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Table of Contents
Intro
Preface
Contents
Notation-Related Comments
1 Introduction
1.1 Fluid Mechanics and Pseudo-Monotone Operator Theory: A Never-Ending Love Affair or Merely an On-Off Relationship?
1.2 Electro-Rheological Fluids
1.3 Aims and Outline of This Manuscript
1.3.1 Main Part
1.3.2 Extensions
2 Preliminaries
2.1 Theory of Pseudo-Monotone Operators
2.2 Variable Exponent Spaces
2.2.1 Classical Function Spaces
2.2.2 Variable Lebesgue Spaces
2.2.3 Duality in Variable Lebesgue Spaces
2.2.4 Variable Sobolev Spaces
2.2.5 The Hardy-Littlewood Maximal Operator and log-Hl̲der Continuity
2.2.6 Mollification in Lp(ʺ)(Rn)
2.3 Banach-Valued Function Spaces
2.3.1 Banach-Valued Classical Function Spaces
2.3.2 Bochner-Lebesgue Spaces
2.3.3 Bochner-Sobolev Spaces
2.3.4 Advanced Theory of Pseudo-Monotone Operators for Evolution Equations
Part I Main Part
3 Variable Bochner-Lebesgue Spaces
3.1 The Spaces Xq,p(QT) and Xq,p(QT)
3.2 Duality in Xq,p(QT)
3.3 Embedding Theorems for Xq,p(QT)
3.4 Smoothing in Xq,p(QT)
3.5 The Spaces q,p(QT) and q,p(QT)
5 Existence Theory for Lipschitz Domains
5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner Coercivity
5.2 The Hirano-Landes Approach
5.3 C0-Bochner Pseudo-Monotonicity, C0-Bochner Condition (M) and C0-Bochner Coercivity
5.4 Abstract Existence Theorem for Lipschitz Domains and p-?2
5.5 Application to Model Problems
5.5.1 Unsteady p(ʺ,ʺ)-Stokes Equations in a Lipschitz Domain with p-?2
5.5.2 Unsteady p(ʺ,ʺ)-Navier-Stokes Equations in a Lipschitz Domain with p- ?3d+2d+2
Part II Extensions
6 Pressure Reconstruction
6.1 Pressure Reconstruction
6.2 Application to Model Problems
6.3 Applicability of Parabolic L?- and Lipschitz Truncation
6.3.1 Parabolic L?- and Lipschitz Truncation
6.3.2 Parabolic Solenoidal Lipschitz Truncation
7 Existence Theory for Irregular Domains
7.1 Bochner-Sobolev Condition (M)
7.2 L1-Monotonicity
7.2.1 Finite Radon Measures
7.2.2 Minty-Trick Like Argument for L1-Monotone Operators
7.3 Anisotropic Variable Exponent Bochner-Lebesgue Spaces
7.3.1 The Space Xq,p,s,div(QT)
7.3.2 Duality in Xq,p,s,div(QT)
7.3.3 Smoothing in Xq,p,s,div(QT)
Preface
Contents
Notation-Related Comments
1 Introduction
1.1 Fluid Mechanics and Pseudo-Monotone Operator Theory: A Never-Ending Love Affair or Merely an On-Off Relationship?
1.2 Electro-Rheological Fluids
1.3 Aims and Outline of This Manuscript
1.3.1 Main Part
1.3.2 Extensions
2 Preliminaries
2.1 Theory of Pseudo-Monotone Operators
2.2 Variable Exponent Spaces
2.2.1 Classical Function Spaces
2.2.2 Variable Lebesgue Spaces
2.2.3 Duality in Variable Lebesgue Spaces
2.2.4 Variable Sobolev Spaces
2.2.5 The Hardy-Littlewood Maximal Operator and log-Hl̲der Continuity
2.2.6 Mollification in Lp(ʺ)(Rn)
2.3 Banach-Valued Function Spaces
2.3.1 Banach-Valued Classical Function Spaces
2.3.2 Bochner-Lebesgue Spaces
2.3.3 Bochner-Sobolev Spaces
2.3.4 Advanced Theory of Pseudo-Monotone Operators for Evolution Equations
Part I Main Part
3 Variable Bochner-Lebesgue Spaces
3.1 The Spaces Xq,p(QT) and Xq,p(QT)
3.2 Duality in Xq,p(QT)
3.3 Embedding Theorems for Xq,p(QT)
3.4 Smoothing in Xq,p(QT)
3.5 The Spaces q,p(QT) and q,p(QT)
5 Existence Theory for Lipschitz Domains
5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner Coercivity
5.2 The Hirano-Landes Approach
5.3 C0-Bochner Pseudo-Monotonicity, C0-Bochner Condition (M) and C0-Bochner Coercivity
5.4 Abstract Existence Theorem for Lipschitz Domains and p-?2
5.5 Application to Model Problems
5.5.1 Unsteady p(ʺ,ʺ)-Stokes Equations in a Lipschitz Domain with p-?2
5.5.2 Unsteady p(ʺ,ʺ)-Navier-Stokes Equations in a Lipschitz Domain with p- ?3d+2d+2
Part II Extensions
6 Pressure Reconstruction
6.1 Pressure Reconstruction
6.2 Application to Model Problems
6.3 Applicability of Parabolic L?- and Lipschitz Truncation
6.3.1 Parabolic L?- and Lipschitz Truncation
6.3.2 Parabolic Solenoidal Lipschitz Truncation
7 Existence Theory for Irregular Domains
7.1 Bochner-Sobolev Condition (M)
7.2 L1-Monotonicity
7.2.1 Finite Radon Measures
7.2.2 Minty-Trick Like Argument for L1-Monotone Operators
7.3 Anisotropic Variable Exponent Bochner-Lebesgue Spaces
7.3.1 The Space Xq,p,s,div(QT)
7.3.2 Duality in Xq,p,s,div(QT)
7.3.3 Smoothing in Xq,p,s,div(QT)