no. 5
A lattice gas model for the incompressible Navier-Stokes equation
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 886-914.

Nous retrouvons l'équation de Navier-Stokes comme limite incompressible d'un gas sur réseau où les particules peuvent sauter sur des distances mésoscopiques. Le résultat est valable en toute dimension supposant l'existence d'une solution lisse de l'équation de Navier-Stokes en un intervale de temps donné. La démonstration ne dépend pas des méthodes non-gradients ou l'analyse multi-échelle grâce aux sauts de longue portée.

We recover the Navier-Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a mesoscopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier-Stokes equation in a fixed time interval. The proof does not use nongradient methods or the multi-scale analysis due to the long range jumps.

DOI : 10.1214/07-AIHP125
Classification : 60K35, 82C22
Mots clés : interacting particle systems, hydrodynamic limit, incompressible Navier-Stokes equation 
@article{AIHPB_2008__44_5_886_0,
     author = {Beltr\'an, J. and Landim, C.},
     title = {A lattice gas model for the incompressible {Navier-Stokes} equation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {886--914},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {5},
     year = {2008},
     doi = {10.1214/07-AIHP125},
     mrnumber = {2453775},
     zbl = {1184.60035},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/07-AIHP125/}
}
TY  - JOUR
AU  - Beltrán, J.
AU  - Landim, C.
TI  - A lattice gas model for the incompressible Navier-Stokes equation
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2008
SP  - 886
EP  - 914
VL  - 44
IS  - 5
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/07-AIHP125/
DO  - 10.1214/07-AIHP125
LA  - en
ID  - AIHPB_2008__44_5_886_0
ER  - 
%0 Journal Article
%A Beltrán, J.
%A Landim, C.
%T A lattice gas model for the incompressible Navier-Stokes equation
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2008
%P 886-914
%V 44
%N 5
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/07-AIHP125/
%R 10.1214/07-AIHP125
%G en
%F AIHPB_2008__44_5_886_0
Beltrán, J.; Landim, C. A lattice gas model for the incompressible Navier-Stokes equation. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 886-914. doi : 10.1214/07-AIHP125. http://archive.numdam.org/articles/10.1214/07-AIHP125/

[1] R. Esposito, R. Marra and H. T. Yau. Diffusive limit of asymmetric simple exclusion. Rev. Math. Phys. 6 (1994) 1233-1267. | MR | Zbl

[2] R. Esposito, R. Marra and H. T. Yau. Navier-Stokes equations for stochastic particle systems on the lattice. Comm. Math. Phys. 182 (1996) 395-456. | MR | Zbl

[3] C. Kipnis and C. Landim. Scaling Limit of Interacting Particle Systems. Fundamental Principles of Mathematical Sciences 320. Springer, Berlin, 1999. | MR | Zbl

[4] V. V. Petrov. Sums of Independent Random Variables. Springer, New York, 1975. | MR | Zbl

[5] J. Quastel. Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. XLV (1992) 623-679. | MR | Zbl

[6] S. R. S. Varadhan. Nonlinear diffusion limit for a system with nearest neighbor interactions II. In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals. Pitman Res. Notes Math. Ser. 283 K. Elworthy and N. Ikeda (Eds), 75-128. Longman Sci. Tech., Harlow, 1993. | MR | Zbl

[7] H. T. Yau. Relative entropy and hydrodynamics of Ginsburg-Landau models. Lett. Math. Phys. 22 (1991) 63-80. | MR | Zbl

Cité par Sources :