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Über dieses Buch

Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view singular perturbations generate in the system under consideration thin layers located often but not always at the boundary of the domains that are called boundary layers or internal layers if the layer is located inside the domain. Important physical phenomena occur in boundary layers. The most common boundary layers appear in fluid mechanics, e.g., the flow of air around an airfoil or a whole airplane, or the flow of air around a car. Also in many instances in geophysical fluid mechanics, like the interface of air and earth, or air and ocean. This self-contained monograph is devoted to the study of certain classes of singular perturbation problems mostly related to thermic, fluid mechanics and optics and where mostly elliptic or parabolic equations in a bounded domain are considered.

This book is a fairly unique resource regarding the rigorous mathematical treatment of boundary layer problems. The explicit methodology developed in this book extends in many different directions the concept of correctors initially introduced by J. L. Lions, and in particular the lower- and higher-order error estimates of asymptotic expansions are obtained in the setting of functional analysis. The review of differential geometry and treatment of boundary layers in a curved domain is an additional strength of this book. In the context of fluid mechanics, the outstanding open problem of the vanishing viscosity limit of the Navier-Stokes equations is investigated in this book and solved for a number of particular, but physically relevant cases.

This book will serve as a unique resource for those studying singular perturbations and boundary layer problems at the advanced graduate level in mathematics or applied mathematics and may be useful for practitioners in other related fields in science and engineering such as aerodynamics, fluid mechanics, geophysical fluid mechanics, acoustics and optics.



Chapter 1. Singular Perturbations in Dimension One

The study of Singular Perturbation Problems (SPP) in dimension one has a great importance since the boundary layer problems are generally one-dimensional problems in the direction normal to the boundary and, as we will see throughout the chapters of this book, many higher dimensional problems (in terms of singular perturbations) will be reduced to solving some Ordinary Differential Equations (ODE) in dimension 1.
Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Chapter 2. Singular Perturbations in Higher Dimensions in a Channel

In this chapter, we will study the extension of the results on singular perturbations to higher dimensions. In dimension d ≥ 2, new problems arise related to the geometry of the domain, and in particular whether the domain is sufficiently regular or it has corners. Even if the domain is smooth, some boundary layers occur which are due to the curvature of the boundary. These issues necessitating some elements of geometry are addressed in Chapters 3 and 5 The case of a domain with corners is particularly complicated and some aspects of it will be studied in Chapter 4; this includes the possible lack of regularity of the inviscid solution and the appearance of the so-called corner boundary layers due to the interaction between the boundary layers that meet at a corner.
Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Chapter 3. Boundary Layers in a Curved Domain in , d = 2, 3

In this chapter, we present some recent progresses, which are based on [GJT16], about the boundary layer analysis in a domain enclosed by a curved boundary.
Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Chapter 4. Corner Layers and Turning Points for Convection-Diffusion Equations

In this chapter and in Chapter 5, we investigate the boundary layers of convection-diffusion equations in space dimension one or two, and discuss additional issues to further develop the analysis performed in the previous Chapters 1 and 2
Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Chapter 5. Convection-Diffusion Equations in a Circular Domain with Characteristic Point Layers

Following the approach introduced in [JT14a, JT11, JT12], we consider in this chapter the convection-diffusion equations in a circular domain where two characteristic points appear. The singular behaviors may occur at these points depending on the behavior of the given data, that is the domain (unit circle D), and f; see (5.1). As explained below, three types of analysis are necessary depending on the structure of f.
Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Chapter 6. The Navier-Stokes Equations in a Periodic Channel

The Navier-Stokes equations appear as a singular perturbation of the Euler equations in which the small parameter ɛ is the viscosity or inverse of the Reynolds number. In many cases the convergence of the solutions of the Navier-Stokes equations to those of the Euler equations remains an outstanding open problem of mathematical physics. The result is not known in the case of the no-slip boundary condition, even in space dimension 2 for which the existence, uniqueness, and regularity of solution for all time is known for both the Navier-Stokes and Euler equations; see, e.g., [Kat84, Kat86, FT79, Tem75, Tem76, Tem01]. Fortunately, and this is the object of Chapters 6 and 7, this problem has been solved in a number of particular situations: special symmetries or boundary conditions other than the no-slip boundary condition.
Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Chapter 7. The Navier-Stokes Equations in a Curved Domain

In fluid dynamics, we often study the flow of liquids and gases inside a region enclosed by a rigid boundary or around such a region. Some interesting applications in this field include analyzing, e.g., the motion of air around airplanes or automobiles to increase the efficiency of motion, the flow of atmosphere and oceans to predict the weather, and the blood flow inside vessels in medicine where the fluid is blood. In all these applications the fluid, such as air, water, or blood, is considered incompressible and its viscosity is usually very small.
Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam


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