×

zbMATH — the first resource for mathematics

A stochastic system of particles modelling the Euler equations. (English) Zbl 0682.76002
Summary: We consider a system of N spheres interacting through elastic collisions at a stochastic distance. In the limit \(N\to \infty\), for a suitable rescaling of the interaction parameters, we prove that the one-particle distribution function converges to a local Maxwellian, whose gross density, velocity, and temperature satisfy the Euler equation.

MSC:
76A05 Non-Newtonian fluids
76Nxx Compressible fluids and gas dynamics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L. Arkeryd, On the Boltzmann equation. Part I: Existence. Part II: The full initial value problem, Arch. Rational Mech. Anal. 45 (1972), pp. 1-34. · Zbl 0245.76060
[2] R. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33 (1980), pp. 651-666. · Zbl 0435.76049
[3] C. Cercignani, The Grad limit for a system of soft spheres, Comm. Pure Appl Math. 36 (1983), pp. 479-494. · Zbl 0529.76080
[4] A. de Masi, N. Ianiro, A. Pellegrinotti & E. Presutti, A survey of the hydrodynamical behaviour of many-particle systems, in Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics, Eds. J. L. Lebowitz & E. W. Montroll, Elsevier Science Publisher 1984, pp. 123-294. · Zbl 0567.76006
[5] R. Illner & M. Pulvirenti, Global validity of the Boltzmann equation for a twodimensional rare gas in vacuum, Comm. Math. Phys. 105 (1986), pp. 189-203. · Zbl 0609.76083
[6] R. Illner & M. Pulvirenti, Global validity of the Boltzmann equation for two and three-dimensional rare gas in vacuum: Erratum and improved result, to appear in Comm. Math. Phys. · Zbl 0850.76600
[7] R. Illner & M. Pulvirenti, A derivation of the BBGKY-hierarchy for hard sphere particle systems, Transport Theory Statist. Phys. 16, n. 7 (1987), pp. 997-1012. · Zbl 0629.76088
[8] M. Lachowicz, On the initial layer and the existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl. Sci. 9, n. 3 (1987), pp. 342-366. · Zbl 0632.45005
[9] M. Lachowicz, On the limit of the nonlinear Enskog equation corresponding with fluid dynamics, Arch. Rational Mech. Anal. 101 (1988), pp. 179-194. · Zbl 0668.76093
[10] O. Lanford III, Time evolution of large classical systems, Ed. E. J. Moser, Lecture Notes in Physics, vol. 38, Springer 1975, pp. 1-111. · Zbl 0329.70011
[11] A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, Springer 1984. · Zbl 0537.76001
[12] A. Ya. Povzner, The Boltzmann equation in the kinetic theory of gases, Amer. Math. Soc. Transl. Ser. 2, 47 (1962), pp. 193-216. · Zbl 0188.21204
[13] M. Shinbrot, Local validity of the Boltzmann equation, Math. Methods Appl. Sci. 6 (1984), pp. 539-549. · Zbl 0661.76078
[14] H. Spohn, Boltzmann hierarchy and Boltzmann equation. Ed. C. Cercignani, Lecture Notes in Mathematics, vol. 1048, Springer 1984, pp. 207-220. · Zbl 0559.76073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.