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E-raamat: Singular Perturbations and Boundary Layers

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  • Formaat: EPUB+DRM
  • Sari: Applied Mathematical Sciences 200
  • Ilmumisaeg: 21-Nov-2018
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030006389
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Singular Perturbations and Boundary LayersSuurem pilt
  • Formaat: EPUB+DRM
  • Sari: Applied Mathematical Sciences 200
  • Ilmumisaeg: 21-Nov-2018
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030006389
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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.

Arvustused

The book as a helpful guide for the construction of (at least the lower order terms of) boundary layers in a variety of singularly perturbed problems. (Hans Babovsky, Mathematical Reviews, November, 2019) This very nice book concerns boundary layer solutions to singularly perturbed boundary and initial-boundary value problems. (Lutz Recke, zbMATH 1411.35002, 2019)

1 Singular Perturbations in Dimension One 1(30)
1.1 Introduction
1(1)
1.2 Regular and Singular Perturbations
1(4)
1.3 Reaction-Diffusion Equations in 1D
5(10)
1.3.1 Convergence by Energy Methods
5(2)
1.3.2 Thickness of the Boundary Layer and the Boundary Layer Correctors
7(4)
1.3.3 Inner and Outer Expansions: The Higher Orders
11(3)
1.3.4 Higher Order Regularity and Convergence
14(1)
1.4 Convection-Diffusion Equations in 1D
15(16)
1.4.1 Asymptotic Expansions at Order epsilonn, n greater or equal to 0
18(3)
1.4.2 Higher Order Regularity and Convergence
21(1)
1.4.3 Problem with a Variable Coefficient b(x)
21(10)
2 Singular Perturbations in Higher Dimensions in a Channel 31(32)
2.1 Introduction
31(1)
2.2 Reaction-Diffusion Equations in a Channel: Ordinary Boundary Layers
32(16)
2.2.1 Energy Method
33(1)
2.2.2 Boundary Layer Analysis
34(2)
2.2.3 Outer and Inner Expansions
36(2)
2.2.4 Some Lemmas
38(3)
2.2.5 Outer and Inner Expansions (Continued)
41(5)
2.2.6 Higher Order Regularity and Convergence
46(2)
2.3 Convection-Diffusion Equations in a Channel: Parabolic Boundary Layers
48(15)
2.3.1 Convection-Diffusion Equations in Higher Dimensions
48(2)
2.3.2 Introduction of the Parabolic Boundary Layers (PBL)
50(1)
2.3.3 Outer Expansions
50(1)
2.3.4 PBL at Order 0: &phi0,epsilon
51(5)
2.3.5 Inner Expansions
56(3)
2.3.5.1 PBL at Order j:φj,epsilon, j greater or equal to 1
56(2)
2.3.5.2 Estimates on the PBLs
58(1)
2.3.6 The Approximation Results
59(2)
2.3.7 Higher Order Regularity and Convergence
61(2)
3 Boundary Layers in a Curved Domain in Rd, d = 2,3 63(46)
3.1 Elements of Differential Geometry
64(5)
3.1.1 A Curvilinear Coordinate System Adapted to the Boundary
64(3)
3.1.2 Examples of the Curvilinear System for Some Special Geometries
67(2)
3.2 Reaction-Diffusion Equations in a Curved Domain
69(15)
3.2.1 Boundary Layer Analysis at Order epsilon0
70(4)
3.2.2 Boundary Layer Analysis at Order epsilon1/2: The Effect of the Curvature
74(4)
3.2.3 Asymptotic Expansions at Arbitrary Orders epsilonn and epsilonn+1/2, n greater or equal to 0
78(6)
3.3 Parabolic Equations in a Curved Domain
84(25)
3.3.1 Boundary Layer Analysis at Orders epsilon0 and epsilon1/2
85(14)
3.3.2 Boundary Layer Analysis at Arbitrary Orders epsilonn and epsilonn+1/2 , n greater or equal to 0
99(6)
3.3.3 Analysis of the Initial Layer: The Case of Ill-Prepared Initial Data
105(4)
4 Corner Layers and Turning Points for Convection-Diffusion Equations 109(66)
4.1 Convection-Diffusion Equations in a Rectangular Domain
110(27)
4.1.1 The Zeroth Order epsilon0
113(6)
4.1.1.1 Parabolic Boundary Layers (PBL)
113(1)
4.1.1.2 Ordinary Boundary Layers (OBL)
114(1)
4.1.1.3 Ordinary Corner Layers (OCL)
115(2)
4.1.1.4 Convergence Theorem
117(2)
4.1.2 The Higher Orders epsilonn, n greater or equal to 1
119(18)
4.1.2.1 Parabolic Boundary Layers (PBL) Near y = 0
119(7)
4.1.2.2 Elliptic Boundary Layers (EBL) Near y = 0 and x = 1
126(4)
4.1.2.3 Ordinary Boundary Layers (OBL) Near x = 0
130(2)
4.1.2.4 Ordinary Corner Layers (OCL) Near y = 0 and x = 0
132(2)
4.1.2.5 Elliptic Corner Layers (ECL) Near y = 0 and x = 0
134(2)
4.1.2.6 Convergence Theorem
136(1)
4.2 Convection-Diffusion Equations in a Bounded Interval with a Turning Point
137(38)
4.2.1 The Outer Expansion
139(1)
4.2.2 Definition of the Correctors at All Orders
140(3)
4.2.3 The Case of f, b Compatible
143(7)
4.2.4 The Case of f, b Noncompatible
150(25)
5 Convection-Diffusion Equations in a Circular Domain with Characteristic Point Layers 175(76)
5.1 The Compatible Case
177(22)
5.1.1 Compatibility Conditions
177(6)
5.1.2 Boundary Fitted Coordinates
183(1)
5.1.3 The Zeroth Order epsilon0
184(3)
5.1.4 The Higher Orders epsilonn, n greater or equal to 1
187(12)
5.2 The Case of the Generic Taylor Monomials
199(18)
5.2.1 The Zeroth Order epsilon0
206(4)
5.2.2 The Higher Orders epsilonn, n greater or equal to 1
210(7)
5.3 Parabolic Boundary Layers at the Characteristic Points
217(20)
5.3.1 Characteristic Point Layers at (plus - minus 1,0)
218(14)
5.3.2 Convergence Analysis
232(5)
5.4 The General Case
237(14)
5.4.1 The Orders epsilonn, n greater or equal to 0
237(2)
5.4.2 Complements for Order epsilon0
239(12)
6 The Navier-Stokes Equations in a Periodic Channel 251(56)
6.1 The Stokes Equations with the No-Slip Boundary Condition
252(9)
6.1.1 Asymptotic Expansion of the Solutions to the Stokes Problem
255(3)
6.1.2 Estimates on the Corrector
258(1)
6.1.3 Convergence Result
259(2)
6.2 The Navier-Stokes Equations with the Non-characteristic Boundary Condition
261(34)
6.2.1 The Linear Case
263(22)
6.2.1.1 Correctors at Order epsilon0 and epsilon1
264(4)
6.2.1.2 Corrector at Order epsilonN , N greater or equal to 2
268(4)
6.2.1.3 Convergence Result
272(4)
6.2.1.4 Estimates on the Pressure
276(2)
6.2.1.5 Existence of Solution of the Linearized Euler Equations
278(7)
6.2.2 The Nonlinear Case
285(10)
6.2.2.1 Corrector at Order epsilon0
287(1)
6.2.2.2 Corrector at Order epsilon1
288(3)
6.2.2.3 Convergence Result
291(4)
6.3 The Navier-Stokes Equations with the Navier-Friction Boundary Condition
295(12)
6.3.1 The Boundary Layer Corrector
297(2)
6.3.2 Estimates on the Corrector
299(1)
6.3.3 Convergence Results
300(4)
6.3.4 Remark on the Uniform Convergence
304(3)
7 The Navier-Stokes Equations in a Curved Domain 307(80)
7.1 Notations and Differential Geometry
308(2)
7.2 The Stokes Equations
310(16)
7.2.1 Well Posedness of the Limit Problem
312(1)
7.2.2 Asymptotic Expansion at Order epsilon0
313(5)
7.2.3 Estimates on the Corrector
318(3)
7.2.4 Error Analysis and Convergence Result at Order epsilon0
321(2)
7.2.5 Remarks on the Higher Order Expansions
323(3)
7.3 The Navier-Stokes Equations Linearized Around a Stationary Euler Flow
326(16)
7.3.1 Asymptotic Expansion of the Solutions to the LNSE
327(4)
7.3.2 Estimates on the Corrector
331(6)
7.3.3 Convergence Results
337(5)
7.4 The Navier-Stokes Equations with the Non-characteristic Boundary Condition
342(25)
7.4.1 Model Equations with the Homogenized Boundary Conditions and Main Result
345(2)
7.4.2 Asymptotic Expansion at Order epsilon0
347(8)
7.4.2.1 Proof of Theorem 7.4 at Order epsilon0
350(5)
7.4.3 Asymptotic Expansion at Order epsilon1
355(12)
7.4.3.1 Outer Expansion at Order epsilon1
355(1)
7.4.3.2 Inner Expansion at Order epsilon1
356(6)
7.4.3.3 Corrector q1,epsilon of the Pressure pepsilon
362(1)
7.4.3.4 Proof of Theorem 7.4 at Order epsilon1
363(4)
7.5 The Navier-Stokes Equations with the Generalized Navier Boundary Conditions
367(9)
7.5.1 Asymptotic Expansion of uepsilon and the Corrector
370(2)
7.5.2 Error Analysis and Convergence Results
372(4)
7.6 Circularly Symmetric Flows in a Disk Domain
376(11)
7.6.1 Asymptotic Expansion of Vepsilon and Convergence Result
379(2)
7.6.2 Proof of Theorem 7.6
381(6)
A Elements of Functional Analysis 387(8)
A.1 Introduction
387(1)
A.2 Function Spaces
387(3)
A.3 Some Useful Inequalities
390(2)
A.3.1 The Holder Inequality
390(1)
A.3.2 The Poincare Inequality
390(1)
A.3.3 The Gronwall Inequality
390(1)
A.3.4 The Hardy Inequalities
391(1)
A.3.5 The Chebyshev Inequality
391(1)
A.3.6 The Jensen Inequality
391(1)
A.3.7 The Korn Inequality
392(1)
A.3.8 The Agmon Inequalities
392(1)
A.4 Existence Results
392(3)
A.4.1 The Lax-Milgram Theorem
392(1)
A.4.2 The Hille-Yosida Theorem
393(2)
References 395(16)
Index 411
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