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Existence Theory for Generalized Newtonian Fluids

Book Description

Existence Theory for Generalized Newtonian Fluids provides a rigorous mathematical treatment of the existence of weak solutions to generalized Navier-Stokes equations modeling Non-Newtonian fluid flows. The book presents classical results, developments over the last 50 years of research, and recent results with proofs.

  • Provides the state-of-the-art of the mathematical theory of Generalized Newtonian fluids
  • Combines elliptic, parabolic and stochastic problems within existence theory under one umbrella
  • Focuses on the construction of the solenoidal Lipschitz truncation, thus enabling readers to apply it to mathematical research
  • Approaches stochastic PDEs with a perspective uniquely suitable for analysis, providing an introduction to Galerkin method for SPDEs and tools for compactness

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. Preface
  7. Acknowledgment
    1. References
  8. Notation
  9. Part 1: Stationary problems
    1. Chapter 1: Preliminaries
      1. Abstract
      2. 1.1. Lebesgue & Sobolev spaces
      3. 1.2. Orlicz spaces
      4. 1.3. Basics on Lipschitz truncation
      5. 1.4. Existence results for power law fluids
      6. References
    2. Chapter 2: Fluid mechanics & Orlicz spaces
      1. Abstract
      2. 2.1. Bogovskiĭ operator
      3. 2.2. Negative norms & the pressure
      4. 2.3. Sharp conditions for Korn-type inequalities
      5. References
    3. Chapter 3: Solenoidal Lipschitz truncation
      1. Abstract
      2. 3.1. Solenoidal truncation – stationary case
      3. 3.2. Solenoidal Lipschitz truncation in 2D
      4. 3.3. A-Stokes approximation – stationary case
      5. References
    4. Chapter 4: Prandtl–Eyring fluids
      1. Abstract
      2. 4.1. The approximated system
      3. 4.2. Stationary flows
      4. References
  10. Part 2: Non-stationary problems
    1. Chapter 5: Preliminaries
      1. Abstract
      2. 5.1. Bochner spaces
      3. 5.2. Basics on parabolic Lipschitz truncation
      4. 5.3. Existence results for power law fluids
      5. References
    2. Chapter 6: Solenoidal Lipschitz truncation
      1. Abstract
      2. 6.1. Solenoidal truncation – evolutionary case
      3. 6.2. A-Stokes approximation – evolutionary case
      4. References
    3. Chapter 7: Power law fluids
      1. Abstract
      2. 7.1. The approximated system
      3. 7.2. Non-stationary flows
      4. References
  11. Part 3: Stochastic problems
    1. Chapter 8: Preliminaries
      1. Abstract
      2. 8.1. Stochastic processes
      3. 8.2. Stochastic integration
      4. 8.3. Itô's Lemma
      5. 8.4. Stochastic ODEs
      6. References
    2. Chapter 9: Stochastic PDEs
      1. Abstract
      2. 9.1. Stochastic analysis in infinite dimensions
      3. 9.2. Stochastic heat equation
      4. 9.3. Tools for compactness
      5. References
    3. Chapter 10: Stochastic power law fluids
      1. Abstract
      2. 10.1. Pressure decomposition
      3. 10.2. The approximated system
      4. 10.3. Non-stationary flows
      5. References
  12. Appendix A: Function spaces
    1. A.1. Function spaces involving the divergence
    2. A.2. Function spaces involving symmetric gradients
    3. References
  13. Appendix B: The A-Stokes system
    1. B.1. The stationary problem
    2. B.2. The non-stationary problem
    3. B.3. The non-stationary problem in divergence form
    4. References
  14. Appendix C: Itô's formula in infinite dimensions
    1. References
  15. References
  16. Index