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Birkhäuser

Small Viscosity and Boundary Layer Methods

Theory, Stability Analysis, and Applications

  • Book
  • © 2004

Overview

Authors:
  1. Guy Métivier

    1. MAB, Université de Bordeaux 1, Talence Cedex, France

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  • Develops mathematical tools relevant to the study of multidimensional problems
  • Useful supplement for the study of mathematical models in the applied sciences

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xxi
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  2. Semilinear Layers

    1. Front Matter

      Pages 1-1
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    2. Introduction and Example

      • Guy Métivier
      Pages 3-20

    3. Hyperbolic Mixed Problems

      • Guy Métivier
      Pages 21-50

    4. Hyperbolic-Parabolic Problems

      • Guy Métivier
      Pages 51-78

    5. Semilinear Boundary Layers

      • Guy Métivier
      Pages 79-93

  3. Quasilinear Layers

    1. Front Matter

      Pages 95-95
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    2. Quasilinear Boundary Layers: The Inner Layer ODE

      • Guy Métivier
      Pages 97-112

    3. Plane Wave Stability

      • Guy Métivier
      Pages 113-129

    4. Stability Estimates

      • Guy Métivier
      Pages 131-147

    5. Kreiss Symmetrizers for Hyperbolic-Parabolic Systems

      • Guy Métivier
      Pages 149-177

    6. Linear and Nonlinear Stability of Quasilinear Boundary Layers

      • Guy Métivier
      Pages 179-187

  4. Back Matter

    Pages 189-194
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Keywords

About this book

This book has evolved from lectures and graduate courses given in Brescia (Italy), Bordeaux and Toulouse (France};' It is intended to serve as an intro­ duction to the stability analysis of noncharacteristic multidimensional small viscosity boundary layers developed in (MZl]. We consider parabolic singular perturbations of hyperbolic systems L(u) - £P(u) = 0, where L is a nonlinear hyperbolic first order system and P a nonlinear spatially elliptic term. The parameter e measures the strength of the diffusive effects. With obvious reference to fluid mechanics, it is referred to as a "viscosity." The equation holds on a domain n and is supplemented by boundary conditions on an.The main goal of this book is to studythe behavior of solutions as etends to O. In the interior of the domain, the diffusive effects are negligible and the nondiffusive or inviscid equations (s = 0) are good approximations. However, the diffusive effects remain important in a small vicinity of the boundary where they induce rapid fluctuations of the solution, called layers. Boundary layers occur in many problems in physics and mechanics. They also occur in free boundary value problems, and in particular in the analysis of shock waves. Indeed, our study of noncharacteristic boundary layers is strongly motivated by the analysis of multidimensional shock waves. At the least, it is a necessary preliminary and important step. We also recall the importance of the viscous approach in the theoretical analysis ofconservation laws (see, e.g., [Lax], (Kru], (Bi-Br]).

Reviews

From the reviews:

"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)

Authors and Affiliations

  • MAB, Université de Bordeaux 1, Talence Cedex, France

    Guy Métivier

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