Small Viscosity and Boundary Layer Methods

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. The main goal of this presentation is to provide basic tools for the understanding of multidimensional boundary layers for systems. Included are self-contained introductions to different topics such as hyperbolic boundary value problems, parabolic systems, BKW methods, construction of profiles, introduction to the theory of Evansa?? functions, and energy methods with Kreissa?? symmetrizers. Part I is devoted to linear and semilinear problems. For simplicity, the analysis restricts its attention to constant coefficients of systemic dissipative systems. An important feature of this section is the derivation of energy estimates independent of viscosity. Part II is a treatment of quasilinear problems; the equation that governs the rapid variation inside the layer is derived and subsequently studied, allowing for the examination of multidimensional stability of planar layers. 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.

Preface

vii

List of Symbols

xix

Part I Semilinear Layers

1 Introduction and Example

3

(18)

1.1 The equation

3

(1)

1.2 Transport equation

4

(2)

1.2.1 The Cauchy problem

4

(1)

1.2.2 The mixed Cauchy boundary value problems

4

(1)

1.2.3 The boundary value problem

5

(1)

1.3 Viscous perturbation: Existence of layers

6

(2)

1.3.1 The Cauchy problem

6

(1)

1.3.2 The mixed Cauchy problem

7

(1)

1.4 WKB expansions

8

(4)

1.5 Laplace Fourier transform

12

(9)

1.5.1 Definitions

12

(2)

1.5.2 Green's functions

14

(2)

1.5.3 The inviscid limit: Layers

16

(1)

1.5.4 Estimates

17

(3)

1.5.5 Solutions of the boundary value problem

20

(1)

2 Hyperbolic Mixed Problems

21

(30)

2.1 The equations

21

(1)

2.2 Hyperbolic boundary value problems

22

(9)

2.2.1 The adjoint problem

23

(5)

2.2.2 Energy estimates and existence of weak solutions

28

(1)

2.2.3 Strong solutions

29

(1)

2.2.4 Regularity of solutions

30

(1)

2.2.5 Solutions of the boundary value problem (2.2.1)

31

(1)

2.3 Solutions on ] - infinity, T] and the causality principle

31

(3)

2.4 The mixed Cauchy problem

34

(9)

2.4.1 L2 solutions

34

(3)

2.4.2 Compatibility conditions

37

(4)

2.4.3 Smooth solutions

41

(2)

2.5 Nonlinear mixed problems

43

(8)

2.5.1 Nonlinear estimates

44

(1)

2.5.2 Compatibility conditions

45

(1)

2.5.3 Existence and uniqueness

46

(2)

2.5.4 A criterion for blow-up

48

(2)

2.5.5 Regularity of solutions

50

(1)

3 Hyperbolic-Parabolic Problems

51

(28)

3.1 The equations

51

(1)

3.2 Linear existence theory

52

(11)

3.2.1 Variational methods

52

(4)

3.2.2 Regularity estimates

56

(3)

3.2.3 Smooth solutions

59

(4)

3.3 Uniform estimates

63

(4)

3.3.1 Long time estimates

63

(3)

3.3.2 Small viscosity estimates

66

(1)

3.4 Tangential and conormal estimates

67

(6)

3.4.1 Tangential regularity

67

(2)

3.4.2 Conormal regularity

69

(4)

3.5 Nonlinear problems

73

(6)

3.5.1 Existence for fixed viscosity

73

(2)

3.5.2 Uniform existence theorem I

75

(2)

3.5.3 Uniform existence theorem II

77

(2)

4 Semilinear Boundary Layers

79

(18)

4.1 Statement of the problem

79

(1)

4.2 Asymptotic boundary layers

80

(3)

4.3 The boundary layer ODE and the hyperbolic boundary conditions

83

(3)

4.3.1 The inner layer ODE

83

(1)

4.3.2 Layers profiles

84

(1)

4.3.3 The hyperbolic boundary conditions

85

(1)

4.4 Solving the WKB equations

86

(2)

4.4.1 The leading term

86

(1)

4.4.2 The full expansion

87

(1)

4.5 Convergence and approximation theorems

88

(9)

4.5.1 Approximate solutions

88

(1)

4.5.2 An equation for the remainder

89

(3)

4.5.3 Exact solutions and their asymptotic expansions

92

(5)

Part II Quasilinear Layers

5 Quasilinear Boundary Layers: The Inner Layer ODE

97

(16)

5.1 The equations

97

(2)

5.2 The inner-layer ODE and the hyperbolic boundary conditions

99

(2)

5.2.1 Example: The Burgers equation

100

(1)

5.2.2 Example: The linear case

101

(1)

5.3 Solutions of the inner layer ODE

101

(5)

5.4 Smooth hyperbolic boundary conditions

106

(2)

5.4.1 Small amplitude layers

107

(1)

5.4.2 Large amplitude layers

107

(1)

5.5 The linearized profile equation

108

(5)

5.5.1 Conjugation to constant coefficients

109

(2)

5.5.2 Transversality and the tangent space to C

111

(2)

6 Plane Wave Stability

113

(18)

6.1 Statement of the problem

113

(2)

6.2 Necessary conditions

115

(4)

6.2.1 General discussion

115

(1)

6.2.2 The hyperbolic case: The Lopatinski determinant

116

(3)

6.3 Evans functions

119

(5)

6.3.1 Reduction to first order and rescaling

119

(1)

6.3.2 Spectral analysis of the symbol

120

(2)

6.3.3 Conjugation to constant coefficients

122

(1)

6.3.4 Stability conditions

123

(1)

6.4 Low frequency analysis of the Evans condition

124

(7)

6.4.1 Detailed spectral analysis of G infinity

124

(2)

6.4.2 Proof of Theorem 6.4.1: Necessary conditions

126

(3)

6.4.3 Proof of Theorem 6.4.1: Sufficient conditions

129

(2)

7 Stability Estimates

131

(18)

7.1 The estimates

131

(2)

7.2 The method of symmetrizers

133

(3)

7.3 High frequencies

136

(4)

7.4 Medium frequencies

140

(1)

7.5 Low frequencies

141

(5)

7.6 Proof of the L2 estimates

146

(3)

8 Kreiss Symmetrizers for Hyperbolic-Parabolic Systems

149

(30)

8.1 Scheme of the construction

149

(4)

8.1.1 Notation

149

(2)

8.1.2 Block reduction

151

(1)

8.1.3 Elliptic modes

152

(1)

8.2 Hyperbolic modes

153

(3)

8.2.1 Preliminaries

153

(1)

8.2.2 Symmetrizers for hyperbolic modes

154

(2)

8.3 The block structure property

156

(12)

8.3.1 The hyperbolic case

157

(6)

8.3.2 The generalized block structure condition

163

(5)

8.4 Construction of symmetrizers near glancing modes

168

(11)

8.4.1 Examples

168

(3)

8.4.2 Proof of the main lemmas

171

(3)

8.4.3 Proof of Proposition 8.1.6 near glancing modes

174

(5)

9 Linear and Nonlinear Stability of Quasilinear Boundary Layers

179

(10)

9.1 Assumptions

179

(2)

9.2 Linear stability

181

(5)

9.2.1 L2 stability

182

(3)

9.2.2 Conormal stability

185

(1)

9.3 Nonlinear stability

186

(3)

References

189

(4)

Index

193

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Small Viscosity and Boundary Layer Methods

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