The first part of this volume presents the basic ideas concerning
perturbation and scaling methods in the mathematical theory of dilute
gases, based on Boltzmann's integro-differential equation. It is of
course impossible to cover the developments of this subject in less than
one hundred pages. Already in 1912 none less than David Hilbert
indicated how to obtain approximate solutions of the scaled Boltzmann
equation in the form of a perturbation of a parameter inversely
proportional to the gas density. His paper is also reprinted as Chapter
XXII of his treatise Grundzuge einer allgemeinen Theorie der linearen
Integralgleichungen. The motive for this circumstance is clearly stated
in the preface to that book ("Recently I have added, to conclude, a new
chapter on the kinetic theory of gases. [ . . . ]. I recognize in the
theory of gases the most splendid application of the theorems concerning
integral equations. ") The mathematically rigorous theory started,
however, in 1933 with a paper [48] by Tage Gillis Torsten Carleman,
who proved a theorem of global exis- tence and uniqueness for a gas of
hard spheres in the so-called space-homogeneous case. Many other results
followed; those based on perturbation and scaling meth- ods will be
dealt with in some detail. Here, I cannot refrain from mentioning that,
when Pierre-Louis Lions obtained the Fields medal (1994), the commenda-
tion quoted explicitly his work with the late Ronald DiPerna on the
existence of solutions of the Boltzmann equation.
