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December  2017, 10(4): 925-955. doi: 10.3934/krm.2017037

Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations

Department of Mathematics and Computer Science, Karlstad University, 65188 Karlstad, Sweden

Received  March 2016 Revised  December 2016 Published  March 2017

We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting with a Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.

Citation: Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic & Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037
References:
[1]

L. Arkeryd, On low temperature kinetic theory; spin diffusion, Bose-Einstein condensates, anyons, J. Stat. Phys., 150 (2013), 1063-1079.  doi: 10.1007/s10955-013-0695-y.  Google Scholar

[2]

L. Arkeryd and A. Nouri, A Milne problem from a Bose condensate with excitations, Kinet. Relat. Models, 6 (2013), 671-686.  doi: 10.3934/krm.2013.6.671.  Google Scholar

[3]

H. Babovsky, Kinetic boundary layers: On the adequate discretization of the Boltzmann collision operator, J. Comp. Appl. Math., 110 (1999), 225-239.  doi: 10.1016/S0377-0427(99)00196-X.  Google Scholar

[4]

C. BardosF. Golse and Y. Sone, Half-space problems for the Boltzmann equation: A survey, J. Stat. Phys., 124 (2006), 275-300.  doi: 10.1007/s10955-006-9077-z.  Google Scholar

[5]

C. Bardos and X. Yang, The classification of well-posed kinetic boundary layer for hard sphere gas mixtures, Comm. Partial Differ. Equ., 37 (2012), 1286-1314.  doi: 10.1080/03605302.2011.624149.  Google Scholar

[6]

N. Bernhoff, On half-space problems for the linearized discrete Boltzmann equation, Riv. Mat. Univ. Parma, 9 (2008), 73-124.   Google Scholar

[7]

N. Bernhoff, On half-space problems for the weakly non-linear discrete Boltzmann equation, Kinet. Relat. Models, 3 (2010), 195-222.  doi: 10.3934/krm.2010.3.195.  Google Scholar

[8]

N. Bernhoff, Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures, Kinet. Relat. Models, 5 (2012), 1-19.  doi: 10.3934/krm.2012.5.1.  Google Scholar

[9]

N. Bernhoff, Half-space problem for the discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas, J. Stat. Phys., 147 (2012), 1156-1181.  doi: 10.1007/s10955-012-0513-y.  Google Scholar

[10]

N. Bernhoff, Half-space problems for a linearized discrete quantum kinetic equation, J. Stat. Phys., 159 (2015), 358-379.  doi: 10.1007/s10955-015-1190-4.  Google Scholar

[11]

N. Bernhoff, Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles, AIP Conference Proceedings, 1786 (2016), 040005.  doi: 10.1063/1.4967543.  Google Scholar

[12]

N. Bernhoff, Discrete velocity models for polyatomic molecules without nonphysical collision invariants, Preprint. Google Scholar

[13]

N. Bernhoff and A. Bobylev, Weak shock waves for the general discrete velocity model of the Boltzmann equation, Commun. Math. Sci., 5 (2007), 815-832.  doi: 10.4310/CMS.2007.v5.n4.a4.  Google Scholar

[14]

N. Bernhoff and M. C. Vinerean, Discrete velocity models for multicomponent mixtures without nonphysical collision invariants, J. Stat. Phys., 165 (2016), 434-453.  doi: 10.1007/s10955-016-1624-7.  Google Scholar

[15]

G. A. Bird, Molecular gas dynamics Clarendon-Press, 1976. Google Scholar

[16]

A. V. Bobylev and N. Bernhoff, Discrete velocity models and dynamical systems, in Lecture Notes on the Discretization of the Boltzmann Equation (eds. N. Bellomo and R. Gatignol), World Scientific, 63 (2003), 203–222. doi: 10.1142/9789812796905_0008.  Google Scholar

[17]

A. V. Bobylev and C. Cercignani, Discrete velocity models without non-physical invariants, J. Stat. Phys., 97 (1999), 677-686.  doi: 10.1023/A:1004615309058.  Google Scholar

[18]

A. V. BobylevA. Palczewski and J. Schneider, On approximation of the Boltzmann equation by discrete velocity models, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 639-644.   Google Scholar

[19]

A. V. Bobylev and M. C. Vinerean, Construction of discrete kinetic models with given invariants, J. Stat. Phys., 132 (2008), 153-170.  doi: 10.1007/s10955-008-9536-9.  Google Scholar

[20]

C. Buet, Conservative and entropy schemes for Boltzmann collision operator of polyatomic gases, Math. Mod. Meth. Appl. Sci., 7 (1997), 165-192.  doi: 10.1142/S0218202597000116.  Google Scholar

[21]

H. Cabannes, The discrete Boltzmann equation, 1980 (2003), Lecture notes given at the University of California at Berkeley, 1980, revised with R. Gatignol and L-S. Luo, 2003. Google Scholar

[22]

C. Cercignani, The Boltzmann Equation and its Applications Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[23]

C. Cercignani, Rarefied Gas Dynamics Cambridge University Press, 2000.  Google Scholar

[24]

C. CercignaniR. IllnerM. Pulvirenti and M. Shinbrot, On nonlinear stationary half-space problems in discrete kinetic theory, J. Stat. Phys., 52 (1988), 885-896.  doi: 10.1007/BF01019733.  Google Scholar

[25]

F. CoronF. Golse and C. Sulem, A classification of well-posed kinetic layer problems, Comm. Pure Appl. Math., 41 (1988), 409-435.  doi: 10.1002/cpa.3160410403.  Google Scholar

[26]

A. Ern and V. Giovangigli, Multicomponent Transport Algorithms Springer-Verlag, 1994.  Google Scholar

[27]

L. FainsilberP. Kurlberg and B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation, Siam J. Math. Anal., 37 (2006), 1903-1922.  doi: 10.1137/040618916.  Google Scholar

[28]

R. Gatignol, Théorie Cinétique des Gaz á Répartition Discréte de Vitesses Springer-Verlag, 1975.  Google Scholar

[29]

F. Golse, Analysis of the boundary layer equation in the kinetic theory of gases, Bull. Inst. Math. Acad. Sin., 3 (2008), 211-242.   Google Scholar

[30]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.   Google Scholar

[31]

S. Kawashima and S. Nishibata, Existence of a stationary wave for the discrete Boltzmann equation in the half space, Comm. Math. Phys., 207 (1999), 385-409.  doi: 10.1007/s002200050730.  Google Scholar

[32]

S. Kawashima and S. Nishibata, Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries, Comm. Math. Phys., 211 (2000), 183-206.  doi: 10.1007/s002200050808.  Google Scholar

[33]

T. R. Kirkpatrick and J. R. Dorfman, Transport in a dilute but condensed nonideal Bose gas: Kinetic equations, J. Low Temp. Phys., 58 (1985), 301-331.  doi: 10.1007/BF00681309.  Google Scholar

[34]

C. MouhotL. Pareschi and T. Rey, Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation, Math. Model. Numer. Anal., 47 (2013), 1515-1531.  doi: 10.1051/m2an/2013078.  Google Scholar

[35]

A. MunafóJ. R. HaackI. M. Gamba and T. E. Magin, A spectral-Lagrangian Boltzmann solver for a multi-energy level gas, J. Comput. Phys., 264 (2014), 152-176.  doi: 10.1016/j.jcp.2014.01.036.  Google Scholar

[36]

L. W. Nordheim, On the kinetic methods in the new statistics and its applications in the electron theory of conductivity, Proc. Roy. Soc. London Ser. A, 119 (1928), 689-698.  doi: 10.1098/rspa.1928.0126.  Google Scholar

[37]

A. PalczewskiJ. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal., 34 (1997), 1865-1883.  doi: 10.1137/S0036142995289007.  Google Scholar

[38]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.  Google Scholar

[39]

Y. Sone, Kinetic Theory and Fluid Dynamics Birkhauser, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar

[40]

Y. Sone, Molecular Gas Dynamics Birkhauser, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[41]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases, Phys. Rev., 43 (1933), 552-561.  doi: 10.1103/PhysRev.43.552.  Google Scholar

[42]

S. Ukai, On the half-space problem for the discrete velocity model of the Boltzmann equation, in Advances in Nonlinear Partial Differential Equations and Stochastics (eds. S. Kawashima and T. Yanagisawa), World Scientific, 48 (1998), 160–174. doi: 10.1142/9789812816481_0005.  Google Scholar

[43]

S. UkaiT. Yang and S.-H. Yu, Nonlinear boundary layers of the Boltzmann equation: Ⅰ. Existence, Comm. Math. Phys., 236 (2003), 373-393.  doi: 10.1007/s00220-003-0822-8.  Google Scholar

[44]

X. Yang, The solutions for the boundary layer problem of Boltzmann equation in a half-space, J. Stat. Phys., 143 (2011), 168-196.  doi: 10.1007/s10955-011-0158-2.  Google Scholar

[45]

E. ZarembaT. Nikuni and A. Griffin, Dynamics of trapped Bose gases at finite temperatures, J. Low Temp. Phys., 116 (1999), 277-345.   Google Scholar

show all references

References:
[1]

L. Arkeryd, On low temperature kinetic theory; spin diffusion, Bose-Einstein condensates, anyons, J. Stat. Phys., 150 (2013), 1063-1079.  doi: 10.1007/s10955-013-0695-y.  Google Scholar

[2]

L. Arkeryd and A. Nouri, A Milne problem from a Bose condensate with excitations, Kinet. Relat. Models, 6 (2013), 671-686.  doi: 10.3934/krm.2013.6.671.  Google Scholar

[3]

H. Babovsky, Kinetic boundary layers: On the adequate discretization of the Boltzmann collision operator, J. Comp. Appl. Math., 110 (1999), 225-239.  doi: 10.1016/S0377-0427(99)00196-X.  Google Scholar

[4]

C. BardosF. Golse and Y. Sone, Half-space problems for the Boltzmann equation: A survey, J. Stat. Phys., 124 (2006), 275-300.  doi: 10.1007/s10955-006-9077-z.  Google Scholar

[5]

C. Bardos and X. Yang, The classification of well-posed kinetic boundary layer for hard sphere gas mixtures, Comm. Partial Differ. Equ., 37 (2012), 1286-1314.  doi: 10.1080/03605302.2011.624149.  Google Scholar

[6]

N. Bernhoff, On half-space problems for the linearized discrete Boltzmann equation, Riv. Mat. Univ. Parma, 9 (2008), 73-124.   Google Scholar

[7]

N. Bernhoff, On half-space problems for the weakly non-linear discrete Boltzmann equation, Kinet. Relat. Models, 3 (2010), 195-222.  doi: 10.3934/krm.2010.3.195.  Google Scholar

[8]

N. Bernhoff, Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures, Kinet. Relat. Models, 5 (2012), 1-19.  doi: 10.3934/krm.2012.5.1.  Google Scholar

[9]

N. Bernhoff, Half-space problem for the discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas, J. Stat. Phys., 147 (2012), 1156-1181.  doi: 10.1007/s10955-012-0513-y.  Google Scholar

[10]

N. Bernhoff, Half-space problems for a linearized discrete quantum kinetic equation, J. Stat. Phys., 159 (2015), 358-379.  doi: 10.1007/s10955-015-1190-4.  Google Scholar

[11]

N. Bernhoff, Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles, AIP Conference Proceedings, 1786 (2016), 040005.  doi: 10.1063/1.4967543.  Google Scholar

[12]

N. Bernhoff, Discrete velocity models for polyatomic molecules without nonphysical collision invariants, Preprint. Google Scholar

[13]

N. Bernhoff and A. Bobylev, Weak shock waves for the general discrete velocity model of the Boltzmann equation, Commun. Math. Sci., 5 (2007), 815-832.  doi: 10.4310/CMS.2007.v5.n4.a4.  Google Scholar

[14]

N. Bernhoff and M. C. Vinerean, Discrete velocity models for multicomponent mixtures without nonphysical collision invariants, J. Stat. Phys., 165 (2016), 434-453.  doi: 10.1007/s10955-016-1624-7.  Google Scholar

[15]

G. A. Bird, Molecular gas dynamics Clarendon-Press, 1976. Google Scholar

[16]

A. V. Bobylev and N. Bernhoff, Discrete velocity models and dynamical systems, in Lecture Notes on the Discretization of the Boltzmann Equation (eds. N. Bellomo and R. Gatignol), World Scientific, 63 (2003), 203–222. doi: 10.1142/9789812796905_0008.  Google Scholar

[17]

A. V. Bobylev and C. Cercignani, Discrete velocity models without non-physical invariants, J. Stat. Phys., 97 (1999), 677-686.  doi: 10.1023/A:1004615309058.  Google Scholar

[18]

A. V. BobylevA. Palczewski and J. Schneider, On approximation of the Boltzmann equation by discrete velocity models, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 639-644.   Google Scholar

[19]

A. V. Bobylev and M. C. Vinerean, Construction of discrete kinetic models with given invariants, J. Stat. Phys., 132 (2008), 153-170.  doi: 10.1007/s10955-008-9536-9.  Google Scholar

[20]

C. Buet, Conservative and entropy schemes for Boltzmann collision operator of polyatomic gases, Math. Mod. Meth. Appl. Sci., 7 (1997), 165-192.  doi: 10.1142/S0218202597000116.  Google Scholar

[21]

H. Cabannes, The discrete Boltzmann equation, 1980 (2003), Lecture notes given at the University of California at Berkeley, 1980, revised with R. Gatignol and L-S. Luo, 2003. Google Scholar

[22]

C. Cercignani, The Boltzmann Equation and its Applications Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[23]

C. Cercignani, Rarefied Gas Dynamics Cambridge University Press, 2000.  Google Scholar

[24]

C. CercignaniR. IllnerM. Pulvirenti and M. Shinbrot, On nonlinear stationary half-space problems in discrete kinetic theory, J. Stat. Phys., 52 (1988), 885-896.  doi: 10.1007/BF01019733.  Google Scholar

[25]

F. CoronF. Golse and C. Sulem, A classification of well-posed kinetic layer problems, Comm. Pure Appl. Math., 41 (1988), 409-435.  doi: 10.1002/cpa.3160410403.  Google Scholar

[26]

A. Ern and V. Giovangigli, Multicomponent Transport Algorithms Springer-Verlag, 1994.  Google Scholar

[27]

L. FainsilberP. Kurlberg and B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation, Siam J. Math. Anal., 37 (2006), 1903-1922.  doi: 10.1137/040618916.  Google Scholar

[28]

R. Gatignol, Théorie Cinétique des Gaz á Répartition Discréte de Vitesses Springer-Verlag, 1975.  Google Scholar

[29]

F. Golse, Analysis of the boundary layer equation in the kinetic theory of gases, Bull. Inst. Math. Acad. Sin., 3 (2008), 211-242.   Google Scholar

[30]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.   Google Scholar

[31]

S. Kawashima and S. Nishibata, Existence of a stationary wave for the discrete Boltzmann equation in the half space, Comm. Math. Phys., 207 (1999), 385-409.  doi: 10.1007/s002200050730.  Google Scholar

[32]

S. Kawashima and S. Nishibata, Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries, Comm. Math. Phys., 211 (2000), 183-206.  doi: 10.1007/s002200050808.  Google Scholar

[33]

T. R. Kirkpatrick and J. R. Dorfman, Transport in a dilute but condensed nonideal Bose gas: Kinetic equations, J. Low Temp. Phys., 58 (1985), 301-331.  doi: 10.1007/BF00681309.  Google Scholar

[34]

C. MouhotL. Pareschi and T. Rey, Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation, Math. Model. Numer. Anal., 47 (2013), 1515-1531.  doi: 10.1051/m2an/2013078.  Google Scholar

[35]

A. MunafóJ. R. HaackI. M. Gamba and T. E. Magin, A spectral-Lagrangian Boltzmann solver for a multi-energy level gas, J. Comput. Phys., 264 (2014), 152-176.  doi: 10.1016/j.jcp.2014.01.036.  Google Scholar

[36]

L. W. Nordheim, On the kinetic methods in the new statistics and its applications in the electron theory of conductivity, Proc. Roy. Soc. London Ser. A, 119 (1928), 689-698.  doi: 10.1098/rspa.1928.0126.  Google Scholar

[37]

A. PalczewskiJ. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal., 34 (1997), 1865-1883.  doi: 10.1137/S0036142995289007.  Google Scholar

[38]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.  Google Scholar

[39]

Y. Sone, Kinetic Theory and Fluid Dynamics Birkhauser, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar

[40]

Y. Sone, Molecular Gas Dynamics Birkhauser, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[41]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases, Phys. Rev., 43 (1933), 552-561.  doi: 10.1103/PhysRev.43.552.  Google Scholar

[42]

S. Ukai, On the half-space problem for the discrete velocity model of the Boltzmann equation, in Advances in Nonlinear Partial Differential Equations and Stochastics (eds. S. Kawashima and T. Yanagisawa), World Scientific, 48 (1998), 160–174. doi: 10.1142/9789812816481_0005.  Google Scholar

[43]

S. UkaiT. Yang and S.-H. Yu, Nonlinear boundary layers of the Boltzmann equation: Ⅰ. Existence, Comm. Math. Phys., 236 (2003), 373-393.  doi: 10.1007/s00220-003-0822-8.  Google Scholar

[44]

X. Yang, The solutions for the boundary layer problem of Boltzmann equation in a half-space, J. Stat. Phys., 143 (2011), 168-196.  doi: 10.1007/s10955-011-0158-2.  Google Scholar

[45]

E. ZarembaT. Nikuni and A. Griffin, Dynamics of trapped Bose gases at finite temperatures, J. Low Temp. Phys., 116 (1999), 277-345.   Google Scholar

Figure 1.  27-velocity model for a mixture of three species with mass ratios 2, 3/2, and 3
Figure 2.  24-velocity supernormal DVM for internal energies E, 2E, and 3E
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