# American Institute of Mathematical Sciences

2012, 5(1): 1-19. doi: 10.3934/krm.2012.5.1

## Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures

Received  January 2011 Revised  June 2011 Published  January 2012

We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for half-space problems for the linearized discrete Boltzmann equation, existence results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation. A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for a class of symmetric models. Some explicit calculations are also made for a simplified 6 + 4 -velocity model.
Citation: Niclas Bernhoff. Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures. Kinetic & Related Models, 2012, 5 (1) : 1-19. doi: 10.3934/krm.2012.5.1
##### References:
 [1] K. Aoki, C. Bardos and S. Takata, Knudsen layer for gas mixtures,, J. Stat. Phys., 112 (2003), 629. doi: 10.1023/A:1023876025363. [2] C. Bardos, R. E. Caflisch and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas,, Comm. Pure Appl. Math., 39 (1986), 323. doi: 10.1002/cpa.3160390304. [3] C. Bardos, F. Golse and Y. Sone, Half-space problems for the Boltzmann equation: A survey,, J. Stat. Phys., 124 (2006), 275. doi: 10.1007/s10955-006-9077-z. [4] N. Bernhoff, On half-space problems for the linearized discrete Boltzmann equation,, Riv. Mat. Univ. Parma (7), 9 (2008), 73. [5] N. Bernhoff, On half-space problems for the weakly non-linear discrete Boltzmann equation,, Kinet. Relat. Models, 3 (2010), 195. doi: 10.3934/krm.2010.3.195. [6] N. Bernhoff, On half-space problems for the discrete Boltzmann equation,, Nuovo Cim. C, 33 (2010), 47. [7] N. Bernhoff, On half-space problems for the non-linear discrete Boltzmann equation in the presence of a non-condensable gas,, preprint., (). [8] N. Bernhoff, Boundary layers for the non-linear discrete Boltzmann equation,, preprint., (). [9] N. Bernhoff and A. Bobylev, Weak shock waves for the general discrete velocity model of the Boltzmann equation,, Commun. Math. Sci., 5 (2007), 815. [10] A. V. Bobylev and N. Bernhoff, Discrete velocity models and dynamical systems,, in, (2003), 203. doi: 10.1142/9789812796905_0008. [11] A. V. Bobylev and C. Cercignani, Discrete velocity models for mixtures,, J. Stat. Phys., 91 (1998), 327. doi: 10.1023/A:1023052423760. [12] A. V. Bobylev and C. Cercignani, Discrete velocity models without nonphysical invariants,, J. Stat. Phys., 97 (1999), 677. doi: 10.1023/A:1004615309058. [13] A. V. Bobylev, A. Palczewski and J. Schneider, On approximation of the Boltzmann equation by discrete velocity models,, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 639. [14] A. V. Bobylev and M. C. Vinerean, Construction and classification of discrete kinetic models without spurious invariants,, Riv. Mat. Univ. Parma (7), 7 (2007), 1. [15] A. V. Bobylev and M. C. Vinerean, Construction of discrete kinetic models with given invariants,, J. Stat. Phys., 132 (2008), 153. doi: 10.1007/s10955-008-9536-9. [16] C. Bose, R. Illner and S. Ukai, On shock wave solutions for discrete velocity models of the Boltzmann equation,, Transp. Th. Stat. Phys., 27 (1998), 35. doi: 10.1080/00411459808205140. [17] J. E. Broadwell, Shock structure in a simple discrete velocity gas,, Phys. Fluids, 7 (1964), 1243. doi: 10.1063/1.1711368. [18] R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation,, Comm. Math. Phys., 86 (1982), 161. doi: 10.1007/BF01206009. [19] C. Cercignani and H. Cornille, Shock waves for a discrete velocity gas mixture,, J. Stat. Phys., 99 (2000), 115. doi: 10.1023/A:1018692522765. [20] H. Cornille and C. Cercignani, A class of planar discrete velocity models for gas mixtures,, J. Stat. Phys., 99 (2000), 967. doi: 10.1023/A:1018603831215. [21] H. Cornille and C. Cercignani, Large size planar discrete velocity models for gas mixtures,, J. Phys. A: Math. Gen., 34 (2001), 2985. doi: 10.1088/0305-4470/34/14/306. [22] L. Fainsilber, P. Kurlberg and B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation,, SIAM J. Math. Anal., 37 (2006), 1903. doi: 10.1137/040618916. [23] S. Kawashima and S. Nishibata, Existence of a stationary wave for the discrete Boltzmann equation in the half space,, Comm. Math. Phys., 207 (1999), 385. doi: 10.1007/s002200050730. [24] S. Kawashima and S. Nishibata, Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries,, Comm. Math. Phys., 211 (2000), 183. doi: 10.1007/s002200050808. [25] T.-P. Liu and S.-H. Yu, Boltzmann equation: Mikro-macro decompositions and positivity of shock profiles,, Comm. Math. Phys., 246 (2004), 133. doi: 10.1007/s00220-003-1030-2. [26] T.-P. Liu and S.-H. Yu, Invariant manifolds for steady Boltzmann flows and applications,, preprint., (). [27] A. Palczewski, J. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation,, SIAM J. Numer. Anal., 34 (1997), 1865. doi: 10.1137/S0036142995289007. [28] Y. Sone, "Kinetic Theory and Fluid Dynamics," Modeling and Simulation in Science, Engineering and Technology,, Birkhäuser Boston, (2002). doi: 10.1007/978-1-4612-0061-1. [29] Y. Sone, "Molecular Gas Dynamics. Theory, Techniques, and Applications,", Modeling and Simulation in Science, (2007). [30] S. Taguchi, K. Aoki and V. Latocha, Vapor flows along a plane condensed phase with weak condensation in the presence of a noncondensable gas,, J. Stat. Phys., 124 (2006), 321. doi: 10.1007/s10955-005-0001-8. [31] S. Takata and F. Golse, Half-space problem of the nonlinear Boltzmann equation for weak evaporation and condensation of a binary mixture of vapors,, Eur. J. Mech. B Fluids., 26 (2007), 105. doi: 10.1016/j.euromechflu.2006.04.003. [32] S. Ukai, On the half-space problem for the discrete velocity model of the Boltzmann equation,, in, (1998), 160. [33] S. Ukai, T. Yang and S.-H. Yu, Nonlinear boundary layers of the Boltzmann equation. I. Existence,, Comm. Math. Phys., 236 (2003), 373. doi: 10.1007/s00220-003-0822-8. [34] X. Yang, The solutions for the boundary layer problem of Boltzmann equation in a half-space,, J. Stat. Phys., 143 (2011), 168. doi: 10.1007/s10955-011-0158-2.

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##### References:
 [1] K. Aoki, C. Bardos and S. Takata, Knudsen layer for gas mixtures,, J. Stat. Phys., 112 (2003), 629. doi: 10.1023/A:1023876025363. [2] C. Bardos, R. E. Caflisch and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas,, Comm. Pure Appl. Math., 39 (1986), 323. doi: 10.1002/cpa.3160390304. [3] C. Bardos, F. Golse and Y. Sone, Half-space problems for the Boltzmann equation: A survey,, J. Stat. Phys., 124 (2006), 275. doi: 10.1007/s10955-006-9077-z. [4] N. Bernhoff, On half-space problems for the linearized discrete Boltzmann equation,, Riv. Mat. Univ. Parma (7), 9 (2008), 73. [5] N. Bernhoff, On half-space problems for the weakly non-linear discrete Boltzmann equation,, Kinet. Relat. Models, 3 (2010), 195. doi: 10.3934/krm.2010.3.195. [6] N. Bernhoff, On half-space problems for the discrete Boltzmann equation,, Nuovo Cim. C, 33 (2010), 47. [7] N. Bernhoff, On half-space problems for the non-linear discrete Boltzmann equation in the presence of a non-condensable gas,, preprint., (). [8] N. Bernhoff, Boundary layers for the non-linear discrete Boltzmann equation,, preprint., (). [9] N. Bernhoff and A. Bobylev, Weak shock waves for the general discrete velocity model of the Boltzmann equation,, Commun. Math. Sci., 5 (2007), 815. [10] A. V. Bobylev and N. Bernhoff, Discrete velocity models and dynamical systems,, in, (2003), 203. doi: 10.1142/9789812796905_0008. [11] A. V. Bobylev and C. Cercignani, Discrete velocity models for mixtures,, J. Stat. Phys., 91 (1998), 327. doi: 10.1023/A:1023052423760. [12] A. V. Bobylev and C. Cercignani, Discrete velocity models without nonphysical invariants,, J. Stat. Phys., 97 (1999), 677. doi: 10.1023/A:1004615309058. [13] A. V. Bobylev, A. Palczewski and J. Schneider, On approximation of the Boltzmann equation by discrete velocity models,, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 639. [14] A. V. Bobylev and M. C. Vinerean, Construction and classification of discrete kinetic models without spurious invariants,, Riv. Mat. Univ. Parma (7), 7 (2007), 1. [15] A. V. Bobylev and M. C. Vinerean, Construction of discrete kinetic models with given invariants,, J. Stat. Phys., 132 (2008), 153. doi: 10.1007/s10955-008-9536-9. [16] C. Bose, R. Illner and S. Ukai, On shock wave solutions for discrete velocity models of the Boltzmann equation,, Transp. Th. Stat. Phys., 27 (1998), 35. doi: 10.1080/00411459808205140. [17] J. E. Broadwell, Shock structure in a simple discrete velocity gas,, Phys. Fluids, 7 (1964), 1243. doi: 10.1063/1.1711368. [18] R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation,, Comm. Math. Phys., 86 (1982), 161. doi: 10.1007/BF01206009. [19] C. Cercignani and H. Cornille, Shock waves for a discrete velocity gas mixture,, J. Stat. Phys., 99 (2000), 115. doi: 10.1023/A:1018692522765. [20] H. Cornille and C. Cercignani, A class of planar discrete velocity models for gas mixtures,, J. Stat. Phys., 99 (2000), 967. doi: 10.1023/A:1018603831215. [21] H. Cornille and C. Cercignani, Large size planar discrete velocity models for gas mixtures,, J. Phys. A: Math. Gen., 34 (2001), 2985. doi: 10.1088/0305-4470/34/14/306. [22] L. Fainsilber, P. Kurlberg and B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation,, SIAM J. Math. Anal., 37 (2006), 1903. doi: 10.1137/040618916. [23] S. Kawashima and S. Nishibata, Existence of a stationary wave for the discrete Boltzmann equation in the half space,, Comm. Math. Phys., 207 (1999), 385. doi: 10.1007/s002200050730. [24] S. Kawashima and S. Nishibata, Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries,, Comm. Math. Phys., 211 (2000), 183. doi: 10.1007/s002200050808. [25] T.-P. Liu and S.-H. Yu, Boltzmann equation: Mikro-macro decompositions and positivity of shock profiles,, Comm. Math. Phys., 246 (2004), 133. doi: 10.1007/s00220-003-1030-2. [26] T.-P. Liu and S.-H. Yu, Invariant manifolds for steady Boltzmann flows and applications,, preprint., (). [27] A. Palczewski, J. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation,, SIAM J. Numer. Anal., 34 (1997), 1865. doi: 10.1137/S0036142995289007. [28] Y. Sone, "Kinetic Theory and Fluid Dynamics," Modeling and Simulation in Science, Engineering and Technology,, Birkhäuser Boston, (2002). doi: 10.1007/978-1-4612-0061-1. [29] Y. Sone, "Molecular Gas Dynamics. Theory, Techniques, and Applications,", Modeling and Simulation in Science, (2007). [30] S. Taguchi, K. Aoki and V. Latocha, Vapor flows along a plane condensed phase with weak condensation in the presence of a noncondensable gas,, J. Stat. Phys., 124 (2006), 321. doi: 10.1007/s10955-005-0001-8. [31] S. Takata and F. Golse, Half-space problem of the nonlinear Boltzmann equation for weak evaporation and condensation of a binary mixture of vapors,, Eur. J. Mech. B Fluids., 26 (2007), 105. doi: 10.1016/j.euromechflu.2006.04.003. [32] S. Ukai, On the half-space problem for the discrete velocity model of the Boltzmann equation,, in, (1998), 160. [33] S. Ukai, T. Yang and S.-H. Yu, Nonlinear boundary layers of the Boltzmann equation. I. Existence,, Comm. Math. Phys., 236 (2003), 373. doi: 10.1007/s00220-003-0822-8. [34] X. Yang, The solutions for the boundary layer problem of Boltzmann equation in a half-space,, J. Stat. Phys., 143 (2011), 168. doi: 10.1007/s10955-011-0158-2.
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