From the Kinetic Theory of Gases to Continuum Mechanics

Abstract

Abstract. Recent results on the fluid dynamic limits of the Boltzmann equation based on the DiPerna-Lions theory of renormalized solutions are reviewed in this paper, with an emphasis on regimes where the velocity field behaves to leading order like that of an incompressible fluid with constant density.Keywords: Hydrodynamic limits, Kinetic models, Boltzmann equation, Entropy production, Euler equations, Navier-Stokes equations PACS: 47.45-n, 51.10.+y, 51.20.+d, 47.10.ad In memory of Carlo Cercignani (1939Cercignani ( -2010 Relating the kinetic theory of gases to their description by the equations of continuum mechanics is a question that finds its origins in the work of Maxwell [30]. It was subsequently formulated by Hilbert as a mathematical problem -specifically, as an example of his 6th problem on the axiomatization of physics [21]. In Hilbert's own words "Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes which lead from the atomistic view to the laws of motion of continua". Hilbert himself studied this problem in [22] as an application of his theory of integral equations. The present paper reviews recent progress on this problem in the past 10 years as a consequence of the DiPerna-Lions global existence and stability theory [12] for solutions of the Boltzmann equation. This Harold Grad lecture is dedicated to the memory of Carlo Cercignani, who gave the first Harold Grad lecture in the 17th Rarefied Gas Dynamics Symposium, in Aachen (1990), in recognition of his outstanding influence on the mathematical analysis of the Boltzmann equation in the past 40 years. THE BOLTZMANN EQUATION: FORMAL STRUCTUREIn kinetic theory, the state of a monatomic gas at time t and position x is described by its velocity distribution function F ≡ F(t, x, v) ≥ 0. It satisfies the Boltzmann equationassuming that gas molecules behave as perfectly elastic hard spheres of diameter d. In this formula, we have denotedMolecular interactions more general than hard sphere collisions can be considered by replacing d 2 2 |(v − v * ) · ω| with appropriate collision kernels of the form b(|v − v * |, | v−v * |v−v * | · ω|). In this paper, we restrict our attention to the case of hard sphere collisions to avoid dealing with more technical conditions on the collision kernel. Properties of the collision integralWhile the collision integral is a fairly intricate mathematical expression, the formulas (1) entail remarkable symmetry properties. As a result, the collision integral satisfies, for each continuous, rapidly decaying f ≡ f (v), the identities

Parabolic temperature profile and second-order temperature jump of a slightly rarefied gas in an unsteady two-surface problem

“…Thanks to the symmetric temperature profile in x, all of τ Hm , P Hm , and u Hm for m ≥ 3 are found to be constant from Eqs. (14a) and (16), and thus the corresponding φ Hm are reduced to linearized Maxwellians [see Eqs. (13) and (15)].…”

“…The connection between the Boltzmann and the NS systems has been studied since the days of Hilbert, and a number of useful results have been obtained in the limit of small Knudsen numbers. [7][8][9][10][11][12][13][14][15][16] Specifically, fluid-dynamic-type sets of equations and appropriate slip and jump boundary conditions for describing the steady gas behavior in the regime of small Knudsen numbers (the so-called slip flow regime) have been established [11][12][13] since the late 1960s and early 1970s. For rigorous and complete descriptions of the general theory of slip flow, the reader is referred to Refs.…”

The energy density distribution of an ideal gas and Bernoulli’s equations

“…Problems in the interpretation or derivation of Bernoulli's equation have been extensively discussed in the literature [5,[7][8][9][10][11][12][13][14][15][16]. Every relation in fluid mechanics is deduced from the continuum hypothesis 7 or from the kinetic theory of fluids [17][18][19], and the terms of the energy density of Bernoulli's equation stimulated the interpretation on an atomic molecular scale [5,12,13]. Regardless of the interpretation of Bernoulli's equation (3) or (4), there must be a relation between Bernoulli's equation and the energy density distribution.…”

Fluid Dynamic Limits of the Kinetic Theory of Gases

“…However, the diffusion coefficient in the temperature equation is 3/5 of its value for an incompressible fluid with the same heat capacity and heat conductivity. The difference comes from the work of the pressure: see the detailed discussion of this subtle point in [30] on pp. 22-23, and especially in [74] (footnote 6 on p. 93) and [75] (footnote 43 on p. 107, together with section 3.7.2).…”

“…This first lecture is a slightly expanded version of the author's Harold Grad Lecture [30], with an emphasis on mathematical tools and methods used in the theory of the Boltzmann equation and of its fluid dynamic limits.…”