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

This book provides a concise introduction to the mathematical analysis of the incompressible Euler and Navier-Stokes equations for beginning graduate students. It covers the fundamentals of the Euler theory, including derivation, Eulerian and Lagrangian perspectives, vorticity, special solutions, existence theory, and blowup criteria. The Navier-Stokes theory is also discussed, including derivation, special solutions, existence theory, Leray theory, weak-strong uniqueness, existence theory of mild solutions, and Prodi-Serrin regularity criteria. Each chapter concludes with exercises, making the text suitable for a one-semester graduate course.

Format: Hardback
Length: 218 pages
Publication date: 01 October 2022
Publisher: American Mathematical Society

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.

Weight: 620g
Dimension: 187 x 263 x 20 (mm)
ISBN-13: 9781470470494