Derivation of the linear Boltzmann equation without cut-off starting from particles
Nathalie Ayi1
1 Sorbonne Universités UPMC Université Paris 06 UMR 7598 Laboratoire Jacques-Louis Lions F-75005, Paris, France
Journées équations aux dérivées partielles (2017), Talk no. 1, 12 p.

We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford’s strategy. Our proof relies then on a combination of Lanford’s strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated to the long-range interaction, leading to some explicit weak estimates.

Published online:
DOI: 10.5802/jedp.651
Nathalie Ayi 1

1 Sorbonne Universités UPMC Université Paris 06 UMR 7598 Laboratoire Jacques-Louis Lions F-75005, Paris, France 
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Nathalie Ayi. Derivation of the linear Boltzmann equation without cut-off starting from particles. Journées équations aux dérivées partielles (2017), Talk no. 1, 12 p. doi : 10.5802/jedp.651. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.651/

[1] R. Alexandre Remarks on 3D Boltzmann linear equation without cutoff, Transport Theory and Statistical Physics, Volume 28 (1999) no. 5, pp. 433-473 | DOI

[2] R. Alexandre; Y. Morimoto; S. Ukai; C.-J. Xu; T. Yang Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff, Communications in Mathematical Physics, Volume 304 (2011) no. 2, pp. 513-581

[3] R. Alexandre; Y. Morimoto; S. Ukai; C.-J. Xu; T. Yang The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential, Analysis and Applications, Volume 09 (2011) no. 02, pp. 113-134 http://www.worldscientific.com/doi/abs/10.1142/S0219530511001777 | DOI

[4] R. Alexandre; Y. Morimoto; S. Ukai; C.-J. Xu; T. Yang The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, Journal of Functional Analysis, Volume 262 (2012) no. 3, pp. 915 -1010 http://www.sciencedirect.com/science/article/pii/S0022123611003752 | DOI

[5] R. Alexandre; C. Villani On The Boltzmann Equation For Long-Range Interactions, Comm. Pure Appl. Math, Volume 55 (2002), pp. 30-70

[6] N. Ayi From Newton’s law to the Boltzmann equation without cut-off, Comm. Math. Phys., Volume 350 (2017) no. 3, pp. 1219-1274

[7] T. Bodineau; I. Gallagher; L. Saint-Raymond The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones mathematicae, Volume 203 (2015) no. 2, pp. 493-553 | DOI

[8] I. Gallagher; L. Saint-Raymond; B. Texier From Newton to Boltzmann: Hard Spheres and Short-range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2013 https://books.google.fr/books?id=K6VCngEACAAJ

[9] H. Grad Principles of the Kinetic Theory of Gases, Handbuch der Physik, Volume 12 (1958), pp. 205-294 | DOI

[10] P. T. Gressman; R. M. Strain; R. V. Kadison Global classical solutions of the Boltzmann equation with long-range interactions, Proceedings of the National Academy of Sciences of the United States of America, Volume 107 (2010) no. 13, pp. 5744-5749 http://www.jstor.org/stable/25665060

[11] F. King BBGKY hierarchy for positive potentials, Dept. Mathematics, Univ. California, Berkeley (1975) (Ph. D. Thesis)

[12] O. E. Lanford Time evolution of large classical systems, Dynamical Systems, Theory and Applications (J. Moser, ed.) (Lecture Notes in Physics), Volume 38, Springer Berlin Heidelberg, 1975, pp. 1-111 | DOI

[13] J. L. Lebowitz; H. Spohn Steady state self-diffusion at low density, Journal of Statistical Physics, Volume 29 (1982) no. 1, pp. 39-55 | DOI

[14] M. Pulvirenti; C. Saffirio; S. Simonella On the validity of the Boltzmann equation for short range potentials, Reviews in Mathematical Physics, Volume 26 (2014) no. 02, 1450001 pages http://www.worldscientific.com/doi/abs/10.1142/S0129055X14500019 | DOI

[15] H. van Beijeren; O. E. Lanford; J. L. Lebowitz; H. Spohn Equilibrium time correlation functions in the low-density limit, Journal of Statistical Physics, Volume 22 (1980) no. 2, pp. 237-257 | DOI

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