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The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations - An Introduction (Paperback)
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The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations - An Introduction (Paperback)
Series: Graduate Studies in Mathematics
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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.
General
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