Dispersive Equations and Nonlinear Waves

This book details key ideas in dispersive equations. It offers coverage of tools and techniques, the energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, scattering, control of interacting waves, and Schrödinger maps.

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.

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

Autorentext
Herbert Koch has been a professor at the University of Bonn, Germany since 2006, Daniel Tataru at the University of California in Berkeley, USA, since 2001 and Monica Vian is an associate professor at UCLA, USA.

Inhalt

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