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Existence Theory for Generalized Newtonian Fluids
1st Edition - March 22, 2017
Author: Dominic Breit
Paperback ISBN:
9 7 8 - 0 - 1 2 - 8 1 1 0 4 4 - 7
eBook ISBN:
9 7 8 - 0 - 1 2 - 8 1 1 0 4 5 - 4
Existence Theory for Generalized Newtonian Fluids provides a rigorous mathematical treatment of the existence of weak solutions to generalized Navier-Stokes equations modeling… Read more
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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
Scientists and graduate students with basic knowledge in nonlinear partial differential equations and interest in mathematical fluid mechanics
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
- No. of pages: 286
- Language: English
- Edition: 1
- Published: March 22, 2017
- Imprint: Academic Press
- Paperback ISBN: 9780128110447
- eBook ISBN: 9780128110454
DB
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