عنوان

Singular perturbations and boundary layers /

3030006387

9783030006389

3030006379

9783030006372

Singular perturbations and boundary layers /

[Book]

Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam.

Cham, Switzerland :

Springer,

[2018]

©2018

1 online resource

Applied mathematical sciences,

volume 200

2196-968X ;

Includes bibliographical references and index.

Intro; Preface; Contents; List of Symbols and Abbreviations; 1 Singular Perturbations in Dimension One; 1.1 Introduction; 1.2 Regular and Singular Perturbations; 1.3 Reaction-Diffusion Equations in 1D; 1.3.1 Convergence by Energy Methods; 1.3.2 Thickness of the Boundary Layer and the Boundary Layer Correctors; 1.3.3 Inner and Outer Expansions: The Higher Orders; 1.3.4 Higher Order Regularity and Convergence; 1.4 Convection-Diffusion Equations in 1D; 1.4.1 Asymptotic Expansions at Order n, n ≥0; 1.4.2 Higher Order Regularity and Convergence; 1.4.3 Problem with a Variable Coefficient b(x)

2 Singular Perturbations in Higher Dimensions in a Channel2.1 Introduction; 2.2 Reaction-Diffusion Equations in a Channel: Ordinary Boundary Layers; 2.2.1 Energy Method; 2.2.2 Boundary Layer Analysis; 2.2.3 Outer and Inner Expansions; 2.2.4 Some Lemmas; 2.2.5 Outer and Inner Expansions (Continued); 2.2.6 Higher Order Regularity and Convergence; 2.3 Convection-Diffusion Equations in a Channel: Parabolic Boundary Layers; 2.3.1 Convection-Diffusion Equations in Higher Dimensions; 2.3.2 Introduction of the Parabolic Boundary Layers (PBL); 2.3.3 Outer Expansions; 2.3.4 PBL at Order 0: 0,

2.3.5 Inner Expansions2.3.5.1 PBL at Order j: j, j e"; 2.3.5.2 Estimates on the PBLs; 2.3.6 The Approximation Results; 2.3.7 Higher Order Regularity and Convergence; 3 Boundary Layers in a Curved Domain in Rd, d = 2, 3; 3.1 Elements of Differential Geometry; 3.1.1 A Curvilinear Coordinate System Adapted to the Boundary; 3.1.2 Examples of the Curvilinear System for Some Special Geometries; 3.2 Reaction-Diffusion Equations in a Curved Domain; 3.2.1 Boundary Layer Analysis at Order 0; 3.2.2 Boundary Layer Analysis at Order 1/2: The Effect of the Curvature

3.2.3 Asymptotic Expansions at Arbitrary Orders n and n+1/2, n ≥03.3 Parabolic Equations in a Curved Domain; 3.3.1 Boundary Layer Analysis at Orders 0 and 1/2; 3.3.2 Boundary Layer Analysis at Arbitrary Orders n and n+1/2, n ≥0; 3.3.3 Analysis of the Initial Layer: The Case of Ill-Prepared Initial Data; 4 Corner Layers and Turning Points for Convection-Diffusion Equations; 4.1 Convection-Diffusion Equations in a Rectangular Domain; 4.1.1 The Zeroth Order 0; 4.1.1.1 Parabolic Boundary Layers (PBL); 4.1.1.2 Ordinary Boundary Layers (OBL); 4.1.1.3 Ordinary Corner Layers (OCL)

4.1.1.4 Convergence Theorem4.1.2 The Higher Orders n, n≥1; 4.1.2.1 Parabolic Boundary Layers (PBL) Near y=0; 4.1.2.2 Elliptic Boundary Layers (EBL) Near y = 0 and x=1; 4.1.2.3 Ordinary Boundary Layers (OBL) Near x=0; 4.1.2.4 Ordinary Corner Layers (OCL) Near y=0 and x=0; 4.1.2.5 Elliptic Corner Layers (ECL) Near y=0 and x=0; 4.1.2.6 Convergence Theorem; 4.2 Convection-Diffusion Equations in a Bounded Interval with a Turning Point; 4.2.1 The Outer Expansion; 4.2.2 Definition of the Correctors at All Orders; 4.2.3 The Case of f, b Compatible; 4.2.4 The Case of f, b Noncompatible

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"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"--Print version, page 4 of cover.

Springer Nature

com.springer.onix.9783030006389

Singular perturbations and boundary layers.

9783030006372

Boundary layer.

Differential equations, Partial.

Singular perturbations (Mathematics)

Boundary layer.

Differential calculus & equations.

Differential equations, Partial.

Functional analysis & transforms.

MATHEMATICS-- Essays.

MATHEMATICS-- Pre-Calculus.

MATHEMATICS-- Reference.

Singular perturbations (Mathematics)

MAT-- 023000

MAT-- 026000

MAT-- 039000

PBKF

PBKF

Gie, Gung-Min

Hamouda, Makram,1973-

Jung, Chang-Yeol

Temam, Roger

20200823072023.0

pn

[Book]

Y