The Mathematical Theory of Dilute Gases

The idea for this book was conceived by the authors some time in 1988, and a first outline of the manuscript was drawn up during a summer school on mathematical physics held in Ravello in September 1988, where all three of us were present as lecturers or organizers. The project was in some sense inherited from our friend Marvin Shinbrot, who had planned a book about recent progress for the Boltzmann equation, but, due to his untimely death in 1987, never got to do it.

As early as 1738 Daniel Bernoulli advanced the idea that gases are formed of elastic molecules rushing hither and thither at large speeds, colliding and rebounding according to the laws of elementary mechanics. Of course, this was not a completely new idea, because several Greek philosophers asserted that the molecules of all bodies are in motion even when the body itself appears to be at rest. The new idea was that the mechanical effect of the impact of these moving molecules when they strike against a solid is what is commonly called the pressure of the gas. In fact if we were guided solely by the atomic hypothesis, we might suppose that the pressure would be produced by the repulsions of the molecules. Although Bernoulli’s scheme was able to account for the elementary properties of gases (compressibility, tendency to expand, rise of temperature in a compression and fall in an expansion, trend toward uniformity), no definite opinion could be passed on it until it was investigated quantitatively. The actual development of the kinetic theory of gases was, accordingly, accomplished much later, in the nineteenth century. Within the scope of this book, the molecules of a gas will be assumed to be perfectly elastic spheres that move according to the laws of classical mechanics. Thus, e. g., if no external forces, such as gravity, are assumed to act on the molecules, each of them will move in a straight line unless it happens to strike another sphere or a solid wall. Systems of this kind are usually called billiards, for obvious reasons.

As indicated in the previous chapter, we shall investigate the hard sphere model of a gas. The reason for choosing such a simple model is based on the expectation that in the asymptotic regimes (hydrodynamic and kinetic) in which we are interested the general features should not depend on the particular type of interaction between the particles.

In this chapter we shall devote ourselves to a study of the main properties of the solutions of the Boltzmann equation. We assume that our solutions are as smooth as required. It will be the purpose of the remaining part of the book to show that sufficiently smooth solutions exist for which the manipulations presented here make sense.

In Chapter 2 we gave a formal derivation of the Boltzmann equation from the basic laws of mechanics. In particular, we introduced the Liouville equation, the BBGKY hierarchy, the Boltzmann hierarchy, and the Boltzmann equation, and we discussed the assumptions that allowed us to make the transitions from each of those to the next. The objective of this chapter is to do all these steps rigocrously, wherever possible. In particular, our discussion will lead to a rigorous validity and existence result for the Boltzmann equation, locally for a general situation and globally for a rare gas cloud in vacuum.

Existence and uniqueness theorems play a very central part in the theory of partial differential equations, particularly in the context of mathematical physics. The well-posedness of a Cauchy or boundary value problem is of tantamount importance for the physical interpretation and/or practical application of the equation under consideration. For instance, numerical calculations become a touchy business in the absence of uniqueness or continuous dependence on the data, and the function spaces in which existence theorems can be proved usually contain intrinsic useful information about the solutions. Moreover, having in mind that the Boltzmann equation is a schematization of the reality (described at a more detailed level by the Newton laws) expected to be valid only in the asymptotic regime when a gas is extremely rarefied, a good existence theorem for the solutions of such an equation is, at least, the first check of the validity of the mathematical model under investigation.

In this chapter we treat the spatially homogeneous Boltzmann equation, i.e., the special case where f does not depend on x. In this case the main difficulty in estimating the collision operator, namely, the pointwise interaction, disappears, and we can develop a rather complete and satisfactory theory. The remaining difficulties are due to large velocities (high energy tails).

Our first aim in this chapter will be to find a global solution f = f(x, ξ, t) of the Cauchy problem for the Boltzmann equation for hard spheres:

If we want to describe a physical situation where a gas flows past a solid body or is contained in a region bounded by one or more solid bodies, the Boltzmann equation must be accompanied by boundary conditions, which describe the interaction of the gas molecules with the solid walls. It is to this interaction that one can trace the origin of the drag and lift exerted by the gas on the body and the heat transfer between the gas and the solid boundary. Hence, in order to write down the correct boundary conditions for the Boltzmann equation we need information that stems from a discipline that may be regarded as a bridge between the kinetic theory of gases and solid-state physics.

Validity analysis, existence and uniqueness theorems, and qualitative results on the behavior of the solutions are certainly central to the understanding of rarefied gases. For real physical situations, however, like the flow pattern around an object that moves inside a rarefied gas, we need methods to actually calculate or approximate solutions of the Boltzmann equation. For most situations, it is hopeless to even look for explicit solutions of the Boltzmann equation. On the other hand, the five-dimensional integral in the collision operator makes numerical approximations a difficult topic. Specifically, recall that Suppose we want to approximate the collision integral by a quadrature formula that requires the evaluation of the integrand at a number of points. Obviously, the integrand must decay fast enough at infinity to give us reasonable accuracy with a finite number of evaluation points.

In Chapter 3 Section 8 we discussed the hydrodynamical limit for the Boltzmann equation in general terms and showed how a pure space-time scaling leads to the asymptotic limit ∈ → 0 of solutions of the Boltzmann equations

The first eleven chapters of this book comprise a collection of much of what we (the authors) know about the Boltzmann equation for hard spheres. In this last chapter, we want to revisit some of the questions addressed in the earlier chapters and discuss some possible further developments.