From the Boltzmann Equation to Hydrodynamic Equations in thin Layers
Golse, François
Bollettino dell'Unione Matematica Italiana, Tome 4 (2011), p. 163-186 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

The present paper discusses an asymptotic theory for the Boltzmann equation leading to either the Prandtl incompressible boundary layer equations, or the incompressible hydrostatic equations. These results are formal, and based on the same moment method used in [C. Bardos, F. Golse, D. Levermore, J. Stat. Phys 63 (1991), pp. 323-344] to derive the incompressible Euler and Navier-Stokes equations from the Boltzmann equation.

Publié le : 2011-06-01
@article{BUMI_2011_9_4_2_163_0,
     author = {Fran\c cois Golse},
     title = {From the Boltzmann Equation to Hydrodynamic Equations in thin Layers},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {4},
     year = {2011},
     pages = {163-186},
     zbl = {1235.35208},
     mrnumber = {2840601},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2011_9_4_2_163_0}
}

Golse, François. From the Boltzmann Equation to Hydrodynamic Equations in thin Layers. Bollettino dell'Unione Matematica Italiana, Tome 4 (2011) pp. 163-186. http://gdmltest.u-ga.fr/item/BUMI_2011_9_4_2_163_0/

[1] Aoki, K., unpublished manuscript, 1998.

[2] Arnold, V. - Khesin, B., Topological methods in hydrodynamics, Springer, New York NY 1998. | MR 1612569 | Zbl 0902.76001

[3] Bardos, C. - Golse, F. - Levermore, D., Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles C. R. Acad. Sci., 309 (1989), 727-732. | MR 1054287 | Zbl 0697.35111

[4] Bardos, C. - Golse, F. - Levermore, D., Fluid Dynamic Limits of the Boltzmann Equation I, J. Stat. Phys., 63 (1991), 323-344. | MR 1115587

[5] Bardos, C. - Golse, F. - Levermore, D., Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation, Comm. Pure and Applied Math., 46 (1993), 667-753. | MR 1213991 | Zbl 0817.76002

[6] Bardos, C. - Golse, F. - Sone, Y., Half-Space Problems for the Boltzmann Equation: A Survey, J. Stat. Phys., 124 (2006), 275-300. | MR 2264610 | Zbl 1125.82013

[7] Bardos, C. - Levermore, D. - Ukai, S. - Yang, T., Kinetic equations: fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin. (N.S.), 3 (2008), 1-49. | MR 2398020 | Zbl 1151.35066

[8] Bardos, C. - Ukai, S., The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models and Methods in the Appl. Sci., 1 (1991), 235-257. | MR 1115292 | Zbl 0758.35060

[9] Batchelor, G. K., An introduction to fluid dynamics, Cambridge Univ. Press, 2000. | MR 1744638

[10] Bouchut, F. - Golse, F. - Pulvirenti, M., Kinetic Equations and Asymptotic Theory, L. Desvillettes & B. Perthame ed., Editions scientifiques et médicales Elsevier, Paris, 2000. | MR 2065070

[11] Brenier, Y., Homogeneous hydrostatic flows with convex velocity profile, Nonlinearity, 12 (1999), 495-512. | MR 1690189 | Zbl 0984.35131

[12] Brenier, Y., Remarks on the derivation of the hydrostatic Euler equations, Bull. Sci. Math., 127 (2003), 585-595. | MR 2004720 | Zbl 1040.35068

[13] Caflisch, R., The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. on Pure and Appl. Math., 33 (1980), 651-666. | MR 586416 | Zbl 0424.76060

[14] Caflisch, R. - Sammartino, M., Existence and singularities for the Prandtl boundary layer equations, Z. Angew. Math. Mech., 80 (2000), 733-744. | MR 1801538 | Zbl 1050.76016

[15] Cercignani, C., Bifurcation problems in fluid mechanics, Meccanica - J. Italian Assoc. Theoret. Appl. Mech., 5 (1970), 7-16. | MR 261850 | Zbl 0195.27401

[16] Cercignani, C., The Boltzmann Equation and Its Applications, Springer-Verlag, New-York NY, 1988. | MR 1313028 | Zbl 0646.76001

[17] Cercignani, C. - Illner, R. - Pulvirenti, M., The Mathematical Theory of Dilute Gases, Springer Verlag, New York NY, 1994. | MR 1307620 | Zbl 0813.76001

[18] Dafermos, C., The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179. | MR 546634 | Zbl 0448.73004

[19] Demasi, A. - Esposito, R. - Lebowitz, J., Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. on Pure and Appl. Math., 42 (1990), 1189-1214. | MR 1029125 | Zbl 0689.76024

[20] Diperna, R., Uniqueness of solutions to hyperbolic conservation laws, Indiana U. Math. J., 28 (1979), 137-188. | MR 523630 | Zbl 0409.35057

[21] Diperna, R. - Lions, P.-L., On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 130 (1989), 321-366. | MR 1014927 | Zbl 0698.45010

[22] E, W., Engquist, B., Blowup of solutions of the unsteady Prandtl equation, Comm. on Pure and Appl. Math., 50 (1997), 1287-1293. | MR 1476316 | Zbl 0908.35099

[23] Golse, F. - Levermore, D., The Stokes-Fourier and Acoustic Limits for the Boltzmann Equation, Comm. on Pure and Appl. Math., 55 (2002), 336-393. | MR 1866367 | Zbl 1044.76055

[24] Golse, F. - Saint-Raymond, L., The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161. | MR 2025302 | Zbl 1060.76101

[25] Golse, F. - Saint-Raymond, L., The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures et Appl., 91 (2009), 508-552. | MR 2517786 | Zbl 1178.35290

[26] Grad, H., Asymptotic theory of the Boltzmann equation II, 1963 Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I, pp. 26-59. | MR 156656

[27] Grenier, E., On the derivation of homogeneous hydrostatic equations, M2AN Math. Modél. Anal. Num., 33 (1999), 965-970. | MR 1726718 | Zbl 0947.76013

[28] Grenier, E., Non dérivation des équations de Prandtl, Séminaire Equations aux Dérivées Partielles1997-1998, Exp. No. XVIII, Ecole Polytech. Palaiseau1998. | MR 1660531

[29] Leray, J., Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. | MR 1555394 | Zbl 60.0726.05

[30] Levermore, D. - Masmoudi, N., From the Boltzmann Equation to an Incompressible Navier-Stokes-Fourier System, Archive for Rational Mech. and Anal., 196 (2010), 753-809. | MR 2644440 | Zbl 1304.35476

[31] Lions, P.-L. - Masmoudi, N., From Boltzmann Equation to the Navier-Stokes and Euler Equations I, Archive Rat. Mech. & Anal., 158 (2001), 173-193. | MR 1842343 | Zbl 0987.76088

[32] Lions, P.-L. - Masmoudi, N., From Boltzmann Equation to the Navier-Stokes and Euler Equations II, Archive Rat. Mech. & Anal., 158 (2001), 195-211. | MR 1842343 | Zbl 0987.76088

[33] Maxwell, J. C., On the dynamical theory of gases, Philosophical Transactions, 157 (1866).

[34] Nishida, T., Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys., 61 (1978), 119-148. | MR 503305 | Zbl 0381.76060

[35] Saint-Raymond, L., Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal., 166 (2003), 47-80. | MR 1952079 | Zbl 1016.76071

[36] Sammartino, M. - Caflisch, R., Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space. I. Existence for Euler and Prandtl equations, Commun. Math. Phys., 192 (1998), 433-461. | MR 1617542 | Zbl 0913.35102

[37] Sone, Y., Kinetic Theory and Fluid Dynamics, Birkhäuser, Boston, 2002. | MR 1919070 | Zbl 1021.76002

[38] Sone, Y., Molecular Gas Dynamics, Theory, Techniques, and Applications, Birkhäuser, Boston, 2007. | MR 2274674 | Zbl 1144.76001

[39] Sone, Y. - Bardos, C. - Golse, F. - Sugimoto, H., Asymptotic theory for 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 (2000), 325-360. | MR 1764652 | Zbl 0973.76076