The Euler limit for kinetic models with Fermi–Dirac statistics

Article type: Research Article

Authors: Zakrevskiy, Timofey

Affiliations: Centre de Mathématiques Laurent Schwartz, UMR 7640, Ecole Polytechnique, 91128 Palaiseau Cedex, France. Tel.: +33 1 69 33 49 16; Fax: +33 1 69 33 49 49; E-mail: [email protected]

Abstract: We are interested in the connection between kinetic models with Fermi–Dirac statistics and fluid dynamics. We establish that moments and parameters of Fermi–Dirac distributions are related by a diffeomorphism. We obtain the macroscopic limits when the fluid is dense enough that particles undergo many collisions per unit of time. This situation is described via a small parameter ε, called the Knudsen number, that represents the ratio of mean free path of particles between collisions to some characteristic length of the flow. We give the conditions that allow us to formally derive the generalized Euler equations from the Boltzmann equation by adopting the formalism proposed in [Advances in Kinetic Theory and Continuum Mechanics, Springer, Berlin, 1991, pp. 57–71]. These conditions are related to the H-theorem and assume a formally consistent convergence for fluid dynamical moments and entropy of the kinetic equation. We also discuss the well-posedness of the obtained Euler equations by using Godunov’s criterion of hyperbolicity.

Keywords: kinetic model, Boltzmann equation, Euler equations, entropy

DOI: 10.3233/ASY-151323

Journal: Asymptotic Analysis, vol. 95, no. 1-2, pp. 59-77, 2015