On hydrodynamic equations at the limit of infinitely many molecules. (English. Russian original) Zbl 1326.35262
J. Math. Sci., New York 205, No. 2, 222-239 (2015); translation from Probl. Mat. Anal. 77, 91-104 (2014).
The authors show that if weak convergence of point measures together with moments of order \(2+\epsilon\) (with \(\epsilon>0\) small) exist, then dynamics of \(N\)-classical particles interacting through suitable 2-body potentials converge to the corresponding hydrodynamic equations for density and momentum.
Reviewer: Bernard Ducomet (Bruyères le Châtel)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 82B40 | Kinetic theory of gases in equilibrium statistical mechanics |
References:
| [1] | J. C. Maxwell, “On the dynamical theory of gases,” Philos. Trans. R. Soc. Lond.157, 49-88 (1867). · doi:10.1098/rstl.1867.0004 |
| [2] | L. Boltzmann, Lectures on Gas Theory, Univ. California Press (1964). |
| [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). |
| [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). · doi:10.1063/1.1747782 |
| [5] | H. Grad, “On the kinetic theory of rarefied gases,” Commun. Pure Appl. Math.2, 331-407 (1949). · Zbl 0037.13104 · doi:10.1002/cpa.3160020403 |
| [6] | C. B. Morrey, “On the derivation of the equations of hydrodynamics from statistical mechanics,” Commun. Pure Appl. Math.8, 279-326 (1955). · Zbl 0065.19406 · doi:10.1002/cpa.3160080206 |
| [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). · Zbl 0329.70011 |
| [8] | R. Esposito and M. Pulvirenti, “From particles to fluids,” in Handbook of Mathematical Fluid Dynamics. III, pp. 1-82, North-Holland, Amsterdam (2004). · Zbl 1221.82080 |
| [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). · Zbl 1294.35052 · doi:10.1090/S0273-0979-2013-01439-3 |
| [10] | O. Reynolds, “On the dynamical theory of incompressible viscous fluids and the determination of the criterion,” Philos. Trans. R. Soc. Lond., A186, 123-164 (1895). · JFM 26.0872.02 · doi:10.1098/rsta.1895.0004 |
| [11] | D. W. Jepsen and D. ter Haar, “A derivation of the hydrodynamic equations from the equations of motion of collective coordinates,” Physica28, 70-75 (1962). · Zbl 0125.23203 · doi:10.1016/0031-8914(62)90093-9 |
| [12] | S. Dostoglou, “On Hydrodynamic averages,” J. Math. Sci., New York189, No. 4, 582-595 (2013). · Zbl 1285.82027 · doi:10.1007/s10958-013-1209-9 |
| [13] | L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel (2005). · Zbl 1090.35002 |
| [14] | L. D. Landau and E. M. Lifshitz, Mechanics. Pergamon Press, (1976). · Zbl 0081.22207 |
| [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). · Zbl 0968.82523 |
| [16] | E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger Pub Co, Malabar (1972). |
| [17] | H. Cartan, Differential Calculus, Hermann, Paris (1971). · Zbl 0223.35004 |
| [18] | R. B. Ash, Real Analysis and Probability, Academic Press, New York etc. (1972). · Zbl 1381.28001 |
| [19] | L. Ambrosio, N. Fusco, and D. Palara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000). · Zbl 0957.49001 |
| [20] | J. Jacod and P. Protter, Probability Essentials, Springer, Berlin (2003). · Zbl 1014.60004 |
| [21] | H. Goldstein, Classical Mechanics, Addison-Wesley, Massachusetts (1980). · Zbl 0491.70001 |
| [22] | I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes I, Springer, Berlin (1974) · Zbl 0291.60019 · doi:10.1007/978-3-642-61943-4 |
| [23] | N. C. Jacob, University of Missouri Ph.D. Thesis (2013). · Zbl 1285.82027 |
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