The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations
An Introduction
Seiten
2022
American Mathematical Society (Verlag)
978-1-4704-7049-4 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-7049-4 (ISBN)
Provides graduate students with exposure to the mathematical analysis of the incompressible Euler and Navier-Stokes equations. The book gives a concise introduction to the fundamental results in the well-posedness theory of these PDEs, leaving aside some of the technical challenges presented by bounded domains or by intricate functional spaces.
The aim of this book is to provide beginning graduate students who completed the first two semesters of graduate-level analysis and PDE courses with a first exposure to the mathematical analysis of the incompressible Euler and Navier-Stokes equations. The book gives a concise introduction to the fundamental results in the well-posedness theory of these PDEs, leaving aside some of the technical challenges presented by bounded domains or by intricate functional spaces.
Chapters 1 and 2 cover the fundamentals of the Euler theory: derivation, Eulerian and Lagrangian perspectives, vorticity, special solutions, existence theory for smooth solutions, and blowup criteria. Chapters 3, 4, and 5 cover the fundamentals of the Navier-Stokes theory: derivation, special solutions, existence theory for strong solutions, Leray theory of weak solutions, weak-strong uniqueness, existence theory of mild solutions, and Prodi-Serrin regularity criteria. Chapter 6 provides a short guide to the must-read topics, including active research directions, for an advanced graduate student working in incompressible fluids. It may be used as a roadmap for a topics course in a subsequent semester. The appendix recalls basic results from real, harmonic, and functional analysis. Each chapter concludes with exercises, making the text suitable for a one-semester graduate course.
Prerequisites to this book are the first two semesters of graduate-level analysis and PDE courses.
The aim of this book is to provide beginning graduate students who completed the first two semesters of graduate-level analysis and PDE courses with a first exposure to the mathematical analysis of the incompressible Euler and Navier-Stokes equations. The book gives a concise introduction to the fundamental results in the well-posedness theory of these PDEs, leaving aside some of the technical challenges presented by bounded domains or by intricate functional spaces.
Chapters 1 and 2 cover the fundamentals of the Euler theory: derivation, Eulerian and Lagrangian perspectives, vorticity, special solutions, existence theory for smooth solutions, and blowup criteria. Chapters 3, 4, and 5 cover the fundamentals of the Navier-Stokes theory: derivation, special solutions, existence theory for strong solutions, Leray theory of weak solutions, weak-strong uniqueness, existence theory of mild solutions, and Prodi-Serrin regularity criteria. Chapter 6 provides a short guide to the must-read topics, including active research directions, for an advanced graduate student working in incompressible fluids. It may be used as a roadmap for a topics course in a subsequent semester. The appendix recalls basic results from real, harmonic, and functional analysis. Each chapter concludes with exercises, making the text suitable for a one-semester graduate course.
Prerequisites to this book are the first two semesters of graduate-level analysis and PDE courses.
Jacob Bedrossian, University of Maryland, College Park, MD. Vlad Vicol, Courant Institute of Mathematical Sciences, New York University, NY.
Ideal incompressible fluids: The Euler equations
Existence of solutions and continuation criteria for Euler
Incompressible viscous fluids: The Navier-Stokes equations
Leray-Hopf weak solutions of Navier-Stokes
Mild solutions of Navier-Stokes
A survey of some advanced topics
Appendix
Bibliography
Index
| Erscheinungsdatum | 03.01.2023 |
|---|---|
| Reihe/Serie | Graduate Studies in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 800 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Naturwissenschaften ► Physik / Astronomie | |
| ISBN-10 | 1-4704-7049-7 / 1470470497 |
| ISBN-13 | 978-1-4704-7049-4 / 9781470470494 |
| Zustand | Neuware |
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