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Nonlinear Dispersive Equations - Local and Global Analysis (Paperback, UK ed.)
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Nonlinear Dispersive Equations - Local and Global Analysis (Paperback, UK ed.)
Series: CBMS Regional Conference Series in Mathematics
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Among nonlinear PDEs, dispersive and wave equations form an
important class of equations. These include the nonlinear
Schrodinger equation, the nonlinear wave equation, the Korteweg de
Vries equation, and the wave maps equation. This book is an
introduction to the methods and results used in the modern analysis
(both locally and globally in time) of the Cauchy problem for such
equations. Starting only with a basic knowledge of graduate real
analysis and Fourier analysis, the text first presents basic
nonlinear tools such as the bootstrap method and perturbation
theory in the simpler context of nonlinear ODE, then introduces the
harmonic analysis and geometric tools used to control linear
dispersive PDE.These methods are then combined to study four model
nonlinear dispersive equations. Through extensive exercises,
diagrams, and informal discussion, the book gives a rigorous
theoretical treatment of the material, the real-world intuition and
heuristics that underlie the subject, as well as mentioning
connections with other areas of PDE, harmonic analysis, and
dynamical systems. As the subject is vast, the book does not
attempt to give a comprehensive survey of the field, but instead
concentrates on a representative sample of results for a selected
set of equations, ranging from the fundamental local and global
existence theorems to very recent results, particularly focusing on
the recent progress in understanding the evolution of
energy-critical dispersive equations from large data. The book is
suitable for a graduate course on nonlinear PDE.
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