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

## Cercignani-Lampis boundary in the Boltzmann theory

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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.

Citation: Hongxu Chen. Cercignani-Lampis boundary in the Boltzmann theory. Kinetic and 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. [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. [3] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Science & Business Media, 2013. [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. [5] H. Chen, C. Kim and Q. Li, Local Well-posedness of Vlasov-Poisson-Boltzmann Equation with Generalized Diffuse Boundary Condition, submitted. [6] T. Cowling, On the Cercignani-Lampis formula for gas-surface interactions, Journal of Physics D: Applied Physics, 7 (1974), 781-785. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [3] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Science & Business Media, 2013. [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. [5] H. Chen, C. Kim and Q. Li, Local Well-posedness of Vlasov-Poisson-Boltzmann Equation with Generalized Diffuse Boundary Condition, submitted. [6] T. Cowling, On the Cercignani-Lampis formula for gas-surface interactions, Journal of Physics D: Applied Physics, 7 (1974), 781-785. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
Maxwell boundary condition with $c = 1/2$
C-L boundary condition with $r_\perp = r_\parallel = 1/2$
C-L boundary condition with $r_\perp = r_\parallel = 1/10$
C-L boundary condition with $r_\perp = r_\parallel = 1/30$
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