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February  2021, 14(1): 175-197. doi: 10.3934/krm.2021001

Navier-Stokes limit of globally hyperbolic moment equations

Zhiting Ma

Department of Mathematical Sciences, Tsinghua University, Beijing, China

Received  May 2020 Revised  November 2020 Published  February 2021 Early access  December 2020

This paper is concerned with the Navier-Stokes limit of a class of globally hyperbolic moment equations from the Boltzmann equation. we show that the Navier-Stokes equations can be formally derived from the hyperbolic moment equations for various different collision mechanisms. Furthermore, the formal limit is justified rigorously by using an energy method. It should be noted that the hyperbolic moment equations are in non-conservative form and do not have a convex entropy function, therefore some additional efforts are required in the justification.

 

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Keywords: Globally hyperbolic moment equations, Navier-Stokes equations, hyperbolic relaxation systems, stability condition, convergence.
Mathematics Subject Classification: 35Q35, 82C40, 35L60.
Citation: Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic and Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001
References:
[1]

L. Arlotti, N. Bellomo and M. Lachowicz, Kinetic equations modelling population dynamics, Transport Theory Statist. Phys., 29 (2000), 125-139.  doi: 10.1080/00411450008205864.

[2]

C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations, II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.

[3]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases, I. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[4]

G. A. Bird, Direct simulation and the Boltzmann equation, Phys. Fluids, 13 (1970), 2676-2681.  doi: 10.1063/1.1692849.

[5]

A. Bobylev and Å. Windfäll, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models, 5 (2012), 237-260.  doi: 10.3934/krm.2012.5.237.

[6]

L. Boltzmann, Vorlesungen über Gastheorie: Th. Theorie van der Waals', Gase mit zusammengesetzten Molekülen, Gasdissociation; Schlussbemerkungen, vol. 2, JA Barth, 1898.

[7]

N. V. Brilliantov, F. Spahn, J.-M. Hertzsch and T. Pöschel, Model for collisions in granular gases, Phys. Rev. E, 53 (1996), 5382. doi: 10.1103/PhysRevE.53.5382.

[8]

Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11 (2013), 547-571.  doi: 10.4310/CMS.2013.v11.n2.a12.

[9]

Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518.  doi: 10.1002/cpa.21472.

[10]

Z. Cai, Y. Fan and R. Li, On hyperbolicity of 13-moment system, Kinet. Relat. Models, 7 (2014), 415-432.  doi: 10.3934/krm.2014.7.415.

[11]

Z. Cai, Y. Fan and R. Li, A framework on moment model reduction for kinetic equation, SIAM J. Appl. Math., 75 (2015), 2001-2023.  doi: 10.1137/14100110X.

[12]

Z. Cai, Y. Fan, R. Li and Z. Qiao, Dimension-reduced hyperbolic moment method for the Boltzmann equation with BGK-type collision, Commun. Comput. Phys., 15 (2014), 1368-1406.  doi: 10.4208/cicp.220313.281013a.

[13]

Z. Cai and M. Torrilhon, Numerical simulation of large hyperbolic moment systems with linear and relaxation production terms, in AIP Conference Proceedings, American Institute of Physics, 1628 (2014), 1040–1047. doi: 10.1063/1.4902708.

[14]

C. Cercignani, The Boltzmann equation, in The Boltzmann Equation and Its Applications, Springer, 1988, 40–103. doi: 10.1007/978-1-4612-1039-9.

[15] S. Chapman, T. G. Cowling and D. Burnett, The Mathematical Theory of Non-Uniform Gases: An Account of The Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge university press, 1960. 
[16]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]., Springer-Verlag, Berlin, second edition, 2005. doi: 10.1007/3-540-29089-3.

[17]

Y. Di, Y. Fan, R. Li and L. Zheng, Linear stability of hyperbolic moment models for Boltzmann equation, Numer. Math. Theory Methods Appl., 10 (2017), 255-277.  doi: 10.4208/nmtma.2017.s04.

[18]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.

[19]

R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 321–366. doi: 10.2307/1971423.

[20]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), 1-119.  doi: 10.1007/s40818-017-0037-5.

[21]

Y. Fan, J. Koellermeier, J. Li, R. Li and M. Torrilhon, Model reduction of kinetic equations by operator projection, J. Stat. Phys., 162 (2016), 457-486.  doi: 10.1007/s10955-015-1384-9.

[22]

Y. Fan, R. Li and L. Zheng, A nonlinear hyperbolic model for radiative transfer equation in slab geometry, SIAM J. Appl. Math., 80 (2020), 2388-2419.  doi: 10.1137/19M126774X.

[23]

L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics, Phys. Rep., 465 (2008), 149-189.  doi: 10.1016/j.physrep.2008.04.010.

[24]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5.

[25]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[26]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121.

[27]

M. Henon, Vlasov equation, Astronom. and Astrophys., 114 (1982), 211-212. 

[28]

L. H. Holway Jr, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673.  doi: 10.1063/1.1761920.

[29]

Q. Huang, S. Li and W.-A. Yong, Stability analysis of quadrature-based moment methods for kinetic equations, SIAM J. Appl. Math., 80 (2020), 206-231.  doi: 10.1137/18M1231845.

[30]

S. Kawashima, Systems of A Hyperbolic-Parabolic Composite Type, with Applications to The Equations of Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1984. doi: 10.14989/doctor.k3193.

[31]

S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J., 40 (1988), 449-464.  doi: 10.2748/tmj/1178227986.

[32]

S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal., 174 (2004), 345-364.  doi: 10.1007/s00205-004-0330-9.

[33]

G. M. Kremer, An Introduction to The Boltzmann Equation and Transport Processes in Gases, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-11696-4.

[34]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[35]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, vol. 53, Springer Science & Business Media, 1984. doi: 10.1007/978-1-4612-1116-7.

[36]

L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149.  doi: 10.1142/S0218202500000562.

[37]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, vol. 37, Springer Science & Business Media, 1998. doi: 10.1007/978-1-4612-2210-1.

[38] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, 1999. 
[39]

E. M. Shakhov, Generalization of the Krook kinetic relaxation equation, Fluid Dynamics, 3 (1968), 95-96.  doi: 10.1007/BF01029546.

[40]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005

[41]

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.

[42]

K. T. Waldeer, The direct simulation Monte Carlo method applied to a Boltzmann-like vehicular traffic flow model, Comput. Phys. Commun., 156 (2003), 1-12.  doi: 10.1016/S0010-4655(03)00368-0.

[43]

Y. Wang and Z. Cai, Approximation of the Boltzmann collision operator based on Hermite spectral method, J. Comput. Phys., 397 (2019), 108815, 23pp. doi: 10.1016/j.jcp.2019.07.014.

[44]

E. P. Wigner, On the quantum correction for thermodynamic equilibrium, in Part I: Physical Chemistry, Part II: Solid State Physics, Springer, (1997), 110–120. doi: 10.1007/978-3-642-59033-7_9.

[45]

Z. Yang and W.-A. Yong, Validity of the Chapman-Enskog expansion for a class of hyperbolic relaxation systems, J. Differential Equations, 258 (2015), 2745–-2766. doi: 10.1016/j.jde.2014.12.024.

[46]

W.-A. Yong, Singular perturbations of first-order hyperbolic systems, in Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects, Springer, (1993), 597–604.

[47]

W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations, 155 (1999), 89-132.  doi: 10.1006/jdeq.1998.3584.

[48]

W.-A. Yong, Basic aspects of hyperbolic relaxation systems, in Advances in The Theory of Shock Waves, Springer, 2001,259–305.

[49]

W.-A. Yong, An interesting class of partial differential equations, J. Math. Phys., 49 (2008), 033503, 21pp. doi: 10.1063/1.2884710.

[50]

W. Zhao, J. Huang and W.-A. Yong, Boundary conditions for kinetic theory based models I: Lattice Boltzmann models, Multiscale Model. Simul., 17 (2019), 854-872.  doi: 10.1137/18M1201986.

[51]

W. Zhao, W.-A. Yong and L.-S. Luo, Stability analysis of a class of globally hyperbolic moment system, Commun. Math. Sci., 15 (2017), 609-633.  doi: 10.4310/CMS.2017.v15.n3.a3.

show all references

References:
[1]

L. Arlotti, N. Bellomo and M. Lachowicz, Kinetic equations modelling population dynamics, Transport Theory Statist. Phys., 29 (2000), 125-139.  doi: 10.1080/00411450008205864.

[2]

C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations, II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.

[3]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases, I. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[4]

G. A. Bird, Direct simulation and the Boltzmann equation, Phys. Fluids, 13 (1970), 2676-2681.  doi: 10.1063/1.1692849.

[5]

A. Bobylev and Å. Windfäll, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models, 5 (2012), 237-260.  doi: 10.3934/krm.2012.5.237.

[6]

L. Boltzmann, Vorlesungen über Gastheorie: Th. Theorie van der Waals', Gase mit zusammengesetzten Molekülen, Gasdissociation; Schlussbemerkungen, vol. 2, JA Barth, 1898.

[7]

N. V. Brilliantov, F. Spahn, J.-M. Hertzsch and T. Pöschel, Model for collisions in granular gases, Phys. Rev. E, 53 (1996), 5382. doi: 10.1103/PhysRevE.53.5382.

[8]

Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11 (2013), 547-571.  doi: 10.4310/CMS.2013.v11.n2.a12.

[9]

Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518.  doi: 10.1002/cpa.21472.

[10]

Z. Cai, Y. Fan and R. Li, On hyperbolicity of 13-moment system, Kinet. Relat. Models, 7 (2014), 415-432.  doi: 10.3934/krm.2014.7.415.

[11]

Z. Cai, Y. Fan and R. Li, A framework on moment model reduction for kinetic equation, SIAM J. Appl. Math., 75 (2015), 2001-2023.  doi: 10.1137/14100110X.

[12]

Z. Cai, Y. Fan, R. Li and Z. Qiao, Dimension-reduced hyperbolic moment method for the Boltzmann equation with BGK-type collision, Commun. Comput. Phys., 15 (2014), 1368-1406.  doi: 10.4208/cicp.220313.281013a.

[13]

Z. Cai and M. Torrilhon, Numerical simulation of large hyperbolic moment systems with linear and relaxation production terms, in AIP Conference Proceedings, American Institute of Physics, 1628 (2014), 1040–1047. doi: 10.1063/1.4902708.

[14]

C. Cercignani, The Boltzmann equation, in The Boltzmann Equation and Its Applications, Springer, 1988, 40–103. doi: 10.1007/978-1-4612-1039-9.

[15] S. Chapman, T. G. Cowling and D. Burnett, The Mathematical Theory of Non-Uniform Gases: An Account of The Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge university press, 1960. 
[16]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]., Springer-Verlag, Berlin, second edition, 2005. doi: 10.1007/3-540-29089-3.

[17]

Y. Di, Y. Fan, R. Li and L. Zheng, Linear stability of hyperbolic moment models for Boltzmann equation, Numer. Math. Theory Methods Appl., 10 (2017), 255-277.  doi: 10.4208/nmtma.2017.s04.

[18]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.

[19]

R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 321–366. doi: 10.2307/1971423.

[20]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), 1-119.  doi: 10.1007/s40818-017-0037-5.

[21]

Y. Fan, J. Koellermeier, J. Li, R. Li and M. Torrilhon, Model reduction of kinetic equations by operator projection, J. Stat. Phys., 162 (2016), 457-486.  doi: 10.1007/s10955-015-1384-9.

[22]

Y. Fan, R. Li and L. Zheng, A nonlinear hyperbolic model for radiative transfer equation in slab geometry, SIAM J. Appl. Math., 80 (2020), 2388-2419.  doi: 10.1137/19M126774X.

[23]

L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics, Phys. Rep., 465 (2008), 149-189.  doi: 10.1016/j.physrep.2008.04.010.

[24]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5.

[25]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[26]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121.

[27]

M. Henon, Vlasov equation, Astronom. and Astrophys., 114 (1982), 211-212. 

[28]

L. H. Holway Jr, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673.  doi: 10.1063/1.1761920.

[29]

Q. Huang, S. Li and W.-A. Yong, Stability analysis of quadrature-based moment methods for kinetic equations, SIAM J. Appl. Math., 80 (2020), 206-231.  doi: 10.1137/18M1231845.

[30]

S. Kawashima, Systems of A Hyperbolic-Parabolic Composite Type, with Applications to The Equations of Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1984. doi: 10.14989/doctor.k3193.

[31]

S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J., 40 (1988), 449-464.  doi: 10.2748/tmj/1178227986.

[32]

S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal., 174 (2004), 345-364.  doi: 10.1007/s00205-004-0330-9.

[33]

G. M. Kremer, An Introduction to The Boltzmann Equation and Transport Processes in Gases, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-11696-4.

[34]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[35]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, vol. 53, Springer Science & Business Media, 1984. doi: 10.1007/978-1-4612-1116-7.

[36]

L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149.  doi: 10.1142/S0218202500000562.

[37]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, vol. 37, Springer Science & Business Media, 1998. doi: 10.1007/978-1-4612-2210-1.

[38] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, 1999. 
[39]

E. M. Shakhov, Generalization of the Krook kinetic relaxation equation, Fluid Dynamics, 3 (1968), 95-96.  doi: 10.1007/BF01029546.

[40]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005

[41]

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.

[42]

K. T. Waldeer, The direct simulation Monte Carlo method applied to a Boltzmann-like vehicular traffic flow model, Comput. Phys. Commun., 156 (2003), 1-12.  doi: 10.1016/S0010-4655(03)00368-0.

[43]

Y. Wang and Z. Cai, Approximation of the Boltzmann collision operator based on Hermite spectral method, J. Comput. Phys., 397 (2019), 108815, 23pp. doi: 10.1016/j.jcp.2019.07.014.

[44]

E. P. Wigner, On the quantum correction for thermodynamic equilibrium, in Part I: Physical Chemistry, Part II: Solid State Physics, Springer, (1997), 110–120. doi: 10.1007/978-3-642-59033-7_9.

[45]

Z. Yang and W.-A. Yong, Validity of the Chapman-Enskog expansion for a class of hyperbolic relaxation systems, J. Differential Equations, 258 (2015), 2745–-2766. doi: 10.1016/j.jde.2014.12.024.

[46]

W.-A. Yong, Singular perturbations of first-order hyperbolic systems, in Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects, Springer, (1993), 597–604.

[47]

W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations, 155 (1999), 89-132.  doi: 10.1006/jdeq.1998.3584.

[48]

W.-A. Yong, Basic aspects of hyperbolic relaxation systems, in Advances in The Theory of Shock Waves, Springer, 2001,259–305.

[49]

W.-A. Yong, An interesting class of partial differential equations, J. Math. Phys., 49 (2008), 033503, 21pp. doi: 10.1063/1.2884710.

[50]

W. Zhao, J. Huang and W.-A. Yong, Boundary conditions for kinetic theory based models I: Lattice Boltzmann models, Multiscale Model. Simul., 17 (2019), 854-872.  doi: 10.1137/18M1201986.

[51]

W. Zhao, W.-A. Yong and L.-S. Luo, Stability analysis of a class of globally hyperbolic moment system, Commun. Math. Sci., 15 (2017), 609-633.  doi: 10.4310/CMS.2017.v15.n3.a3.

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Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235

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