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Journal of Mathematical Sciences

, Volume 205, Issue 2, pp 222–239 | Cite as

On Hydrodynamic Equations at the Limit of Infinitely Many Molecules

  • S. Dostoglou
  • N. C. Jacob
  • Jianfei Xue
Article

We show that the weak convergence of point measures and (2 + )-moment conditions imply hydrodynamic equations at the limit of infinitely many interacting molecules. The conditions are satisfied whenever the solutions of the classical equations for N interacting molecules obey uniform in N bounds. As an example, we show that this holds when the initial conditions are bounded and the molecule interaction, a certain N-rescaling of potentials that include all r p for 1 < p, is weak enough at the initial time. In this case, the hydrodynamic equations coincide with the macroscopic Maxwell equations. Bibliography: 23 titles.

Keywords

Probability Measure Boltzmann Equation Momentum Equation Weak Convergence Hydrodynamic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J. C. Maxwell, “On the dynamical theory of gases,” Philos. Trans. R. Soc. Lond. 157, 49–88 (1867).CrossRefGoogle Scholar
  2. 2.
    L. Boltzmann, Lectures on Gas Theory, Univ. California Press (1964).Google Scholar
  3. 3.
    S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge Univ. Press, London (1970).Google Scholar
  4. 4.
    J. H Irving and J. G. Kirkwood “The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics,” J. Chem. Phys. 18, No. 6, 817–829 (1950).CrossRefMathSciNetGoogle Scholar
  5. 5.
    H. Grad, “On the kinetic theory of rarefied gases,” Commun. Pure Appl. Math. 2, 331–407 (1949).CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    C. B. Morrey, “On the derivation of the equations of hydrodynamics from statistical mechanics,” Commun. Pure Appl. Math. 8, 279–326 (1955).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    O. E. Lanford, “Time evolution of large classical systems,” Dyn. Syst., Theor. Appl., Battelle Seattle 1974 Renc., Lect. Notes Phys. 38, 1–111 (1975).MathSciNetGoogle Scholar
  8. 8.
    R. Esposito and M. Pulvirenti, “From particles to fluids,” in Handbook of Mathematical Fluid Dynamics. III, pp. 1–82, North-Holland, Amsterdam (2004).Google Scholar
  9. 9.
    A. N. Gorban and I. Karlin, “Hilbert’s 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations,” Bull. Am. Math. Soc. 51, No. 2, 187-246 (2014).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    O. Reynolds, “On the dynamical theory of incompressible viscous fluids and the determination of the criterion,” Philos. Trans. R. Soc. Lond., A 186, 123–164 (1895).CrossRefMATHGoogle Scholar
  11. 11.
    D. W. Jepsen and D. ter Haar, “A derivation of the hydrodynamic equations from the equations of motion of collective coordinates,” Physica 28, 70–75 (1962).CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    S. Dostoglou, “On Hydrodynamic averages,” J. Math. Sci., New York 189, No. 4, 582-595 (2013).CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel (2005).MATHGoogle Scholar
  14. 14.
    L. D. Landau and E. M. Lifshitz, Mechanics. Pergamon Press, (1976).Google Scholar
  15. 15.
    K. Aoki, C. Bardos, F. Golse, and Y. Sone, “Derivation of hydrodynamic limits from either the Liouville equation or kinetic models: Study of an example,” RIMS Kokyuroku No. 1146, 154–181 (2000).MATHMathSciNetGoogle Scholar
  16. 16.
    E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger Pub Co, Malabar (1972).Google Scholar
  17. 17.
    H. Cartan, Differential Calculus, Hermann, Paris (1971).MATHGoogle Scholar
  18. 18.
    R. B. Ash, Real Analysis and Probability, Academic Press, New York etc. (1972).Google Scholar
  19. 19.
    L. Ambrosio, N. Fusco, and D. Palara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000).MATHGoogle Scholar
  20. 20.
    J. Jacod and P. Protter, Probability Essentials, Springer, Berlin (2003).Google Scholar
  21. 21.
    H. Goldstein, Classical Mechanics, Addison-Wesley, Massachusetts (1980).MATHGoogle Scholar
  22. 22.
    I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes I, Springer, Berlin (1974)CrossRefMATHGoogle Scholar
  23. 23.
    N. C. Jacob, University of Missouri Ph.D. Thesis (2013).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.University of MissouriColumbiaUSA

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