American Institute of Mathematical Sciences

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January  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  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.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

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

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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

[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.  Google Scholar

[48]

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

[49]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

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

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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

[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.  Google Scholar

[48]

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

[49]

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

[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.  Google Scholar

[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.  Google Scholar

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