Entropy production for ellipsoidal BGK model of the Boltzmann equation

Seok-Bae  Yun

doi:10.3934/krm.2016009

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

Entropy production for ellipsoidal BGK model of the Boltzmann equation

Seok-Bae Yun ^1,

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

Received October 2015 Revised January 2016 Published May 2016

Abstract
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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.

Keywords: BGK model, Boltzmann equation, Kinetic theory of gases, ellipsoidal BGK model, entropy production..

Mathematics Subject Classification: Primary: 35Q20, 82C40; Secondary: 35B4.

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), 3369-3390. 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), 813-830. 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), p1144. 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), 1807-1835. 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), 157-184. 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), 511-525. 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, The Clarendon Press, Oxford University Press, New York, 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), 1028-1069. 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), 603-618. 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), 297-330. doi: 10.1007/s10955-009-9782-5. Google Scholar
[11]	S. Brull, An ellipsoidal statistical model for gas mixtures, Comm. Math Sci., 13 (2015), 1-13. 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), 63-74. 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, Cambridge, 1990. doi: 10.1119/1.1942035. Google Scholar
[16]	W. M. Chan, An Energy Method for the BGK Model, M. Phil thesis, City University of Hong Kong, 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, Anal., 186 (2007), 133-158. 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), 321-366. 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), 204-224. 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), 231-250. 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), 030601. 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., Univ. Toronto, 1964), Academic Press, New York, (1966), 193-215. 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), 2099-2119. 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), 514-521. 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), 493-525. 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), 1119-1138. 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), 1121-1149. 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), p2797. 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), 191-205. 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), 289-295. doi: 10.1007/BF00383223. Google Scholar
[33]	S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations, J. Sci. Comput., 32 (2007), 1-28. 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), 1111-1135. 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), 271-317. 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), 149-184. 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. 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), 667-706. 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), 487-500. doi: 10.1080/00411459208203795. Google Scholar
[42]	S. Ukai and T. Yang, Mathematical Theory of Boltzmann equation, Lecture Notes Series. no. 8, Liu Bie Ju Center for Math. Sci, City University of Hong Kong, 2006. Google Scholar
[43]	C. Villani, A Review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I. North-Holland. Amsterdam, 2002, 71-305. 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), 455-490. doi: 10.1007/s00220-002-0777-1. Google Scholar
[45]	P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), 507-553. 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), 10-25. 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), 1053-1066. 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), 123514, 24pp. 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), 6009-6037. 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), 2324-2354. 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), 566-588. 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), 113304, 17pp. doi: 10.1063/1.2816261. Google Scholar

show all references

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), 3369-3390. 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), 813-830. 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), p1144. 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), 1807-1835. 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), 157-184. 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), 511-525. 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, The Clarendon Press, Oxford University Press, New York, 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), 1028-1069. 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), 603-618. 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), 297-330. doi: 10.1007/s10955-009-9782-5. Google Scholar
[11]	S. Brull, An ellipsoidal statistical model for gas mixtures, Comm. Math Sci., 13 (2015), 1-13. 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), 63-74. 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, Cambridge, 1990. doi: 10.1119/1.1942035. Google Scholar
[16]	W. M. Chan, An Energy Method for the BGK Model, M. Phil thesis, City University of Hong Kong, 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, Anal., 186 (2007), 133-158. 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), 321-366. 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), 204-224. 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), 231-250. 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), 030601. 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., Univ. Toronto, 1964), Academic Press, New York, (1966), 193-215. 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), 2099-2119. 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), 514-521. 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), 493-525. 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), 1119-1138. 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), 1121-1149. 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), p2797. 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), 191-205. 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), 289-295. doi: 10.1007/BF00383223. Google Scholar
[33]	S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations, J. Sci. Comput., 32 (2007), 1-28. 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), 1111-1135. 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), 271-317. 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), 149-184. 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. 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), 667-706. 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), 487-500. doi: 10.1080/00411459208203795. Google Scholar
[42]	S. Ukai and T. Yang, Mathematical Theory of Boltzmann equation, Lecture Notes Series. no. 8, Liu Bie Ju Center for Math. Sci, City University of Hong Kong, 2006. Google Scholar
[43]	C. Villani, A Review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I. North-Holland. Amsterdam, 2002, 71-305. 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), 455-490. doi: 10.1007/s00220-002-0777-1. Google Scholar
[45]	P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), 507-553. 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), 10-25. 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), 1053-1066. 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), 123514, 24pp. 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), 6009-6037. 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), 2324-2354. 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), 566-588. 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), 113304, 17pp. doi: 10.1063/1.2816261. Google Scholar

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Entropy production for ellipsoidal BGK model of the Boltzmann equation

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Entropy production for ellipsoidal BGK model of the Boltzmann equation

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