Dispersive Equations and Nonlinear Waves: Generalized Korteweg–de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps

✍ Scribed by Herbert Koch, Daniel Tataru, Monica Vişan (auth.)

• Publisher: Birkhäuser Basel
• Year: 2014
• Tongue: English
• Leaves: 310
• Series: Oberwolfach Seminars 45
• Edition: 1
• Category: Library

✦ Synopsis

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.​

✦ Table of Contents

Front Matter....Pages i-xii
Front Matter....Pages 1-1
Introduction....Pages 3-3
Stationary phase and dispersive estimates....Pages 5-22
Strichartz estimates and small data for the nonlinear Schrödinger equation....Pages 23-39
Functions of bounded p -variation....Pages 41-71
Convolution of measures on hypersurfaces, bilinear estimates, and local smoothing....Pages 73-85
Well-posedness for nonlinear dispersive equations....Pages 87-109
Appendix A: Young’s inequality and interpolation....Pages 111-122
Appendix B: Bessel functions....Pages 123-126
Appendic C: The Fourier transform....Pages 127-134
Back Matter....Pages 135-137
Front Matter....Pages 139-139
Introduction....Pages 141-142
Maps into manifolds....Pages 143-150
Geometric pde’s....Pages 151-160
Wave maps....Pages 161-199
Schrödinger maps....Pages 201-218
Back Matter....Pages 219-222
Front Matter....Pages 223-224
Notation....Pages 225-225
Dispersive and Strichartz estimates....Pages 227-238
An inverse Strichartz inequality....Pages 239-243
A linear profile decomposition....Pages 245-250
Stability theory for the energy-critical NLS....Pages 251-257
A large data critical problem....Pages 259-260
Back Matter....Pages 309-312
Front Matter....Pages 223-224
A Palais–Smale type condition....Pages 261-269
Existence of minimal blowup solutions and their properties....Pages 271-279
Long-time Strichartz estimates and applications....Pages 281-289
Frequency-localized interaction Morawetz inequalities and applications....Pages 291-302
Appendix A: Background material....Pages 303-308
Back Matter....Pages 309-312

✦ Subjects

Partial Differential Equations