Heat transfer problem for the Boltzmann equation in a channel with diffusive boundary condition. (English) Zbl 1503.35156
Summary: In this paper, the authors study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures \(\theta_0\) and \(\theta_1\). This problem was studied by R. Esposito et al. [Commun. Math. Phys. 160, No. 1, 49–80 (1994; Zbl 0790.76072); J. Stat. Phys. 78, No. 1–2, 389–412 (1995; Zbl 1078.82525)] where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition. However, a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary. In the regime where the Knudsen number is reasonably small, the slip phenomenon is significant near the boundary. Thus, they revisit this problem by taking into account the slip boundary conditions. Following the lines of [F. Coron, J. Stat. Phys. 54, No. 3–4, 829–857 (1989; Zbl 0666.76103)], the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points. Then they will establish a uniform \(L^\infty\) estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q20 | Boltzmann equations |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 76N06 | Compressible Navier-Stokes equations |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
Keywords:
Boltzmann equation; compressible Navier-Stokes approximation; slip boundary conditions; Chapman-Enskog expansionReferences:
| [1] | Aoki, K.; Baranger, C.; Hattori, M., Slip boundary conditions for the compressible Navier-Stokes equations, J. Stat. Phys., 169, 4, 744-781 (2017) · Zbl 1383.82045 · doi:10.1007/s10955-017-1886-8 |
| [2] | Arkeryd, L.; Esposito, R.; Marra, R.; Nouri, A., Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Ration. Mech. Anal., 198, 1, 125-187 (2010) · Zbl 1333.35150 · doi:10.1007/s00205-010-0292-z |
| [3] | Bardos, C., Caflisch, R.-E. and Nicolaenko, B.-N., The Milne and Kramers problems for the Boltzmann equations of a hard sphere gas, Comm. Pure Appl. Math., 39(3), 323-352. · Zbl 0612.76088 |
| [4] | Bardos, C.; Golse, F.; Levermore, C. D., Fluid dynamic limits of kinetic equations, I, Formal derivations, J. Stat. Phys., 63, 1-1, 323-344 (1991) · doi:10.1007/BF01026608 |
| [5] | Bardos, C.; Golse, F.; Levermore, C. D., Fluid dynamic limits of kinetic equations, II, Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46, 5, 667-753 (1993) · Zbl 0817.76002 · doi:10.1002/cpa.3160460503 |
| [6] | Bardos, C.; Ukai, S., The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1, 2, 235-157 (1991) · Zbl 0758.35060 · doi:10.1142/S0218202591000137 |
| [7] | Caflisch, R. E., The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33, 5, 651-666 (1980) · Zbl 0424.76060 · doi:10.1002/cpa.3160330506 |
| [8] | Cercignani, C., The Boltzmann Equation and Its Applications, Applied Mathematical Sciences (1988), New York: Springer-Verlag, New York · Zbl 0646.76001 · doi:10.1007/978-1-4612-1039-9 |
| [9] | Coron, F., Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation, J. Stat. Phys., 54, 3-4, 829-857 (1989) · Zbl 0666.76103 · doi:10.1007/BF01019777 |
| [10] | Duan, R-J; Liu, S-Q, Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains, Trans. Amer. Math. Soc., 374, 11, 7867-7924 (2021) · Zbl 1479.35661 · doi:10.1090/tran/8437 |
| [11] | Esposito, R.; Guo, Y.; Kim, C.; Marra, R., Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm. Math. Phys., 323, 177-139 (2013) · Zbl 1280.82009 · doi:10.1007/s00220-013-1766-2 |
| [12] | Esposito, R., Guo, Y., Kim, C. and Marra, R., Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4(1), 2018, Paper No. 1, 119 pp. · Zbl 1403.35195 |
| [13] | Esposito, R.; Lebowitz, J. L.; Marra, R., Hydrodynamic limit of the stationary Boltzmann equation in a slab, Comm. Math. Phys., 160, 1, 49-80 (1994) · Zbl 0790.76072 · doi:10.1007/BF02099789 |
| [14] | Esposito, R.; Lebowitz, J. L.; Marra, R., The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, J. Stat. Phys., 78, 1-1, 389-412 (1995) · Zbl 1078.82525 · doi:10.1007/BF02183355 |
| [15] | Esposito, R.; Marra, R., Stationary non equilibrium states in kinetic theory, J. Stat. Phys., 180, 1-6, 773-809 (2020) · Zbl 1447.82026 · doi:10.1007/s10955-020-02528-w |
| [16] | Golse, F.; Perthame, B.; Sulem, C., On a boundary layer problem for the nonlinear Boltzmann equation, Arch. Ration. Mech. Anal., 103, 81-96 (1986) · Zbl 0668.76089 · doi:10.1007/BF00292921 |
| [17] | Golse, F.; Saint-Raymond, L., The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155, 1, 81-161 (2004) · Zbl 1060.76101 · doi:10.1007/s00222-003-0316-5 |
| [18] | Grad, H.; Laurmann, J. A., Asymptotic theory of the Boltzmann equation, Rarefied Gas Dynamics, 26-59 (1963), New York: Academic Press, New York |
| [19] | Grad, H., Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, 154-183 (1965), New York: Amer. Math. Soc., New York · Zbl 0144.48203 |
| [20] | Guo, Y., Boltzmann diffuse limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59, 5, 626-687 (2006) · Zbl 1089.76052 · doi:10.1002/cpa.20121 |
| [21] | Guo, Y., Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197, 3, 713-809 (2010) · Zbl 1291.76276 · doi:10.1007/s00205-009-0285-y |
| [22] | Guo, Y.; Huang, F.; Wang, Y., Hilbert expansion of the Boltzmann equation with specular boundary condition in half-space, Arch. Ration. Mech. Anal., 241, 1, 231-309 (2021) · Zbl 1468.76059 · doi:10.1007/s00205-021-01651-6 |
| [23] | Guo, Y.; Jang, J., Global Hilbert expansion for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 299, 2, 469-501 (2010) · Zbl 1198.35273 · doi:10.1007/s00220-010-1089-5 |
| [24] | Guo, Y.; Jang, J.; Jiang, N., Local Hilbert expansion for the Boltzmann equation, Kinet. Relat. Models, 2, 1, 205-114 (2009) · Zbl 1372.76089 · doi:10.3934/krm.2009.2.205 |
| [25] | Huang, F.; Wang, Y.; Wang, Y.; Yang, T., The limit of the Boltzmann equation to the Euler equations for Riemann problems, SIAM J. Math. Anal., 45, 3, 1741-1811 (2013) · Zbl 1367.76055 · doi:10.1137/120898541 |
| [26] | Ann. PDE202172Paper No. 22, 103pp |
| [27] | Jiang, N.; Masmoudi, N., Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain I, Comm. Pure Appl. Math., 70, 1, 90-171 (2017) · Zbl 1362.35233 · doi:10.1002/cpa.21631 |
| [28] | Lachowitz, M., Solutions of nonlinear kinetic equations on the level of the Navier-Stokes dynamics, J. Math. Kyoto Univ., 32, 1, 31-43 (1992) · Zbl 0785.76004 |
| [29] | Liu, S-Q; Yang, T.; Zhao, H-J, Compressible Navier-Stokes approximation to the Boltzmann equation, J. Differential Equations, 256, 11, 3770-3816 (2014) · Zbl 1441.76105 · doi:10.1016/j.jde.2014.02.020 |
| [30] | Liu, T-P; Yang, T.; Yu, S-H; Zhao, H-J, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181, 2, 333-371 (2006) · Zbl 1095.76024 · doi:10.1007/s00205-005-0414-1 |
| [31] | Nishida, T., Fluid dynamical limit of the nonlinear Boltzmann equations to the level of the compressible Euler equation, Comm. Math. Phys., 61, 2, 119-148 (1978) · Zbl 0381.76060 · doi:10.1007/BF01609490 |
| [32] | Saint-Raymond, L., Hydrodynamic Limits of the Boltzmann Equation (2009), Berlin: Springer-Verlag, Berlin · Zbl 1171.82002 |
| [33] | Sone, Y., Kinetic Theory and Fluid Dynamics (2002), Boston: Birkhäuser Boston Inc., Boston · Zbl 1021.76002 · doi:10.1007/978-1-4612-0061-1 |
| [34] | Sone, Y., Molecular Gas Dynamics: Theory, Techniques and Applications (2007), Boston: Birkhäuser Boston Inc., Boston · Zbl 1144.76001 · doi:10.1007/978-0-8176-4573-1 |
| [35] | Sone, Y.; Bardos, C.; Golse, F.; Sugimoto, H., Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number: General theory, Eur J. Mech. B Fluids, 19, 3, 325-360 (2000) · Zbl 0973.76076 · doi:10.1016/S0997-7546(00)00110-2 |
| [36] | Ukai, S.; Asano, K., The Euler limit and the initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12, 303-324 (1983) · Zbl 0525.76062 · doi:10.14492/hokmj/1470081009 |
| [37] | Wu, L., Hydrodynamic limit with geometric correction of stationary Boltzmann equation, J. Differential Equations, 260, 10, 7152-7249 (2016) · Zbl 1336.35272 · doi:10.1016/j.jde.2016.01.024 |
| [38] | Xin, Z. P.; Zeng, H. H., Convergence to the rarefaction waves for the nonlinear Boltzmann equation, J. Differential Equations, 249, 827-871 (2010) · Zbl 1203.35184 · doi:10.1016/j.jde.2010.03.011 |
| [39] | Yu, S-H, Hydrodynamic limits with shock waves of the Boltzmann equations, Comm. Pure Appl. Math., 58, 409-443 (2005) · Zbl 1088.82022 · doi:10.1002/cpa.20027 |
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