Nonlinear dispersive equations : local and global analysis
著者
書誌事項
Nonlinear dispersive equations : local and global analysis
（Regional conference series in mathematics, no. 106）
Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society with support from the National Science Foundation, c2006
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注記
"This monograph is based on ... a lecture series given at the NSF-CBMS regional conference on nonlinear and dispersive wave equations at New Mexico State University, held in June 2005."--Pref
Bibliography: p. 363-373
内容説明・目次
内容説明
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.
目次
Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.
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