. Mech. Thermodyn.
**1989**, 1, 3–20. [Google Scholar] [CrossRef] - Müller, I. On the entropy inequality. Arch. Rat. Mech. Anal.
**1967**, 26, 118–141. [Google Scholar] [CrossRef] - Ruggeri, T.; Strumia, A. Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincaré
**1981**, 34, 65–84. [Google Scholar] - Hanouzet, B.; Natalini, R. Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy. Arch. Ration. Mech. Anal.
**2003**, 169, 89–117. [Google Scholar] [CrossRef] - Godunov, S.K. An interesting class of quasilinear systems. Sov. Math. Dokl.
**1961**, 139, 521–523. [Google Scholar] - Friedrichs, K.O.; Lax, P.D. Systems of conservation equations with a convex extension. Proc. Natl. Acad. Sci. USA
**1971**, 68, 1686–1688. [Google Scholar] [CrossRef] [Green Version] - Boillat, G.; Ruggeri, T. Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions. Arch. Rat. Mech. Anal.
**1997**, 137, 305–320. [Google Scholar] [CrossRef] - Pavelka, M.; Peshkov, I.; Klika, V. On Hamiltonian continuum mechanics. Phys. D Nonlinear Phenom.
**2020**, 408, 132510. [Google Scholar] [CrossRef] [Green Version] - Grmela, M.; Hong, L.; Jou, D.; Lebon, G.; Pavelka, M. Hamiltonian and Godunov Structures of the Grad Hierarchy. Phys. Rev. E
**2017**, 95, 033121. [Google Scholar] [CrossRef] [Green Version] - Yong, W.-A. Entropy and Global Existence for Hyperbolic Balance Laws. Arch. Rational Mech. Anal.
**2004**, 172, 247–266. [Google Scholar] [CrossRef] - Arima, T.; Taniguchi, S.; Ruggeri, T.; Sugiyama, M. Extended thermodynamics of dense gases. Contin. Mech. Thermodyn.
**2012**, 24, 271–292. [Google Scholar] [CrossRef] - Brini, F. Hyperbolicity region in extended thermodynamics with 14 moments. Contin. Mech. Thermodyn.
**2001**, 13, 1–8. [Google Scholar] [CrossRef] - Kogan, M.N. Rarefied Gas Dynamics; Plenum Press: New York, NY, USA, 1969. [Google Scholar]
- Dreyer, W. Maximisation of the entropy in non-equilibrium. J. Phys. A Math. Gen.
**1987**, 20, 6505–6517. [Google Scholar] [CrossRef] - Levermore, C.D. Moment Closure Hierarchies for Kinetic Theories. J. Stat. Phys.
**1996**, 83, 1021–1065. [Google Scholar] [CrossRef] - Borgnakke, C.; Larsen, P.S. Statistical Collision Model for Monte Carlo Simulation of Polyatomic Gas Mixture. J. Comput. Phys.
**1975**, 18, 405–420. [Google Scholar] [CrossRef] - Bourgat, J.-F.; Desvillettes, L.; Le Tallec, P.; Perthame, B. Microreversible collisions for polyatomic gases. Eur. J. Mech. B Fluids
**1994**, 13, 237–254. [Google Scholar] - Desvillettes, L.; Monaco, R.; Salvarani, F. A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. Eur. J. Mech. B Fluids
**2005**, 24, 219. [Google Scholar] [CrossRef] - Pavić, M.; Ruggeri, T.; Simić, S. Maximum entropy principle for rarefied polyatomic gases. Phys. A Stat. Mech. Its Appl.
**2013**, 392, 1302–1317. [Google Scholar] [CrossRef] [Green Version] - Pavić-Čolić, M.; Simić, S. Moment Equations for Polyatomic Gases. Acta Applic. Math.
**2014**, 132, 469–482. [Google Scholar] [CrossRef] - Ruggeri, T.; Simić, S. On the hyperbolic system of a mixture of Eulerian fluids: A comparison between single- and multi-temperature models. Math. Methods Appl. Sci.
**2007**, 30, 827–849. [Google Scholar] [CrossRef] - Ruggeri, T.; Simić, S. Average temperature and maxwellian iteration in multitemperature mixtures of fluids. Phys. Rev. E
**2009**, 80, 026–317. [Google Scholar] [CrossRef] [PubMed] - Madjarević, D.; Simić, S. Shock structure in helium-argon mixture—A comparison of hyperbolic multi-temperature model with experiment. EPL (Europhys. Lett.)
**2013**, 102, 44002. [Google Scholar] [CrossRef] - Madjarević, D.; Ruggeri, T.; Simić, S. Shock structure and temperature overshoot in macroscopic multi-temperature model of mixtures. Phys. Fluids
**2014**, 26, 106102. [Google Scholar] [CrossRef] - Pavić-Čolić, M. Multi-velocity and multi-temperature model of the mixture of polyatomic gases issuing from kinetic theory. Phys. Lett. A
**2019**, 383, 2829–2835. [Google Scholar] [CrossRef] - Brini, F.; Ruggeri, T. Entropy principle for the moment systems of degree α associated to the Boltzmann equation. Critical derivatives and non controllable boundary data. Contin. Mech. Thermodyn.
**2002**, 14, 165–189. [Google Scholar] [CrossRef] - Brini, F.; Ruggeri, T. Second-order approximation of extended thermodynamics of a monatomic gas and hyperbolicity region. Contin. Mech. Thermodyn.
**2020**, 32, 23–39. [Google Scholar] [CrossRef] - Struchtrup, H. Macroscopic Transport Equations for Rarefied Gas Flows; Springer-Verlag: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Struchtrup, H.; Torrilhon, M. H Theorem, Regularization, and Boundary Conditions for Linearized 13 Moment Equations. Phys. Rev. Lett.
**2007**, 99, 014502. [Google Scholar] [CrossRef] - Barbera, E.; Müller, I.; Reitebuch, D.; Zhao, N.-R. Determination of boundary conditions in extended thermodynamics via fluctuation theory. Contin. Mech. Thermodyn.
**2004**, 16, 411–425. [Google Scholar] [CrossRef] - Ruggeri, T. Can constitutive relations be represented by non-local equations? Quart. Appl. Math.
**2012**, 70, 597–611. [Google Scholar] [CrossRef] - Ván, P.; Berezovski, A.; Fülöp, T.; Gróf, Gy.; Kovács, R.; Lovas, Á.; Verhás, J. Guyer-Krumhansl-type heat conduction at room temperature. EPL
**2017**, 118, 50005. [Google Scholar] [CrossRef] [Green Version] - Kovács, R.; Rogolino, P.; Jou, D. When theories and experiments meet: Rarefied gases as a benchmark of non-equilibrium thermodynamic models. arXiv
**2019**, arXiv:1912.02158. [Google Scholar] - Kovács, R.; Ván, P. Second sound and ballistic heat conduction: NaF experiments revisited. Int. J. Heat Mass Transf.
**2018**, 117, 682–690. [Google Scholar] [CrossRef] [Green Version] - Boillat, G.; Ruggeri, T. On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn.
**1998**, 10, 285–292. [Google Scholar] [CrossRef] - Taniguchi, S.; Arima, T.; Ruggeri, T.; Sugiyama, M. Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: Beyond the Bethe-Teller theory. Phys. Rev. E
**2014**, 89, 013025. [Google Scholar] [CrossRef] [PubMed] - Johannesen, N.H.; Zienkiewicz, H.K.; Blythe, P.A.; Gerrard, J.H. Experimental and theoretical analysis of vibrational relaxation regions in carbon dioxide. J. Fluid Mech.
**1962**, 13, 213–224. [Google Scholar] [CrossRef] - Kosuge, S.; Aoki, K. Shock-wave structure for a polyatomic gas with large bulk viscosity. Phys. Rev. Fluids
**2018**, 3, 023401. [Google Scholar] [CrossRef] - Harnett, L.N.; Muntz, E.P. Experimental investigation of normal shock wave velocity distribution functions in mixtures of argon and helium. Phys. Fluids
**1972**, 15, 565–572. [Google Scholar] [CrossRef] - Bird, G. The structure of normal shock waves in a binary gas mixture. J. Fluid Mech.
**1968**, 31, 657–668. [Google Scholar] [CrossRef]

Material/Substantial Description | Spatial/Local Description | |
---|---|---|

Integral form | $\underset{\mathcal{B}}{\int}\rho \dot{x}\mathrm{d}V=-\underset{\partial \mathcal{B}}{\oint}{\left({J}_{X}\right)}_{i}{n}_{i}\mathrm{d}A+\underset{\mathcal{B}}{\int}{\Sigma}_{X}\mathrm{d}V$ | $\underset{\Omega}{\int}{\partial}_{t}\left(\rho x\right)\mathrm{d}V=-\underset{\partial \Omega}{\oint}\left[{\left({J}_{X}\right)}_{i}+\rho x{v}_{i}\right]{n}_{i}\mathrm{d}A+\underset{\Omega}{\int}{\Sigma}_{X}\mathrm{d}V$ |

Differential form | $\rho \dot{x}=-{\partial}_{i}{\left({J}_{X}\right)}_{i}+{\Sigma}_{X}$ | ${\partial}_{t}\left(\rho x\right)=-{\partial}_{i}\left[{\left({J}_{X}\right)}_{i}+\rho x{v}_{i}\right]+{\Sigma}_{X}$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Szücs, M.; Kovács, R.; Simić, S.
Open Mathematical Aspects of Continuum Thermodynamics: Hyperbolicity, Boundaries and Nonlinearities. *Symmetry* **2020**, *12*, 1469.
https://doi.org/10.3390/sym12091469

**AMA Style**

Szücs M, Kovács R, Simić S.
Open Mathematical Aspects of Continuum Thermodynamics: Hyperbolicity, Boundaries and Nonlinearities. *Symmetry*. 2020; 12(9):1469.
https://doi.org/10.3390/sym12091469

**Chicago/Turabian Style**

Szücs, Mátyás, Róbert Kovács, and Srboljub Simić.
2020. "Open Mathematical Aspects of Continuum Thermodynamics: Hyperbolicity, Boundaries and Nonlinearities" *Symmetry* 12, no. 9: 1469.
https://doi.org/10.3390/sym12091469