Existence Theory for Generalized Newtonian Fluids - 1st Edition - ISBN: 9780128110447, 9780128110454

Existence Theory for Generalized Newtonian Fluids

1st Edition

Authors: Dominic Breit
eBook ISBN: 9780128110454
Paperback ISBN: 9780128110447
Imprint: Academic Press
Published Date: 23rd March 2017
Page Count: 286
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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.

Key Features

  • 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

Readership

Scientists and graduate students with basic knowledge in nonlinear partial differential equations and interest in mathematical fluid mechanics

Table of Contents

Part 1: Stationary problems

Chapter 1: Preliminaries

  • Abstract
  • 1.1. Lebesgue & Sobolev spaces
  • 1.2. Orlicz spaces
  • 1.3. Basics on Lipschitz truncation
  • 1.4. Existence results for power law fluids
  • References

Chapter 2: Fluid mechanics & Orlicz spaces

  • Abstract
  • 2.1. Bogovskiĭ operator
  • 2.2. Negative norms & the pressure
  • 2.3. Sharp conditions for Korn-type inequalities
  • References

Chapter 3: Solenoidal Lipschitz truncation

  • Abstract
  • 3.1. Solenoidal truncation – stationary case
  • 3.2. Solenoidal Lipschitz truncation in 2D
  • 3.3. A-Stokes approximation – stationary case
  • References

Chapter 4: Prandtl–Eyring fluids

  • Abstract
  • 4.1. The approximated system
  • 4.2. Stationary flows
  • References

Part 2: Non-stationary problems

Chapter 5: Preliminaries

  • Abstract
  • 5.1. Bochner spaces
  • 5.2. Basics on parabolic Lipschitz truncation
  • 5.3. Existence results for power law fluids
  • References

Chapter 6: Solenoidal Lipschitz truncation

  • Abstract
  • 6.1. Solenoidal truncation – evolutionary case
  • 6.2. A-Stokes approximation – evolutionary case
  • References

Chapter 7: Power law fluids

  • Abstract
  • 7.1. The approximated system
  • 7.2. Non-stationary flows
  • References

Part 3: Stochastic problems

Chapter 8: Preliminaries

  • Abstract
  • 8.1. Stochastic processes
  • 8.2. Stochastic integration
  • 8.3. Itô's Lemma
  • 8.4. Stochastic ODEs
  • References

Chapter 9: Stochastic PDEs

  • Abstract
  • 9.1. Stochastic analysis in infinite dimensions
  • 9.2. Stochastic heat equation
  • 9.3. Tools for compactness
  • References

Chapter 10: Stochastic power law fluids

  • Abstract
  • 10.1. Pressure decomposition
  • 10.2. The approximated system
  • 10.3. Non-stationary flows
  • References

Appendix A: Function spaces

  • A.1. Function spaces involving the divergence
  • A.2. Function spaces involving symmetric gradients
  • References

Appendix B: The A-Stokes system

  • B.1. The stationary problem
  • B.2. The non-stationary problem
  • B.3. The non-stationary problem in divergence form
  • References

Appendix C: Itô's formula in infinite dimensions

  • References

Details

No. of pages:
286
Language:
English
Copyright:
© Academic Press 2017
Published:
Imprint:
Academic Press
eBook ISBN:
9780128110454
Paperback ISBN:
9780128110447

About the Author

Dominic Breit

Dominic Breit is currently Assistant Professor in the Department of Mathematics at Heriot Watt University, Edinburgh. In 2009, Breit finished his PhD study at Saarland University in Saarbrücken (Germany). In 2014, he was awarded a price for the best habilitation thesis at LMU Munich for a thesis which is the basis for this book.

Affiliations and Expertise

Heriot Watt University, Edinburgh

Reviews

"The main tools used in the book are related with Sobolev, Lebesgue and Orlicz spaces, with Bogovskii operator and with some special Korn-type inequalities...A large number of proofs and details are given, very useful for those interested in this field." --Zentralblatt MATH

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