Small Viscosity and Boundary Layer Methods: Theory, Stability Analysis, and Applications / Edition 1

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This book is an introduction to the stability analysis of noncharacteristic boundary layers, emphasizing selected topics and developing mathematical tools relevant to the study of multidimensional problems. Boundary layers are present in problems from physics, engineering, mechanics, and fluid mechanics and typically appear for problems with small diffusion. Boundary layers also occur in free boundary value problems, particularly in the analysis of shock waves. The main goal of this presentation is to provide basic tools for the understanding of multidimensional boundary layers for systems. Included are self-contained introductions to different topics such as hyperbolic boundary value problems, parabolic systems, WKB methods, construction of profiles, introduction to the theory of Evans' functions, and energy methods with Kreiss' symmetrizers. Part I is devoted to linear and semilinear problems. For simplicity, the analysis restricts its attention to constant coefficients of systemic dissipative systems. An important feature of this section is the derivation of energy estimates independent of viscosity. Part II is a treatment of quasilinear problems; the equation that governs the rapid variation inside the layer is derived and subsequently studied, allowing for the examination of multidimensional stability of planar layers. This monograph is a valuable text for researchers, practitioners, and graduate students in applied mathematics, mathematical physics, and engineering and will be a useful supplement for the study of mathematical models in the applied sciences. Prerequisites for the reader include standard courses in analysis, integration theory, and PDEs.

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Editorial Reviews

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"The main aim of this book is to provide a self-contained introduction to the topic together with a large exposition of the recent results … . The book is very well written with an interesting level of difficulty which makes it easy to read. It is recommended to everyone interested in this area, beginners and specialists, since it starts with a good introduction and the presentation of rather simple results and finishes with a nice exposition of current research." (Frédéric Rousset, Mathematical Reviews, Issue 2007 b)

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Product Details

Table of Contents

List of Symbols
Pt. I Semilinear Layers
1 Introduction and Example 3
2 Hyperbolic Mixed Problems 21
3 Hyperbolic-Parabolic Problems 51
4 Semilinear Boundary Layers 79
Pt. II Quasilinear Layers
5 Quasilinear Boundary Layers: The Inner Layer ODE 97
6 Plane Wave Stability 113
7 Stability Estimates 131
8 Kreiss Symmetrizers for Hyperbolic-Parabolic Systems 149
9 Linear and Nonlinear Stability of Quasilinear Boundary Layers 179
References 189
Index 193
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