Details

Subject(s)

• Monotone operators [Browse]
• Navier-Stokes equations [Browse]

Series

• Lecture Notes in Mathematics, 2329 [More in this series]
• Lecture Notes in Mathematics Series ; Volume 2329 [More in this series]

Summary note

This book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(.,.)-Navier–Stokes equations describing the motion of an incompressible electro-rheological fluid. Due to the variable dependence of the power-law index p(.,.) in this system, the classical weak existence analysis based on the pseudo-monotone operator theory in the framework of Bochner–Lebesgue spaces is not applicable. As a substitute for Bochner–Lebesgue spaces, variable Bochner–Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute, the theory of pseudo-monotone operators is extended to unsteady problems with variable exponents, leading to the weak solvability of the unsteady p(.,.)-Navier–Stokes equations under general assumptions. Aimed primarily at graduate readers, the book develops the material step-by-step, starting with the basics of PDE theory and non-linear functional analysis. The concise introductions at the beginning of each chapter, together with illustrative examples, graphics, detailed derivations of all results and a short summary of the functional analytic prerequisites, will ease newcomers into the subject.

Bibliographic references

Includes bibliographical references.

Source of description

Description based on print version record.

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-Hölder Continuity
• 2.2.6 Mollification in Lp(·)(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)
• 3.6 Generalized Time Derivative in q,p(QT)*
• 3.7 Formula of Integration-by-Parts for Wq,p(QT)
• 3.8 Abstract Existence Result for Lipschitz Domains
• 3.9 Application to Model Problem
• 4 Solenoidal Variable Bochner-Lebesgue Spaces
• 4.1 The Spaces Vq,p(QT) and Vq,p(QT)
• 4.2 Duality in Vq,p(QT)
• 4.3 Smoothing in Vq,p(QT)
• 4.3.1 Failure of Smoothing via Bogovskiĭ Correction
• 4.3.2 Smoothing via Transversal Expansion of LipschitzDomains
• 4.4 The Spaces q,p(QT) and q,p(QT)
• 4.5 Generalized Time Derivative in q,p(QT)*
• 4.6 Formula of Integration-by-Parts for Wq,p,σ(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(·,·)-Stokes Equations in a Lipschitz Domain with p-≥2
• 5.5.2 Unsteady p(·,·)-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)
• 7.3.4 Generalized Time Derivative in Xq,p,s,div(QT)* and Formula of Integration-by-Parts for Wq,p,s,div(QT)
• 7.4 First Parabolic Compensated Compactness Principle
• 7.5 Abstract Existence Result for Irregular Domains and p-≥2
• 7.6 Application to Model Problems
• 7.6.1 Unsteady p(·,·)-Stokes Equations in an Irregular Domain with p-≥2
• 7.6.2 Unsteady p(·,·)-Navier-Stokes Equations in an Irregular Domain with p- >
• 3d+2d+2
• 8 Existence Theory for p-<
• 2
• 8.1 Second Parabolic Compensated Compactness Principle
• 8.2 Unsteady p(·,·)-Navier-Stokes Equations in an Irregular Domain with p->
• 3dd+2
• 8.3 Unsteady p(·,·)-Stokes Equations in an Irregular Domain with p->
• 2dd+2
• 9 Appendix
• 9.1 Point-Wise Poincaré Inequality for the Symmetric Gradient
• 9.2 Generalized Lebesgue Differentiation Theorem.
• 9.3 Portemanteau Theorem for Weak-* Convergence
• References.

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ISBN

3-031-29670-2

Doi

• 10.1007/978-3-031-29670-3

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