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2024 | OriginalPaper | Chapter

The Extended Phase Space Method in Kinetic Theory

Author : Francesco Salvarani

Published in: From Particle Systems to Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

This article describes the extended phase space technique in the study of kinetic equations and provides a review of some relevant contributions appeared in the literature.

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previous chapter Uniqueness Criteria for the Vlasov–Poisson System and Applications to Semiclassical Analysis

next chapter Mesoscale Mode Coupling Theory for the Weakly Asymmetric Simple Exclusion Process

The literature reports some examples in which the supplementary variables have other physical meanings: for instance, the two additional variables in [4, 5] are the rescaled distance to the next collision and the corresponding impact parameter.

A family of functions $\sigma _\varepsilon {(v)}$ converges in the sense of Young measures to a measure $\mu _{ {v}}\in \mathscr {M}(\mathscr {V})$ if and only if

$$\begin{aligned}\forall {g}\in C_{0}(\mathbb {R}_{+}),\quad {g}(\sigma _\varepsilon ) \overset{*}{\rightharpoonup }\ \int _{0}^{+\infty } {g}(s) \mu _{ {v}}(\textrm{d}s) \text{ in } L^{\infty }(\boldsymbol{T}^d\times \mathscr {V}) \text{ weak- }*.\end{aligned}$$

Bernard, E., Caglioti, E., Golse, F.: Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes. SIAM J. Math. Anal. 42(5), 2082–2113 (2010)MathSciNetCrossRef

Bernard, E., Golse, F., Salvarani, F.: Homogenization of transport problems and semigroups. Math. Methods Appl. Sci. 33(10), 1228–1234 (2010)MathSciNetCrossRef

Bernard, E., Salvarani, F.: Homogenization of the linear Boltzmann equation with a highly oscillating scattering term in extended phase space. Appl. Math. Lett. 143, 108672 (2023)

Caglioti, E., Golse, F.: The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions. C. R. Math. Acad. Sci. Paris 346(7–8), 477–482 (2008)MathSciNetCrossRef

Caglioti, E., Golse, F.: On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas. J. Stat Phys 141(2), 264–317 (2010)MathSciNetCrossRef

Dautray, R.: Méthodes probabilistes pour les équations de la physique. In: Collection du Commissariat à l’Énergie Atomique. Série Synthèses. Commissariat à lÉnergie Atomique, Paris (1989)

Golse, F.: On the periodic Lorentz gas in the Boltzmann-Grad scaling. Ann. Faculté des Sci. Toulouse (6), 17(4), 735–749 (2008)

Golse, F.: Homogenization and kinetic models in extended phase space. Riv. Math. Univ. Parma (N.S.) 3(1), 71–89 (2012)

Hutridurga, H., Mula, O., Salvarani, F.: Homogenization in the energy variable for a neutron transport model. Asymptot. Anal. 117(1–2), 1–25 (2020)MathSciNet

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Marklof, J., Strömbergsson, A.: The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. Math. (2) 174(1), 225–298 (2011)

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Mathiaud, J., Salvarani, F.: A numerical strategy for radiative transfer problems with higly oscillating opacities. Appl. Math. Comput. 221, 249–256 (2013)MathSciNet

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Papanicolaou, G.C.: Asymptotic analysis of transport processes. Bull. Am. Math. Soc. 81, 330–392 (1975)MathSciNetCrossRef

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Tartar, L.: Nonlocal effects induced by homogenization. In: Partial Differential Equations and the Calculus of Variations, Volume 2 of Progress in Nonlinear Differential Equations and their Applications, pp. 925–938. Birkhäuser Boston, Boston, MA (1989)

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Tartar, L.: An Introduction to Navier-Stokes Equation and Oceanography. Lecture Notes of the Unione Matematica Italiana, vol. 1. Springer-Verlag, Berlin, Heidelberg (2006)

Title: The Extended Phase Space Method in Kinetic Theory
Author: Francesco Salvarani
Publisher: Springer International Publishing
Book: From Particle Systems to Partial Differential Equations
Print ISBN: 978-3-031-65194-6

Electronic ISBN: 978-3-031-65195-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-65195-3_15

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