# American Institute of Mathematical Sciences

September  2016, 9(3): 605-619. doi: 10.3934/krm.2016009

## Entropy production for ellipsoidal BGK model of the Boltzmann equation

 1 Department of mathematics, Sungkyunkwan University, Suwon 440-746, South Korea

Received  October 2015 Revised  January 2016 Published  May 2016

The ellipsoidal BGK model (ES-BGK) is a generalized version of the original BGK model, designed to yield the correct Prandtl number in the Navier-Stokes limit. In this paper, we make two observations on the entropy production functional of the ES-BGK model. First, we show that the Cercignani type estimate holds for the ES-BGK model in the whole range of relaxation parameter $-1/2<\nu<1$. Secondly, we observe that the ellipsoidal relaxation operator satisfies an unexpected sign-definite property. Some implications of these observations are also discussed.
Citation: Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009
##### References:
 [1] P. Andries, J.-F. Bourgat, P. Le Tallec and B. Perthame, Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases,, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3369. doi: 10.1016/S0045-7825(02)00253-0. Google Scholar [2] P. Andries, P. Le Tallec, J.-P.Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number,, Eur. J. Mech. B Fluids, 19 (2000), 813. doi: 10.1016/S0997-7546(00)01103-1. Google Scholar [3] K. Aoki, K. Kanba and S. Takata, Numerical analysis of a supersonic rarefied gas flow past a flat plate,, Phys. Fluids, 9 (1997). doi: 10.1063/1.869204. Google Scholar [4] F. Berthelin and A. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks,, SIAM J. Math. Anal., 36 (2005), 1807. doi: 10.1137/S0036141003431554. Google Scholar [5] A. Bellouquid, Global existence and large-time behavior for BGK model for a gas with non-constant cross section,, Transport Theory Statist. Phys., 32 (2003), 157. doi: 10.1081/TT-120019041. Google Scholar [6] P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. Small amplitude process in charged and neutral one-component systems,, Phys. Rev., 94 (1954), 511. doi: 10.1103/PhysRev.94.511. Google Scholar [7] G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows,, Oxford Engineering Science, (1995). Google Scholar [8] M. Bisi, J. A. Cañizo and B. Lods, Entropy dissipation estimates for the linear Boltzmann operator,, J. Funct. Anal., 269 (2015), 1028. doi: 10.1016/j.jfa.2015.05.002. Google Scholar [9] A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation,, J. Statist. Phys., 94 (1999), 603. doi: 10.1023/A:1004537522686. Google Scholar [10] R. Bosi and M. J. Cáceres, The BGK model with external confining potential: Existence, long-time behaviour and time-periodic Maxwellian equilibria,, J. Stat. Phys., 136 (2009), 297. doi: 10.1007/s10955-009-9782-5. Google Scholar [11] S. Brull, An ellipsoidal statistical model for gas mixtures,, Comm. Math Sci., 13 (2015), 1. doi: 10.4310/CMS.2015.v13.n1.a1. Google Scholar [12] S. Brull and J. Schneider, A new approach of the ellipsoidal statistical model,, Cont. Mech. Thermodyn., 20 (2008), 63. doi: 10.1007/s00161-008-0068-y. Google Scholar [13] C. Cercignani, The Boltzmann Equation and Its Application,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1039-9. Google Scholar [14] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer-Verlag, (1994). doi: 10.1007/978-1-4419-8524-8. Google Scholar [15] C. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases,, Cambridge University Press, (1990). doi: 10.1119/1.1942035. Google Scholar [16] W. M. Chan, An Energy Method for the BGK Model,, M. Phil thesis, (2007). Google Scholar [17] J. Dolbeault, P. Markowich, D. Oelz and C. Schmeiser, Non linear diffusions as limit of kinetic equations with relaxation collision kernels,, Arch. Ration. Mech, 186 (2007), 133. doi: 10.1007/s00205-007-0049-5. Google Scholar [18] R. DiPerna and P.-L. Lions, On the Cauchy problem for the Boltzmann equation: Global existence and weak stability., Ann. Math., 130 (1989), 321. doi: 10.2307/1971423. Google Scholar [19] F. Filbet and S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation,, J. Sci. Comput., 46 (2011), 204. doi: 10.1007/s10915-010-9394-x. Google Scholar [20] F. Filbet and G. Russo, Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics,, Kinet. Relat. Models, 2 (2009), 231. doi: 10.3934/krm.2009.2.231. Google Scholar [21] M. A. Galli and R. Torczynski, Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls,, Phys. Fluids, 23 (2011). doi: 10.1063/1.3558869. Google Scholar [22] R. Glassey, The Cauchy Problems in Kinetic Theory,, SIAM, (1996). doi: 10.1137/1.9781611971477. Google Scholar [23] L. H. Holway, Kinetic theory of shock structure using and ellipsoidal distribution function. Rarefied Gas Dynamics, Vol. I, (Proc. Fourth Internat. Sympos., (1966), 193. Google Scholar [24] D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (B.G,K.) equaiton,, SIAM Journal on Numerical Analysis, 33 (1996), 2099. doi: 10.1137/S0036142994266856. Google Scholar [25] S. K. Loyalka, N. Petrellis and T. S. Storvick, Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates,, Z. Angew. Math. Phys., 30 (1979), 514. doi: 10.1007/BF01588895. Google Scholar [26] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Ration. Mech. Anal., 199 (2011), 493. doi: 10.1007/s00205-010-0354-2. Google Scholar [27] S. Mischler, Uniqueness for the BGK-equation in $R^n$ and the rate of convergence for a semi-discrete scheme,, Differential integral Equations, 9 (1996), 1119. Google Scholar [28] L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics., Math. Models Methods Appl. Sci., 10 (2000), 1121. doi: 10.1142/S0218202500000562. Google Scholar [29] L. Mieussens and H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number,, Phys. Fluids, 16 (2004). doi: 10.1063/1.1758217. Google Scholar [30] S. Park and S.-B. Yun, Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation,, submitted., (). Google Scholar [31] B. Perthame, Global existence to the BGK model of Boltzmann equation,, J. Differential Equations, 82 (1989), 191. doi: 10.1016/0022-0396(89)90173-3. Google Scholar [32] B. Perthame and M. Pulvirenti, Weighted $L^{\infty}$ bounds and uniqueness for the Boltzmann BGK model,, Arch. Rational Mech. Anal., 125 (1993), 289. doi: 10.1007/BF00383223. Google Scholar [33] S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations,, J. Sci. Comput., 32 (2007), 1. doi: 10.1007/s10915-006-9116-6. Google Scholar [34] G. Russo, P. Santagati and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation,, SIAM J. Numer. Anal., 50 (2012), 1111. doi: 10.1137/100800348. Google Scholar [35] L. Saint-Raymond, From the BGK model to the Navier-Stokes equations,, Ann. Sci. Ecole Norm. Sup., 36 (2003), 271. doi: 10.1016/S0012-9593(03)00010-7. Google Scholar [36] L. Saint-Raymond, Discrete time Navier-Stokes limit for the BGK Boltzmann equation,, Comm. Partial Differential Equations, 27 (2002), 149. doi: 10.1081/PDE-120002785. Google Scholar [37] Y. Sone, Kinetic Theory and Fluid Mechanics,, Boston: Birkhäuser, (2002). doi: 10.1007/978-1-4612-0061-1. Google Scholar [38] Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications,, Boston: Brikhäuser, (2006). doi: 10.1007/978-0-8176-4573-1. Google Scholar [39] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory,, Springer. 2005., (2005). doi: 10.1007/3-540-32386-4. Google Scholar [40] G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation,, Comm. Math. Phys., 203 (1999), 667. doi: 10.1007/s002200050631. Google Scholar [41] S. Ukai, Stationary solutions of the BGK model equation on a finite interval with large boundary data,, Transport theory Statist. Phys., 21 (1992), 487. doi: 10.1080/00411459208203795. Google Scholar [42] S. Ukai and T. Yang, Mathematical Theory of Boltzmann equation,, Lecture Notes Series. no. 8, (2006). Google Scholar [43] C. Villani, A Review of mathematical topics in collisional kinetic theory,, Handbook of mathematical fluid dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar [44] C. Villani, Cercignani's conjecture is sometimes true and always almost true,, Comm. Math. Phys., 234 (2003), 455. doi: 10.1007/s00220-002-0777-1. Google Scholar [45] P. Welander, On the temperature jump in a rarefied gas,, Ark. Fys., 7 (1954), 507. Google Scholar [46] J. Wei and X. Zhang, The Cauchy problem for the BGK equation with an external force,, J. Math. Anal. Appl., 391 (2012), 10. doi: 10.1016/j.jmaa.2012.02.039. Google Scholar [47] B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation,, J. Statist. Phys., 86 (1997), 1053. doi: 10.1007/BF02183613. Google Scholar [48] S.-B. Yun, Cauchy problem for the Boltzmann-BGK model near a global Maxwellian,, J. Math. Phy., 51 (2010). doi: 10.1063/1.3516479. Google Scholar [49] S.-B. Yun, Classical solutions for the ellipsoidal BGK model with fixed collision frequency,, J. Differential Equations, 259 (2015), 6009. doi: 10.1016/j.jde.2015.07.016. Google Scholar [50] S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian,, SIAM J. Math. Anal., 47 (2015), 2324. doi: 10.1137/130932399. Google Scholar [51] X. Zhang, On the Cauchy problem of the Vlasov-Poisson-BGK system: global existence of weak solutions., J. Stat. Phys., 141 (2010), 566. doi: 10.1007/s10955-010-0064-z. Google Scholar [52] X. Zhang and S, Hu, $L^p$ solutions to the Cauchy problem of the BGK equation,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2816261. Google Scholar

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##### References:
 [1] P. Andries, J.-F. Bourgat, P. Le Tallec and B. Perthame, Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases,, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3369. doi: 10.1016/S0045-7825(02)00253-0. Google Scholar [2] P. Andries, P. Le Tallec, J.-P.Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number,, Eur. J. Mech. B Fluids, 19 (2000), 813. doi: 10.1016/S0997-7546(00)01103-1. Google Scholar [3] K. Aoki, K. Kanba and S. Takata, Numerical analysis of a supersonic rarefied gas flow past a flat plate,, Phys. Fluids, 9 (1997). doi: 10.1063/1.869204. Google Scholar [4] F. Berthelin and A. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks,, SIAM J. Math. Anal., 36 (2005), 1807. doi: 10.1137/S0036141003431554. Google Scholar [5] A. Bellouquid, Global existence and large-time behavior for BGK model for a gas with non-constant cross section,, Transport Theory Statist. Phys., 32 (2003), 157. doi: 10.1081/TT-120019041. Google Scholar [6] P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. Small amplitude process in charged and neutral one-component systems,, Phys. Rev., 94 (1954), 511. doi: 10.1103/PhysRev.94.511. Google Scholar [7] G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows,, Oxford Engineering Science, (1995). Google Scholar [8] M. Bisi, J. A. Cañizo and B. Lods, Entropy dissipation estimates for the linear Boltzmann operator,, J. Funct. Anal., 269 (2015), 1028. doi: 10.1016/j.jfa.2015.05.002. Google Scholar [9] A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation,, J. Statist. Phys., 94 (1999), 603. doi: 10.1023/A:1004537522686. Google Scholar [10] R. Bosi and M. J. Cáceres, The BGK model with external confining potential: Existence, long-time behaviour and time-periodic Maxwellian equilibria,, J. Stat. Phys., 136 (2009), 297. doi: 10.1007/s10955-009-9782-5. Google Scholar [11] S. Brull, An ellipsoidal statistical model for gas mixtures,, Comm. Math Sci., 13 (2015), 1. doi: 10.4310/CMS.2015.v13.n1.a1. Google Scholar [12] S. Brull and J. Schneider, A new approach of the ellipsoidal statistical model,, Cont. Mech. Thermodyn., 20 (2008), 63. doi: 10.1007/s00161-008-0068-y. Google Scholar [13] C. Cercignani, The Boltzmann Equation and Its Application,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1039-9. Google Scholar [14] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer-Verlag, (1994). doi: 10.1007/978-1-4419-8524-8. Google Scholar [15] C. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases,, Cambridge University Press, (1990). doi: 10.1119/1.1942035. Google Scholar [16] W. M. Chan, An Energy Method for the BGK Model,, M. Phil thesis, (2007). Google Scholar [17] J. Dolbeault, P. Markowich, D. Oelz and C. Schmeiser, Non linear diffusions as limit of kinetic equations with relaxation collision kernels,, Arch. Ration. Mech, 186 (2007), 133. doi: 10.1007/s00205-007-0049-5. Google Scholar [18] R. DiPerna and P.-L. Lions, On the Cauchy problem for the Boltzmann equation: Global existence and weak stability., Ann. Math., 130 (1989), 321. doi: 10.2307/1971423. Google Scholar [19] F. Filbet and S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation,, J. Sci. Comput., 46 (2011), 204. doi: 10.1007/s10915-010-9394-x. Google Scholar [20] F. Filbet and G. Russo, Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics,, Kinet. Relat. Models, 2 (2009), 231. doi: 10.3934/krm.2009.2.231. Google Scholar [21] M. A. Galli and R. Torczynski, Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls,, Phys. Fluids, 23 (2011). doi: 10.1063/1.3558869. Google Scholar [22] R. Glassey, The Cauchy Problems in Kinetic Theory,, SIAM, (1996). doi: 10.1137/1.9781611971477. Google Scholar [23] L. H. Holway, Kinetic theory of shock structure using and ellipsoidal distribution function. Rarefied Gas Dynamics, Vol. I, (Proc. Fourth Internat. Sympos., (1966), 193. Google Scholar [24] D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (B.G,K.) equaiton,, SIAM Journal on Numerical Analysis, 33 (1996), 2099. doi: 10.1137/S0036142994266856. Google Scholar [25] S. K. Loyalka, N. Petrellis and T. S. Storvick, Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates,, Z. Angew. Math. Phys., 30 (1979), 514. doi: 10.1007/BF01588895. Google Scholar [26] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Ration. Mech. Anal., 199 (2011), 493. doi: 10.1007/s00205-010-0354-2. Google Scholar [27] S. Mischler, Uniqueness for the BGK-equation in $R^n$ and the rate of convergence for a semi-discrete scheme,, Differential integral Equations, 9 (1996), 1119. Google Scholar [28] L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics., Math. Models Methods Appl. Sci., 10 (2000), 1121. doi: 10.1142/S0218202500000562. Google Scholar [29] L. Mieussens and H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number,, Phys. Fluids, 16 (2004). doi: 10.1063/1.1758217. Google Scholar [30] S. Park and S.-B. Yun, Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation,, submitted., (). Google Scholar [31] B. Perthame, Global existence to the BGK model of Boltzmann equation,, J. Differential Equations, 82 (1989), 191. doi: 10.1016/0022-0396(89)90173-3. Google Scholar [32] B. Perthame and M. Pulvirenti, Weighted $L^{\infty}$ bounds and uniqueness for the Boltzmann BGK model,, Arch. Rational Mech. Anal., 125 (1993), 289. doi: 10.1007/BF00383223. Google Scholar [33] S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations,, J. Sci. Comput., 32 (2007), 1. doi: 10.1007/s10915-006-9116-6. Google Scholar [34] G. Russo, P. Santagati and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation,, SIAM J. Numer. Anal., 50 (2012), 1111. doi: 10.1137/100800348. Google Scholar [35] L. Saint-Raymond, From the BGK model to the Navier-Stokes equations,, Ann. Sci. Ecole Norm. Sup., 36 (2003), 271. doi: 10.1016/S0012-9593(03)00010-7. Google Scholar [36] L. Saint-Raymond, Discrete time Navier-Stokes limit for the BGK Boltzmann equation,, Comm. Partial Differential Equations, 27 (2002), 149. doi: 10.1081/PDE-120002785. Google Scholar [37] Y. Sone, Kinetic Theory and Fluid Mechanics,, Boston: Birkhäuser, (2002). doi: 10.1007/978-1-4612-0061-1. Google Scholar [38] Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications,, Boston: Brikhäuser, (2006). doi: 10.1007/978-0-8176-4573-1. Google Scholar [39] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory,, Springer. 2005., (2005). doi: 10.1007/3-540-32386-4. Google Scholar [40] G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation,, Comm. Math. Phys., 203 (1999), 667. doi: 10.1007/s002200050631. Google Scholar [41] S. Ukai, Stationary solutions of the BGK model equation on a finite interval with large boundary data,, Transport theory Statist. Phys., 21 (1992), 487. doi: 10.1080/00411459208203795. Google Scholar [42] S. Ukai and T. Yang, Mathematical Theory of Boltzmann equation,, Lecture Notes Series. no. 8, (2006). Google Scholar [43] C. Villani, A Review of mathematical topics in collisional kinetic theory,, Handbook of mathematical fluid dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar [44] C. Villani, Cercignani's conjecture is sometimes true and always almost true,, Comm. Math. Phys., 234 (2003), 455. doi: 10.1007/s00220-002-0777-1. Google Scholar [45] P. Welander, On the temperature jump in a rarefied gas,, Ark. Fys., 7 (1954), 507. Google Scholar [46] J. Wei and X. Zhang, The Cauchy problem for the BGK equation with an external force,, J. Math. Anal. Appl., 391 (2012), 10. doi: 10.1016/j.jmaa.2012.02.039. Google Scholar [47] B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation,, J. Statist. Phys., 86 (1997), 1053. doi: 10.1007/BF02183613. Google Scholar [48] S.-B. Yun, Cauchy problem for the Boltzmann-BGK model near a global Maxwellian,, J. Math. Phy., 51 (2010). doi: 10.1063/1.3516479. Google Scholar [49] S.-B. Yun, Classical solutions for the ellipsoidal BGK model with fixed collision frequency,, J. Differential Equations, 259 (2015), 6009. doi: 10.1016/j.jde.2015.07.016. Google Scholar [50] S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian,, SIAM J. Math. Anal., 47 (2015), 2324. doi: 10.1137/130932399. Google Scholar [51] X. Zhang, On the Cauchy problem of the Vlasov-Poisson-BGK system: global existence of weak solutions., J. Stat. Phys., 141 (2010), 566. doi: 10.1007/s10955-010-0064-z. Google Scholar [52] X. Zhang and S, Hu, $L^p$ solutions to the Cauchy problem of the BGK equation,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2816261. Google Scholar
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