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**Quasi-hydrodynamic semiconductor equations.**
*(English)*
Zbl 0969.35001

Progress in Nonlinear Differential Equations and their Applications. 41. Basel: Birkhäuser. x, 293 p. (2001).

The book under review is devoted to the following three different quasi-hydrodynamic models in semiconductor theory: the isentropic drift-diffusion model, the energy-transport model, and the quantum hydrodynamic model. The author presents the (formal) derivation of each of the models and develops both mathematical analysis and numerical approximation for the three classes of models.

Chapter 1 is concerned with the discussion of a hierarchy of kinetic and quasi-hydrodynamic semiconductor equations. The connections between the various {(semi-)} classical and quantum models are explained. Then the three quasi-hydrodynamic models are considered in more detail.

The second chapter presents a short summary of the physics and properties of semiconductors. The author explains the basic notions of the structure of homogeneous and inhomogeneous semiconductor devices.

The main part of the book are the Chapters 3-5. Chapter 3 is devoted to the study of the isentropic drift-diffusion equations. These equations form a degenerate parabolic-elliptic system. It is derived from a drift-diffusion system incorporating general Fermi-Dirac statistics. The author proves results on the existence of transient solutions, the uniqueness of transient solutions, the localization of vacuum solutions and numerical approximations by mixed finite element discretization in one and two space dimensons.

Chapter 4 starts with the derivation of the energy-transport model from the Boltzmann equation by using a Hilbert expansion method. The following discussion is then concerned with a slightly more general energy-transport model. Then a transformation of the variables which symmetrizes the equations, and an entropy function are discussed. Using these tools the author proves results on the existence of transient solutions, long-time behavior of the transient solution, regularity and uniqueness of transient solutions, existence and uniqueness of steady-state solutions and numerical approximation by mixed finite element discretization in one space dimension.

Chapter 5 starts with the (formal) derivation of the quantum hydrodynamic model from the many-particle Schrödinger equation. The discussion is then concerned with existence, positivity and uniqueness of steady-state solutions, a non-existence result, the classical limits, current-voltage characteristics and a positivity-preserving numerical scheme. Most of the results presented in the book, are obtained by the author (resp. in collaboration with co-authors) in recent years. The presentation of the material in this research monograph is clear throughout and contains numerous hints to the literature (378 items).

Chapter 1 is concerned with the discussion of a hierarchy of kinetic and quasi-hydrodynamic semiconductor equations. The connections between the various {(semi-)} classical and quantum models are explained. Then the three quasi-hydrodynamic models are considered in more detail.

The second chapter presents a short summary of the physics and properties of semiconductors. The author explains the basic notions of the structure of homogeneous and inhomogeneous semiconductor devices.

The main part of the book are the Chapters 3-5. Chapter 3 is devoted to the study of the isentropic drift-diffusion equations. These equations form a degenerate parabolic-elliptic system. It is derived from a drift-diffusion system incorporating general Fermi-Dirac statistics. The author proves results on the existence of transient solutions, the uniqueness of transient solutions, the localization of vacuum solutions and numerical approximations by mixed finite element discretization in one and two space dimensons.

Chapter 4 starts with the derivation of the energy-transport model from the Boltzmann equation by using a Hilbert expansion method. The following discussion is then concerned with a slightly more general energy-transport model. Then a transformation of the variables which symmetrizes the equations, and an entropy function are discussed. Using these tools the author proves results on the existence of transient solutions, long-time behavior of the transient solution, regularity and uniqueness of transient solutions, existence and uniqueness of steady-state solutions and numerical approximation by mixed finite element discretization in one space dimension.

Chapter 5 starts with the (formal) derivation of the quantum hydrodynamic model from the many-particle Schrödinger equation. The discussion is then concerned with existence, positivity and uniqueness of steady-state solutions, a non-existence result, the classical limits, current-voltage characteristics and a positivity-preserving numerical scheme. Most of the results presented in the book, are obtained by the author (resp. in collaboration with co-authors) in recent years. The presentation of the material in this research monograph is clear throughout and contains numerous hints to the literature (378 items).

Reviewer: Joachim Naumann (Berlin)

### MSC:

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

35Q99 | Partial differential equations of mathematical physics and other areas of application |

35J60 | Nonlinear elliptic equations |

35K55 | Nonlinear parabolic equations |

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

35A35 | Theoretical approximation in context of PDEs |

35Q60 | PDEs in connection with optics and electromagnetic theory |

82D37 | Statistical mechanics of semiconductors |

81V99 | Applications of quantum theory to specific physical systems |