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Über dieses Buch

This monograph is dedicated to the derivation and analysis of fluid models occurring in plasma physics. It focuses on models involving quasi-neutrality approximation, problems related to laser propagation in a plasma, and coupling plasma waves and electromagnetic waves. Applied mathematicians will find a stimulating introduction to the world of plasma physics and a few open problems that are mathematically rich. Physicists who may be overwhelmed by the abundance of models and uncertain of their underlying assumptions will find basic mathematical properties of the related systems of partial differential equations. A planned second volume will be devoted to kinetic models.

First and foremost, this book mathematically derives certain common fluid models from more general models. Although some of these derivations may be well known to physicists, it is important to highlight the assumptions underlying the derivations and to realize that some seemingly simple approximations turn out to be more complicated than they look. Such approximations are justified using asymptotic analysis wherever possible. Furthermore, efficient simulations of multi-dimensional models require precise statements of the related systems of partial differential equations along with appropriate boundary conditions. Some mathematical properties of these systems are presented which offer hints to those using numerical methods, although numerics is not the primary focus of the book.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction: Some Plasma Characteristic Quantities

Abstract
The first chapter is devoted to a heuristic presentation of some basic concepts in plasma physics and the definition of some plasma characteristic quantities.
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Chapter 2. Quasi-Neutrality and Magneto-Hydrodynamics

Abstract
In this chapter, we justify firstly the massless-electron approximation from the general ion–electron electrodynamic model. Secondly, we present the quasi-neutrality approximation, which is the heart of most of the fluid models presented in this book; this approximation is rigorously proved by an asymptotic analysis where a small parameter related to the Debye length goes to zero. We then present the two-temperature Euler system which is the basic model for quasi-neutral plasmas; in this framework we deal also with thermal conduction and radiative coupling. Lastly, we introduce the well-known model called electron magneto-hydrodynamics (MHD) which is the fundamental model for all magnetized plasmas. We give some details about the related boundary conditions.Some crucial mathematical properties related to the “ideal part” of the previous models are displayed at the end of this chapter.
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Chapter 3. Laser Propagation: Coupling with Ion Acoustic Waves

Abstract
This chapter contains most of fundamental concepts for the laser–plasma interaction. We first derive the paraxial approximation for the laser propagation from the full Maxwell equations. This is done by using a time envelope model and performing the Wenzel–Kramer–Brillouin (WKB) expansion. By the way, we compare the geometrical optics approximation and the paraxial approximation. In the second part of this chapter, we focus on the modelling of the Brillouin instability which corresponds to a coupling of the laser waves and an ion acoustic wave. This leads to the so-called three-wave coupling system which was introduced 40 years ago by Kadomstev. We give some crucial mathematical properties of this system, which are new according to our knowledge and which enable a better understanding of the structure of the three-wave coupling system.
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Chapter 4. Langmuir Waves and Zakharov Equations

Abstract
Here we address for the sake of completeness the modelling of the electron plasma waves, also called Langmuir waves. We recall how the coupling of these waves with the ion population leads to the system of Zakharov equations and the different approximations that are made for this derivation.
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Chapter 5. Coupling Electron Waves and Laser Waves

Abstract
In this chapter, we go back to laser–plasma interaction by addressing the coupling of the laser waves with the electron plasma waves. So we derive the so-called Raman instability model. In the case of the fixed-ion assumption, it leads to a three-wave coupling system that shows the same structure as the system of Brillouin instability. In the second part of this chapter, we deal with the modelling of the interaction of an ultra-intense laser pulse and a plasma. This leads to the so-called Euler–Maxwell system. We give some mathematical properties of this system and we show how an envelope description may be useful in some cases.
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Chapter 6. Models with Several Species

Abstract
The first part of this chapter is devoted to the modelling of hot plasmas with different species of ions in the framework of the quasi-neutrality approximation. Our aim is to show how to derive models describing the averaged ion fluid and its coupling with the electron temperature equation. In the second part, we are concerned by a different framework: the weakly ionized plasmas. We show how the quasi-neutrality approximation works (in the case where the Debye length is small enough) and we justify the so-called ambipolar diffusion approximation.
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Backmatter

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