Golse, François:
From the Boltzmann Equation to Hydrodynamic Equations in thin Layers
Bollettino dell'Unione Matematica Italiana Serie 9 4 (2011), fasc. n.2, p. 163-186, (English)
pdf (335 Kb), djvu (216 Kb). | MR 2840601 | Zbl 1235.35208
Sunto
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.
Referenze Bibliografiche
[1] K. AOKI, unpublished manuscript, 1998.
[2]
V. ARNOLD -
B. KHESIN,
Topological methods in hydrodynamics,
Springer, New York NY
1998. |
MR 1612569 |
Zbl 0902.76001[3]
C. BARDOS -
F. GOLSE -
D. LEVERMORE,
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]
C. BARDOS -
F. GOLSE -
D. LEVERMORE,
Fluid Dynamic Limits of the Boltzmann Equation I,
J. Stat. Phys.,
63 (
1991), 323-344. |
fulltext (doi) |
MR 1115587[5]
C. BARDOS -
F. GOLSE -
D. LEVERMORE,
Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation,
Comm. Pure and Applied Math.,
46 (
1993), 667-753. |
fulltext (doi) |
MR 1213991 |
Zbl 0817.76002[7]
C. BARDOS -
D. LEVERMORE -
S. UKAI -
T. YANG,
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]
C. BARDOS -
S. UKAI,
The classical incompressible Navier-Stokes limit of the Boltzmann equation,
Math. Models and Methods in the Appl. Sci.,
1 (
1991), 235-257. |
fulltext (doi) |
MR 1115292 |
Zbl 0758.35060[9]
G. K. BATCHELOR,
An introduction to fluid dynamics,
Cambridge Univ. Press,
2000. |
MR 1744638[10]
F. BOUCHUT -
F. GOLSE -
M. PULVIRENTI,
Kinetic Equations and Asymptotic Theory,
L. Desvillettes &
B. Perthame ed., Editions scientifiques et médicales
Elsevier, Paris,
2000. |
MR 2065070[14]
R. CAFLISCH -
M. SAMMARTINO,
Existence and singularities for the Prandtl boundary layer equations,
Z. Angew. Math. Mech.,
80 (
2000), 733-744. |
fulltext (doi) |
MR 1801538 |
Zbl 1050.76016[15]
C. CERCIGNANI,
Bifurcation problems in fluid mechanics,
Meccanica - J. Italian Assoc. Theoret. Appl. Mech.,
5 (
1970), 7-16. |
MR 261850 |
Zbl 0195.27401[19]
A. DEMASI -
R. ESPOSITO -
J. LEBOWITZ,
Incompressible Navier-Stokes and Euler limits of the Boltzmann equation,
Comm. on Pure and Appl. Math.,
42 (
1990), 1189-1214. |
fulltext (doi) |
MR 1029125 |
Zbl 0689.76024[21]
R. DIPERNA -
P.-L. LIONS,
On the Cauchy problem for Boltzmann equations: global existence and weak stability,
Ann. Math.,
130 (
1989), 321-366. |
fulltext (doi) |
MR 1014927 |
Zbl 0698.45010[23]
F. GOLSE -
D. LEVERMORE,
The Stokes-Fourier and Acoustic Limits for the Boltzmann Equation,
Comm. on Pure and Appl. Math.,
55 (
2002), 336-393. |
fulltext (doi) |
MR 1866367 |
Zbl 1044.76055[24]
F. GOLSE -
L. SAINT-RAYMOND,
The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels,
Invent. Math.,
155 (
2004), 81-161. |
fulltext (doi) |
MR 2025302 |
Zbl 1060.76101[25]
F. GOLSE -
L. SAINT-RAYMOND,
The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials,
J. Math. Pures et Appl.,
91 (
2009), 508-552. |
fulltext (doi) |
MR 2517786 |
Zbl 1178.35290[26]
H. GRAD,
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[28]
E. GRENIER,
Non dérivation des équations de Prandtl,
Séminaire Equations aux Dérivées Partielles 1997-1998, Exp. No. XVIII, Ecole Polytech. Palaiseau
1998. |
fulltext EuDML |
MR 1660531[30]
D. LEVERMORE -
N. MASMOUDI,
From the Boltzmann Equation to an Incompressible Navier-Stokes-Fourier System,
Archive for Rational Mech. and Anal.,
196 (
2010), 753-809. |
fulltext (doi) |
MR 2644440 |
Zbl 1304.35476[31]
P.-L. LIONS -
N. MASMOUDI,
From Boltzmann Equation to the Navier-Stokes and Euler Equations I,
Archive Rat. Mech. & Anal.,
158 (
2001), 173-193. |
fulltext (doi) |
MR 1842343 |
Zbl 0987.76088[32]
P.-L. LIONS -
N. MASMOUDI,
From Boltzmann Equation to the Navier-Stokes and Euler Equations II,
Archive Rat. Mech. & Anal.,
158 (
2001), 195-211. |
fulltext (doi) |
MR 1842343 |
Zbl 0987.76088[33] J. C. MAXWELL, On the dynamical theory of gases, Philosophical Transactions, 157 (1866).
[34]
T. NISHIDA,
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]
L. SAINT-RAYMOND,
Convergence of solutions to the Boltzmann equation in the incompressible Euler limit,
Arch. Ration. Mech. Anal.,
166 (
2003), 47-80. |
fulltext (doi) |
MR 1952079 |
Zbl 1016.76071[36]
M. SAMMARTINO -
R. CAFLISCH,
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. |
fulltext (doi) |
MR 1617542 |
Zbl 0913.35102[39]
Y. SONE -
C. BARDOS -
F. GOLSE -
H. SUGIMOTO,
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. |
fulltext (doi) |
MR 1764652 |
Zbl 0973.76076