Abstract
A finite system of stochastic interacting particles is considered. The system approximates the solutions of the kinetic equations (the Boltzmann equation, the Boltzmann-Enskog equation) as well as the solutions describing the macroscopic evolution of fluids: the Euler and the Navier-Stokes hydrodynamic equations.
Similar content being viewed by others
References
Caprino, S., DeMasi, A., Presutti, E., and Pulvirenti, M., ‘A stochastic particle system modeling the Carleman equation’, J. Statist. Phys. 55, 1989, 625–638.
Mürmann, M., ‘On the derivation of hydrodynamics from molecular dynamics’, J. Math. Phys., 25, 1984, 1356–1363.
Bellomo, N. and Riganti, R., Nonlinear Stochastic Systems in Physics and Mechanics, World Scientific, Singapore 1987.
Skorohod, A. V., Stochastic Equations for Complex Systems, Nauka, Moscow 1983, in Russian and Reidel Pub. Co., Dordrecht, 1988.
Lachowicz, M., ‘A system of stochastic differential equations modeling the Euler and the Navier-Stokes hydrodynamic equations’, Japan J. Industr. Appl. Math., 10, n.1, 1993, 109–131.
Arsen'ev, A. A., ‘Approximation of the Boltzmann equation by stochastic equations’, Zh. Vychisl. Mat. i Mat. Fiz. 28, 1988, 560–567, in Russian.
Lachowicz, M. and Pulvirenti, M., ‘A stochastic system of particles modeling the Euler equations’, Arch. Rational Mech. Anal. 109, 1990, 81–93.
Gerasimenko, V. I. and Petrina, D. Ya., ‘Existence of the Boltzmann-Grad limit for infinite system of hard spheres’, Teoret, i Mat. Fiz. 83, 1990, 92–114, in Russian.
Lachowicz, M., ‘On the symptotic behaviour of solutions of nonlinear kinetic equations’, Ann. Mat. Pura Apl., CLX, IV, 1991, 41–62.
Lachowicz, M., ‘Solutions of nonlinear kinetic equations at the level of the Navier-Stokes dynamics’, J. Math. Kyoto Univ. 32, 1992, 31–43.
Lachowicz, M., ‘On the Enskog equation and its hydrodynamic limit’, Kinetic Theory and Hyperbolic Systems, V. Boffi. F. Bampi, and G. Toscani, eds., World Sci., 1992.
Arsen'ev, A. A., ‘On the approximation of the solution of the Boltzmann equation by solutions of Itô stochastic differential equations’, Zh. Vychisl. Mat. i Mat. Fiz. 271, 1987, 400–410, in Russian.
Horowitz, J. and Karandikar, R. L., ‘Martingale problems associated with the Boltzmann equation’, Progress in Probability 18, Birkhäuser, 1990.
Sznitman, A. S., ‘Equations de type de Boltzmann, spatialement homogènes’, Z. Warscheinlickeitstheorie verw. Geb. 66, 1984, 559–592.
Tanaka, H., ‘Probabilistic treatment of the Boltzmann equation of Maxwellian molecules’, Z. Warscheinlichkeitstheorie verw. Geb. 46, 1978, 67–105.
Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland and Kodansha, Amsterdam, Tokyo, 1981.
Ukai, S., Point, N., and Ghidouche, H., ‘Sur la solution globale du problème mixte de l'équation de Boltzmann non-linéaire’, J. Math. Pures Appl. 57, 1978, 203–229.
Bellomo, N., Lachowicz, M., Polewczak, J., and Toscani, G., Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation, World Scientific, Singapore, 1991.
Majda, A., Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984.
Valli, A. and Zajaczkowski, W., ‘Navier-Stokes equations for compressible fluids: global existence and quantitative properties of the solutions in the general case’, Comm. Math. Phys. 103, 1986, 259–296.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lachowicz, M. Stochastic differential equations for the kinetic and hydrodynamic equations. Nonlinear Dyn 5, 393–399 (1994). https://doi.org/10.1007/BF00045344
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00045344