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Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents

  • Book
  • © 2023

Overview

Authors:
  1. Alex Kaltenbach

    1. Department of Applied Mathematics, University of Freiburg, Freiburg, Germany

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  • Includes the first proof of the existence of weak solutions of the unsteady p(t,x)-Navier-Stokes equations
  • Provides a comprehensive review of the rapidly expanding field of unsteady problems with variable >exponents
  • Requires only a basic knowledge of functional analysis

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2329)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    • Alex Kaltenbach
    Pages 1-14

  3. Preliminaries

    • Alex Kaltenbach
    Pages 15-39

  4. Main Part

    1. Front Matter

      Pages 41-41
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    2. Variable Bochner–Lebesgue Spaces

      • Alex Kaltenbach
      Pages 43-111

    3. Solenoidal Variable Bochner–Lebesgue Spaces

      • Alex Kaltenbach
      Pages 113-176

    4. Existence Theory for Lipschitz Domains

      • Alex Kaltenbach
      Pages 177-222

  5. Extensions

    1. Front Matter

      Pages 223-223
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    2. Pressure Reconstruction

      • Alex Kaltenbach
      Pages 225-243

    3. Existence Theory for Irregular Domains

      • Alex Kaltenbach
      Pages 245-296

    4. Existence Theory for p− < 2

      • Alex Kaltenbach
      Pages 297-330

    5. Appendix

      • Alex Kaltenbach
      Pages 331-345

  6. Back Matter

    Pages 347-358
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Applied Mathematics, University of Freiburg, Freiburg, Germany

    Alex Kaltenbach

About the author


Bibliographic Information

  • Book Title: Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents

  • Authors: Alex Kaltenbach

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-29670-3

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Softcover ISBN: 978-3-031-29669-7Published: 12 August 2023

  • eBook ISBN: 978-3-031-29670-3Published: 11 August 2023

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XIII, 358

  • Number of Illustrations: 11 illustrations in colour

  • Topics: Functional Analysis, Engineering Fluid Dynamics, Operator Theory, Analysis

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