In this note, we review the recent work [23] on the boundary layer and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation with a general cut-off collision in a bounded domain. Appropriately scaled families of DiPerna-Lions-(Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to zero. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number.
The main new result is that this convergence is strong in the case of Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately, namely they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling.
@article{SLSEDP_2015-2016____A2_0,
author = {Jiang, Ning and Masmoudi, Nader},
title = {Boundary layers and incompressible {Navier-Stokes-Fourier} limit of the {Boltzmann} equation in a bounded domain},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:2},
pages = {1--16},
publisher = {Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique},
year = {2015-2016},
doi = {10.5802/slsedp.95},
language = {en},
url = {http://www.numdam.org/articles/10.5802/slsedp.95/}
}
TY - JOUR AU - Jiang, Ning AU - Masmoudi, Nader TI - Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in a bounded domain JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:2 PY - 2015-2016 SP - 1 EP - 16 PB - Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique UR - http://www.numdam.org/articles/10.5802/slsedp.95/ DO - 10.5802/slsedp.95 LA - en ID - SLSEDP_2015-2016____A2_0 ER -
%0 Journal Article %A Jiang, Ning %A Masmoudi, Nader %T Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in a bounded domain %J Séminaire Laurent Schwartz — EDP et applications %Z talk:2 %D 2015-2016 %P 1-16 %I Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique %U http://www.numdam.org/articles/10.5802/slsedp.95/ %R 10.5802/slsedp.95 %G en %F SLSEDP_2015-2016____A2_0
Jiang, Ning; Masmoudi, Nader. Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in a bounded domain. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Exposé no. 2, 16 p. doi : 10.5802/slsedp.95. http://www.numdam.org/articles/10.5802/slsedp.95/
[1] C. Bardos, R. Caflisch, and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Comm. Pure Appl. Math. 39 (1986), no. 3, 323-352.
[2] C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamic limits of kinetic equations I: formal derivations. J. Stat. Phys. 63 (1991), 323-344.
[3] C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamic limits of kinetic equations II: convergence proof for the Boltzmann equation. Comm. Pure Appl. Math. 46 (1993), 667-753.
[4] C. Bardos, F. Golse, and C. D. Levermore, The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal. 153 (2000), no. 3, 177-204.
[5] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4, Gauthier-Villars, Paris, 2000, 41-126.
[6] C. Cercignani, The Boltzmann equation and its applications. Springer, New York, 1988.
[7] C. Cercignani, R. Illner and M. Pulvirenti. The mathematical theory of dilute gases. Springer, New York, 1994.
[8] F. Coron, F. Golse, and C. Sulem, A classification of well-posed kinetic layer problems. Comm. Pure Appl. Math. 41 (1988), no. 4, 409–435.
[9] R. Dalmasso. A new result on the Pompeiu problem. Trans. Amer. Math. Soc. 352 (2000), no. 6, 2723–2736.
[10] B. Desjardins, E. Grenier, P.-L. Lions, N. Masmoudi, Incompressible limit for solutions to the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 1999, 461-471.
[11] R. DiPerna and P.-L. Lions, On the Cauchy problem for the Boltzmann equation: global existence and weak stability. Ann. of Math. 130 (1989), 321-366.
[12] F. Golse and C. D. Levermore, The Stokes-Fourier and acoustic limits for the Boltzmann equation. Comm. Pure Appl. Math. 55 (2002), 336-393.
[13] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solutions to a transport equation. J. Funct. Anal. 76, 1988, 110-125.
[14] F. Golse, B. Perthame, and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rational Mech. Anal. 103 (1988), no. 1, 81-96.
[15] F. Golse, F. Poupaud, Stationary solutions of the linearized Boltzmann equation in a half-space. Math. Methods Appl. Sci. 11 (1989), no. 4, 483-502.
[16] R. Glassey, The Cauchy Problems in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1996.
[17] F. Golse and L. Saint-Raymond, Velocity averaging in for the transport equation. C. R. Acas. Sci. Paris Sr. I Math. 334 (2002), 557-562.
[18] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155 (2004), no. 1, 81-161.
[19] F. Golse and L. Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. 91 (2009), no. 5, 508–552.
[20] H. Grad, Principles of the kinetic theory of gases. Handbuch der Physik, vol. 12, 205-294. Springer, Berlin, 1958.
[21] D, Hilbert, Begründung der kinetischen Gastheorie. Math. Annalen. 72 (1912), 562-577. English translation: Foundations of the kinetic theory of gases. Kinetic theory, vol. 3, 89-101. Pergamon, Oxford-New York, 1965-1972.
[22] N. Jiang, C. D. Levermore and N. Masmoudi, Remarks on the acoustic limit for the Boltzmann equation. Comm. Partial Differential Equations 35 (2010), no. 9, 1590-1609.
[23] N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann Equation in Bounded Domain I. To appear Comm. Pure Appl. Math. 2016.
[24] N. Jiang and N. Masmoudi, On the construction of boundary layers in the incompressible limit. J. Math. Pures Appl. 103 (2015), 269-290.
[25] N. Jiang and N. Masmoudi, Boundary layers and hydrodynamic limits of Boltzmann equation (II): higher order acoustic Aaproximation. In preparation, 2014.
[26] J. Leray, Sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63 (1934), 193-248.
[27] C. D. Levermore and N. Masmoudi, From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 196 (2010), no. 3, 753-809.
[28] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77 (1998), 585-627.
[29] P.-L. Lions and N. Masmoudi, Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris Sr. I Math. 329 (1999), 387-392.
[30] P.-L. Lions and N. Masmoudi, From Boltzmann equation to Navier-Stokes and Euler equations I. Arch. Ration. Mech. Anal. 158 (2001), 173-193.
[31] P.-L. Lions and N. Masmoudi, From Boltzmann equation to Navier-Stokes and Euler equations II. Arch. Ration. Mech. Anal. 158 (2001), 195-211.
[32] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol 1: Incompressible Models, Oxford Lecture Series in Mathematics and its Applications 3. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.
[33] N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Comm. Pure Appl. Math. 56 (2003), 1263-1293.
[34] J.-C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. Roy. Soc. London 170 (1879), Appendix 231-256.
[35] S. Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Comm. Math. Phys. 210 (2000), no. 2, 447-466.
[36] S. Mischler, On the trace problem for solutions to the Vlasov equation. Comm. Partial Differential Equations 25 (2000), no. 7-8, 1415-1443.
[37] S. Mischler, Kinetic equation with Maxwell boundary condition. Ann. Sci. Ec. Norm. Super. (4) 43 (2010), no. 5, 719-760.
[39] L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Arch. Ration. Mech. Anal. 166 (2003), 47-80.
[40] L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation. Lecture Notes in Mathematics, 1971. Springer-Verlag, Berlin, 2009.
[41] S. Schochet, Fast singular limits of hyperbolic PDE’s. J. Diff. equations 114 (1994), 476-512.
[42] L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Based on lecture notes by Norbert Hungerbühler. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996.
[43] Y. Sone, Asymptotic theory of flow of a rarefied gas over a smooth boundary. II. IXth International Symposium on Rarefied Gas Dynamics, 737-749. Editrice Tecnico Scientifica, Pisa, 1971.
[44] Y. Sone, Kinetic Theory and Fluid Dynamics , Birkhauser, Boston 2002.
[45] S. Ukai, Asymptotic analysis of fluid equations. Mathematical foundation of turbulent viscous flows, 189-250, Lecture Notes in Math., 1871, Springer, Berlin, 2006.
[46] S. Ukai, T. Yang, and S-H. Yu, Nonlinear boundary layers of the Boltzmann equation. I. Existence. Comm. Math. Phys. 236 (2003), no. 3, 373–393.
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