##
**The mathematical theory of dilute gases.**
*(English)*
Zbl 0813.76001

Applied Mathematical Sciences. 106. New York, NY: Springer-Verlag. vii, 347 p. (1994).

This book provides a general overview of the mathematical results on the Boltzmann equation. The classical results are presented as well as a lot of developments that have been published during the last twenty years.

The Boltzmann equation describes the evolution of the distribution function \(f(t,x,\xi)\) of a dilute gas with collisions \({{\partial f} \over {\partial t}} +\xi\cdot \nabla_ x f= Q(f,f)\). Here \(f\) is a probability density defined on the one-particle phase space (\(t\), \(x\) and \(\xi\) respectively represent the time, position and velocity of a particle). For hard-sphere collisions, the collision operator is proportional to

\[ Q(f,f) = \int_{\mathbb{R}^3} \int_{n\in S^2, n\cdot\xi_*>0} (f' f_*^ \prime- ff_*) |(\xi- \xi_*)\cdot n| \,d\xi_* \,dn, \]

where \(f'\), \(f'_*\) and \(f_*\) respectively stand for \(f(t,x, \xi')\), \(f(t,x, \xi'_*)\) and \(f(t,x, \xi_*)\), and where \(\xi' = \xi- n[n\cdot (\xi- \xi_*)]\), \(\xi'_* = \xi+ n[n\cdot (\xi- \xi_ *)]\).

Most of the results of the book are concerned with the hard-sphere case. When the particles interact through a more general two-bodies potential, the collision operator takes the form

\[ Q(f,f)= \alpha\int_{ \mathbb{R}^3} \int_{n\in S^2} q(\xi- \xi_*,n) (f' f'_*- ff_*) \,d\xi_* \,dn, \]

where \(q\) is a (nonnegative) cross-section.

The first chapter contains a brief historical introduction. Chapter 2 provides an informal derivation of the Boltzmann equation and introduces the BBGKY hierarchy. Elementary properties of the solutions such as invariants or the \(H\)-theorem are presented in chapter 3. Chapter 4 is concerned with a rigorous and detailed derivation of the Boltzmann equation from the BBGKY hierarchy in the hard-sphere case. Lanford’s result is precised, according to the work of two of the authors. The question of irreversibility is also examined in details.

Existence results are the subject of chapter 5. These results are of two kinds: the first one is deduced from the dynamical approach (BBGKY hierarchy) and provides local existence results and uniqueness. The second one is given by the theorem proved in 1988 by DiPerna and Lions, which ensures the global existence, but leaves the problem of uniqueness open. Chapter 6 is concerned with the existence and uniqueness theory for the spatially homogeneous Boltzmann equation.

Chapter 7 is devoted to the study of perturbations of equilibria and space homogeneous solutions. Most of the work is done here in the case of the linearized Boltzmann equation. This chapter contains among many properties and extensions the proof of asymptotic stability of Maxwellian equilibria due to Ukai and Asano. Chapter 8 and chapter 9 contain a discussion of boundary conditions. The first of these chapters deals with the modelisation. The second is concerned with the rigorous results on the important problem of the initial-boundary value problem, including the results of Hamdache and many other people. A brief coverage of the simulations techniques is given in chapter 10 for simple situations. This approach does not pretend to be exhaustive. Chapter 11 is devoted to the presentation of hydrodynamical limits. The Hilbert expansion and the very few mathematical results on that subject are briefly discussed. The book ends with a list of fundamental open problems in the theory of the Boltzmann equation.

The main interest of this book is the fact that it provides a unified presentation of the mathematical results known up to now. It is a very useful guide to the existing literature. The most important results of the domain are, in general, given with at least a sketch of the proof. No book had been published on the subject for years, while the domain was developing a lot: there was a real need of it. For all these reasons, this work should become a reference book.

The Boltzmann equation describes the evolution of the distribution function \(f(t,x,\xi)\) of a dilute gas with collisions \({{\partial f} \over {\partial t}} +\xi\cdot \nabla_ x f= Q(f,f)\). Here \(f\) is a probability density defined on the one-particle phase space (\(t\), \(x\) and \(\xi\) respectively represent the time, position and velocity of a particle). For hard-sphere collisions, the collision operator is proportional to

\[ Q(f,f) = \int_{\mathbb{R}^3} \int_{n\in S^2, n\cdot\xi_*>0} (f' f_*^ \prime- ff_*) |(\xi- \xi_*)\cdot n| \,d\xi_* \,dn, \]

where \(f'\), \(f'_*\) and \(f_*\) respectively stand for \(f(t,x, \xi')\), \(f(t,x, \xi'_*)\) and \(f(t,x, \xi_*)\), and where \(\xi' = \xi- n[n\cdot (\xi- \xi_*)]\), \(\xi'_* = \xi+ n[n\cdot (\xi- \xi_ *)]\).

Most of the results of the book are concerned with the hard-sphere case. When the particles interact through a more general two-bodies potential, the collision operator takes the form

\[ Q(f,f)= \alpha\int_{ \mathbb{R}^3} \int_{n\in S^2} q(\xi- \xi_*,n) (f' f'_*- ff_*) \,d\xi_* \,dn, \]

where \(q\) is a (nonnegative) cross-section.

The first chapter contains a brief historical introduction. Chapter 2 provides an informal derivation of the Boltzmann equation and introduces the BBGKY hierarchy. Elementary properties of the solutions such as invariants or the \(H\)-theorem are presented in chapter 3. Chapter 4 is concerned with a rigorous and detailed derivation of the Boltzmann equation from the BBGKY hierarchy in the hard-sphere case. Lanford’s result is precised, according to the work of two of the authors. The question of irreversibility is also examined in details.

Existence results are the subject of chapter 5. These results are of two kinds: the first one is deduced from the dynamical approach (BBGKY hierarchy) and provides local existence results and uniqueness. The second one is given by the theorem proved in 1988 by DiPerna and Lions, which ensures the global existence, but leaves the problem of uniqueness open. Chapter 6 is concerned with the existence and uniqueness theory for the spatially homogeneous Boltzmann equation.

Chapter 7 is devoted to the study of perturbations of equilibria and space homogeneous solutions. Most of the work is done here in the case of the linearized Boltzmann equation. This chapter contains among many properties and extensions the proof of asymptotic stability of Maxwellian equilibria due to Ukai and Asano. Chapter 8 and chapter 9 contain a discussion of boundary conditions. The first of these chapters deals with the modelisation. The second is concerned with the rigorous results on the important problem of the initial-boundary value problem, including the results of Hamdache and many other people. A brief coverage of the simulations techniques is given in chapter 10 for simple situations. This approach does not pretend to be exhaustive. Chapter 11 is devoted to the presentation of hydrodynamical limits. The Hilbert expansion and the very few mathematical results on that subject are briefly discussed. The book ends with a list of fundamental open problems in the theory of the Boltzmann equation.

The main interest of this book is the fact that it provides a unified presentation of the mathematical results known up to now. It is a very useful guide to the existing literature. The most important results of the domain are, in general, given with at least a sketch of the proof. No book had been published on the subject for years, while the domain was developing a lot: there was a real need of it. For all these reasons, this work should become a reference book.

Reviewer: Jean Dolbeault (Paris)

### MSC:

35Q20 | Boltzmann equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

82C40 | Kinetic theory of gases in time-dependent statistical mechanics |

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |

82D05 | Statistical mechanics of gases |