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A Gaussian beam approach for computing Wigner measures in convex domains
A simple particle model for a system of coupled equations with absorbing collision term
1. | Université de Lyon and CNRS, UMPA, UMR-CNRS 5669, ENS-Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07, France |
2. | Dipartimento di Metodi e Modelli Matematici, Università di Palermo, Viale delle Scienze Edificio 8, I90128 Palermo, Italy |
References:
[1] |
C. Boldrighini, L. A. Bunimovich and Ya. G. Sinaĭ, On the Boltzmann equation for the Lorentz gas, J. Stat. Phys., 32 (1983), 477-501.
doi: 10.1007/BF01008951. |
[2] |
J. Bourgain, F. Golse and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas, Commun. Math. Phys., 190 (1998), 491-508.
doi: 10.1007/s002200050249. |
[3] |
L. Desvillettes and V. Ricci, Non-Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field, Bull. Sci. Math., 128 (2004), 39-46.
doi: 10.1016/j.bulsci.2003.09.003. |
[4] |
L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic of the Fokker-Planck type in the limit of grazing collisions, J. Stat. Phys., 104 (2001), 1173-1189.
doi: 10.1023/A:1010461929872. |
[5] |
G. Gallavotti, "Rigorous Theory of the Boltzmann Equation in the Lorentz Gas," Nota interna n. 358, Istituto di Fisica, Università di Roma, 1972. |
[6] |
F. Golse, The mean-field limit for the dynamics of large particle systems, in Journées "Equations aux Dérivées Partielles," Exp. No. IX, 47 pp., Univ. Nantes, Nantes, 2003. |
[7] |
F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse, Ser.6, 17 (2008), 735-749. |
[8] |
F. Golse, Recent results on the periodic Lorentz gas, preprint, HAL: hal-00390895, (2009), 1-62. |
[9] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 2, Functional and Variational Methods, Springer-Verlag, Berlin, 1988. |
[10] |
P. Malliavin and H. Airault, "Intégration et Analyse de Fourier, Probabilités et Analyse Gaussienne," 2nd edition, Collection Maîtrise de Mathématiques Pures, Masson, Paris, 1994. |
[11] |
D. Mihalas and B. Weibel Mihalas, "Foundations of Radiation Hydrodynamics," Oxford University Press, New York, 1984, Reprint, Dover 1999. |
[12] |
G. Nappo, E. Orlandi and H. Rost, A reaction-diffusion model for moderately interacting particles, J. Stat. Phys., 55 (1989), 579-600.
doi: 10.1007/BF01041598. |
[13] |
V. Ricci and B. Wennberg, On the derivation of a linear Boltzmann equation from a periodic lattice gas, Stochastic Process. Appl., 111 (2004), 281-315.
doi: 10.1016/j.spa.2004.01.002. |
[14] |
L. Schwartz, "Théorie des Distributions," Publications de l'institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966. |
[15] |
H. Spohn, The Lorentz process converges to a random flight process, Commun. Math. Phys., 60 (1978), 277-290.
doi: 10.1007/BF01612893. |
[16] |
C. I. Steefel, D. J. De Paolo and P. C. Lichtner, Reactive transport modeling: An essential tool and a new research approach for the Earth sciences, Earth and Planetary Science Letters, 240 (2005), 539-558. |
[17] |
A. S. Sznitman, Propagation of chaos for a system of annihilating Brownian spheres, CPAM, 40 (1987), 663-690. |
show all references
References:
[1] |
C. Boldrighini, L. A. Bunimovich and Ya. G. Sinaĭ, On the Boltzmann equation for the Lorentz gas, J. Stat. Phys., 32 (1983), 477-501.
doi: 10.1007/BF01008951. |
[2] |
J. Bourgain, F. Golse and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas, Commun. Math. Phys., 190 (1998), 491-508.
doi: 10.1007/s002200050249. |
[3] |
L. Desvillettes and V. Ricci, Non-Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field, Bull. Sci. Math., 128 (2004), 39-46.
doi: 10.1016/j.bulsci.2003.09.003. |
[4] |
L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic of the Fokker-Planck type in the limit of grazing collisions, J. Stat. Phys., 104 (2001), 1173-1189.
doi: 10.1023/A:1010461929872. |
[5] |
G. Gallavotti, "Rigorous Theory of the Boltzmann Equation in the Lorentz Gas," Nota interna n. 358, Istituto di Fisica, Università di Roma, 1972. |
[6] |
F. Golse, The mean-field limit for the dynamics of large particle systems, in Journées "Equations aux Dérivées Partielles," Exp. No. IX, 47 pp., Univ. Nantes, Nantes, 2003. |
[7] |
F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse, Ser.6, 17 (2008), 735-749. |
[8] |
F. Golse, Recent results on the periodic Lorentz gas, preprint, HAL: hal-00390895, (2009), 1-62. |
[9] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 2, Functional and Variational Methods, Springer-Verlag, Berlin, 1988. |
[10] |
P. Malliavin and H. Airault, "Intégration et Analyse de Fourier, Probabilités et Analyse Gaussienne," 2nd edition, Collection Maîtrise de Mathématiques Pures, Masson, Paris, 1994. |
[11] |
D. Mihalas and B. Weibel Mihalas, "Foundations of Radiation Hydrodynamics," Oxford University Press, New York, 1984, Reprint, Dover 1999. |
[12] |
G. Nappo, E. Orlandi and H. Rost, A reaction-diffusion model for moderately interacting particles, J. Stat. Phys., 55 (1989), 579-600.
doi: 10.1007/BF01041598. |
[13] |
V. Ricci and B. Wennberg, On the derivation of a linear Boltzmann equation from a periodic lattice gas, Stochastic Process. Appl., 111 (2004), 281-315.
doi: 10.1016/j.spa.2004.01.002. |
[14] |
L. Schwartz, "Théorie des Distributions," Publications de l'institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966. |
[15] |
H. Spohn, The Lorentz process converges to a random flight process, Commun. Math. Phys., 60 (1978), 277-290.
doi: 10.1007/BF01612893. |
[16] |
C. I. Steefel, D. J. De Paolo and P. C. Lichtner, Reactive transport modeling: An essential tool and a new research approach for the Earth sciences, Earth and Planetary Science Letters, 240 (2005), 539-558. |
[17] |
A. S. Sznitman, Propagation of chaos for a system of annihilating Brownian spheres, CPAM, 40 (1987), 663-690. |
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