From Hard Sphere Dynamics to the Stokes–Fourier Equations: An [InlineEquation not available: see fulltext.] Analysis of the Boltzmann–Grad Limit

被引：7

作者：

Bodineau T. ^{[1
]}

Gallagher I. ^{[2
]}

Saint-Raymond L. ^{[3
]}

机构：

[1] CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, Palaiseau

[2] Université Paris-Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Université Paris 06, Paris

[3] Département de Mathématiques et Applications, UPMC Université Paris 06 & Ecole Normale Supérieure, 45 rue d’Ulm, Paris Cedex 05

来源：

Annals of PDE | / 3卷 / 1期

关键词：

Cumulant expansion; Fluid limits; Hard sphere dynamics; Stokes-Fourier equations; Symmetry; Weak chaos property;

D O I：

10.1007/s40818-016-0018-0

中图分类号：

学科分类号：

摘要：

We derive the linear acoustic and Stokes–Fourier equations as the limiting dynamics of a system of N hard spheres of diameter ε in two space dimensions, when N→ ∞, ε→ 0 , Nε= α→ ∞, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford’s strategy (Time evolution of large classical systems, Springer, Berlin, 1975), and on the pruning procedure developed in Bodineau et al. (Invent Math 203:493–553, 2016) to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform L2 a priori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions. © 2017, Springer International Publishing AG.

引用

共 29 条

[1]

Alexander R., The Infinite Hard Sphere System, Ph.D. Dissertation, Dept. Mathematics, Univ. California, Berkeley, (1975)

[2]

Ayi N., From Newton’s law to the linear Boltzmann equation without cut-off, Commun. Math. Phys.

[3]

Bardos C., Golse F., Levermore C.D., Fluid dynamic limits of kinetic equations II: convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math., 46, pp. 667-753, (1993)

[4]

Bodineau T., Gallagher I., Saint-Raymond L., The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. Math., 203, pp. 493-553, (2016)

[5]

Caflisch R.E., The fluid dynamic limit of the nonlinear Boltzmann equation, Commun. Pure Appl. Math., 33, pp. 651-666, (1980)

[6]

Cercignani C., Illner R., Pulvirenti M., The Mathematical Theory of Dilute Gases, (1994)

[7]

DiPerna R., Lions P.-L., On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math. (2), 130, 2, pp. 321-366, (1989)

[8]

Gallagher I., Saint-Raymond L., Texier B., From Newton to Boltzmann: The case of hard-spheres and short-range potentials, ZLAM, (2014)

[9]

Golse F., Levermore D., The Stokes-Fourier and acoustic limits for the Boltzmann equation, Commun. Pure Appl. Math., 55, pp. 336-393, (2002)

[10]

Golse F., Saint-Raymond L., The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl., 91, pp. 508-552, (2009)

← 1 2 3 →