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

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]).

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)