Chapter 1 From particles to fluids

Handbook of Mathematical Fluid Dynamics

Volumn 3, Issue , 2005, Pages 1-82

  Esposito, Raffaele ^a,b   Pulvirenti, Mario ^c  

^a Centro Linceo Interdisciplinare 'Beniamino Segre'   (Italy)

^b UNIVERSITY OF L'AQUILA   (Italy)

^c SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67649382229     PISSN: 18745792     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5792(05)80004-7     Document Type: Chapter

References (90)

1. Alexander R.K. The infinite hard sphere system. Ph.D. Thesis (1975), Dept. of Math., University of California, Berkeley
2. Arkeryd L., and Nouri A. The stationary Boltzmann equation in the slab with given weighted mass for hard and soft forces. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 27 (1998) 533-556
3. Arkeryd L., and Nouri A. On the Milne problem and the hydrodynamic limit for a steady Boltzmann equation model. J. Statist. Phys. 99 3/4 (2000) 993-1019
4. Asano K., and Ukai S. The Euler limit and the initial layer of the nonlinear Boltzmann equation. Hokkaido Math. J. 12 (1983) 303-324
5. Bardos C., Caflisch R.E., and Nicolaenko B. Thermal layer solutions of the Boltzmann equation. In: Fritz J., Jaffe A., and Szasz D. (Eds). Random Fields: Rigorous Results in Statistical Physics, Koszeg 1984 (1985), Birkhäuser, Boston
6. Bardos C., Golse F., and Levermore D. Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles. C. R. Acad. Sci. Paris Sér. I Math. 309 (1989) 727-732
7. Bardos C., Golse F., and Levermore D. Fluid dynamical limits of kinetic equations. I. Formal derivations. J. Statist. Phys. 63 (1991) 323-344
8. Bardos C., Golse F., and Levermore D. Fluid dynamical limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46 (1993) 667-753
9. Bardos C., Golse F., and Levermore D. Acoustic and Stokes limits for the Boltzmann equation. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 323-328
10. Bardos C., Golse F., and Levermore D. The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal. 153 (2000) 177-204
11. Bardos C., and Nicolaenko B. Navier-Stokes equations and dynamical systems. In: Fieldler B. (Ed). Handbook of Dynamical Systems Vol. 2 (2002), North-Holland, Amsterdam 503-597
12. Benois O., Esposito R., and Marra R. Navier-Stokes limit for a thermal stochastic lattice gas. J. Statist. Phys. 96 (1999) 653-713
13. Boldrighini C., Bunimovich L.A., and Sinai Ya.G. On the Boltzmann equation for the Lorentz gas. J. Statist. Phys. 32 (1983) 477-501
14. Boldrighini C., Dobrushin R.L., and Suhov Y.M. One-dimensional hard rod caricatures of hydrodynamics. J. Statist. Phys. 31 (1983) 577-616
15. Bourgain J., Golse F., and Wennberg B. On the distribution of free path length for the periodic Lorentz gas. Comm. Math. Phys. 190 (1998) 491-508
16. Boussinesq J. Theorie analytique de la chaleur (1903), Gauthier-Villars, Paris
17. Bunimovich L.A., and Sinai Ya.G. Statistical properties of the Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 (1981) 478-497
18. Caflisch R.E. The fluid dynamical limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math. 33 (1980) 651-666
19. Caflisch R.E. The Boltzmann equation with a soft potential. Comm. Math. Phys. 74 (1980) 71-95
20. Caflisch R.E., and Nicolaenko B. Shock waves the Boltzmann equation. Nonlinear partial differential equations. In: Durham N.H. (Ed). Contemp. Math. 17 (1982-1983) 35-44
21. Caflisch R.E., and Papanicolaou G. The fluid dynamical limit of a nonlinear Boltzmann equation. Comm. Pure Appl. Math. 32 (1979) 589-616
22. Caglioti E., Marchioro C., and Pulvirenti M. Non-equilibrium dynamics of three-dimensional infinite particle systems. Comm. Math. Phys. 215 (2000) 25-43
23. Caglioti E., Pulvirenti M., and Ricci V. Derivation of a linear Boltzmann equation for a periodic Lorentz gas, Markov Process. Related Fields 3 (1900) 265-285
24. Caprino S., Pulvirenti M., and Wagner W. A particle systems approximating stationary solutions to the Boltzmann equation. SIAM J. Math. Anal. 4 (1998) 913-934
25. Cercignani C. On the Boltzmann equation for rigid spheres. Transport Theory Statist. Phys. 2 (1972) 211-225
26. Cercignani C. The Grad limit for a system of soft spheres. Comm. Pure Appl. Math. 36 (1983) 479-494
27. Cercignani C. The Boltzmann Equation and Its Applications (1988), Springer-Verlag, New York
28. Cercignani C. Ludwig Boltzmann. The Man Who Trusted Atoms (1998), Oxford University Press, Oxford
29. Cercignani C., Illner R., and Pulvirenti M. The Mathematical Theory of Dilute Gases (1994), Springer-Verlag, New York
30. Chapman S., and Cowling T.G. The Mathematical Theory of Nonuniform Gases (1952), Cambridge University Press, Cambridge
31. Costantini C., and Marra R. Hydrodynamic limits for the Boltzmann process. J. Statist. Phys. 67 (1992) 229-249
32. Masi A.D., Esposito R., and Lebowitz J.L. Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math. 42 (1989) 1189-1214
33. Masi A.D., Ianiro N., Pellegrinotti S., and Presutti E. A survey of the hydrodynamical behavior of many-particle systems. In: Lebowitz J.L., and Montroll E.W. (Eds). Nonequilibrium Phenomena, II. From Stochastics to Hydrodynamics (1984), North-Holland, Amsterdam 123-294
34. Desvillettes L., and Pulvirenti M. The linear Boltzmann equation for long-range forces: A derivation from particle systems. Models Methods Appl. Sci. 9 (1999) 1123-1145
35. Desvillettes L., and Ricci V. Rigorous derivation of a linear kinetic equation of Fokker-Plank type in the limit of grazing collisions. J. Statist. Phys. 104 (2001) 1173-1189
36. Desvillettes L., and Villani C. Entropic methods for the study of the long time behavior of kinetic equations. The Sixteenth International Conference on Transport Theory. Part I (Atlanta, GA, 1999). Transport Theory Statist. Phys. 30 (2001) 155-168
37. DiPerna R.J., and Lions P.-L. On the Cauchy problem for the Boltzmann equations: Global existence and weak stability. Ann. Math. 130 (1989) 321-366
38. Dobrushin R.L., and Fritz J. Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction. Comm. Math. Phys. 57 (1977) 67-81
39. Dürr D., Goldstein S., and Lebowitz J.L. Asymptotic motion of a classical particle in a random potential in two dimensions: Landau Model. Comm. Math. Phys. 113 (1987) 209-230
40. Esposito R., Lebowitz J.L., and Marra R. Hydrodynamic limit of the stationary Boltzmann equation in a slab. Comm. Math. Phys. 160 (1994) 49-80
41. Esposito R., Lebowitz J.L., and Marra R. The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation. J. Statist. Phys. 78 (1995) 389-412
42. Esposito R., Lebowitz J.L., and Marra R. Solutions to the Boltzmann equation in the Boussinesq regime. J. Statist. Phys. 90 (1998) 1129-1178
43. Esposito R., and Marra R. On the derivation of the incompressible Navier-Stokes equation for Hamiltonian particle systems. J. Statist. Phys. 74 (1993) 981-1004
44. Esposito R., and Marra R. Hydrodynamics as scaling limit of kinetic systems and stochastic particle systems. Trends Statist. Phys. 3 (2000) 1-43
45. Esposito R., Marra R., and Yau H.T. Diffusive limit of asymmetric simple exclusion. Rev. Math. Phys. 6 (1994) 1233-1267
46. Esposito R., Marra R., and Yau H.T. Navier-Stokes equations for stochastic particle systems on the lattice. Comm. Math. Phys. 182 (1996) 395-456
47. Gallavotti G. Divergences and approach to equilibrium in the Lorentz and Wind-tree models. Phys. Rev. 185 (1969) 308-322
48. Gallavotti G. Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota interna, n. 358 (1972), Istituto di Fisica Università di Roma
49. reprinted in. Gallavotti G. Meccanica statistica. Quaderni del CNR, n. 50 (1995) 191-204
50. Golse F., and Levermore C.D. The Stokes-Fourier limit for the Boltzmann equation. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 145-150
51. Golse F., and Levermore C.D. The Stokes and acoustic limits for the Boltzmann equation. Comm. Pure Appl. Math. 55 (2002) 336-393
52. Golse F., and Saint-Raymond L. The Navier-Stokes limit of the Boltzmann equation: Convergence proof. Preprint; The Navier-Stokes limit for the Boltzmann equation. C R. Acad. Sci. Paris Sér. I Math. 333 (2001) 897-902
53. Grad H. On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331-407
54. Grad H. Asymptotic theory of the Boltzmann equation, II. In: Laurmann J.A. (Ed). Rarefied Gas Dynamics Vol. I (1963), Academic Press, New York 26-59
55. Grad A. Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann operator. Proc. Symp. Appl. Math. 17 (1965) 154-183
56. Green M.S. Markoff random processes and the statistical mechanics of time-dependent phenomena, II. Irreversible processes in fluids. J. Chem. Phys. 22 (1954) 398-413
57. Hilbert D. Mathematical Problems. Göttinges Nachr. (1900) 253-297
58. 1917. Hilbert D. Begründung der kinetischen Gastheorie. Math. Ann. 72 (1916) 331-407
59. Hilbert D. Grundzügeiner allgemeinen Theorie der linearen Integralgleichungen (1953), Chelsea, New York
60. Illner R., and Pulvirenti M. Global validity of the Boltzmann equation for two and three-dimensional rare gas in vacuum. Comm. Math. Phys. 121 (1989) 143-146
61. Kac M. Probability and Related Topics (1959), Interscience, New York
62. Klainerman S., and Majda A. Compressible and incompressible fluids. Comm. Pure Appl. Math. 35 (1982) 629-651
63. Kubo R. Statistical-mechanical theory of irreversible processes, I. General theory and simple applications in magnetic and conduction problems. J. Phys. Soc. Jap. 12 (1957) 570-586
64. Lachowicz M. On the initial layer and the existence theorem for the nonlinear Boltzmann equation. Math. Methods Appl. Sci. 9 3 (1987) 27-70
65. Lachowicz M., and Pulvirenti M. A stochastic system of particles modelling the Euler equation. Arch. Ration. Mech. Anal. 109 (1990) 81-93
66. Lanford O.E. Time Evolution of large classical systems. In: Moser J. (Ed). Lecture Notes in Phys. Vol. 38 (1976), Springer-Verlag, Berlin-Heidelberg 1-111
67. Lions P.-L., and Masmoudi N. Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 387-392
68. Lions P.-L., and Masmoudi N. From Boltzmann equations to incompressible fluid mechanics equations, I. Arch. Ration. Mech. Anal. 158 (1901) 173-193
69. Lions P.-L., and Masmoudi N. From Boltzmann equations to incompressible fluid mechanics equations, II. Arch. Ration. Mech. Anal. 158 (2001) 195-211
70. Liu T.-P., and Yu S.-H. Boltzmann equation: Micro-macro decompositions and positivity of shock profiles. Comm. Math. Phys. 246 (1904) 133-179
71. Maxwell J.C. On the dynamical theory of gases. Philos. Trans. Roy. Soc. London 157 (1867) 49-88
72. Méléard S. Probabilistic models for nonlinear partial differential equations. In: Talay D., and Tubaro L. (Eds). Lecture Notes in Math. Vol. 1627 (1995), Springer-Verlag, New York 42-95
73. Morrey C.B. On the derivation of the equations of hydrodynamics from statistical mechanics. Comm. Pure Appl. Math. 8 (1955) 279-326
74. Mürmann M.G. The hydrodynamic limit of a deterministic particle system with conservation of mass and momentum. J. Statist. Phys. 105 (1901) 483-510
75. Nishida T. Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Comm. Math. Phys. 61 (1978) 119-148
76. Oelschlager K. On the connection between Hamiltonian many-particle systems and the hydrodynamical equations. Arch. Ration. Mech. Anal. 115 (1991) 297-310
77. Olla S., Varadhan S.R.S., and Yau H.T. Hydrodynamical limit for a Hamiltonian system with weak noise. Comm. Math. Phys. 155 (1993) 523
78. Pulvirenti M. Kinetic limits for stochastic particle systems. In: Talay D., and Tubaro L. (Eds). Lecture Notes in Math. Vol. 1627 (1995), Springer-Verlag, New York 96-126
79. Pulvirenti M., Wagner W., and Rossi M.B.Z. Convergence of particle schemes for the Boltzmann equation. Eur. J. Mech. B Fluids 13 (1994) 339-351
80. Quastel J., and Yau H.T. Lattice gases, large deviations and the incompressible Navier-Stokes equations. Ann. Math. 148 (1998) 51-108
81. d. Comm. Math. Phys. 140 (1991) 417-448
82. Ruelle D. Statistical Mechanics (1969), Benjamin, Reading, MA
83. Sone Y., Aoki K., Tabata S., Sugimoto H., and Bobylev A.V. Inappropriateness of the heat-conduction equation for the description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation. Phys. Fluids 8 (1996) 628-638
84. Spohn H. The Lorentz flight process converges to a random flight process. Comm. Math. Phys. 60 (1978) 277-290
85. Toda M., Kubo R., and Saitô N. Statistical Physics I: Equilibrium Statistical Mechanics (1992), Springer-Verlag, Berlin-Heidelberg
86. Uchiyama K. On the Boltzmann-Grad limit for the Broadwell model of the Boltzmann equation. J. Statist. Phys. 52 (1988) 331-3557
87. Varadhan S.R.S. Nonlinear diffusion limit for a system with nearest neighbor interactions, II. In: Elworthy K.D., and Ikeda N. (Eds). Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals. Pitman Research Notes in Math. Vol. 283 (1994), Wiley, New York 75-128
88. Villani C. A review of mathematical topics in collisional kinetic theory. In: Friedlander S., and Serre D. (Eds). Handbook of Mathematical Fluid Dynamics Vol. 1 (2002), North-Holland, Amsterdam 71-306
89. Wagner W. A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equa-tion. J. Statist. Phys. 66 (1992) 1011-1044
90. Yau H.T. Relative entropy and hydrodynamics of Ginsburg-Landau models. Math. Phys. Lett. 22 (1991) 63-80