Mehr lesen
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]).
Inhaltsverzeichnis
I Semilinear Layers.- 1 Introduction and Example.- 2 Hyperbolic Mixed Problems.- 3 Hyperbolic-Parabolic Problems.- 4 Semilinear Boundary Layers.- II Quasilinear Layers.- 5 Quasilinear Boundary Layers: The Inner Layer ODE.- 6 Plane Wave Stability.- 7 Stability Estimates.- 8 Kreiss Symmetrizers for Hyperbolic-Parabolic Systems.- 9 Linear and Nonlinear Stability of Quasilinear Boundary Layers.- References.
Zusammenfassung
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.
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.
Zusatztext
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)
Bericht
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)
