# American Institute of Mathematical Sciences

March  2012, 5(1): 185-201. doi: 10.3934/krm.2012.5.185

## H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory

 1 Center for Computational Engineering Science, RWTH Aachen University, Schinkelstr.2, 52062 Aachen, Germany

Received  May 2011 Revised  July 2011 Published  January 2012

The regularized 13-moment equations (R13) are a successful macroscopic model to describe non-equilibrium gas flows in rarefied or micro situations. Even though the equations have been derived for the nonlinear case and many examples demonstrate the usefulness of the equations, sofar, the important property of an accompanying entropy law could only be shown for the linearized equations [Struchtrup&Torrilhon, Phys. Rev. Lett. 99, (2007), 014502]. Based on an approach suggested by Öttinger [Phys. Rev. Lett. 104, (2010), 120601], this paper presents a nonlinear entropy law for the R13 system. In the derivation the variables and equations of the R13 system are nonlinearily extended such that an entropy law with non-negative production can be formulated. It is then demonstrated that the original R13 system is included in the new equations.
Citation: Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic and Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185
##### References:
 [1] K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686. [2] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. [3] S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics," North Holland, Amsterdam, 1962. [4] X.-J. Gu and D. Emerson, A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions, J. Comput. Phys., 225 (2007), 263-283. doi: 10.1016/j.jcp.2006.11.032. [5] G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases," Springer, Berlin, 2010. doi: 10.1007/978-3-642-11696-4. [6] C. D. Levermore and W. J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1999), 72-96. [7] I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," Second edition, With supplementary chapters by H. Struchtrup and Wolf Weiss, Springer Tracts in Natural Philosophy, 37, Springer-Verlag, New York, 1998. [8] H. C. Öttinger, "Beyond Equilibrium Thermodynamics," Wiley, Hoboken, 2005. [9] H. C. Öttinger, Reply to the comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Lett., 105 (2010), 128902. [10] H. C. Öttinger, Thermodynamically admissible 13 moment equations from the Boltzmann equation, Phys. Rev. Lett., 104 (2010), 120601. doi: 10.1103/PhysRevLett.104.120601. [11] H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 221.  doi: 10.1137/040603115. [12] H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory," Interaction of Mechanics and Mathematics, Springer, Berlin, 2005. [13] H. Struchtrup and M. Torrilhon, Regularization of Grad's 13-moment-equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472. [14] H. Struchtrup and M. Torrilhon, H-theorem, regularization, and boundary conditions for linearized 13 moment equations, Phys. Rev. Letters, 99 (2007), 014502. doi: 10.1103/PhysRevLett.99.014502. [15] H. Struchtrup and M. Torrilhon, Comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Letters, 105 (2010) 128901. doi: 10.1103/PhysRevLett.105.128901. [16] P. Taheri, A. S. Rana, M. Torrilhon and H. Struchtrup, Macroscopic presentation of steady and unsteady rarefaction effects in the fundamental boundary value problems of gas dynamics, Continuum Mech. Thermodyn., 21 (2009), 423-443. doi: 10.1007/s00161-009-0115-3. [17] P. Taheri, M. Torrilhon and H. Struchtrup, Couette and poiseuille microflows: Analytical solutions for regularized 13-moment equations, Phys. Fluids, 21 (2009), 017102. doi: 10.1063/1.3064123. [18] M. Torrilhon, Two-dimensional bulk microflow simulations based on regularized Grad's 13-moment-equations, Multiscale Model. Simul., 5 (2006), 695-728. doi: 10.1137/050635444. [19] M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Comm. Comput. Phys., 7 (2010), 639-673. [20] M. Torrilhon, Slow rarefied flow past a sphere: Analytical solutions based on moment equations, Phys. Fluids, 22 (2010), 072001. doi: 10.1063/1.3453707. [21] M. Torrilhon and H. Struchtrup, Regularized 13-moment-equations: Shock structure calculations and comparison to Burnett models, J. Fluid Mech., 513 (2004), 171-198. doi: 10.1017/S0022112004009917. [22] M. Torrilhon and H. Struchtrup, Boundary conditions for regularized 13-moment-equations for micro-channel-flows, J. Comput. Phys., 227 (2008), 1982-2011. doi: 10.1016/j.jcp.2007.10.006.

show all references

##### References:
 [1] K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686. [2] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. [3] S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics," North Holland, Amsterdam, 1962. [4] X.-J. Gu and D. Emerson, A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions, J. Comput. Phys., 225 (2007), 263-283. doi: 10.1016/j.jcp.2006.11.032. [5] G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases," Springer, Berlin, 2010. doi: 10.1007/978-3-642-11696-4. [6] C. D. Levermore and W. J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1999), 72-96. [7] I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," Second edition, With supplementary chapters by H. Struchtrup and Wolf Weiss, Springer Tracts in Natural Philosophy, 37, Springer-Verlag, New York, 1998. [8] H. C. Öttinger, "Beyond Equilibrium Thermodynamics," Wiley, Hoboken, 2005. [9] H. C. Öttinger, Reply to the comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Lett., 105 (2010), 128902. [10] H. C. Öttinger, Thermodynamically admissible 13 moment equations from the Boltzmann equation, Phys. Rev. Lett., 104 (2010), 120601. doi: 10.1103/PhysRevLett.104.120601. [11] H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 221.  doi: 10.1137/040603115. [12] H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory," Interaction of Mechanics and Mathematics, Springer, Berlin, 2005. [13] H. Struchtrup and M. Torrilhon, Regularization of Grad's 13-moment-equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472. [14] H. Struchtrup and M. Torrilhon, H-theorem, regularization, and boundary conditions for linearized 13 moment equations, Phys. Rev. Letters, 99 (2007), 014502. doi: 10.1103/PhysRevLett.99.014502. [15] H. Struchtrup and M. Torrilhon, Comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Letters, 105 (2010) 128901. doi: 10.1103/PhysRevLett.105.128901. [16] P. Taheri, A. S. Rana, M. Torrilhon and H. Struchtrup, Macroscopic presentation of steady and unsteady rarefaction effects in the fundamental boundary value problems of gas dynamics, Continuum Mech. Thermodyn., 21 (2009), 423-443. doi: 10.1007/s00161-009-0115-3. [17] P. Taheri, M. Torrilhon and H. Struchtrup, Couette and poiseuille microflows: Analytical solutions for regularized 13-moment equations, Phys. Fluids, 21 (2009), 017102. doi: 10.1063/1.3064123. [18] M. Torrilhon, Two-dimensional bulk microflow simulations based on regularized Grad's 13-moment-equations, Multiscale Model. Simul., 5 (2006), 695-728. doi: 10.1137/050635444. [19] M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Comm. Comput. Phys., 7 (2010), 639-673. [20] M. Torrilhon, Slow rarefied flow past a sphere: Analytical solutions based on moment equations, Phys. Fluids, 22 (2010), 072001. doi: 10.1063/1.3453707. [21] M. Torrilhon and H. Struchtrup, Regularized 13-moment-equations: Shock structure calculations and comparison to Burnett models, J. Fluid Mech., 513 (2004), 171-198. doi: 10.1017/S0022112004009917. [22] M. Torrilhon and H. Struchtrup, Boundary conditions for regularized 13-moment-equations for micro-channel-flows, J. Comput. Phys., 227 (2008), 1982-2011. doi: 10.1016/j.jcp.2007.10.006.
 [1] Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic and Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004 [2] Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic and Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003 [3] Florian Schneider, Jochen Kall, Graham Alldredge. A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry. Kinetic and Related Models, 2016, 9 (1) : 193-215. doi: 10.3934/krm.2016.9.193 [4] Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic and Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317 [5] Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control and Related Fields, 2020, 10 (1) : 113-140. doi: 10.3934/mcrf.2019032 [6] Vladimir Djordjić, Milana Pavić-Čolić, Nikola Spasojević. Polytropic gas modelling at kinetic and macroscopic levels. Kinetic and Related Models, 2021, 14 (3) : 483-522. doi: 10.3934/krm.2021013 [7] Pierre Degond, Amic Frouvelle, Jian-Guo Liu. From kinetic to fluid models of liquid crystals by the moment method. Kinetic and Related Models, 2022, 15 (3) : 417-465. doi: 10.3934/krm.2021047 [8] Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic and Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429 [9] Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic and Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621 [10] Jessy Mallet, Stéphane Brull, Bruno Dubroca. General moment system for plasma physics based on minimum entropy principle. Kinetic and Related Models, 2015, 8 (3) : 533-558. doi: 10.3934/krm.2015.8.533 [11] Christian Bourdarias, Marguerite Gisclon, Stéphane Junca. Kinetic formulation of a 2 × 2 hyperbolic system arising in gas chromatography. Kinetic and Related Models, 2020, 13 (5) : 869-888. doi: 10.3934/krm.2020030 [12] Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic and Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255 [13] Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic and Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235 [14] Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic and Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030 [15] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [16] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 4035-4067. doi: 10.3934/dcdss.2020458 [17] Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, Shlomo Ta'asan. An entropy based theory of the grain boundary character distribution. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 427-454. doi: 10.3934/dcds.2011.30.427 [18] Zbigniew Banach, Wieslaw Larecki. Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries. Kinetic and Related Models, 2017, 10 (4) : 879-900. doi: 10.3934/krm.2017035 [19] Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic and Related Models, 2015, 8 (2) : 201-214. doi: 10.3934/krm.2015.8.201 [20] Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025

2020 Impact Factor: 1.432