Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents / Alex Kaltenbach.

Author
Kaltenbach, Alex [Browse]
Format
Book
Language
English
Εdition
First edition.
Published/​Created
  • Cham, Switzerland : Springer Nature Switzerland AG, [2023]
  • ©2023
Description
1 online resource (364 pages)

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Subject(s)
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.
ISBN
3-031-29670-2
Doi
  • 10.1007/978-3-031-29670-3
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