Advanced Search
Article Contents
Article Contents

Entropy production for ellipsoidal BGK model of the Boltzmann equation

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35Q20, 82C40; Secondary: 35B45.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    S. Brull, An ellipsoidal statistical model for gas mixtures, Comm. Math Sci., 13 (2015), 1-13.doi: 10.4310/CMS.2015.v13.n1.a1.


    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.


    C. Cercignani, The Boltzmann Equation and Its Application, Springer-Verlag, 1988.doi: 10.1007/978-1-4612-1039-9.


    C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, 1994.doi: 10.1007/978-1-4419-8524-8.


    C. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1990.doi: 10.1119/1.1942035.


    W. M. Chan, An Energy Method for the BGK Model, M. Phil thesis, City University of Hong Kong, 2007.


    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.


    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.


    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.


    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.


    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.


    R. Glassey, The Cauchy Problems in Kinetic Theory, SIAM, 1996.doi: 10.1137/1.9781611971477.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    S. Park and S.-B. Yun, Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation, submitted.


    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.


    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.


    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.


    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.


    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.


    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.


    Y. Sone, Kinetic Theory and Fluid Mechanics, Boston: Birkhäuser, 2002.doi: 10.1007/978-1-4612-0061-1.


    Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Boston: Brikhäuser, 2006.doi: 10.1007/978-0-8176-4573-1.


    H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer. 2005.doi: 10.1007/3-540-32386-4.


    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.


    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.


    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.


    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.


    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.


    P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), 507-553.


    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.


    B. Wennberg, Entropy dissipation and moment production for the Boltzmann equation, J. Statist. Phys., 86 (1997), 1053-1066.doi: 10.1007/BF02183613.


    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.


    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.


    S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.doi: 10.1137/130932399.


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(228) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint