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June  2020, 13(3): 549-597. doi: 10.3934/krm.2020019

Cercignani-Lampis boundary in the Boltzmann theory

Hongxu Chen

Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, USA

Received  June 2019 Revised  December 2019 Published  March 2020

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an $ L^\infty $ estimate. In particular, for the $ L^\infty $ bound we develop a new decomposition on the boundary term combining with repeated interaction through the characteristic. Moreover, under some constraints on the wall temperature and the accommodation coefficients, we construct a unique steady solution of the Boltzmann equation.

Keywords: Kinetic theory, Boltzmann equation, Cercignani-Lampis boundary, boundary value problem, local well-posedness, steady solution.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Hongxu Chen. Cercignani-Lampis boundary in the Boltzmann theory. Kinetic & Related Models, 2020, 13 (3) : 549-597. doi: 10.3934/krm.2020019
References:
[1]

Y. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov–Poisson–Boltzmann system in bounded domains, Archive for Rational Mechanics and Analysis, 233 (2019), 1027-1130.  doi: 10.1007/s00205-019-01374-9.  Google Scholar

[2]

C. Cercignani, The boltzmann equation, in The Boltzmann Equation and Its Applications, Springer, (1988), 40–103. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[3]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Science & Business Media, 2013. Google Scholar

[4]

C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions, transport Theory and Statistical Physics, 1 (1971), 101-114.  doi: 10.1080/00411457108231440.  Google Scholar

[5]

H. Chen, C. Kim and Q. Li, Local Well-posedness of Vlasov-Poisson-Boltzmann Equation with Generalized Diffuse Boundary Condition, submitted. Google Scholar

[6]

T. Cowling, On the Cercignani-Lampis formula for gas-surface interactions, Journal of Physics D: Applied Physics, 7 (1974), 781-785.   Google Scholar

[7]

R. Duan, F. Huang, Y. Wang and Z. Zhang, Effects of soft interaction and non-isothermal boundary upon long-time dynamics of rarefied gas, Archive for Rational Mechanics and Analysis, 234 (2019), 925-1006.  doi: 10.1007/s00205-019-01405-5.  Google Scholar

[8]

R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Communications in Mathematical Physics, 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2.  Google Scholar

[9]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Annals of PDE, 4 (2018), Art. 1,119 pp. doi: 10.1007/s40818-017-0037-5.  Google Scholar

[10]

R. Garcia and C. Siewert, The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: Basic flow problems in a plane channel, European Journal of Mechanics-B/Fluids, 28 (2009), 387-396.  doi: 10.1016/j.euromechflu.2008.12.001.  Google Scholar

[11]

R. Garcia and C. Siewert, Viscous-slip, thermal-slip, and temperature-jump coefficients based on the linearized Boltzmann equation (and five kinetic models) with the Cercignani–Lampis boundary condition, European Journal of Mechanics-B/Fluids, 29 (2010), 181-191.   Google Scholar

[12]

Y. Guo, Bounded solutions for the Boltzmann equation, Quarterly of Applied Mathematics, 68 (2010), 143-148.  doi: 10.1090/S0033-569X-09-01180-4.  Google Scholar

[13]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Archive for Rational Mechanics and Analysis, 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[14]

Y. Guo, C. Kim, D. Tonon and A. Trescases, BV-regularity of the Boltzmann equation in non-convex domains, Archive for Rational Mechanics and Analysis, 220 (2016), 1045-1093.  doi: 10.1007/s00205-015-0948-9.  Google Scholar

[15]

Y. Guo, C. Kim, D. Tonon and A. Trescases, Regularity of the Boltzmann equation in convex domains, Inventiones Mathematicae, 207 (2017), 115-290.  doi: 10.1007/s00222-016-0670-8.  Google Scholar

[16]

C. Kim, Formation and propagation of discontinuity for Boltzmann equation in non-convex domains, Communications in Mathematical Physics, 308 (2011), 641-701.  doi: 10.1007/s00220-011-1355-1.  Google Scholar

[17]

C. Kim and D. Lee, Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains, Archive for Rational Mechanics and Analysis, 230 (2018), 49-123.  doi: 10.1007/s00205-018-1241-5.  Google Scholar

[18]

C. Kim and D. Lee, The Boltzmann equation with specular boundary condition in convex domains, Communications on Pure and Applied Mathematics, 71 (2018), 411-504.  doi: 10.1002/cpa.21705.  Google Scholar

[19]

R. Knackfuss and L. Barichello, Surface effects in rarefied gas dynamics: An analysis based on the Cercignani–Lampis boundary condition, European Journal of Mechanics-B/Fluids, 25 (2006), 113-129.  doi: 10.1016/j.euromechflu.2005.04.003.  Google Scholar

[20]

R. Knackfuss and L. Barichello, On the temperature-jump problem in rarefied gas dynamics: The effect of the Cercignani–Lampis boundary condition, SIAM Journal on Applied Mathematics, 66 (2006), 2149-2186.  doi: 10.1137/050643209.  Google Scholar

[21]

R. Lord, Some extensions to the Cercignani–Lampis gas–surface scattering kernel, Physics of Fluids A: Fluid Dynamics, 3 (1991), 706-710.  doi: 10.1063/1.858076.  Google Scholar

[22]

R. Lord, Some further extensions of the Cercignani–Lampis gas–surface interaction model, Physics of Fluids, 7 (1995), 1159-1161.  doi: 10.1063/1.868557.  Google Scholar

[23]

S. Lorenzani, Higher order slip according to the linearized Boltzmann equation with general boundary conditions, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369 (2011), 2228-2236.  doi: 10.1098/rsta.2011.0059.  Google Scholar

[24]

S. Mischler, Kinetic equations with Maxwell boundary conditions, Annales Scientifiques de l'Ecole Normale Superieure, 43 (2010), 719-760.  doi: 10.24033/asens.2132.  Google Scholar

[25]

F. Sharipov, Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. Ⅰ. Plane flow between two parallel plates, European Journal of Mechanics-B/Fluids, 21 (2002), 113-123.  doi: 10.1016/S0997-7546(01)01160-8.  Google Scholar

[26]

F. Sharipov, Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. Ⅱ. Slip and jump coefficients, European Journal of Mechanics-B/Fluids, 22 (2003), 133-143.  doi: 10.1016/S0997-7546(03)00017-7.  Google Scholar

[27]

F. Sharipov, Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. Ⅲ. Poiseuille flow and thermal creep through a long tube, European Journal of Mechanics-B/Fluids, 22 (2003), 145-154.  doi: 10.1016/S0997-7546(03)00018-9.  Google Scholar

[28]

C. Siewert, Generalized boundary conditions for the S-model kinetic equations basic to flow in a plane channel, Journal of Quantitative Spectroscopy and Radiative Transfer, 72 (2002), 75-88.  doi: 10.1016/S0022-4073(01)00057-7.  Google Scholar

[29]

C. Siewert, Viscous-slip, thermal-slip, and temperature-jump coefficients as defined by the linearized Boltzmann equation and the Cercignani–Lampis boundary condition, Physics of Fluids, 15 (2003), 1696-1701.  doi: 10.1063/1.1567284.  Google Scholar

[30]

M. Woronwicz and D. Rault, Cercignani-lampis-lord gas surface interaction model-comparisons between theory and simulation, Journal of Spacecraft and Rockets, 31 (1994), 532-534.  doi: 10.2514/3.26474.  Google Scholar

show all references

References:
[1]

Y. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov–Poisson–Boltzmann system in bounded domains, Archive for Rational Mechanics and Analysis, 233 (2019), 1027-1130.  doi: 10.1007/s00205-019-01374-9.  Google Scholar

[2]

C. Cercignani, The boltzmann equation, in The Boltzmann Equation and Its Applications, Springer, (1988), 40–103. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[3]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Science & Business Media, 2013. Google Scholar

[4]

C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions, transport Theory and Statistical Physics, 1 (1971), 101-114.  doi: 10.1080/00411457108231440.  Google Scholar

[5]

H. Chen, C. Kim and Q. Li, Local Well-posedness of Vlasov-Poisson-Boltzmann Equation with Generalized Diffuse Boundary Condition, submitted. Google Scholar

[6]

T. Cowling, On the Cercignani-Lampis formula for gas-surface interactions, Journal of Physics D: Applied Physics, 7 (1974), 781-785.   Google Scholar

[7]

R. Duan, F. Huang, Y. Wang and Z. Zhang, Effects of soft interaction and non-isothermal boundary upon long-time dynamics of rarefied gas, Archive for Rational Mechanics and Analysis, 234 (2019), 925-1006.  doi: 10.1007/s00205-019-01405-5.  Google Scholar

[8]

R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Communications in Mathematical Physics, 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2.  Google Scholar

[9]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Annals of PDE, 4 (2018), Art. 1,119 pp. doi: 10.1007/s40818-017-0037-5.  Google Scholar

[10]

R. Garcia and C. Siewert, The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: Basic flow problems in a plane channel, European Journal of Mechanics-B/Fluids, 28 (2009), 387-396.  doi: 10.1016/j.euromechflu.2008.12.001.  Google Scholar

[11]

R. Garcia and C. Siewert, Viscous-slip, thermal-slip, and temperature-jump coefficients based on the linearized Boltzmann equation (and five kinetic models) with the Cercignani–Lampis boundary condition, European Journal of Mechanics-B/Fluids, 29 (2010), 181-191.   Google Scholar

[12]

Y. Guo, Bounded solutions for the Boltzmann equation, Quarterly of Applied Mathematics, 68 (2010), 143-148.  doi: 10.1090/S0033-569X-09-01180-4.  Google Scholar

[13]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Archive for Rational Mechanics and Analysis, 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[14]

Y. Guo, C. Kim, D. Tonon and A. Trescases, BV-regularity of the Boltzmann equation in non-convex domains, Archive for Rational Mechanics and Analysis, 220 (2016), 1045-1093.  doi: 10.1007/s00205-015-0948-9.  Google Scholar

[15]

Y. Guo, C. Kim, D. Tonon and A. Trescases, Regularity of the Boltzmann equation in convex domains, Inventiones Mathematicae, 207 (2017), 115-290.  doi: 10.1007/s00222-016-0670-8.  Google Scholar

[16]

C. Kim, Formation and propagation of discontinuity for Boltzmann equation in non-convex domains, Communications in Mathematical Physics, 308 (2011), 641-701.  doi: 10.1007/s00220-011-1355-1.  Google Scholar

[17]

C. Kim and D. Lee, Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains, Archive for Rational Mechanics and Analysis, 230 (2018), 49-123.  doi: 10.1007/s00205-018-1241-5.  Google Scholar

[18]

C. Kim and D. Lee, The Boltzmann equation with specular boundary condition in convex domains, Communications on Pure and Applied Mathematics, 71 (2018), 411-504.  doi: 10.1002/cpa.21705.  Google Scholar

[19]

R. Knackfuss and L. Barichello, Surface effects in rarefied gas dynamics: An analysis based on the Cercignani–Lampis boundary condition, European Journal of Mechanics-B/Fluids, 25 (2006), 113-129.  doi: 10.1016/j.euromechflu.2005.04.003.  Google Scholar

[20]

R. Knackfuss and L. Barichello, On the temperature-jump problem in rarefied gas dynamics: The effect of the Cercignani–Lampis boundary condition, SIAM Journal on Applied Mathematics, 66 (2006), 2149-2186.  doi: 10.1137/050643209.  Google Scholar

[21]

R. Lord, Some extensions to the Cercignani–Lampis gas–surface scattering kernel, Physics of Fluids A: Fluid Dynamics, 3 (1991), 706-710.  doi: 10.1063/1.858076.  Google Scholar

[22]

R. Lord, Some further extensions of the Cercignani–Lampis gas–surface interaction model, Physics of Fluids, 7 (1995), 1159-1161.  doi: 10.1063/1.868557.  Google Scholar

[23]

S. Lorenzani, Higher order slip according to the linearized Boltzmann equation with general boundary conditions, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369 (2011), 2228-2236.  doi: 10.1098/rsta.2011.0059.  Google Scholar

[24]

S. Mischler, Kinetic equations with Maxwell boundary conditions, Annales Scientifiques de l'Ecole Normale Superieure, 43 (2010), 719-760.  doi: 10.24033/asens.2132.  Google Scholar

[25]

F. Sharipov, Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. Ⅰ. Plane flow between two parallel plates, European Journal of Mechanics-B/Fluids, 21 (2002), 113-123.  doi: 10.1016/S0997-7546(01)01160-8.  Google Scholar

[26]

F. Sharipov, Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. Ⅱ. Slip and jump coefficients, European Journal of Mechanics-B/Fluids, 22 (2003), 133-143.  doi: 10.1016/S0997-7546(03)00017-7.  Google Scholar

[27]

F. Sharipov, Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. Ⅲ. Poiseuille flow and thermal creep through a long tube, European Journal of Mechanics-B/Fluids, 22 (2003), 145-154.  doi: 10.1016/S0997-7546(03)00018-9.  Google Scholar

[28]

C. Siewert, Generalized boundary conditions for the S-model kinetic equations basic to flow in a plane channel, Journal of Quantitative Spectroscopy and Radiative Transfer, 72 (2002), 75-88.  doi: 10.1016/S0022-4073(01)00057-7.  Google Scholar

[29]

C. Siewert, Viscous-slip, thermal-slip, and temperature-jump coefficients as defined by the linearized Boltzmann equation and the Cercignani–Lampis boundary condition, Physics of Fluids, 15 (2003), 1696-1701.  doi: 10.1063/1.1567284.  Google Scholar

[30]

M. Woronwicz and D. Rault, Cercignani-lampis-lord gas surface interaction model-comparisons between theory and simulation, Journal of Spacecraft and Rockets, 31 (1994), 532-534.  doi: 10.2514/3.26474.  Google Scholar

Figure 1.  Maxwell boundary condition with $ c = 1/2 $
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Figure 2.  C-L boundary condition with $ r_\perp = r_\parallel = 1/2 $
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Figure 3.  C-L boundary condition with $ r_\perp = r_\parallel = 1/10 $
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Figure 4.  C-L boundary condition with $ r_\perp = r_\parallel = 1/30 $
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