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

• Record Nr.: UNINA9910739446303321
• Autore: Kaltenbach Alex
• Pubbl/distr/stampa: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2023
• ISBN: 3-031-29670-2
• Edizione: [1st ed. 2023.]
• Descrizione fisica: 1 online resource (364 pages)
• Collana: Lecture Notes in Mathematics, , 1617-9692 ; ; 2329
• Disciplina: 515.353
• Soggetti: Functional analysis; Fluid mechanics; Operator theory; Differential equations; Functional Analysis; Engineering Fluid Dynamics; Operator Theory; Differential Equations
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: 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- &gt -- 3d+2d+2 -- 8 Existence Theory for p-&lt -- 2 -- 8.1 Second Parabolic Compensated Compactness Principle -- 8.2 Unsteady p(·,·)-Navier-Stokes Equations in an Irregular Domain with p-&gt -- 3dd+2 -- 8.3 Unsteady p(·,·)-Stokes Equations in an Irregular Domain with p-&gt -- 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.
• Sommario/riassunto: 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 andnon-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.