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Birkhäuser - Birkhäuser Mathematics | Dispersive Equations and Nonlinear Waves - Generalized Korteweg–de Vries, Nonlinear Schrödinger,

Dispersive Equations and Nonlinear Waves

Generalized Korteweg–de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps

Series: Oberwolfach Seminars, Vol. 45

Koch, Herbert, Tataru, Daniel, Vişan, Monica

2014, XII, 312 p. 1 illus.

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  • Exposition of central ideas in dispersive equations
  • Basic techniques and function spaces
  • Coherent introduction to induction on energy, minimal blow up solutions and interaction Morawetz estimates
  • Introduction to gauge transform, choice of functions spaces, and control of interacting waves
The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.

Content Level » Graduate

Keywords » Fourier transform - dispersion - wave interaction - wave propagation

Related subjects » Birkhäuser Mathematics

Table of contents 

Local existence of solutions to the initial value problem for dispersive equations.- The energy critical nonlinear Schrödinger equation.- Wave maps and Schrödinger maps.

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