Modeling of collisions. Summer school, Saint-Malo, France, September 1994.

*(English)*Zbl 0924.76002
Series in Applied Mathematics (Paris). 2. Paris: Gauthier-Villars/ North-Holland. 222 p. (1998).

[The articles of this volume will not be indexed individually.]

This book consists of three parts, in which three (groups of) authors give introductions to special parts of kinetic theory and summaries of recent original results.

The first part, written by A. Decoster, gives an introduction to the hydrodynamics of plasmas. Transport equations for plasmas consisting of several ion species are considered, and hydrodynamic equations for such systems are derived by perturbation analysis around local thermodynamical equilibria. This is done in a two-step process: first, a set of multi-fluid equations for various ion species is set up and closed by equilibrium assumptions. Then one- and two-temperature hydrodynamic limit equations are derived under the assumption of a unique mean velocity. Formulas for collision frequencies between Maxwellian distributions based on the Fokker-Planck-Landau collision term are given and interpreted. Small perturbations from the local thermodynamical equilibria are then governed by simpler plasma transport equations, whose analysis and solution yields transport coefficients, e.g., the temperature equalization rate. Many examples are given. The chapter is a very solid review of the vast literature on this subject.

Part 2 of the book, authored by B. Perthame and containing also contributions by L. Desvillettes, contains a concise yet interesting introduction into the modern theory of the Boltzmann equation, including the discussion of classical properties, a Fourier transformation based proof of the uniqueness of Maxwellian equilibria, and a statement on the global existence of renormalized solutions. A discussion of derivations of kinetic equations from hierarchies (including the Boltzmann-Grad limit) is also given. The most original part of this section is the material on other collision models, such as the generalization to polyatomic gases, the Enskog equation, the case of inelastic collisions and the inclusion of rotational degrees of freedom. The Euler limits in the polyatomic and Enskog scenarios are also discussed.

Finally, in the third part of the book, I. Gasser, P. Markowich and A. Unterreiter present a selection of results on quantum hydrodynamics (QHD). The QHD equations are first motivated by a suitable ansatz for the (nonlinear) Schrödinger equation, and then obtained for mixed quantum mechanical states as lowest order moment equations of the Wigner equation. The classical limit of the zero temperature QHD equations is discussed, and some care is taken in addressing the possibility of shock formation (and the ensuing problems with weak limits) in the nonlinear conservation laws which arise as limit equations in this situation. Results on stationary quantum hydrodynamics, based on energy minimization, are also given. In the last section of this part, existence and uniqueness of thermal equilibria solutions of the bipolar quantum hydrodynamic equations (i.e., coupled QHD equations for the dependent variables of both electrons and holes) are proved, and semiclassical and zero space charge limits for this scenario are discussed.

Remark: This review is submitted to both the Mathematical Reviews and the Zentralblatt für Mathematik.

This book consists of three parts, in which three (groups of) authors give introductions to special parts of kinetic theory and summaries of recent original results.

The first part, written by A. Decoster, gives an introduction to the hydrodynamics of plasmas. Transport equations for plasmas consisting of several ion species are considered, and hydrodynamic equations for such systems are derived by perturbation analysis around local thermodynamical equilibria. This is done in a two-step process: first, a set of multi-fluid equations for various ion species is set up and closed by equilibrium assumptions. Then one- and two-temperature hydrodynamic limit equations are derived under the assumption of a unique mean velocity. Formulas for collision frequencies between Maxwellian distributions based on the Fokker-Planck-Landau collision term are given and interpreted. Small perturbations from the local thermodynamical equilibria are then governed by simpler plasma transport equations, whose analysis and solution yields transport coefficients, e.g., the temperature equalization rate. Many examples are given. The chapter is a very solid review of the vast literature on this subject.

Part 2 of the book, authored by B. Perthame and containing also contributions by L. Desvillettes, contains a concise yet interesting introduction into the modern theory of the Boltzmann equation, including the discussion of classical properties, a Fourier transformation based proof of the uniqueness of Maxwellian equilibria, and a statement on the global existence of renormalized solutions. A discussion of derivations of kinetic equations from hierarchies (including the Boltzmann-Grad limit) is also given. The most original part of this section is the material on other collision models, such as the generalization to polyatomic gases, the Enskog equation, the case of inelastic collisions and the inclusion of rotational degrees of freedom. The Euler limits in the polyatomic and Enskog scenarios are also discussed.

Finally, in the third part of the book, I. Gasser, P. Markowich and A. Unterreiter present a selection of results on quantum hydrodynamics (QHD). The QHD equations are first motivated by a suitable ansatz for the (nonlinear) Schrödinger equation, and then obtained for mixed quantum mechanical states as lowest order moment equations of the Wigner equation. The classical limit of the zero temperature QHD equations is discussed, and some care is taken in addressing the possibility of shock formation (and the ensuing problems with weak limits) in the nonlinear conservation laws which arise as limit equations in this situation. Results on stationary quantum hydrodynamics, based on energy minimization, are also given. In the last section of this part, existence and uniqueness of thermal equilibria solutions of the bipolar quantum hydrodynamic equations (i.e., coupled QHD equations for the dependent variables of both electrons and holes) are proved, and semiclassical and zero space charge limits for this scenario are discussed.

Remark: This review is submitted to both the Mathematical Reviews and the Zentralblatt für Mathematik.

Reviewer: R.Illner (Victoria)

##### MSC:

76-06 | Proceedings, conferences, collections, etc. pertaining to fluid mechanics |

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

00B15 | Collections of articles of miscellaneous specific interest |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76Mxx | Basic methods in fluid mechanics |

76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |

76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |

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