The Mathematical Analysis of the Incompressible Euler and Navier-stokes Equations: An Introduction

Description

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. Cover Title page Contents Preface Chapter 1. Ideal Incompressible Fluids: The Euler Equations 1.1. Eulerian vs Lagrangian representations 1.2. Incompressibility and transport 1.3. The incompressible homogeneous Euler equations 1.4. Vorticity 1.5. Symmetries and conservation laws 1.6. Special explicit solutions 1.7. Exercises Chapter 2. Existence of Solutions and Continuation Criteria for Euler 2.1. Local existence of 𝐻^{𝑠} solutions 2.2. The Lipschitz continuation criterion 2.3. The Beale-Kato-Majda theorem 2.4. The global existence of strong solutions in 2D 2.5. The Constantin-Fefferman-Majda criterion 2.6. Exercises Chapter 3. Incompressible Viscous Fluids: The Navier-Stokes Equations 3.1. Viscosity 3.2. Nondimensionalization 3.3. Vorticity, symmetries, and balance laws 3.4. Special explicit solutions 3.5. Local existence of 𝐻^{𝑠} solutions 3.6. Strong solutions with initial data in 𝐻¹: Local and global 3.7. Exercises Chapter 4. Leray-Hopf Weak Solutions of Navier-Stokes 4.1. Weak solutions 4.2. Existence of weak solutions on the whole space via mollification 4.3. The uniqueness of weak solutions in 2D 4.4. Weak-strong uniqueness and the Prodi-Serrin class 4.5. Partial regularity in time for Leray-Hopf weak solutions 4.6. Existence of weak solutions on the periodic box via Galerkin 4.7. Exercises Chapter 5. Mild Solutions of Navier-Stokes 5.1. Mild formulation 5.2. Scaling criticality 5.3. Local-in-time well-posedness in 𝐻̇^{1/2} 5.4. Local-in-time well-posedness in 𝐿³ 5.5. Local regularization 5.6. Continuation of smooth solutions 5.7. Exercises Chapter 6. A Survey of Some Advanced Topics 6.1. Local regularity and the Prodi-Serrin conditions 6.2. Partial regularity of suitable weak solutions in 3D 6.3. Bounded domains 6.4. Stationary solutions of the Navier-Stokes equations 6.5. Ruling out backward self-similar finite-time singularities 6.6. Critical and supercritical well-posedness for Navier-Stokes 6.7. Yudovich theory and 2D Euler with 𝐿^{𝑝} vorticity 6.8. Gradient growth in the 2D Euler equations 6.9. The search for finite-time singularties in 3D Euler 6.10. Hydrodynamic stability: Euler 6.11. Hydrodynamic stability: Navier-Stokes 6.12. The energy balance and Onsager’s conjecture Appendix A.1. The contraction mapping principle A.2. Existence and uniqueness for ODEs A.3. Fourier transform A.4. Integral operators A.5. Sobolev spaces A.6. Basic properties of the Poisson and heat equations A.7. Mollifiers A.8. Sobolev and Gagliardo-Nirenberg inequalities A.9. Compactness Bibliography Index Back Cover