Skip to main content
Log in

Stochastic differential equations for the kinetic and hydrodynamic equations

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Caprino, S., DeMasi, A., Presutti, E., and Pulvirenti, M., ‘A stochastic particle system modeling the Carleman equation’, J. Statist. Phys. 55, 1989, 625–638.

    Google Scholar 

  2. Mürmann, M., ‘On the derivation of hydrodynamics from molecular dynamics’, J. Math. Phys., 25, 1984, 1356–1363.

    Google Scholar 

  3. Bellomo, N. and Riganti, R., Nonlinear Stochastic Systems in Physics and Mechanics, World Scientific, Singapore 1987.

    Google Scholar 

  4. Skorohod, A. V., Stochastic Equations for Complex Systems, Nauka, Moscow 1983, in Russian and Reidel Pub. Co., Dordrecht, 1988.

    Google Scholar 

  5. 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.

    Google Scholar 

  6. Arsen'ev, A. A., ‘Approximation of the Boltzmann equation by stochastic equations’, Zh. Vychisl. Mat. i Mat. Fiz. 28, 1988, 560–567, in Russian.

    Google Scholar 

  7. Lachowicz, M. and Pulvirenti, M., ‘A stochastic system of particles modeling the Euler equations’, Arch. Rational Mech. Anal. 109, 1990, 81–93.

    Google Scholar 

  8. 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.

    Google Scholar 

  9. Lachowicz, M., ‘On the symptotic behaviour of solutions of nonlinear kinetic equations’, Ann. Mat. Pura Apl., CLX, IV, 1991, 41–62.

    Google Scholar 

  10. Lachowicz, M., ‘Solutions of nonlinear kinetic equations at the level of the Navier-Stokes dynamics’, J. Math. Kyoto Univ. 32, 1992, 31–43.

    Google Scholar 

  11. 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.

  12. 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.

    Google Scholar 

  13. Horowitz, J. and Karandikar, R. L., ‘Martingale problems associated with the Boltzmann equation’, Progress in Probability 18, Birkhäuser, 1990.

  14. Sznitman, A. S., ‘Equations de type de Boltzmann, spatialement homogènes’, Z. Warscheinlickeitstheorie verw. Geb. 66, 1984, 559–592.

    Google Scholar 

  15. Tanaka, H., ‘Probabilistic treatment of the Boltzmann equation of Maxwellian molecules’, Z. Warscheinlichkeitstheorie verw. Geb. 46, 1978, 67–105.

    Google Scholar 

  16. Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland and Kodansha, Amsterdam, Tokyo, 1981.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. Bellomo, N., Lachowicz, M., Polewczak, J., and Toscani, G., Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation, World Scientific, Singapore, 1991.

    Google Scholar 

  19. Majda, A., Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984.

    Google Scholar 

  20. 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.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints 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

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00045344

Key words

Navigation