Overview
- Authors:
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Herbert Koch
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Institute of Mathematics, University of Bonn, Bonn, Germany
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Daniel Tataru
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Department of Mathematics, University of California, Berkeley, USA
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Monica Vişan
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Department of Mathematics, University of California, Los Angeles, USA
- 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
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Table of contents (25 chapters)
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Nonlinear Dispersive Equations
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 3-3
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 5-22
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 23-39
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 41-71
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 73-85
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 87-109
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 111-122
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 123-126
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 127-134
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Back Matter
Pages 135-137
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Geometric Dispersive Evolutions
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Front Matter
Pages 139-139
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 141-142
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 143-150
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 151-160
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 161-199
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- Herbert Koch, Daniel Tataru, Monica Vişan
Pages 201-218
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Back Matter
Pages 219-222
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Dispersive Equations
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Front Matter
Pages 223-224
About this book
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.
Authors and Affiliations
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Institute of Mathematics, University of Bonn, Bonn, Germany
Herbert Koch
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Department of Mathematics, University of California, Berkeley, USA
Daniel Tataru
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Department of Mathematics, University of California, Los Angeles, USA
Monica Vişan
About the authors
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 Vişan is an associate professor at UCLA, USA.